control, computation and information systems

Communications in Computer and Information Science

140

P. Balasubramaniam (Ed.)

Control, Computation and Information Systems First International Conference on Logic, Information, Control and Computation, ICLICC 2011 Gandhigram, India, February 25-27, 2011 Proceedings

13

Volume Editor P. Balasubramaniam Department of Mathematics Gandhigram Rural Institute Deemed University Gandhigram, Tamil Nadu, India E-mail: [email protected]

ISSN 1865-0929 e-ISSN 1865-0937 ISBN 978-3-642-19262-3 e-ISBN 978-3-642-19263-0 DOI 10.1007/978-3-642-19263-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Applied for CR Subject Classification (1998): F.1-2, I.2, I.4-5, H.3, J.3, C.2

© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Computational science is a main pillar of most of the recent research, industrial and commercial activities. We should keep track of the advancements in control, computation and information sciences as these domains touch every aspect of human life. Information and communication technologies are the innovative thrust areas of research to study stability analysis and control problems related to human needs. With this in mind the International Conference on Logic, Information, Computation & Control 2011 (ICLICC 2011) was organized by the Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram, Tamil Nadu, India, during February 25–27, 2011. This volume contains papers that were peer reviewed and presented at ICLICC 2011. The conference provided a platform to academics from diﬀerent parts of the world, industrial experts, researchers and students to share their knowledge, discuss various issues and ﬁnd solutions to problems. Based on the scientiﬁc committee’s reviews that composed of 80 researchers from all over the world we accepted only 52 papers for presentation at ICLICC 2011 out of 278 research papers submitted. Out of 19% of accepted papers only 14% of the research contributions were selected for publication. This volume comprises three parts consisting of 39 accepted papers. Control theory and its real-time applications are presented in Chap. 1. Chapter 2 consists of papers related to information sciences, especially image processing-cumcryptography, and neural networks. Finally, papers that involve mathematical computations and its applications to various ﬁelds are presented in Chap. 3.

Acknowledgements As a Conference Chair of ICLICC 2011, I thank all the national funding agencies for their grant support that contributed toward the successful completion of the international conference. ICLICC 2011 was supported in part by the following funding agencies. 1. University Grant Commission (UGC) - Special Assisstanceship Programme (SAP) - Departmental Research Support II (DRS II) & UGC - Unassigned grant, Government of India, New Delhi. 2. Council for Scientiﬁc and Industrial Research (CSIR), New Delhi, India. 3. Department of Science & Technology (DST), Government of India, New Delhi, India. I would like to express my sincere thanks to S.M. Ramasamy, Vice-Chancellor, Gandhigram Rural Institute (GRI) - Deemed University (DU), Gandhigram, Tamil Nadu, India, for his motivation and support. I also extend my profound

VI

Preface

thanks to all faculty members of the Department of Mathematics, the Department of Computer Applications, and the administrative staﬀ members of GRI DU, Gandhigram. I especially thank the Honorary Chairs, Co-chairs, Technical Program Chairs, Organizing Chair, and all the committee members who worked as a team by investing their time to make ICLICC 2011 a great success. I am grateful to the keynote speakers who kindly accepted our invitation. Especially, I would like to thank Jong Yeoul Park (Pusan National University, Pusan, Republic of Korea), Francesco Zirilli (Univerist´ a de Roma la Sapienza, Italy), Lim Chee Peng (University Sains Malaysia, Malaysia), Ong Seng Huat, Wan Ainun M.O., Kumaresan Nallasamy, and Kurunathan Ratnavelu (Univesity of Malaya, Malaysia), for presenting plenary talks and making ICLICC 2011 a grand event. A total of 80 experts on various topics from around the world reviewed the submissions. I express my greatest gratitude for spending their valuable time to review and sort out the papers for presentation at ICLICC 2011. I thank Microsoft Research CMT for providing a wonderful tool Conference Management Toolkit for managing the papers. I also thank Alfred Hofmann for accepting our proposal to publish the papers in Springer series. Last but not the least, I express my sincere gratitude to S. Jeeva Sathya Theesar and the team of research scholars of the Department of Mathematics, GRI-DU for designing the website http://iclicc2011.co.in and managing the entire publication related work. February 2011

P. Balasubramaniam

ICLICC 2011 Organization

ICLICC 2011 was organized by the Department of Mathematics, Gandhigram Rural Institute - Deemed University, with the support of UGC - SAP (DRS II).

Organization Structure Honorary Chairs: Patron: Program Chair: Co-chairs: Technical Committee Chairs: Organizing Chair: Organizing Team:

V. Lakshmikantham, Xuerong Mao, S.K. Ntouyas, R.P. Agarwal, J.H. Kim S.M. Ramasamy (Vice Chancellor, GRI - DU) P. Balasubramaniam M.O. Wan Ainun, M. Sambandham, N.U. Ahmed S. Parthasarathy, R. Uthayakumar, R. Rajkumar P. Muthukumar J. Sahabtheen, R. Rakkiappan, R. Chandran, R. Sathy, T. Senthil Kumar, C. Vidhya, S. Lakshmanan, S. Jeeva Sathya Theesar, V. Vembarasan, G. Nagamani, M. Kalpana, D. Eswaramoorthy

Referees J.Y. Park C. Alaca M.S. Mahmoud V.N. Phat W. Feng X. Li Z. Wang C.K. Ahn H. Akca W. Zhang A. Dvorak I.G. Tsoulos S. Chen Z. Cen L.H. Lan N.F. Law

W.C. Siu J.P. Fang K. Balachandran K. Somasundaram A. Boulkroune N. Kumaresan K. Sakthivel S. Prasath S. Ponnusamy E. Karthikeyan A.V.A. Kumar S. Muralisankar S. Baskar C. Chandrasekar T. Kathirvalavakumar D. AshokKumar

A.K.B. Chand G. Mahadevan P. Pandian A. Thangam G. Ganesan P. Shanmugavadivu I. Kasparraj P.B. VinodKumar V.L.G. Nayagam A. Vadivel S. Domnic A.S.V. Murthy D. Senthilkumar K. Ramakrishnan M.C. Monica R. Raja

Table of Contents

1. Control Theory and Its Applications Existence and Uniqueness Results for Impulsive Functional Integro-Diﬀerential Inclusions with Inﬁnite Delay . . . . . . . . . . . . . . . . . . . . Park Jong Yeoul and Jeong Jae Ug

1

An Improved Delay-Dependent Robust Stability Criterion for Uncertain Neutral Systems with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . K. Ramakrishnan and G. Ray

11

Design of PID Controller for Unstable System . . . . . . . . . . . . . . . . . . . . . . . Ashish Arora, Yogesh V. Hote, and Mahima Rastogi

19

New Delay-Dependent Stability Criteria for Stochastic TS Fuzzy Systems with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathiyalagan Kalidass

27

H∞ Fuzzy Control of Markovian Jump Nonlinear Systems with Time Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Jeeva Sathya Theesar and P. Balasubramaniam

35

Assessing Morningness of a Group of People by Using Fuzzy Expert System and Adaptive Neuro Fuzzy Inference Model . . . . . . . . . . . . . . . . . . Animesh Biswas, Debasish Majumder, and Subhasis Sahu

47

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kumaresan Nallasamy, Kuru Ratnavelu, and Bernardine R. Wong

57

Multi-objective Optimization in VLSI Floorplanning . . . . . . . . . . . . . . . . . P. Subbaraj, S. Saravanasankar, and S. Anand Approximate Controllability of Fractional Order Semilinear Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Sukavanam and Surendra Kumar

65

73

2. Computational Mathematics Using Genetic Algorithm for Solving Linear Multilevel Programming Problems via Fuzzy Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bijay Baran Pal, Debjani Chakraborti, and Papun Biswas Intuitionistic Fuzzy Fractals on Complete and Compact Spaces . . . . . . . . D. Easwaramoorthy and R. Uthayakumar

79 89

X

Table of Contents

Research on Multi-evidence Combination Based on Mahalanobis Distance Weight Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Si Su and Runping Xu

97

Mode Based K-Means Algorithm with Residual Vector Quantization for Compressing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Somasundaram and M. Mary Shanthi Rani

105

Approximation Studies Using Fuzzy Logic in Image Denoising Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sasi Gopalan, Madhu S. Nair, Souriar Sebastian, and C. Sheela

113

An Accelerated Approach of Template Matching for Rotation, Scale and Illumination Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanmoy Mondal and Gajendra Kumar Mourya

121

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained Histogram Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Shanmugavadivu and K. Balasubramanian

129

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.T. Jaya Christa and P. Venkatesh

137

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P.C. Pal and Dinbandhu Mandal

145

Medical Image Binarization Using Square Wave Representation . . . . . . . . K. Somasundaram and P. Kalavathi

152

Solving Two Stage Transportation Problems . . . . . . . . . . . . . . . . . . . . . . . . . P. Pandian and G. Natarajan

159

Tumor Growth in the Fractal Space-Time with Temporal Density . . . . . . P. Paramanathan and R. Uthayakumar

166

The Use of Chance Constrained Fuzzy Goal Programming for Long-Range Land Allocation Planning in Agricultural System . . . . . . . . . Bijay Baran Pal, Durga Banerjee, and Shyamal Sen

174

A Fuzzy Goal Programming Method for Solving Chance Constrained Programming with Fuzzy Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animesh Biswas and Nilkanta Modak

187

Fractals via Ishikawa Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhagwati Prasad and Kuldip Katiyar

197

Table of Contents

Numerical Solution of Linear and Non-linear Singular Systems Using Single Term Haar Wavelet Series Method . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Prabakaran and S. Sekar

XI

204

3. Information Sciences Cryptographic Image Fusion for Personal ID Image Authentication . . . . . K. Somasundaram and N. Palaniappan Artiﬁcial Neural Network for Assessment of Grain Losses for Paddy Combine Harvester a Novel Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sharanakumar Hiregoudar, R. Udhaykumar, K.T. Ramappa, Bijay Shreshta, Venkatesh Meda, and M. Anantachar Implementation of Elman Backprop for Dynamic Power Management . . . P. Rajasekaran, R. Prabakaran, and R. Thanigaiselvan Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Abdul Hameed and M. Bagavandas A Secure Key Distribution Protocol for Multicast Communication . . . . . . P. Vijayakumar, S. Bose, A. Kannan, and S. Siva Subramanian

213

221

232

241 249

Security-Enhanced Visual Cryptography Schemes Based on Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Monoth and P. Babu Anto

258

Context-Aware Based Intelligent Surveillance System for Adaptive Alarm Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ji-hoon Lim and Seoksoo Kim

266

Modiﬁed Run-Length Encoding Method and Distance Algorithm to Classify Run-Length Encoded Binary Data . . . . . . . . . . . . . . . . . . . . . . . . . . T. Kathirvalavakumar and R. Palaniappan

271

A Fast Fingerprint Image Alignment Algorithms Using K-Means and Fuzzy C-Means Clustering Based Image Rotation Technique . . . . . . . . . . . P. Jaganathan and M. Rajinikannan

281

A Neuro Approach to Solve Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . J. Abdul Samath, P. Ambika Gayathri, and A. Ayisha Begum

289

Research on LBS-Based Context-Aware Platform for Weather Information Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jae-gu Song and Seoksoo Kim

297

A New Feature Reduction Method for Mammogram Mass Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Surendiran and A. Vadivel

303

XII

Table of Contents

Chaos Based Image Encryption Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Shyamala

312

Weighted Matrix for Associating High-Level Features with Images in Web Documents for Image Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Sumathy and P. Shanmugavadivu

318

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

Existence and Uniqueness Results for Impulsive Functional Integro-Diﬀerential Inclusions with Inﬁnite Delay Park Jong Yeoul1 and Jeong Jae Ug2 1

Department of Mathematics, Pusan National University, Busan 609-735, South Korea [email protected] 2 Department of Mathematics, Dongeui University, Busan 614-714, South Korea

Abstract. In this paper, we discuss the existence of mild solutions of ﬁrst-order impulsive functional integro-diﬀerential inclusions with scalar multiple delay and inﬁnite delay. The result is obtained by using a ﬁxed point theorem for condensing maps due to Martelli. Keywords: Impulsive functional diﬀerential equation; Uniqueness; Inﬁnite delay.

1

Introduction

This paper is concerned with the existence of solutions for impulsive functional integro-diﬀerential inclusions with scalar multiple delay and inﬁnite delay y (t) ∈ F (t, yt ,

t

k(t, s, ys )ds) + 0

− − y(t+ k ) − y(tk ) = Ik (y(tk )),

y(t) = φ(t),

n∗

y(t − Ti ) a.e. t ∈ J\{t1 , t2 · · · , tn },(1)

i=1

k = 1, 2, · · · , m,

t ∈ (−∞, 0],

(2) (3)

where J = [0, b], n∗ ∈ {1, 2, · · ·}, k : J × J × B → Rn , F : J × B × Rn → P(Rn ) is a multivalued map, P(Rn ) is the family of all subsets of Rn , φ ∈ B, B is called a phase space that will be deﬁned later and Ik ∈ C(Rn , Rn ), k = 1, 2, · · · , m are given functions satisfying some assumptions that will be speciﬁed later. For every t ∈ [0, b], the history function yt : (−∞, 0] → Rn is deﬁned by yt (θ) = y(t+ θ) for θ ∈ (−∞, 0]. We assume that the histories yt : (−∞, 0] → Rn , yt (θ) = y(t + θ), belong to abstract phase space B. Impulsive diﬀerential and partial diﬀerential equations have become important in recent years in mathematical models of real processes and they arise in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. See the monographs of Bainov and Simeonov [1], Lakshmikantham et al. [4], Samoilenko and Perestyuk [11] and the references P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 1–10, 2011. c Springer-Verlag Berlin Heidelberg 2011

2

J.Y. Park and J.U. Jeong

therein. The boundary value problems for impulsive functional diﬀerential equations with unbounded delay were studied by Benchohra et al. [2]. The problems of functional integro-diﬀerential inclusions with inﬁnite delay have been studied by many researcher’s. Recently, the existence and uniqueness results for impulsive functional diﬀerential equations with scalar multiple delay and inﬁnite delay was studied by Ouahab [10]. Nieto and Rodr´ıguez-L´opez [9] considered the ﬁrst-order impulsive integro-diﬀerential equations with periodic boundary value conditions and proved some maximum principles. Luo and Nieto [7] improved and extended the results of Nieto and Rodr´ıguez-L´opez [9]. Chang and Nieto [3] proved the existence of the solutions of the impulsive neutral integral-diﬀerential inclusions with nonlocal initial conditions in α-norm by a ﬁxed point theorem for condensing multi-valued maps. Lin and Hu [6] proved the existence of mild solutions for a class of impulsive neutral stochastic functional integro-diﬀerential inclusions with nonlocal initial conditions and resulvent operators. They generalized the results of Chang and Nieto [3] to the stochastic setting. Motivated by Ouahab [10], we consider the existence of solutions for impulsive functional integro-diﬀerential inclusions with scalar multiple delay and inﬁnite delay. 1.1

Preliminaries

Let C([0, b], Rn ) be the Banach space of all continuous functions from [0, b] into Rn with norm y∞ = sup{y(t) : 0 ≤ t ≤ b}. A measurable function y : [0, b] → Rn is Bochner integrable if and only if |y| is Lebesgue integrable. Let L1 ([0, b], Rn ) be the Banach space of continuous functions y : [0, b] → Rn which are Lebesgue integrable and normed by b ||y||L1 = |y(t)|dt 0

for all y ∈ L ([0, b], R ). A multivalued mapping G : Rn → P(Rn ) is convex(closed) valued if G(x) is convex(closed) for all x ∈ Rn . G is bounded on bounded sets if G(B) = ∪x∈B G(x) is bounded in Rn for any bounded set B of Rn , i.e., sup {sup{|y| : y ∈ G(x)}} < ∞. 1

n

x∈B

G is called upper semicontinuous(u.s.c.) on Rn if for each x0 ∈ Rn the set G(x0 ) is a nonempty closed subset of Rn and if for each open set B of Rn containing G(x0 ) there exists an open neighborhood V of x0 such that G(V ) ⊆ B. G is said to be completely continuous if G(B) is relatively compact for every bounded subset B of Rn . If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c if and only if G has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ Gxn imply y∗ ∈ Gx∗ . In the following BCC(Rn ) denotes the set of all nonempty bounded closed and convex subset of Rn . A multivalued map G : [0, b] → BCC(Rn ) is said to be measurable if for each x ∈ Rn the function h : [0, b] → R deﬁned by h(t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)},

Impulsive Functional Integro-Diﬀerential Inclusions

3

belongs to L1 ([0, b], R). An upper semicontinuous map G : Rn → P(Rn ) is said to be condensing if for any subset B ⊆ Rn with α(B) = 0, we have α(G(B)) < α(B), where α denotes the Kuratowski measure of noncompactness. G has a ﬁxed point if there is x ∈ Rn such that x ∈ G(x). Lemma 2.1([8]). Let X be a Banach space and N : X → BCC(X) be a condensing map. If the set Ω = {y ∈ X : λy ∈ N y for some λ > 1} is bounded, then N has a ﬁxed point.

2

Existence Result

In order to deﬁne the phase space and the solution of (1)-(3) we shall consider the space + − n P C = {y : (−∞, b] → Rn , y(t− k ), y(tk ) exist with y(tk ) = y(tk ), yk ∈ C(Jk , R )},

where yk is the restriction of y to Jk = (tk , tk+1 ), k = 1, 2, · · · , m. Let · P C be the norm in P C deﬁned by ||y||P C = sup{|y(s)| : 0 ≤ s ≤ b}. We will assume that B satisﬁes the following axioms: + (A) If y : (−∞, b] → Rn , b > 0 and y0 ∈ B and y(t− k ), y(tk ) exist with y(tk ) = − y(tK ), k = 1, 2, · · · , m, then for every t ∈ [0, b]\{t1, t2 , · · · , tm } the following conditions hold: (i) yt is in B and yt is continuous on [0, b]\{t1 , t2 , · · · , tm }, (ii) ||yt ||B ≤ k(t) sup{|y(s)| : 0 ≤ s ≤ t} + M (t)||y0 ||B , (iii) |y(t)| ≤ H||yt ||B , where H ≥ 0 is a constant, k : [0, ∞) → [0, ∞) is continuous, M : [0, ∞) → [0, ∞) is locally bounded and k, H, M are independent of y(·). (A1) For y(·) in A, yt is B-valued continuous function on [0, b]\{t1 , t2 , · · · , tm }, (A2) The space B is complete. Set Bb = {y : (−∞, b] → Rn |y ∈ P C ∩ B}. Let us state by deﬁning what we mean by a solution of problem (1)-(3). Definition 3.1. We say that the function y ∈ Bb is a mild solution of problem (1)-(3) if y(t) = φ(t) for all t ∈ (−∞, 0], the restriction of y(·) to the interval [0, ∞) is continuous and there exists v(·) ∈ L1 ([0, ∞), Rn ) such that v(t) ∈ t F (t, yt , 0 k(t, s, ys )ds) and such that y satisﬁes the integral equation y(t) = φ(0)+

n∗ i=1

0

−Ti

φ(s)ds +

t

v(s)ds + 0

n∗ i=1

0

t−Ti

y(s)ds+

0
Ik (y(t− k )).

4

J.Y. Park and J.U. Jeong

We are now in a position to state and prove our existence result for the problem (1.1)-(1.3). For the study of this problem we ﬁrst list the following hypotheses: (H1) F : J × B × Rn → BCC(Rn ); (t, φ, x) → F (t, φ, x) is measurable with respect to (φ, x) for each t ∈ [0, b] and for each ﬁxed φ ∈ B the set t SF,φ = {v ∈ L1 ([0, b], Rn ) : v(t) ∈ F (t, φ, k(t, s, φ)ds) for a.e. t ∈ [0, b]} 0

is nonempty. (H2) There exists a function q ∈ L1 ([0, b], R+ ) such that t || k(t, s, φ)ds|| ≤ q(t)ϕ(||φ||B ) 0

for a.e. t ∈ [0, b) and all φ ∈ B, where ϕ : R+ → (0, ∞) is a continuous and nondecreasing function. (H3) There exists a function p ∈ L1 ([0, b], R+ ) such that ||F (t, φ, x)|| = sup{|v| : v(t) ∈ F (t, φ, x)} ≤ p(t)ϕ(||φ||B ) + |x| for a.e. t ∈ [0, b] and all φ ∈ B and b M (s)ds < 0

∞ c

ds , s + ϕ(s)

where M (t) =

c=

1−

1 m

k=1 ck

1−

k bm

k=1 ck

(p(t) + q(t) + n∗ ),

(bn∗ kb +kb )|φ(0)|+Mb ||φ||B +kb

n∗

Ti ||φ||∞ +kb

i=1

m

ck |Ik (0)| .

k=1

(H4) There exist constants ck ≥ 0, k = 1, 2, · · · such that |Ik (x) − Ik (¯ x)| ≤ ck |x − x ¯|

for all x, x ¯ ∈ Rn .

Lemma 3.1([5]). Let I be a compact real interval and X be a Banach space. Let F be a multivalued map satisfying (H3) and let Γ be a linear continuous mapping from L1 (I, X) to C(I, X). Then the operator Γ ◦SF : C(I, X) → BCC(C(I, X)), y → (Γ ◦ SF )(y) = Γ (SF,y ) is a closed graph operator in C(I, X) × C(I, X). Theorem 3.1. Assume that hypotheses (H1)-(H4) hold. Then the problem (1)(3) has st least one solution on (−∞, b]. Proof. Transform the problem (1)-(3) into a ﬁxed point problem. Consider the multivalued map P : Bb → 2Bb deﬁned by P (y) = {h : h ∈ Bb } for every y ∈ Bb , where ⎧ t ∈ (−∞, 0], ⎪ ⎨ φ(t) if t n∗ 0 h(t) = φ(0) + i=1 −Ti φ(s)ds + 0 v(s)ds ⎪ ⎩ + n∗ t−Ti y(s)ds + − i=1 0 0
Impulsive Functional Integro-Diﬀerential Inclusions

and

n

v ∈ SF,y = {v ∈ L ([0, b], R ) : v(t) ∈ F (t, yt , 1

t

k(t, s, ys )ds)

5

for a.e. t ∈ [0, b]}.

0

Let x(·) : (−∞, b] → Rn be the function deﬁned by φ(0) if t ∈ [0, b], x(t) = φ(t) if t ∈ (−∞, 0]. Then x0 = φ(0). For each z ∈ P C with z(0) = 0, we denote by z¯ the function deﬁned by z(t) if t ∈ [0, b], z¯(t) = 0 if t ∈ (−∞, 0]. If y(·) satisﬁes the integral equation, t n∗ 0 n∗ φ(s)ds+ v(s)ds + y(t) = φ(0) + i=1

−Ti

0

i=1

t−Ti

y(s)ds + 0

0
Ik (y(t− k )),

where v ∈ SF,y . Then we can decompose y(·) as y(t) = z¯(t) + x(t), 0 ≤ t ≤ b, which implies yt = z¯t + xt for every 0 ≤ t ≤ b and the function z(·) satisﬁes t n∗ 0 φ(s)ds + v(s)ds (4) z(t) = i=1 −Ti n∗ t−Ti

+

i=1

0

(¯ z (s) + x(s))ds +

0

0
− Ik (¯ z (t− k ) + x(tk )),

where v ∈ SF,z

t = {v ∈ L1 ([0, b], Rn ) : v(t) ∈ F (t, z¯t + xt , k(t, s, z¯s + xs )ds)

for a.e.t ∈ [0, b]}.

0

Set B0 = {z ∈ Bb : z0 = 0 ∈ B}, ||z||∞ = ||z0 ||B + sup{|z(t)| : t ∈ [0, b]} = sup{|z(t)| : t ∈ [0, b]}. Thus (B0 , || · ||∞ ) is a Banach space. Let Br = {z ∈ B0 : ||z||∞ ≤ r} for some r ≥ 0. Then Br ⊆ B0 is uniformly bounded and for some r ≥ 0 we have ||¯ zt + xt ||B ≤ ||¯ zt ||B + ||xt ||B

(5)

≤ k(t) sup{|z(s)| : 0 ≤ s ≤ t} + M (t)||z0 ||B + k(t) sup{|x(s)| : 0 ≤ s ≤ t} + M (t)||x0 ||B ≤ kb (r + |φ(0)|) + Mb ||φ||B ,

6

J.Y. Park and J.U. Jeong

where kb = maxt∈[0,b] k(t) and Mb = maxt∈[0,b] M (t). Deﬁne the multivalued map N : B0 → 2B0 by N (z) = {ρ ∈ B0 }, where ⎧ if t ∈ (−∞, 0), ⎨0 t n∗ t−Ti n∗ 0 ρ(t) = φ(s)ds + 0 v(s)ds + i=1 [¯ z (s) + x(s)]ds i=1 −T 0 i ⎩ − − + 0
n∗

0

t

φ(s)ds +

i=1 −Ti n∗ t−Ti

+

vi (s)ds 0

[¯ z (s) + x(s)]ds +

0

i=1

0
− Ik (¯ z (t− k ) + x(tk )).

Let 0 ≤ α ≤ 1. Then for each t ∈ [0, b] we see that (αρ1 + (1 − α)ρ2 )(t) =

n∗

0

i=1 −Ti n∗ t−Ti

+

i=1

t

φ(s)ds +

[αv1 (s) + (1 − α)v2 (s)]ds

0

[¯ z (s) + x(s)]ds +

0

0
− Ik (¯ z (t− k ) + x(tk )).

Since SF,z is convex( because F has convex values), then completing the proof. Step 2. N maps bounded sets into bounded set in B0 . In fact, we need only to show that for any r > 0 there exists a positive constant l such that for each z ∈ Br = {z ∈ B0 : ||z||∞ ≤ r} we have ||ρ||∞ ≤ l. If ρ ∈ N (z), then there exists v ∈ SF,z such that for each t ∈ [0, b] we have ρ(t) =

n∗

0

t

φ(s)ds +

i=1 −Ti n∗ t−Ti

+

v(s)ds 0

0

i=1

(¯ z (s) + x(s))ds +

0
From (H1)-(H2) and (5) we have |ρ(t)| ≤

n∗ i=1

0

−Ti

t

|φ(s)|ds + 0

|v(s)|ds

− Ik (¯ z (t− k ) + x(tk )).

Impulsive Functional Integro-Diﬀerential Inclusions

t−Ti

+n∗

(|¯ z (s) + x(s)|)ds + 0

≤

n∗

Ti ||φ||∞ + ϕ(r )

i=1 m

+

m i=1

b

7

− |Ik (¯ z (t− k ) + x(tk ))|

(p(t) + q(t))dt + bn∗ r + bn∗ |φ(0)|

0

sup{|Ik (x)| : x ∈ (0, r∗ )} = l,

k=1

where z (tk )| + x(tk )| ≤ r + |φ(0)| = r∗ . |¯ z (tk ) + x(tk )| ≤ |¯

(6)

Thus for each ρ ∈ N (Br ) we obtain ||ρ||∞ ≤ l. Step 3. N maps bounded sets into equicontinuous sets of B0 . Let t1 , t2 ∈ [0, b], t1 < t2 and let Br be a bounded set of B0 as in Step 2. Let z ∈ Br and ρ ∈ N (z), according to (5) and (6), we deduce that |ρ(t2 ) − ρ(t1 )| ≤

t2

t1

|v(s)|ds + r∗ n∗ |t2 − t1 | +

sup{|Ik (x)| : |x| ≤ r∗ }.

0≤t≤t2 −t1

So, as t2 → t1 , the right hand side of the above inequality is independent and tends to zero. The equicontinuous for the cases t1 < t2 < 0 and t1 ≤ 0 ≤ t2 are obvious. Thus the set {N (z) : z ∈ Br } is equicontinuous. As a consequence of Step 2 and Step 3 together with the Ascoli-Arzela theorem, we conclude that N : B0 → 2B0 is completely continuous and therefore is a condensing map. Step 4. N has a closed graph. Let zn → z∗ , ρn ∈ N (zn ) and ρn → ρ∗ . We shall prove that ρ∗ ∈ N (z∗ ). In fact ρn ∈ N (zn ) means that there is vn ∈ SF,zn such that t n∗ 0 n∗ t−Ti ρn (t) = φ(s)ds + vn (s)ds + (¯ zn (s) + x(s))ds i=1

+

−Ti

0
0

Ik (¯ zn (t− k)

+

i=1

x(t− k ))

0

for t ∈ [0, b].

We must show that there exists v∗ ∈ SF,z∗ such that ρ∗ (t) =

n∗

0

φ(s)ds +

i=1 −Ti n∗ t−T

t 0

v∗ (s)ds

i

+

i=1

0

(¯ z∗ (s) + x(s))ds +

0
− Ik (z∗ (t− k ) + x(tk )) for any t ∈ [0, b].

8

J.Y. Park and J.U. Jeong

Clearly, we see that, as n → ∞ n∗ | ρn (t) − i=1

0 −Ti

n∗ − ρ∗ (t) − i=1

φ(s)ds −

0 −Ti

n∗

φ(s)ds −

t−Ti

i=1 0 n∗ t−Ti 0

i=1

(¯ zn (t) + x(s))ds −

0

(¯ z∗ (s) + x(s))ds −

0
− Ik (¯ zn (t− k ) + x(tk ))

− |∞ → 0. Ik (¯ z∗ (t− ) + x(t )) k k

Consider the linear and continuous operator Γ : L1 ([0, b], Rn ) → C([0, b], Rn ) t deﬁned by Γ (v)(t) = 0 v(s)ds. In view of Lemma 3.1, we deduce that Γ ◦ SF is a closed graph operator. Moreover, we get ρn (t) −

n∗ i=1

−

0
0

−Ti

φ(s)ds −

n∗

t−Ti

(¯ zn (s) + x(s))ds

0

i=1

− Ik (¯ zn (t− k ) + x(tk )) ∈ Γ SF,zn .

Since z¯n → z∗ , it follows from Lemma 3.1 that ρ∗ (t) −

n∗ i=1

−

0
0

−Ti

φ(s)ds −

n∗

t−Ti

(¯ z∗ (s) + x(s))ds

0

i=1

− Ik (¯ z∗ (t− k ) + x(tk ))

t

v∗ (s)ds 0

for some v∗ ∈ SF.z∗ . Therefore N is completely continuous multivalued map with convex closed values. The proof is complete. Step 5. The set Ω = {z ∈ B0 : λz ∈ N (z) for some λ > 1} is bounded. Let z ∈ Ω. Then λz ∈ N (z) for some λ > 1. Thus there exists v ∈ SF,z such that

t n∗ n∗ t−Ti −1 (¯ z (s) + x(s))ds z(t) = λ i=1 φ(s)ds + 0 v(s)ds + i=1 0 − + 0
≤

|z(t)| n∗

0

i=1 −Ti n∗

|φ(s)|ds +

Ti ||φ||∞ +

≤

i=1

+bn∗ |φ(0)| +

t 0

t 0

m k=1

(7)

|v(s)|ds + n∗

t 0

|¯ z (s) + x(s)|ds +

ck |¯ z (tk ) + x(tk )| +

m k=1

ck |Ik (0)|.

− |Ik (¯ z (t− k ) + x(tk ))|

i=1

(p(s) + q(s))ϕ(||¯ zs + xs ||B ) + n∗

m

t 0

|z(s)|ds

Impulsive Functional Integro-Diﬀerential Inclusions

9

But ||¯ zs + xs ||B ≤ ||¯ zs ||B + ||xs ||B

(8)

≤ k(t) sup{|z(s)| : 0 ≤ s ≤ t} + M (t)||z0 ||B +k(t) sup{|x(s)| : 0 ≤ s ≤ t} + M (t)||x0 ||B ≤ kb sup{|z(s)| : 0 ≤ s ≤ t} + kb |φ(0)| + Mb ||φ||B , where kb = maxt∈[0,b] k(t) and Mb = maxt∈[0,b] M (t). If we name w(t) the right hand side of (8), then we see that ||¯ zs + xs ||B ≤ w(t) and therefore (7) becomes t t ∗ |z(t)| ≤ bn∗ |φ(0)| + ni=1 Ti ||φ||∞ + n∗ 0 |z(s)|ds + 0 (p(s) + q(s))ϕ(w(s))ds m m + k=1 ck |¯ z (tk ) + x(tk )| + k=1 ck |Ik (0)|, t ∈ [0, b]. (9) Using (10) in the deﬁnition of w we obtain w(t) ≤ (bn∗ kb + kb )|φ(0)| + Mb ||φ||B + kb +

t

kb (p(s) + q(s))ϕ(w(s))ds + n∗ 0

Ti ||φ||∞ + kb

i=1

t

n∗

kb w(s)ds + 0

m

m

ck |Ik (0)| (10)

k=1

ck w(t),

t ∈ [0, b].

k=1

Thus, from (10), we get ∗ 1 w(t) ≤ 1− m ck (bn∗ kb + kb )|φ(0)| + Mb ||φ||B + kb ni=1 Ti ||φ||∞ k=1 t m +kb k=1 ck |Ik (0)| + 0 kb [(p(s) + q(s))ϕ(w(s)) + n∗ w(s)]ds . (11) From (11) we derive that

t

w(t) ≤ c +

M (s)[w(s) + ϕ(w(s))]ds,

(12)

0

where c=

1−

1 m

k=1 ck

[(bn∗ kb + kb )|φ(0)| + Mb ||φ||B + kb

n∗

Ti ||φ||∞ + kb

i=1

and M (t) =

1−

k bm

k=1 ck

(p(t) + q(t) + n∗ ).

m k=1

ck |Ik (0)|]

10

J.Y. Park and J.U. Jeong

Denoting by β(t) the right hand side of (3.9) we have w(t) ≤ β(t),

t ∈ [0, b], β(0) = c

and β (t) ≤ M (t)(β(t) + ϕ(β(t))),

t ∈ [0, b].

This implies that for each t ∈ [0, b] b ∞ β(t) ds ds ≤ . M (s)ds < s + ϕ(s) s + ϕ(s) β(0) 0 c Thus from (H3) there exists a constant M∗ such that β(t) ≤ M∗ , t ∈ [0, b]. From the deﬁnition of w we conclude that sup{|z(t)| : t ∈ [0, b]} ≤ M∗ . This shows that Ω is bounded. As a consequence of Lemma 2.1 we deduce that Γ has a ﬁxed point y which is a solution to problem (1)-(3).

References 1. Bainov, D.D., Simeonov, P.S.: Systems with Impulsive Eﬀect. Horwood, Chichester (1989) 2. Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Boundary value problems for impulsive functional diﬀerential equations with inﬁnite delay. International Jour. Math. Comput. Sci. 1, 27–39 (2006) 3. Chang, Y.K., Nieto, J.J.: Existence of solutions for impulsive neutral integrodiﬀerential inclusions with nonlocal initial conditions via fractional operators. Numer. Funct. Anal. Optim. 30, 227–244 (2009) 4. Laksmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Diﬀerential Equations. World Scientiﬁc, Singapora (1989) 5. Lasota, A., Opial, Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary diﬀerential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronorm. Phys. 13, 781–786 (1965) 6. Lin, A., Hu, L.: Existence results for impulsive neutral stochastic functional integrodiﬀerential inclusions with nonlocal initial conditions. Comp. Math. Appl. 59, 64–73 (2010) 7. Luo, Z., Nieto, J.J.: New results for the periodic boundary value problem for impulsive integro-diﬀerential equations. Nonlinear Anal. 70, 2248–2260 (2009) 8. Martelli, M.A.: A Rothe’s type theorem for noncompact acyclic-valued maps. Bollettino della Unione Mathematica Italiana 4, 70–76 (1975) 9. Nieto, J.J., Rodr´ıguez-L´ opez, R.: New comparison results for impulsive integrodiﬀerential equations and applications. J. Math. Anal. Appl. 328, 1343–1368 (2007) 10. Ouahab, A.: Existence and uniqueness results for impulsive functional diﬀerential equations with scalar multiple delay and inﬁnite delay. Nonlinear Anal. 67, 1027– 1041 (2007) 11. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Diﬀerential Euqations. World Scientiﬁc, Singapore (1995)

An Improved Delay-Dependent Robust Stability Criterion for Uncertain Neutral Systems with Time-Varying Delays K. Ramakrishnan and G. Ray Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India

Abstract. In this paper, we consider the problem of delay-dependent robust stability of a class of linear uncertain neutral systems with time-varying delays using Lyapunov-Krasovskii functional approach. By constructing a candidate Lyapunov-Krasovskii (LK) functional, a less conservative stability criterion is derived in terms of linear matrix inequalities (LMIs). In addition to the LK functional candidate, conservativeness in the proposed stability analysis is further reduced by imposing tighter bounding on the time-derivative of the functional using minimal number of slack matrix variables. The proposed analysis, subsequently, yields a stability criterion in convex LMI framework, and is solved non-conservatively at boundary conditions using standard LMI solvers. The eﬀectiveness of the proposed stability criterion is demonstrated through a standard numerical example. Keywords: Neutral systems, Delay-dependent stability, LK functional, Linear Matrix Inequality, Robust stability.

1

System Description and Problem Statement

Consider a class of uncertain neutral systems with time-varying delays given by x(t) ˙ = (A + ΔA(t))x(t) + (B + ΔB(t))x(t − h(t)) + (C + ΔC(t))x(t ˙ − τ (t)), x(θ) = φ(θ), x(θ) ˙ = ϕ(θ), ∀θ ∈ [−max(h, τ ), 0],

(1)

where x(t) ∈ Rn is the state vector; A, B and C are constant matrices of appropriate dimensions; the time-varying uncertain matrices ΔA(t), ΔB(t) and ΔC(t) are expressed as follows: ΔA(t) ΔB(t) ΔC(t) = DF (t) Ea Eb Ec , (2) where D, Ea , Eb and Ec are known matrices of appropriate dimensions, and the time-varying matrix F (t) satisﬁes F T (t)F (t) ≤ I, ∀t ≥ 0. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 11–18, 2011. c Springer-Verlag Berlin Heidelberg 2011

(3)

12

K. Ramakrishnan and G. Ray

The delays h(t) and τ (t) are time-varying functions satisfying the following conditions: ˙ 0 ≤ h(t) ≤ h, h(t) ≤ hD < ∞; (4) 0 ≤ τ (t) ≤ τ, τ˙ (t) ≤ τD < 1. This paper investigates the delay-dependent robust stability of the uncertain system (1) under norm-bounded uncertainty (2) and time-varying delays (4), and formulates a less conservative stability criterion in terms of linear matrix inequalities for computing the maximum allowable bound h of the time-delay h(t), within which, the system under consideration remains asymptotically stable, provided conditions imposed on τ (t) as given in (4) is satisﬁed. Notations: Notations used in this paper are fairly standard: Rn denotes the n-dimensional Euclidian space, Rn×m is the set of n × m real matrices, I and 0 represents the identity matrix and null matrix of appropriate dimensions; the superscript T stands for the matrix transposition; The notation X > 0 (respectively X ≥ 0), for X ∈ Rn×n means that the matrix is real symmetric positive deﬁnite (respectively, positive semi deﬁnite); R (Z) denotes the set of real numbers (integers); λmin (.) and λmax (.) denote the minimum and maximum eigenvalue of the matrix (.). The ‘’ represents the symmetric elements in a symmetric matrix. Unless speciﬁed, the matrices considered for multiplication operation are assumed to have compatible dimensions. Assumption 1. [1] Throughout the paper, the results will be derived based on the assumption that all the eigen values of (C + ΔC(t)) are inside the unit circle i.e. λmax (C + ΔC(t)) < 1, ∀t. Following lemmas and facts are indispensable in deriving the proposed stability criterion, and they are stated below: Lemma 1. [2] For any symmetric positive deﬁnite matrix W ∈ Rn×n , a scalar γ > 0, and vector function x˙ : [−γ, 0] → Rn such that the integration t x˙ T (s)W x(s)ds ˙ is well deﬁned, then the following inequality holds: t−γ −γ

t

t−γ

x˙ T (s)W x(s)ds ˙ ≤

x(t) x(t − γ)

T

−W W −W

x(t) . x(t − γ)

(5)

Fact 1 Schur complement: Given constant symmetric matrices Σ1 , Σ2 and Σ3 where Σ1 = Σ1T and 0 < Σ2 = Σ2T , then Σ1 + Σ3T Σ2−1 Σ3 < 0, if and only if Σ1 Σ3T −Σ2 Σ3 < 0, or < 0. Σ3 −Σ2 Σ3T Σ1 Fact 2 Jenson Integral Inequality: [3] For any symmetric positive deﬁnite matrix M ∈ Rn×n , scalars γ1 and γ2 satisfying γ1 < γ2 , vector function ω : [γ1 , γ2 ] → Rn such that following integrals are well deﬁned, then

γ2

γ1

ω T (s)M ω(s)ds ≥

1 γ2 − γ1

T

γ2

γ1

ω(s)ds

M

γ2 γ1

ω(s)ds .

(6)

Improved Delay-Dependent Robust Stability Criterion for Neutral Systems

13

Lemma 2. Suppose r1 ≤ r(t) ≤ r2 , where r(.) : R+ (or Z+ ) → R+ (or Z+ ). Then, for any R = RT > 0, following integral inequality holds: t−r1 − x˙ T (s)Rx(s)ds ˙ ≤ δ T (t)[(r2 − r(t))T R−1 T T + (r(t) − r1 )Y R−1 Y T t−r2

+ [Y − Y + T − T ] + [Y

−Y +T

− T ]T ]δ(t), (7)

where δ(t) = [xT (t − r1 ) xT (t − r(t)) xT (t − r2 )]T ; T = [T1T T2T T3T ]T , and Y = [Y1T Y2T Y3T ]T are free matrices of appropriate dimensions. Proof. For any vectors z, y, and symmetric, positive deﬁnite matrix X, the following inequality holds: −2z T y ≤ z T X −1 z + y T Xy. t−r(t) R ˙ X = (r2 −r(t)) , and subsequently Substituting z = T T δ(t), y = t−r2 x(s)ds, using Jenson integral inequality, we get, t−r(t) t−r(t) T T −1 T −2δ (t)T x(s)ds ˙ ≤ (r2 −r(t))δ (t)T R T δ(t)+ x˙ T (s)Rx(s)ds, ˙ t−r2

t−r2

which, by Newton-Leibniz formula, is expressed as follows: t−r(t) − x˙ T (s)Rx(s)ds ˙ ≤ δ T (t) [(r2 − r(t))T R−1 T T + [0 T − T ] t−r2

+ [0 T − T ]T ] δ(t). By analogy, we can deduce the following inequality as well: t−r1 − x˙ T (s)Rx(s)ds ˙ ≤ δ T (t) [(r(t) − r1 )Y R−1 Y T + [Y − Y 0] t−r(t)

+ [Y − Y 0]T ] δ(t). Summation of the last two equations completes the proof of the lemma.

2

Lemma 3. [4] Suppose γ1 ≤ γ(t) ≤ γ2 , where γ(.) : R+ (or Z+ ) → R+ (or Z+ ). Then, for any constant matrices Ξ1 , Ξ2 , and Ξ with proper dimensions, the following matrix inequality Ξ + (γ(t) − γ1 )Ξ1 + (γ2 − γ(t))Ξ2 < 0 holds, if and only if Ξ + (γ2 − γ1 )Ξ1 < 0 and Ξ + (γ2 − γ1 )Ξ2 < 0.

2

Main Result

The main result of the paper i.e. robust stability criterion for the uncertain system (1) satisfying norm-bounded uncertainties (2) and time-varying delays (4) is presented in the form of a theorem:

14

K. Ramakrishnan and G. Ray

Theorem 1. The uncertain system (1) satisfying norm-bounded uncertainties (2) and time-varying delays (4) is robustly stable, if there exists real symmetric positive deﬁnite matrices P, Q, R, Z1 and Z2 ; matrices Q11 , Q12 and Q22 such Q12 > 0; scalars i ≥ 0, i = 1 to 4; slack matrices Ti , Yi , Mi , Ni , i = that Q11 Q 22 1, 2, 3 of appropriate dimensions such that the following LMIs hold: ⎤ ⎡ ¯ T h Ta Π1 + Π + Π1 + Π1T A¯T U 1 E 2 ⎢ −U 0 0 ⎥ ⎥ < 0, ⎢ (8) ⎣ −1 I 0 ⎦ h − 2 Z1 ⎤ ⎡ ¯ T h Ya Π2 + Π + Π1 + Π1T A¯T U 2 E 2 ⎢ −U 0 0 ⎥ ⎥ < 0, ⎢ (9) ⎣ −2 I 0 ⎦ − h2 Z1 ⎡ ⎤ ¯ T h Ma Π3 + Π + Π2 + Π2T A¯T U 3 E 2 ⎢ −U 0 0 ⎥ ⎢ ⎥ < 0, (10) ⎣ −3 I 0 ⎦ − h2 Z2 ⎡ ⎤ ¯ T h Na Π4 + Π + Π2 + Π2T A¯T U 4 E 2 ⎢ −U 0 0 ⎥ ⎢ ⎥ < 0, (11) ⎣ −4 I 0 ⎦ − h2 Z2 where

⎡

AT P + P A + Q + Q11 PB Q12 0 PC ⎢ 0 0 0 −(1 − hD )Q ⎢ 0 Q22 − Q11 −Q12 ⎢ Π = ⎢ 0 −Q22 ⎢ ⎣ −(1 − τD )R

2 0 0 I −I 0 0 , Z2 h T 2 I 0 −I 0 0 0 , Π = I 0 −I 0 0 0 − Z1 h Π1 = Ya −Ya + Ta −Ta 0 0 0 , Π2 = 0 −Na + Ma Na −Ma 0 0 , A¯ = A B 0 0 C D , E¯ = Ea Eb 0 0 Ec 0 , h U = (Z1 + Z2 ) + R, 2 with T T Ya = Y1T Y2T Y3T 0 0 0 , Ta = T1T T2T T3T 0 0 0 , T T Ma = 0 M1T M2T M3T 0 0 , Na = 0 N1T N2T N3T 0 0 . T Π = 0 0 I −I 0 0

−

⎤

PD 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ 0 ⎦ −I

Improved Delay-Dependent Robust Stability Criterion for Neutral Systems

15

Proof. The uncertain system (1) can be re-written as follows: x(t) ˙ = Ax(t) + Bx(t − h(t)) + C x(t ˙ − τ (t)) + Dp(t), p(t) = F (t)(Ea x(t) + Eb x(t − h(t)) + Ec x(t ˙ − τ (t))), x(θ) = φ(θ), x(θ) ˙ = ϕ(θ),

(12) (13)

∀θ ∈ [−max(h, τ ), 0].

Consider the following Lyapunov-Krasovskii functional candidate: t t V (xt ) = xT (t)P x(t) + xT (s)Qx(s)ds + x˙ T (s)Rx(s)ds ˙ +

t− h 2

+

t

0

−h 2

t−h(t) T

t

t+θ

t−τ (t)

x(s) ds, x(s − h2 ) − h2 t x˙ T (s)Z1 x(s)dsdθ ˙ + x˙ T (s)Z2 x(s)dsdθ. ˙

x(s) x(s − h2 )

Q11 Q12 Q22

−h

(14)

t+θ

The time-derivative of functional V (xt ) along the trajectory of (12) is given by V˙ (xt ) ≤ 2xT (t)P x(t) ˙ + xT (t)Qx(t) − (1 − hD )xT (t − h(t))Qx(t − h(t)) ˙ − (1 − τD )x˙ T (t − τ (t))Rx(t ˙ − τ (t)) + x˙ T (t)Rx(t) ⎤T ⎡ ⎤ ⎡ ⎤⎡ x(t) x(t) Q12 0 Q11 + ⎣ x(t − h2 ) ⎦ ⎣ Q22 − Q11 −Q12 ⎦ ⎣ x(t − h2 ) ⎦ −Q22 x(t − h) x(t − h) h + x˙ T (t) (Z1 + Z2 ) x(t) ˙ 2 t t− h2 − x˙ T (s)Z1 x(s)ds ˙ − x˙ T (s)Z2 x(s)ds. ˙ (15) t− h 2

t−h

Now, for any ≥ 0, from (13), it follows that − pT (t)p(t) + (Ea x(t) + Eb x(t − h(t)) + Ec x(t ˙ − τ (t)))T × (Ea x(t) + Eb x(t − h(t)) + Ec x(t ˙ − τ (t))) ≥ 0.

(16)

Now, by adding the positive quantity in (2), time-derivative of the LK functional (15) is bounded as follows: ¯ ¯ T E)Σ(t) V˙ (xt ) ≤ Σ T (t)(Π + A¯T U A¯ + E t t− h2 − x˙ T (s)Z1 x(s)ds ˙ − x˙ T (s)Z2 x(s)ds. ˙ t− h 2

(17)

t−h

˙ − τ (t)) pT (t)]T . Now, where Σ(t) = [xT (t) xT (t − h(t)) xT (t − h2 ) x(t − h) x(t t ˙ and using Lemma 1 and Lemma 2, the integral terms − t− h x˙ T (s)Z1 x(s)ds 2 t− h2 T − t−h x˙ (s)Z2 x(s)ds ˙ in V˙ (xt ) are handled in the delay intervals h(t) ∈ [0, h2 ]

16

K. Ramakrishnan and G. Ray

and h(t) ∈ [ h2 , h] as follows: when 0 ≤ h(t) ≤ h2 , we have t h T T − − h(t) T Z1−1 T T + h(t)Y Z1−1 Y T x˙ (s)Z1 x(s)ds ˙ ≤ δ1 (t) 2 t− h 2 + [Y

−Y +T

− T ] + [Y

−Y +T

− T ]T δ1 (t), (18)

and −

t− h 2

t−h

and in a similar manner, when − and −

t− h 2

t−h

t

t− h 2

T

h 2

≤ h(t) ≤ h, we have

T

x(t − h2 ) x˙ (s)Z2 x(s)ds ˙ ≤ x(t − h) T

x(t) x˙ (s)Z1 x(s)ds ˙ ≤ x(t − h2 ) T

− h2 Z2 h2 Z2 − h2 Z2

− h2 Z1 h2 Z1 − h2 Z1

x(t − h2 ) ; (19) x(t − h)

x(t) , x(t − h2 )

(20)

h x˙ T (s)Z2 x(s)ds ˙ ≤ δ2T (t) (h − h(t))M Z2−1 M T + h(t) − N Z2−1 N T 2 + [−N + M N − M ] + [−N + M N − M ]T δ2 (t); (21)

where T δ1 (t) = xT (t) xT (t − h(t)) xT (t − h2 ) , T δ2 (t) = xT (t − h(t)) xT (t − h2 ) xT (t − h) , T T Y = Y1T Y2T Y3T , T = T1T T2T T3T , T T T T T T T T M = M1 M2 M3 , N = N1 N2 N3 . Now, when 0 ≤ h(t) ≤ h2 , we use the relationships stated in (18) and (19) in (17) and obtain the following inequality: h T T T T ¯ ˙ ¯ ¯ ¯ − h(t) Ta Z1−1 TaT V (xt ) ≤ Σ (t)[Π + Π + Π1 + Π1 + A U A + E E + 2 + h(t)Ya Z1−1 YaT ]Σ(t).

(22)

Similarly, when h2 ≤ h(t) ≤ h, we use the relationships stated in (20) and (21) in (17) and obtain the following inequality: ¯ + (h − h(t))Ma Z −1 M T ¯T E V˙ (xt ) ≤ Σ T (t)[Π + Π + Π2 + Π2T + A¯T U A¯ + E a 2 h + h(t) − (23) Na Z2−1 NaT ]Σ(t). 2

Improved Delay-Dependent Robust Stability Criterion for Neutral Systems

17

By Lyapunov-Krasovskii stability theorem [5], the system (1) is asymptotically stable, if V˙ (xt , t) ≤ −λ x(t) 2 for some λ > 0, if, for 0 ≤ h(t) ≤ h2 , h T T T ¯ ¯ ¯ ¯ − h(t) Ta Z1−1 TaT Π + Π + Π1 + Π1 + A U A + E E + 2 + h(t)Ya Z1−1 YaT < 0; and, for

h 2

(24)

≤ h(t) ≤ h,

¯ + (h − h(t)Ma Z −1 M T ¯T E Π + Π + Π2 + Π2T + A¯T U A¯ + E a 2 h + h(t) − Na Z2−1 NaT < 0. 2

(25)

By successively applying Lemma 3 and Schur complement to (24) and (25), we deduce the LMIs stated in Theorem 1. This completes the proof. 2 Remark 1. In situations where there is no restriction on the derivative of the time-varying delay h(t), the delay-rate-independent stability criterion can be deduced readily by letting Q ≡ 0 in Theorem 1.

3

Numerical Example

Consider the following uncertain neutral system: −2 + δ1 0 0 −1 + δ3 c0 x(t) ˙ = x(t)+ x(t−h(t))+ x(t−τ ˙ (t)) 0 −1 + δ2 0 −1 + δ4 0c with 0 ≤ |c| < 1; δi , i = 1 to 4 are unknown parameters satisfying: |δ1 | ≤ 1.6, |δ2 | ≤ 0.05, |δ3 | ≤ 0.1, |δ4 | ≤ 0.3. Table 1. Maximum allowable delay bound h for hD = 0.1 and τD = 0 with diﬀerent values of c Method [6] [7] [8] [9] [10] Theorem 1

c h h h h h h

0 0.92 0.97 1.1072 1.2723 1.3209 1.3418

0.1 0.73 0.78 0.9208 1.0360 1.0777 1.0841

0.2 0.55 0.60 0.7516 0.8302 0.8701 0.8585

0.3 0.41 0.45 0.5991 0.6512 0.6827 0.6664

0.4 0.29 0.31 0.4625 0.4958 0.5123 0.5033

0.5 0.19 0.19 0.3402 0.3607 0.3596 0.3649

0.6 0.11 0.10 0.2310 0.2409 0.2419 0.2438

0.7 0.04 0.02 0.1227 0.1312 0.1450 0.1320

For hD = 0.1 and τD = 0, the maximum allowable bound h provided by the proposed robust stability criterion for diﬀerent values of c is listed in Table 1 along with the existing results. Similarly, the maximum allowable h for c = 0.1 and τD = 0.1 is listed in Table 2 for diﬀerent values of hD . From the tables, it is clear that the proposed stability criterion, owing to the candidate LK functional, and tighter bounding on its time-derivative, is less conservative than the recently reported results [6, 7, 8, 9, 10].

18

K. Ramakrishnan and G. Ray

Table 2. Maximum allowable delay bound h for c = 0.1 and τD = 0.1 with diﬀerent values of hD Method [7] [8] [10] Theorem 1

4

hD 0 0.1 0.2 h 0.81 0.77 0.73 h 0.9404 0.9109 0.8825 h 1.1454 1.0657 0.9974 h 1.1026 1.0711 1.0397

0.3 0.68 0.8537 0.9426 1.0086

0.4 0.62 0.8253 0.9007 0.9802

0.5 0.57 0.7980 0.8732 0.9751

0.6 0.50 0.7711 0.8592 0.9751

0.7 0.42 0.7472 0.8558 0.9751

0.8 0.32 0.7287 0.8642 0.9751

0.9 >1 0.17 0.7199 0.7216 0.8943 0.9376 0.9751 0.9751

Conclusion

In this paper, we have proposed a less conservative robust delay-dependent stability criterion for a class of neutral systems with time-varying delays using Lyapunov-Krasovskii functional approach. The reduction in conservatism of the proposed stability criterion over recently reported results is attributed to the candidate LK functional used in the stability analysis, and to the tighter bounding of time-derivative of the functional without neglecting any useful terms using minimal number of slack matrix variables. A numerical example is employed to demonstrate the eﬀectiveness of the proposed stability criterion.

References 1. Yue, D., Han, Q.L.: A Delay-dependent Stability Criterion for Neutral Systems and its Application to a Partial Element Equivalent Circuit Model. IEEE Trans. Circuits. Syst. II Exp. Briefs 51, 685–689 (2004) 2. Han, Q.L.: Absolute Stablity of Time-delay Systems with Sector-bounded Nonlinearity. Automatica 41, 2171–2176 (2005) 3. Gu, K.: An Integral Inequality in the Stability Problem of Time-delay Systems. In: Proceedings of 39th IEEE Conference on Decision and Control, pp. 2805–2810. IEEE Press, Sydney (2000) 4. Yue, D., Tian, E., Zhang, Y.: A Piecewise Analysis Method to Stability Analysis of Linear Continuous/Discrete Systems with Time-varying Delay. Int. J. Robust Nonlinear Control 19, 1493–1518 (2009) 5. Gu, K., Kharitonov, V.L., Chen, J.: Stability Analysis of Time-delay Systems. Birkh¨ auser, Boston (2003) 6. Zhao, Z., Wang, W., Yang, B.: Delay and its Time-derivative Dependent Robust Stability of Neutral Control System. Appl. Math. Comput. 187, 1326–1332 (2007) 7. Han, Q.L., Yu, L.: Robust Stability of Linear Neutral Systems with Nonlinear Parameter Perturbations. IEE Proc. Control Theory Appl. 151, 539–546 (2004) 8. Qiu, F., Cui, B., Ji, Y.: Further Results on Robust Stability of Neutral System with Mixed Time-varying Delays and Nonlinear Perturbations. Nonlinear Analysis: Real World Appl. 11, 895–906 (2010) 9. Ramakrishnan, K., Ray, G.: An Improved Delay-dependent Stability Criterion for Neutral Systems with Mixed Time-delays and Nonlinear Perturbations. In: Proceedings of Annual IEEE India Conference INDICON 2010 (2010) 10. Rakkiappan, R., Balasubramanium, P., Krishnaswamy, R.: Delay-dependent Stability Analysis of Neutral Systems with Mixed Time-varying Delays and Nonlinear Perturbations. J. Comput. Appl. Math. (2010), doi:10.1016/j.cam.2010.10.011

Design of PID Controller for Unstable System Ashish Arora, Yogesh V. Hote, and Mahima Rastogi Department of Instrumentation and Control Engineering, Netaji Subhas Institute of Technology, Dwarka Sector-3, New Delhi -110078, India [email protected], [email protected], [email protected]

Abstract. This paper describes the design of a PID Controller for an unstable electronic circuit using various tuning techniques like Ziegler Nichols (Z-N), Chien Hrones Reswick (CHR), and Genetic Algorithm (GA). Further, comparison of various performance parameters for these techniques has been done. The results are verified in MATLAB environment. Keywords: Overshoot, PID Controller, Step Response, Tuning Techniques, Third-Order System.

1 Introduction In industry, PID controller (Proportional-Integral-Derivative) is widely used to make unstable systems stable [6]. This is because of its simple design, and furthermore very few parameters need to be tuned in it. In PID controller design, u (t ) which is proportional to the error between the reference signal and the actual output, can be written as: t

u (t ) = Kp[e(t ) +

1 d e(u )du + Td (e(t ) )] . ∫ Ti 0 dt

(1)

In above Eq.(1), u (t ) is combination of Proportional action, Integral action and Derivative action. To control these actions, various PID Controller tuning techniques such as Zeigler-Nichols (Z-N) Method, Chien Hrones Reswick (CHR) Method, and Genetic Algorithm (GA) Method etc are used [2, 3, 6, 7]. Our main focus is to design a PID controller in such a way that it not only stabilizes the unstable system but also satisfy the performance criterion. Therefore, in order to show the application of PID Controllers, first of all, an unstable electronic circuit is designed [4,5]. Then, PID Controller using analog circuits and various tuning techniques such as Z-N, CHR, and Genetic Algorithm (GA) are designed. Finally based on the results of above techniques, various performance parameters such as overshoot, rise time and settling time are calculated and compared. It is observed that Genetic Algorithm (GA), being a soft computing technique is prone to less error than Z-N and CHR both. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 19–26, 2011. © Springer-Verlag Berlin Heidelberg 2011

20

A. Arora, Y.V. Hote, and M. Rastogi

In summary, the rest of this paper is organized as follows. Section 2 deals with the need for the controller tuning and various tuning techniques such as Z-N, CHR, and Genetic Algorithm (GA). In section 3, the proposed design of PID controller for an unstable electronic circuit is explained with the necessary calculations and results are compared graphically. Finally, conclusion is written in section 4.

2 Need for Controller Tuning Controller tuning means selecting the tuning parameters to ensure the best response of the controller. The system shows a poor performance in terms of characteristic parameters and it even shows an unsatisfactory performance if incorrect controller tuning constants are used. If a control system is properly tuned, the process variability is reduced, efficiency is maximized and also the production rates can be increased. So it becomes necessary to tune the PID controller properly to achieve good system performance having accurately tuned constant values [6]. In control literature there exists various tuning techniques which are as given below: 2.1 Ziegler Nichols (Z-N) [1, 2] Ziegler Nichols is the most employed PID design technique in the industry. This technique avoids the need of the model of the plant which has to be controlled and relies solely on the step response of the plant. This method is used only for type 0 plant. The parameter setting, according to Z-N method is carried out in the following steps: •

• •

Increase the gain Kp from 0 to Kc (decrease the proportional band Kp until the process starts to oscillate). At this critical gain Kc the closed-loop system is marginally stable. Note the value of Kc and the period of oscillation T. The recommended settings of Kp, Ti, and Td are shown in Table-1. Table 1. Ziegler-Nichols tuning rule based on critical gain Kc and critical period T

Type of Controller P

Kp 0.5Kc

Ti Infinity

Td Zero

PI PID

0.45Kc 0.60Kc

0.83T 0.50T

Zero 0.125T

2.2 Chien Hrones Reswick (CHR) [3] To select a PID Controller, Chien Hrones Reswick method provides a better way for process control applications as compared to Z-N method. The shape of the step response of the open-loop plant is taken into consideration to calculate the parameters. The CHR technique works for type 0 plant. The steps involved in implementing a CHR tuning as follows:

Design of PID Controller for Unstable System

• • •

21

Obtain a unit step response of the open-loop plant. A line is drawn through the linear portion of the open-loop plant unit step response just after t=0 which gives values of Tg and Tu. The value for the compensator chosen depends on the ratio R.

R=

Tg Tu

(2)

The controllers are of two types, one that should provide over damped closed-loop behavior and one that should provide 20% overshoot. The formulae proposed by Chien Hrones Reswick for the calculation of various tuning parameters are shown in Table-2. Table 2. Chien Hrones Reswick tuning rule Type of Controller P PI

PID

Over damped Kp=0.3R/Kg Kp=0.35R/Kg Ti=1.2Tg

20% Overshoot Kp=0.7R/Kg Kp=0.6R/Kg Ti=Tg

Kp=0.6R/Kg Ti=Tg Td=0.5Tu

Kp=0.95R/Kg Ti=1.35Tg Td=0.47Tu

2.3 Genetic Algorithm (GA) [6, 7] The Genetic Algorithm is a soft computing technique used for solving both constrained, unconstrained and optimization problems that are based on natural selection [7]. This technique works for all types of plants.GA generates a population of points at each iteration and the best point in the population approaches to an accurate solution. The steps involved in implementing a Genetic Algorithm are as follows: • • • • • • •

Generate an initial, random population of individuals for a fixed size (according to conventional methods Kp, Ti, Td. Evaluate their fitness( to minimize integral square error) Select the fittest members of the population. Reproduce using a probabilistic method. Implement cross over operation on the reproduced chromosomes. Execute mutation operation with low probability. Repeat steps until a pre defined convergence criterion is met.

Program Code for Genetic Algorithm. Following is the Matlab code for generating the required values. The first program contains the function that has been called in second program which has sample ranges for calculating tuning parameters.

22

A. Arora, Y.V. Hote, and M. Rastogi

Program 1: function ise=pidise(x) sys=tf(1,[0.001 0.04 0.6 -3]);%system’s transfer function s=tf([1 0],1);%definition of s pidi=x(1)+x(2)/s+x(3)*s;%x(1)=Kp, x(2)=Ki, x(3)=Kd sysnew=series(sys,pidi); final=feedback(sysnew,1); time=0:0.1:30; [y,t]=step(final,time); for i=1:301 error(i)=1-y(i);%error calculation end ise=abs(sum(error));

Program 2: options=gaoptimset('Populationsize',100,’Generations', 100,'StallGenLimit',200,'Tolfun', 1e-10,’CrossoverFraction’,0.65, ‘CreationFcn’,@gaCreationuniform, 'SelectionFcn',@Selectionstochunif, 'Plotfcns',@gaplotbestf); [x,fval]=ga(@pidise,3,[],[],[],[],[0 0 0],[13.8 107.8 .4416],[],options)

3 Proposed PID Controller Design for Unstable Electronic Circuit In this paper, we have designed an unstable electronic circuit. It is shown in Figure 1.

Fig. 1. Proposed Plant

Design of PID Controller for Unstable System

23

In Figure 1, H(s) is a positive feedback circuit with gain factor of 4. Hence the transfer function of the proposed plant is calculated as:

G1( s) = =

P( s) 1 − P( s) H ( s )

(3)

1 0.001s + 0.04 s 2 + 0.6 s − 3

(4)

3

Fig. 2. Step Response of above unstable electronic Circuit

After, analyzing Figure 2, we came to know that the system is unstable. So there is a need for some kind of controller to make this system stable. So we have designed a PID Controller as mentioned below: The transfer function of a PID Controller is:

C ( s ) = Kp +

Ki + Kds s

(5)

Where Kp, Ki, and Kd respectively denotes the proportional gain, integral gain and derivative gain. The block diagram of PID Controller with unstable electronic circuit is given in Fig. 3. r(s)

Σ

e(s)

C(s)

u(s)

G1(s)

y(s)

+

Fig. 3. Block Diagram showing Plant and PID Controller

24

A. Arora, Y.V. Hote, and M. Rastogi

The following control law has been derived:

u ( s ) = C ( s )e( s )

(6)

Here, e(s) is the control error signal, which is given by:

e( s ) = r ( s ) − y ( s )

(7)

Where, r(s) is the reference signal. Electronic Design of PID Controller C(s) and plant G1(s) using operational amplifier is shown below in Fig. 4.

Fig. 4. Circuit Diagram of PID Controller with Plant

The recommended equations for the calculation of electronic components of the PID Controller are [2]:

Kp = (R1C1 + R 2C 2 )

R4 R1R3C 2

(8)

Kd =

R 2 R 4C1 R3

(9)

Ki =

R4 R1R3C 2

(10)

Using above Eq.(8), Eq.(9) and Eq.(10), the tuning parameters have been calculated using the various tuning techniques as shown in Table-3. By taking values of tuning parameters from Table-3, step responses of the system for all the techniques are determined and shown in Figure 5. From Figure 5, various performance parameters are determined and are compared. Table-4 shows the comparative results of various parameters which are being calculated for various methods.

Design of PID Controller for Unstable System

25

Table 3. Tuning Parameters

Parameters

Z-N

CHR damped

over 20%

Kp

13.8

7.6

4.8

10.9827

Ki

107.8

7.04

6

12.7863

Kd

0.4416

0.36

0.3

0.3403

R1 (KΩ)

68

105

75

83

R2 (KΩ)

60

5

5

3

R3 (KΩ)

10

10

10

10

R4 (KΩ)

73

74

45

106

C1 (µF)

1

10

10

10

C2 (µF)

1

10

10

10

GA

Fig. 5. Comparison of response from various techniques

Remark. From Table-4, it is observed that performance parameters i.e. rise time, overshoot, and settling time are improved in GA method. Table 4. Comparison of Performance Parameters for Various Techniques Parameters to be compared

Genetic Algorithm

20% Overshoot

Over Damped

Z-N

Rise Time (sec)

0.17

0.18

0.16

0.17

Overshoot(O) / Undershoot(U)

0.14(O)

0.5(U)

0.14(U)

0.5(O)

Settling Time ( sec)

1.5

3

2.5

2

26

A. Arora, Y.V. Hote, and M. Rastogi

4 Conclusion In this paper, the above mentioned electrical circuit is being stabilized using Z-N method, CHR method, and GA method. Further, it is proven that GA method is far more superior in comparison to Z-N method and CHR method. In future, hardware implementation of the proposed design will be carried out. Also, real time results and simulated results will be compared.

References 1. Basilio, J.C., Matos, S.R.: Design of PID Controllers with Transient performance Specification. IEEE Transactions on Education 45(4), 364–370 (2002) 2. Roy Choudhury, D.: Modern Control Theory. Prentice Hall of India, Englewood Cliffs (2008) 3. Stefani, S., Savant, H.: Design of feedback control systems, 4th edn. Oxford University Press, Oxford (2002) 4. Shen, J.-C., Chiang, H.-K.: PID Tuning Rules for Second Order Systems. In: 5th Asian Control Conference, pp. 472–477 (2004) 5. Gayakward, R.A.: Op-Amps and Linear Integrated Circuits, 4th edn. Pearson Education, London (2004) 6. Nagaraj, B., Subha, S., Rampriya, B.: Tuning Algorithms for PID controllers using Soft Computing Techniques. International Journal of Computer Science and Network Security 8(4), 278–281 (2008) 7. Jain, A., Dutt, A., Saraiya, A., Pandey, P.: Genetic Algorithm based tuning of PID controller for unstable system, B.E Project, University of Delhi (2006)

New Delay-Dependent Stability Criteria for Stochastic TS Fuzzy Systems with Time-Varying Delays Mathiyalagan Kalidass Department of Mathematics, Anna University of Technology Coimbatore, Coimbatore-641 047, India [email protected] Abstract. The problem of asymptotic stability analysis for a class of stochastic Takagi-Sugeno fuzzy systems with time varying delays is investigated. By employing improved free weighting matrix approach and Lyapunov stability theory, new set of suﬃcient conditions are derived in terms of linear matrix inequalities which ensure the asymptotic stability of the considered stochastic fuzzy system with time varying delays. Further the result is extended to the delay independent that is, for constant time delay case. Finally, a numerical example is provided to illustrate the eﬀectiveness of the proposed results. Keywords: Stochastic Takagi-Sugeno fuzzy systems; Asymptotic stability; Delay-dependent stability; Linear matrix inequality (LMI).

1

Introduction

Fuzzy logic theory has been eﬃciently developed to many applications and shown to be an eﬀective approach to model a complex nonlinear systems and deal with its stability. The stability analysis for Takagi Sugeno (TS) [8] fuzzy systems has been attracted an increasing attention since it can combine the ﬂexibility of fuzzy logic theory and rigorous mathematical theory of linear or nonlinear system into a uniﬁed framework. A great number of stability results for TS fuzzy systems have been reported in the literature [1–4, 9]. It is well known that time delays are unavoidable in hardware implementations, the existence of time delays may lead to oscillation, divergence, and even instability in dynamical systems. Therefore, the stability of delayed systems has become a topic of great theoretical and practical importance [6, 7, 11]. Moreover, all successful applications greatly depend on the dynamic behaviors of systems, such as the stability, the attractivity and the oscillation. The stability problems have attracted increasing interests, and some results related to stochastic disturbances have been studied in [2, 10]. All these stability criteria can be generally classiﬁed as delay-independent and delay-dependent types; In the works of stability analysis of delayed systems, delay-dependent stability criteria, which are less conservative than delay-independent ones, have been paid much attentions.

The work of the author was supported by the Senior Engineering Research Fellowship (SERF), Anna University of Technology, Coimbatore, Tamilnadu, INDIA.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 27–34, 2011. c Springer-Verlag Berlin Heidelberg 2011

28

K. Mathiyalagan

With this motivation, in this paper, the problem of asymptotical stability of stochastic fuzzy systems with time varying delay is considered. Based on Lyapunov stability theory, new delay-dependent stability conditions are established in the form of linear matrix inequalities by using the improved free-weighting matrix approach [6], some improved LMI-based stability criteria for stochstic TS fuzzy system with time-varying delay are obtained without ignoring any useful terms in the derivative of a Lyapunov functional, which can be easily calculated by MATLAB LMI control toolbox. Numerical example is also provided to show the eﬀectiveness of the obtained results.

2

Problem Formulation and Preliminaries

In this section, we start by introducing some notations, deﬁnitions and basic results that will be employed throughout the paper. P > 0 means that P is real symmetric and positive deﬁnite. In symmetric block matrices or long matrix expressions, we use an asterisk (*) to represent a term that is induced by symmetry and for a matrix A, λmax (A) represents the maximum eigenvalues of A. Moreover, let (Ω, F , {Ft }t≥0 , P ) be a complete probability space with a ﬁltration {Ft }t≥0 satisfying the usual conditions. E(·) stands for the mathematical expectation operator with respect to the given probability measure P . Consider the stochastic system with time-varying delays described by dx(t) = [Ax(t) + Ad x(t − d(t))]dt + σ(t, x(t), x(t − d(t)))dw(t) x(t) = φ(t),

(1)

t ∈ [−h, 0],

where x(t) ∈ Rn is the state vector; A and Ad are constant matrices with appropriate dimensions; the positive integer d(t) denotes the time varying delay ˙ ≤ μ, where h1 and h2 represents the minimum satifying 0 ≤ h1 ≤ d(t) ≤ h2 , d(t) and maximum delays, respectively. In the stochastic system (1), w(t) is a scalar Wiener process (Brownian Motion) on (Ω, F , {Ft }t≥0 , P ) with E[w(t)] = 0, E[w2 (t)] = t, E[w(i)w(j)] = 0 (i = j). The function σ(t, x, y) : R+ × Rn × Rn → Rn is Borel measurable and is locally Lipschitz continuous, satiﬁying the following assumption: (A1) There exist two positive constants ρ1 and ρ2 such that σ T (t, x, y) σ(t, x, y) ≤ ρ1 xT x + ρ2 y T y,

∀ t ≥ 0, x, y ∈ Rn

TS fuzzy systems are nonlinear systems described by a set of IF-THEN rules. The model of stochastic TS fuzzy system with time varying delays is described as follows. Plant Rule k: IF θ1 (t) is M1k and . . . and θr (t) is Mrk THEN dx(t) = [Ak x(t) + Adk x(t − d(t))]dt + σk (t, x(t), x(t − d(t)))dw(t), where θ(t) = (θ1 (t), θ2 (t), θ2 (t), · · · , θr (t))T are known premise variables. Mjk (k ∈ {1, 2, . . . , n} , j ∈ {1, 2, . . . , r}) are the fuzzy sets and n is the number of model

New Stability Criteria for Stochastic TS Fuzzy Systems

29

rules. By inferring from the fuzzy models, the ﬁnal output of stochastic TS fuzzy system with time varying delays is obtained by dx(t) =

n

ηk (θ(t)) [Ak x(t) + Adk x(t − d(t))]dt + σk (t, x(t), x(t − d(t)))dw(t) .(2)

k=1

Let ηk (θ(t)) be the normalized membership function of the inferred fuzzy set , where ωk : Rn → [0, 1], k = 1, 2, . . . , n, is ωk (ψ(t)). ie., ηk (θ(t)) = mωk (θ(t)) k=1 ωk (θ(t)) the membership of the system with respect to the plant rule: k and r function k M (θ(t)); It is obvious that the fuzzy weighting functions ωk (θ(t)) = j j=1 n ωk (θ(t)) satisfy ωk (θ(t)) ≥ 0, k ∈ {1, 2, . . . , n} and k=1 ηk (θ(t)) = 1, for all t ≥ 0.

3

Main Results

In this section, we consider the stability criteria for stochastic TS fuzzy system with time varying delays. Further, this result is extended to obtain a stability condition for stochastic TS fuzzy system with constant time delays. The following lemma will be employed in the proofs of our results. Lemma 1. [5]. Given constant matrices Ω1 , Ω2 and Ω3 with appropriate dimensions, where Ω1T = Ω1 and Ω2T = Ω2 , then Ω1 + Ω3T Ω2−1 Ω3 < 0 if and only T Ω1 Ω3 if <0 ∗ −Ω2 Let us denote g(t) =

n

n ηk (θ(t)) Ak x(t)+Adk x(t−d(t)) , σ(t) = ηk (θ(t))σk (t, x(t), x(t−d(t))),

k=1

k=1

Then system (2) can be rewritten as, dx(t) = g(t)dt + σ(t)dw(t).

(3)

Theorem 1. Under the assumption (A1), the equilibrium point of system (3) is asymptotically stable, if there exist a positive scalar λ∗ > 0, symmetric positive definite matrices P > 0, Qj > 0, j = 1, 2, 3, Zj > 0, j = 1, 2, 3, 4, M = Mij ≥ 0, N = Nij ≥ 0, i, j = 1, 2, 3, 4 and any appropriately dimensioned matrices S1 , S2 , S3 such that the following LMIs are feasible for k = 1, 2, . . . , n : P + h1 Z3 + (h2 − h1 )Z4 < λ∗ I, ⎤ ⎡ Σ11 Σ12 Σ13 Σ14 S1 S2 S3 ⎢ ∗ Σ22 Σ23 Σ24 0 0 0 ⎥ ⎥ ⎢ ⎢ ∗ ∗ Σ33 Σ34 0 0 0 ⎥ ⎥ ⎢ ∗ Σ44 0 0 0 ⎥ Σk = ⎢ ⎥ < 0, ⎢∗ ∗ ⎥ ⎢∗ ∗ 0 0 ∗ ∗ −Z 3 ⎥ ⎢ ⎣∗ ∗ ∗ ∗ ∗ −Z4 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −Z4 M S1 N S2 N S3 Π1 = ≥ 0, Π2 = ≥ 0, Π3 = ≥ 0, ∗ Z1 ∗ Z2 ∗ Z2

(4)

(5)

(6)

30

K. Mathiyalagan

where T Σ11 = P Ak + ATk P +h1 ATk Ak +(h2 − h1 )ATk Ak + h1 M11 + (h2 − h1 )N11 + S11 + S11

+λ∗ ρ1 I + Q1 + Q2 + Q3 ,

T Σ12 = h1 M12 + (h2 − h1 )N12 − S11 + S21 + S12 ,

T + P Adk + h1 ATk Adk + (h2 − h1 )ATk Adk , Σ13 = h1 M13 +(h2 − h1 )N13 − S21 +S31 +S13 T , Σ14 = h1 M14 + (h2 − h1 )N14 − S31 + S14 T T ) + S22 + S22 − Q1 , Σ22 = h1 M22 + (h2 − h1 )N22 − (S12 − S12 T T Σ23 = h1 M23 + (h2 − h1 )N23 − S13 − S22 + S32 + S23 , T T − S32 + S24 , Σ24 = h1 M24 + (h2 − h1 )N24 − S14 T T + S33 + S33 − (1 − μ)Q2 + h1 ATdk Adk Σ33 = h1 M33 + (h2 − h1 )N33 − S23 − S23

+(h2 − h1 )ATdk Adk + λ∗ ρ2 I,

T T Σ34 = h1 M34 + (h2 − h1 )N34 − S33 − S24 + S34 ,

T − Q3 , Σ44 = h1 M44 + (h2 − h1 )N44 − S34 − S34 T T T T T T T T T T T T T T T , S12 , S13 , S14 ] , S2 = [S21 , S22 , S23 , S24 ] , S3 = [S31 , S32 , S33 , S34 ] . S1 = [S11

Proof. In order to prove the stability result, we consider the following LyapunovKrasovskii functional for model (3): t t V (t, x(t)) = xT (t)P x(t) + xT (s)Q1 x(s)ds + xT (s)Q2 x(s)ds

t−h1

t

+ t−h2 −h1

xT (s)Q3 x(s)ds +

t

+ −h2 −h1

t+θ t

+ −h2

t+θ

0

−h1

g T (s)Z2 g(s)dsdθ +

t−d(t)

t

g T (s)Z1 g(s)dsdθ

t+θ 0

−h1

t

t+θ

σ T (s)Z3 σ(s)dsdθ

σ T (s)Z4 σ(s)dsdθ.

Then, it can be obtained by Ito’s formula that dV (t, x(t)) = LV (t, x(t))dt + 2xT (t)P σ(t)dw(t)

(7)

We can calculate LV (t, x(t)) along the trajectories of the system (3), then LV (t, x(t)) ≤

n

ηk (θ(t)) 2xT (t)P Ak x(t) + 2xT (t)P Adk x(t − d(t)) + σ T (t)P σ(t)

k=1

+xT (t)(Q1 + Q2 + Q3 )x(t) − xT (t − h1 )Q1 x(t − h1 ) −(1 − μ)xT (t − d(t))Q2 x(t − d(t)) − xT (t − h2 )Q3 x(t − h2 ) +h1 [Ak x(t) + Adk x(t − d(t))]T Z1 [Ak x(t) + Adk x(t − d(t))] +(h2 − h1 )[Ak x(t) + Adk x(t − d(t))]T Z2 [Ak x(t) + Adk x(t − d(t))] t t−h1 − g T (s)Z1 g(s)ds − g T (s)Z2 g(s)ds + (h2 − h1 )σ T (t)Z4 σ(t) t−h1

+h1 σ (t)Z3 σ(t) − T

t−h2 t t−h1

σ (s)Z3 σ(s)ds − T

t−h1

T

σ (s)Z4 σ(s)ds t−h2

(8)

New Stability Criteria for Stochastic TS Fuzzy Systems

31

by using the assumption (A1) and condition (4), we have σ T (t)P σ(t) + h1 σ T (t)Z3 σ(t) + (h2 − h1 )σ T (t)Z4 σ(t) ≤ σkT (t, x(t), x(t − d(t))) (P + h1 Z3 + (h2 − h1 )Z4 ) σk (t, x(t), x(t − d(t))) ≤ λmax (P + h1 Z3 + (h2 − h1 )Z4 ) σkT (t, x(t), x(t − d(t)))σk (t, x(t), x(t − d(t))) ≤ λ∗ (ρ1 xT (t)x(t) + ρ2 xT (t − d(t))x(t − d(t)))

(9)

By the Newton-Leibnitz formula, for any arbitrary matrices S1 , S2 , S3 with compatible dimensions, we have, t t Γ1 = 2αT (t)S1 x(t) − x(t − h1 ) − t−h1 g(s)ds − t−h1 σ(s)dw(s) = 0, t−d(t) t−d(t) Γ2 = 2αT (t)S2 x(t − d(t)) − x(t − h2 ) − t−h2 g(s)ds − t−h2 σ(s)dw(s) = 0, t−h1 t−h1 g(s)ds − t−d(t) σ(s)dw(s) = 0. Γ3 = 2αT (t)S3 x(t − h1 ) − x(t − d(t)) − t−d(t) On the other hand, formatrices M = M T , N = N T the following hold: t Γ4 = h1 αT (t)M α(t) − t−h1 αT (t)M α(t)ds = 0 t−h Γ5 = (h2 − h1 )αT (t)N α(t) − t−h21 αT (t)N α(t)ds = 0. From (8),(9) and Γ1 -Γ5 , we have t n T 1 ηk (θ(t)) α (t)Σk α(t) − g T (s)Z1 g(s)ds LV (t, x(t)) ≤ k=1

−

t−d(t)

t−h2

g T (s)Z2 g(s)ds −

T

−2α (t)S1 −2αT (t)S3 −2αT (t)S2 − −

t−d(t)

T

T

g T (s)Z2 g(s)ds

g(s)ds − 2αT (t)S1

t−d(t)

t−d(t)

α (t)N α(t)ds −

t−d(t)

t−h2 t t−h1

g(s)ds

t−h2 t t−h1

σ(s)dw(s) − 2αT (t)S3

σ T (s)Z4 σ(s)ds −

t−d(t)

t−h2

t−d(t)

t−h2

t−d(t)

g(s)ds − 2α (t)S2

t−h1 t−h1

t−h1

t−h1

T

σ (s)Z3 σ(s)ds −

t−h1 t−h1

−

t

t

σ(s)dw(s)

t−h1 t−d(t)

σ(s)dw(s)

σ T (s)Z4 σ(s)ds

αT (t)M α(t)ds

t−h1

t−d(t)

T

α (t)N α(t)ds .

(10)

also we have

T t t t σ(s)dw(s) ≤ αT (t)S1 Z3−1 S1T α(t) + σ(s)dw(s) Z3 σ(s)dw(s) , t−h1 t−h1 t−h1 T t−d(t) t−d(t) t−d(t) σ(s)dw(s) Z4 σ(s)dw(s) −2αT (t)S2 t−h σ(s)dw(s) ≤ αT (t)S2 Z4−1 S2T α(t)+ t−h t−h2 2 2 t−h1 t−h1 t−h1 T σ(s)dw(s) ≤ αT (t)S3 Z4−1 S3T α(t)+ t−d(t) σ(s)dw(s) Z4 σ(s)dw(s) − 2αT (t)S3 t−d(t) t−d(t)

−2αT (t)S1

, .

32

K. Mathiyalagan

Then by using the above inequalities to (10), we get t n ηk (θ(t)) αT (t)Σk1 α(t) − ϕT (t, s)Π1 ϕ(t, s)ds LV (t, x(t)) ≤ t−h1

k=1

−

t−d(t)

t−h2 T

T

ϕ (t, s)Π2 ϕ(t, s)ds −

t−h1

t−h2

⎡

t−d(t)

t−h1

Z4

T σ(s)dw(s)

t

Z4

+ αT (t)S3 Z4−1 S3T α(t) σ(s)dw(s)

t−d(t) t−h2

σ(s)dw(s)

t−h1

t−d(t)

σ(s)dw(s)

t−d(t) σ T (s)Z3 σ(s)ds − σ T (s)Z4 σ(s)ds t−h1 t−h2

−

σ(s)dw(s)

t−h1

+ −

T

t−d(t)

+

ϕT (t, s)Π3 ϕ(t, s)ds

t−d(t) T α (t)S2 Z4−1 S2T α(t)

+α (t)S1 Z3−1 S1T α(t) + t T + σ(s)dw(s) Z3

t−h1

t

t−h1

t−d(t)

σ T (s)Z4 σ(s)ds .

(11)

⎤ Σ14 Σ24 ⎥ ⎥ , and Πi , i = 1, 2, 3 are deﬁned in (6) with where Σ34 ⎦ Σ44 T T T T T α(t) = x (t) x (t−h1 ) x (t − d(t)) xT (t − h2 ) , ϕ(t, s) = αT (t) g T (s) , On the other hand, from Ito’s isometry in [5],we can know T t t t E σ(s)dw(s) Z σ(s)dw(s) = E t−h1 σ T (s)Z3 σ(s)ds, 3 t−h1 t−h1 T t−d(t) t−d(t) t−d(t) E Z4 t−h2 σ(s)dw(s) = E t−h2 σ T (s)Z4 σ(s)ds, t−h2 σ(s)dw(s) T t−h1 T t−h1 t−h1 Z4 t−d(t) σ(s)dw(s) = E t−d(t) σ (s)Z4 σ(s)ds. E t−d(t) σ(s)dw(s) Σ11 ⎢ ∗ Σk1 = ⎢ ⎣∗ ∗

Σ12 Σ22 ∗ ∗

Σ13 Σ23 Σ33 ∗

Taking the mathematical expectation on both sides of (11) and by using the above equalities, we get (5) and (6) holds then d V (t, x(t)) E ≤ E[LV (t, x(t))] < 0. dt Then it follows from the Lyapunov stability theory that the dynamics of the stochastic fuzzy system is asymptotically stable in the mean square. This completes the proof.

New Stability Criteria for Stochastic TS Fuzzy Systems

33

In the following theorem, we extend the stability criteria for the constant time delay case by replacing the time varying delay d(t) by h. We can derive the following theorem for the constant time delay case. Theorem 2. Under the assumption (A1), the equilibrium point of system (3) with constant time delay is asymptotically stable, if there exists a positive scalar λ∗ > 0, symmetric positive definite matrices P > 0, Q > 0, Zj > 0, j = 1, 2, M = Mij , i, j = 1, 2 and any appropriately dimensioned matrix S1 such that the following LMIs are feasible for i = 1, 2, . . . , n : ⎤ ⎡ Σ11 Σ12 S1 M S1 ≥ 0. P + hZ2 < λ∗ I, Σk = ⎣ ∗ Σ22 0 ⎦ < 0, Π = ∗ Z1 ∗ ∗ −Z2 T where Σ11 = P Ak + ATk P + hATk Ak + hM11 + S11 + S11 + λ∗ ρ1 I + Q T + P Adk + hATk Adk Σ12 = hM12 − S11 + S12 T Σ22 = hM22 − S12 − S12 − Q + hATdk Adk + λ∗ ρ2 I

Proof. To get the LMI based suﬃcient condition for the stochastic TakagiSugeno fuzzy system (3) with constant time delay, we choose the following Lyapunov-Krasovskii functional t 0 t V (t, x(t)) = xT (t)P x(t) + xT (s)Qx(s)ds + g T (s)Z1 g(s)dsdθ

0

t−h

t

+ −h

t+θ

σ T (s)Z2 σ(s)dsdθ,

−h

t+θ

and α(t) = xT (t) xT (t − h) ,

The proof of the theorem follows from the same procedure as in Theorem 3.2.

4

Numerical Simulation

In this section, a numerical example with simulation result is presented to illustrate the eﬀectiveness and advantages of the obtained theory. Example 1. Consider the stochastic TS fuzzy system with time varying delays (3) as follows: −2.0 1.0 −1.0 0 −2.0 1 −1.6 0 A1 = , Ad1 = , A2 = , Ad2 = , 0.5 −1.0 −0.5 −1.0 0 −1.0 0 −1.0 The noise diﬀusion coeﬃcient vector satisﬁes assumption (A1) with ρ1 = 0.05 and ρ2 = 0.05, the membership functions for Rule 1 and Rule 2 are M11 = e−2θ11 (t) and M22 = 1 − M11. By solving LMIs in Theorem 3.2 via Matlab LMI toolbox, for μ = 0.6 it is found that, the time delay parameters d(t) satiﬁes 0 ≤ d(t) ≤ 1.03, and the feasible solutions are not given here due to the page limit. Also the simulation results for the considered stochastic fuzzy system for the obtained time delay upper bound is presented in Fig 1. Fig 1 shows the time response of the state variables x1 (t) and x2 (t) of systems for the considered stochastic fuzzy system with initial values (1.5, -0.8). The simulation results reveal that the considered fuzzy system with time varying delays are asymptotically stable.

34

K. Mathiyalagan

2

2 x1 x2

1

1

0.5

0.5

0 −0.5

0 −0.5

−1

−1

−1.5

−1.5

−2

x1 x2

1.5

x(t)

x(t)

1.5

0

10

20

30

40

50

−2

0

10

20

30

40

50

Time t

Time t , Fig. 1. Trajectories of state variable x(t) of the stochastic fuzzy system for k=1 and k=2

5

Conclusion

In this paper, by utilizing a improved free weighting matrix approach and linear matrix inequality (LMI) technique, we derive a set of suﬃcient conditions which guarantees the asymptotical stability of equilibrium point for the stochastic TS fuzzy systems with time delays. Finally, a numerical example with simulation result has been exploited to illustrate the obtained theoretical results.

References 1. Chen, B., Liu, X.P., Tong, S.C.: New delay-dependent stabilization conditions of TS fuzzy systems with constant delay. Fuzzy Sets and Syst. 158, 2209–2224 (2007) 2. Hu, L., Yang, A.: Fuzzy model-based control of nonlinear stochastic systems with time-delay. Nonlinear Analysis 71, e2855–e2865 (2009) 3. Lien, C.H., Yu, K.W., Chen, W.D., Wan, Z.L., Chung, Y.J.: Stability criteria for uncertain Takagi-Sugeno fuzzy systems with interval time-varying delay. IET Control Theory and Applications 1, 746–769 (2007) 4. Liu, F., Wu, M., He, Y., Yokoyama, R.: New delay dependent stability criteria for TS fuzzy systems with time-varying delay. Fuzzy Sets and Syst. 161, 2033–2042 (2010) 5. Mao, X.R.: Stochastic Diﬀerential Equations and Applications. Horwood, West Sussex (1997) 6. Meng, X., Lam, J., Du, B., Gao, H.: A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 46, 610–614 (2010) 7. Peng, C., Tian, Y.C.: Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay. J. Comput. Appl. Math. 214, 480–494 (2008) 8. Takagi, T., Sugeno, S.: Fuzzy identiﬁcation of systems and its applications to modeling and control. IEEE Tran. Sys., Man, Cyber. 15, 116–132 (1985) 9. Tian, E.G., Peng, C.: Delay-dependent stability analysis and synthesis of uncertain TS fuzzy systems with time-varying delay. Fuzzy Sets and Syst. 157, 544–559 (2006) 10. Wu, L., Wang, Z.: Fuzzy ﬁltering of nonlinear fuzzy stochastic systems with timevarying delay. Signal Processing 89, 1739–1753 (2009) 11. Xu, S., Lam, J., Zou, Y.: New results on delay-dependent robust H∞ control for systems with time-varying delay. Automatica 42, 343–348 (2006)

H∞ Fuzzy Control of Markovian Jump Nonlinear Systems with Time Varying Delay S. Jeeva Sathya Theesar and P. Balasubramaniam Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India 624302 [email protected]

Abstract. In this paper, delay dependent H∞ control problem of Markovian jumping nonlinear systems (MJNS) with time varying delay based on Takagi-Sugeno (TS) fuzzy modeling is studied. Suﬃcient criterion for H∞ performance are assured by deriving convex linear matrix inequalities from nonlinear matrix inequalites utilizing novel matrix transforms. Based on the delay decomposition approach, new set of Lyapunov candidates are constructed in order to derive very less conservative and easily testable criterions. Numerical simulation on single link robot arm model is illustrated to adequate the theory. Keywords: Delay dependent, H∞ control, Markovian jump systems, T-S fuzzy systems.

1

Introduction

Many practical dynamical systems are driven by discrete events such as random component failures, sudden environmental disturbance etc., The complex dynamics of these systems can be modeled as Markovian jumping nonlinear sytsems (MJNS) which belongs to stochastic hybrid systems category. Over the past decade, study of stability and control problems of MJNS have been a thrust area of research, see eg. [1,2,3], and reference therein. Fuzzy systems in the form of Takagi-Sugeno (TS) model has been paid considerable research attention due to its potential and eﬀective way of representing complex nonlinear dynamic system. Also delay in systems often leads to instability and degrades system performance. In recent years, control problems of Markovian jumping Fuzzy systems (MJFS) with or with out delay involving linear matrix inequalities (LMIs) became hot research topic. This approach provides a powerful tool for solving practical problems of complex nonlinear systems numerically by computer simulations. For instance, some of the recent works for MJFS about robust stability analysis cum controller design, delay independent robust stabilization cum H∞ control, and delay dependent stabilization have

The work of authors was supported by UGC-SAP (DRS-II) grant No. F.510/2/DRS/ 2009(SAP-I). Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 35–46, 2011. c Springer-Verlag Berlin Heidelberg 2011

36

S. Jeeva Sathya Theesar and P. Balasubramaniam

been given respectively in [3], [4], and [5]. Recently, authors in [6], derived delay dependent H∞ control of MJFS with constant delay using slack variables in Lyapunov function. In this paper, we consider H∞ control problem of MJFS with time varying delay. Hence, we are considering more general complex problem than the one discussed in [6]. Also we have applied the recently developed method of constructing Lyapunov-Krasovskii functional (LKF) known as delay decomposition [7] to derive less conservative results. The aim of this paper is to derive mode-dependent fuzzy controller such that the resulting closed-loop system is stochastically stable and satisﬁes a prescribed H∞ performance level for all admissible disturbance attenuations. The delay-dependent criterions are derived to ensure the H∞ control of MJFS with time varying delay. The results presented in this paper is less conservative due to the fact that no free weighting or slack matrices are used for esuring less number of matrix variables usage. The derived criterions have been applied numerically on single link robot arm model in order to see the performance of the proposed method. Notations: The notations in this paper is quite standard. The superscript T denotes the transposition and the notation X ≥ Y (respectively, X > Y ), where X and Y are symmetric matrices, means that X − Y is positive semi-deﬁnite (respectively, positive deﬁnite). n and n×n denote n-dimensional Euclidean space and the set of all n × n real matrices, respectively. I is the identity matrix. The notation always denotes the symmetric block in one symmetric matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions. We let τ > 0 and C([−τ, 0]; n ) denotes the family of continuous functions ϕ from [−τ, 0] to n with norm ϕ = sup−τ θ0 ϕ. L2 [0, ∞) stands for the space of square integrable functions on [0, ∞). (Ω, F, ℘) is the complete probability space, Ω is the sample space, F is the σ- algebra of subsets of sample space, and ℘ is the probability measure on F. E{·} denotes the expectation operator with respect to some probability measure ℘.

2

Model Formulation

Fix the probability space (Ω, F, ℘) and due to universal approximation property of TS fuzzy systems, any nonlinear system can be modeled by using the commonly used center-average defuzziﬁer, product inference and singleton fuzziﬁer. Based on this the model of delayed MJFS is given as follows: x(t) ˙ = z(t) =

r j=1 r

μj (x(t)) [Aj (rt )x(t) + Adj (rt )x(t − τ (t)) + Bj (rt )u(t) + Bwj (rt )w(t)] μj (x(t)) [Gj (rt )x(t) + Gdj (rt )x(t − τ (t)) + Hwj (rt )w(t)]

(1)

j=1

x(θ) = ϕ(t), ∀θ ∈ [−τ, 0], where x(t) ∈ n is the state, u(t) ∈ m is the control input, w(t) ∈ p is the disturbance input that belongs to L2 [0, ∞), z(t) ∈ s is the controlled output,

H∞ Fuzzy Control of Delayed MJFS

37

τ (t) is the time varying state delay. ϕ(t) is the compatible vector valued initial function, and rt , t ≥ 0, be a right-continuous Markov chain on the probability space taking values in a ﬁnite state space S = {1, 2, . . . , s} with generator (transition rate matrix) Π = [πkk ], k, k ∈ S given by πkk h + o(h) k = k P r {rt+h = k |rt = k} = 1 + πkk h + o(h) k = k with h > 0 and limh→0 o(h) 0. Here, πkk ≥ 0 is the transition rate from k h = to k , if k = k while πkk = − sk =1,k =k πkk , k, k ∈ S. Let τ (t) is a bounded continuous function such that 0 τ (t) τ . τ = max {τ (t) : t > 0}. Aj (rt ), Adj (rt ), Bj (rt ), Bwj (rt ), Gj (rt ), Gdj (rt ), and Hwj (rt ) are the known real constant matrices for each rt ∈ S. For notational simplicity, for each possible rt = k, k ∈ S, a matrix Mj (rt ) will be denoted by Mjk ; for example, Aj (rt ) is denoted by Ajk , Adj (rt ) is denoted by Adjk , and so on. Also μj (x(t)) = s sωj (x(t)) , where ωj (x(t)) = l=1 hjl (xl (t)), x(t) = [x1 (t), . . . , xs (t)], in which l=1 ωl (x(t)) r hjl (xl (t)) is the grade of membership of xl (t) and satisﬁes j=1 μj (x(t)) = 1, and μj (x(t)) 0. Throughout the paper we shall use the following concepts. Definition 1. The MJFS (1) is said to be stochastically stable, if there exists a scalar M (r0 , ϕ(·)) for all finite ϕ(·) defined on [−τ, 0] and initial jumping mode r0 such that t 2 lim E x(s) ds|r0 , x(s) = ϕ(s), s ∈ [−τ, 0] M (r0 , ϕ(·)). (2) t→∞

0

Definition 2. The MJFS (1) is said to be stochastically stable with disturbance attenuation level γ > 0 if for all w(t) ∈ L2 [0, ∞) the system is stochastically stable and the output response z(t) under zero initial condition satisfies ∞

∞ T 2 T E z (t)z(t) dt γ w (t)w(t) dt (3) 0

0

Definition 3. The MJFS (1) is stochastically stabilizable with prescribed attenuation level γ > 0 with H∞ performance level (3), if there exists a state feedback control law such that u(t) =

r

μj (x(t))Kjk x(t)

(4)

j=1

where Kjk is the mode-dependent control gain matrix determined to be later and k ∈ S. The state feedback TS fuzzy model base controller is designed via parallel distributed compensation (PDC) approach. Remark 1. In most of the practical situations such as computer simulated robot arm, continuous stirred tank reactor systems etc., there arises time varying delay

38

S. Jeeva Sathya Theesar and P. Balasubramaniam

due to various eﬀects. The H∞ control problem of MJFS with delays is still an open problem due to its complexity and nontrivial. Zhang et al [6] have considered H∞ control problem of constant delay. But in this paper time varying delay is considered without any strict restrictions such as 0 τ˙ (t) μ < 1. Initially we assume that few information about time varying delay and the derivatives of the delay are unknown. In order to derive the relaxed conditions we need the following lemmas. Lemma 1. For any constant matrix X ∈ Rn×n , X = X T > 0, scalar h with 0 ≤ τ (t) ≤ τ and a vector-valued function x˙ : [t − τ, t] → Rn , the following integration is well defined,

t

xT (t) − τ (t) x˙ (s)X x(s)ds ˙ ≤ T x (t − τ (t)) t−τ (t) T

T

−X X −X

x(t) (5) x(t − τ (t))

S11 S12 > 0, where Lemma 2. (Schur complement). For a given matrix S = T S12 S22 T T , S12 = S12 , is equivalent to any one of the following conditions: S11 = S11

(1) (2)

3

−1 T S12 > 0 S22 > 0, S11 − S12 S22 T −1 S11 > 0, S11 − S12 S11 S12 > 0

Main Results

In this section, we’ll give a suﬃcient condition ensuring the stochastic stabilizability of MJFS with H∞ disturbance attenuation level γ. As there have been enoguh literature [5,6] for H∞ performance of open loop free MJFS (u(t) ≡ w(r) ≡ 0), the H∞ state feedback control problem of (1) is directly considered here. Utilizing (1), and (4), we get the following closed-loop MJFS, x(t) ˙ =

r r

μj μj (Ajk + Bjk Kj k ) x(t) + Adjk x(t − τ (t)) + Bwjk w(t)

j=1 j =1

z(t) =

r

μj Gjk x(t) + Gdjk x(t − τ (t)) + Hwjk w(t)

(6)

j=1

x(θ) = ϕ(t), ∀θ ∈ [−τ, 0],

k ∈ S.

Let δ > 0, choose the positive integer N such that δ = τ /N . Hence the delay interval is decomposed as [−iδ, −(i − 1)δ], where i = 1, 2, . . . , N . It is clear from the above closed loop systems (6), {(x(t), rt ), t 0} is not a Markov process. In order to cast our model involved into framework of Markov processes, we deﬁne a new process {(xt , rt ), t 0} by xt (s) = x(t + s), t − τ s t. We can easily verify that {(xt , rt ), t 0} is a Markov process with initial

H∞ Fuzzy Control of Delayed MJFS

39

state or mode (ϕ(·), r0 ). Now for t 0, let us construct stochastic LKF on the decomposed inerval as V (xt , rt , t) V1 (xt , rt , t) + V2 (xt , rt , t),

(7)

where V1 = xT (t)P (rt )x(t) N −(i−1)δ xT (t+ξ) Qi x(t+ξ)dξ+δ V2 = i=1

−iδ

−(i−1)δ

−iδ

t t+θ

T

x˙ (ξ) Ri x(ξ)dξdθ ˙

with P (rt ) = Pk > 0, Qi > 0, and Ri > 0. Let J be the weak inﬁnitesimal operator deﬁned by JV (xt , rt , t) = lim

→0+

1 {[EV (xt+ , rt+ , t + |xt , rt = k)] − V (xt , k, t)} (8)

Now we give suﬃcient condtions for delay dependent H∞ control of MJFS with time varying delay in the following theorem. Theorem 1. For a given scalar γ > 0, the closed loop MJLS (6) is stochastically asymptotically stable with disturbance attenuation level γ with H∞ perfomance ˜ i > 0, and R ˜ i > 0, with index defined in (3) , if there exist matrices P˜k > 0, Q compatible dimensions for i = 1, 2, . . . , N such that the following LMI holds for u ∈ {1, 2, . . . , N } (jj) Φ˜ku < 0

(jj ) (j j) Φ˜ku + Φ˜ku < 0

⎡

(jj Φ˜ku

)

ψ11 ⎢ ⎢ =⎢ ⎢ ⎣

ψ12 ψ22

0 ψ23 ψ33

1 j r,

(9)

j < j r

(10)

ψ14 ψ24 ψ34 ψ44

⎤ ψ15 ψ25 ⎥ ⎥ ψ35 ⎥ ⎥ 0 ⎦ ψ55

k ∈ S,

˜ u · · · 0 0 ; ψ14 = δ P˜k AT ; ˜ u ; ψ12 = P˜k AT 0 · · · R ˜u R where ψ11 = −2R djk djk

˜k AT + Y T B T ) T δ( P k jk j jk ˜ ; ψ15 = Pk Gdjk ; ψ23 = Bwjk 0 · · · 0 0 · · · 0 0 ; ψ24 = 0(N ×1) T T ; ψ35 = Hwjk ; ψ25 = P˜k Gjk 0 · · · 0 0 · · · 0 0 ; ψ33 = −γ 2 ; ψ34 = δBwk ˜ ˜ ψ44 = R − 2P ; ψ55 = −I;

40

S. Jeeva Sathya Theesar and P. Balasubramaniam

⎤ (1) ˜ 0 0 ··· 0 0 ω1 R 1 ··· ⎢ ω (1) · · · 0 0 ··· 0 0 ⎥ ⎥ ⎢ 2 ⎥ ⎢ . .. . . .. .. .. .. .. ⎥ ⎢ .. . . . . . . . ⎥ ⎢ ⎥ ⎢ (1) ⎢ · · · ωu 0 ··· 0 0 ⎥ ⎥; ψ22 = ⎢ (1) ⎢ 0 ⎥ · · · ω(u+1) · · · 0 ⎥ ⎢ ⎥ ⎢ . .. . . .. .. .. .. .. ⎥ ⎢ . . . . . . . . ⎥ ⎢ . ⎥ ⎢ (1) ⎣ ··· · · · ωN 0 ⎦ (1) ··· · · · ω(N +1) ⎧ T + Bjk Yj k + Ajk P˜k + P˜k ATjk + YjT k Bjk ⎪ ⎪ ⎪ ⎨ ¯ ˜ ˜ ˜ P − πkk Pk + Q1 − R1 i=1 (1) ; ωi = ˜ ˜ ˜ ˜ ⎪ Q − Q − R − R i = 1, 2, . . . , N i i−1 i i−1 ⎪ ⎪ ⎩ ˜ ˜N i=N +1 −QN − R s N ˜ ˜ ˜ ¯ R= Ri ;P = πkk Pk . ⎡

k =1;k=k

i=1

Moreover the suitable H∞ Markovian jump state feecback controllers can be given as u(t) = Kj k x(t) = Yj k P˜k−1 x(t) . Proof. Consider the stochastic LKF deﬁned in (7). Applying the weak inﬁnitesimal operator J on (8 ), we get ˙ ˙ + JV1 = x(t)P k x(t) + x(t)Pk x(t)

s

(11)

πkk Pk

k =1

JV2 =

N T x (t − (i − 1)δ)Qi x(t − (i − 1)δ) − xT (t − iδ)Qi xT (t − iδ) i=1

˙ − +δ 2 x˙ T (t)Rx(t)

N δ i=1

t−(i−1)δ

t−iδ

x˙ T (ξ)Ri x(ξ)dξ ˙

(12)

N where R = i=1 Ri . For time varying delay we can exhibit u such that u ∈ {1, 2, . . . , N }, for τ (t) ∈ [−uδ, −(u − 1)δ]. Utilizing Lemma 2, we have

t−(u−1)δ

−δ

t−uδ

T

x(t − τ (t)) −Ru Ru x(t − τ (t)) x˙ (ξ)Ru x(ξ)dξ ˙ ≤ x(t − uδ) −Ru x(t − uδ)

T −Ru Ru x(t − (u − 1)δ) x(t − (u − 1)δ) . + −Ru x(t − τ (t)) x(t − τ (t)) T

When i = u, again by the Lemma 2, we have −δ

t−(i−1)δ t−iδ

x˙ T (ξ)Ri x(ξ)dξ ˙ ≤

x(t − (i − 1)δ) x(t − iδ)

T

respectively. Now from (6), and (11)-(12), we get

−Ri Ri −Ri

x(t − (i − 1)δ) x(t − iδ)

H∞ Fuzzy Control of Delayed MJFS

JV (xt , k, t) ≤

r r j=1

41

T μj μj {xT (t)[(ATjk + KjT k Bjk )Pk + Pk (Ajk + Bjk Kj k )]x(t)

j =1

T Pk x(t) + xT (t)Pk Bwjk w(t)} +2xT (t)Pk Adjk x(t − τ (t)) + wT (t)Bwjk

−xT (t − N δ)QN−1 x(t − N δ) +

N−1

[xT (t − iδ)(Qi+1 − Qi )x(t − iδ)]

i=1

T

T −Ru Ru x(t − τ (t)) x(t − (u − 1)δ) x(t − τ (t)) + × + −Ru x(t − uδ) x(t − τ (t)) x(t − uδ)

−Ru Ru x(t − (u − 1)δ) ˙ + xT (t)Q1 x(t) + δ12 x˙ T (t)Rx(t) −Ru x(t − τ (t))

T

N x(t − (i − 1)δ) −Ri Ri x(t − (i − 1)δ) (13) + −Ri x(t − iδ) x(t − iδ)

i=1,i=u

Let ζ (t) = x(t − τ (t)) x(t) x(t − δ1 ) . . . x(t − N δ) . . . w(t) . The above inequality (13) can be rewritten as

JV (xt , k, t) ≤

r r j=1 j =1

(jj ) μj μj ζ T (t)[Φku + ΘT δ 2 RΘ]ζ (t) ,

where ⎡

(jj ) Φku

ψ˜11 ⎣ =

ψ˜12 ψ˜22

⎤ 0 ψ˜23 ⎦ , 0

k ∈ S 1 j, j r

where ψ˜11 = −2Ru ; ψ˜12 = ATdk Pk 0 · · · Ru Ru · · · 0 0 ; T ψ˜23 = Pk Bwjk 0 · · · 0 0 · · · 0 0 ; ⎤ ⎡ (1) ω ˜ 1 R1 · · · 0 0 ··· 0 0 (1) ⎢ ω 0 ··· 0 0 ⎥ ˜2 · · · 0 ⎥ ⎢ ⎥ ⎢ . . . . . . .. ⎥ ⎢ .. .. . . . .. .. .. .. . ⎥ ⎢ ⎥ ⎢ (1) ⎢ 0 ··· 0 0 ⎥ ··· ω ˜u ˜ ⎥; ⎢ ψ22 = ⎢ (1) 0 ⎥ ··· ω ˜ (u+1) · · · 0 ⎥ ⎢ ⎥ ⎢ . .. . . .. .. .. .. .. ⎥ ⎢ . . . . . . . . ⎥ ⎢ . ⎥ ⎢ (1) ⎣ 0 ⎦ ··· ··· ω ˜N (1) ··· ··· ω ˜ (N +1) ⎧ T T Ajk Pk + Pk Ajk + KjT k Bjk + Bjk Kj k ⎪ ⎪ ⎪ ⎨+ s i=1 (1) k =1 πkk Pk + Q1 − R1 ωi = ; ⎪ Qi − Qi−1 − Ri − Ri−1 i = 1, 2, . . . , N ⎪ ⎪ ⎩ i=N +1 −QN − RN

(14)

42

S. Jeeva Sathya Theesar and P. Balasubramaniam

T T T 0 Bwk Ri ; Θ = ATdjk ATjk + KjT k Bjk . (jj ) ˆ Now JV (xt , k, t) 0 if and only if Φku 0, where R=

N1

i=1

(jj ) (jj ) Φˆku = Φku + ΘT δ 2 RΘ < 0

By the Schur complement formula, it is readily seen that ⎤ ⎡˜ ˜ ψ11 ψ12 0 ψ˜14 ⎢ ψ˜22 ψ˜23 ψ˜24 ⎥ ⎥ Φˆkju = ⎢ ⎣ 0 ψ˜34 ⎦ < 0 ψ˜44

δ(ATjk + KjT k BkT )R T R; ψ˜44 = −R; ; ψ˜34 = δBwjk where ψ˜14 = δATdjk R; ψ˜24 = 0(N1 ×1) Therefore, JV (xt , k, t) −axT (t)x(t)

ˆ(jj ) )) > 0. By Dynkin’s formulat, we have for each where a = mink∈S (λmin (−Φ ku rt = k, k ∈ S E [V (xt , k, t) − V (ϕ(t), r0 )] −a hence

E

0

T

T

x (t)x(t) dt

0

T

E xT (t)x(t) dt

1 V (ϕ(t), r0 ). a

However, from (7) it is easy to see that for any t 0, E {V (xt , k, t)} b E xT (t)x(t) , where b = mink∈S (λmin (Pk )) > 0. From the above two inequalites, we get T T xT (t)x(t) dt + b−1 V (ϕ(t), r0 ). E x (t)x(t) ab−1 E 0

Thus applying Gronwall-Bellman lemma to this inequality yields, E xT (t)x(t) a−1 (1 − exp(−ab−1 T ))V (ϕ(t), r0 ). As T → ∞, there exists c such that lim E

t→∞

T

0

xT (t)x(t)|(ϕ(t), r0 )

c V (ϕ(t), r0 ) c

sup

−τ t0

|ϕ(t)|2 .

H∞ Fuzzy Control of Delayed MJFS

43

From (2), the closed loop MJFS deﬁned in (6) is stochastically asymptotically stable. In order to cater the H∞ control problem, let us consider the H∞ performance index for given γ > 0 as J =E

0

∞

z T (t)z(t) − γ 2 wT (t)w(t) dt

(15)

On the other hand, z T (t)z(t) − γ 2 wT (t)w(t) =

r

T μj (xT (t) GTjk + xT (t − τ (t))GTdjk + wT (t)Hwjk )×

j=1

(Gjk x(t) + Gdjk x(t − τ (t)) + Hwjk w(t)) − γ 2 wT (t)w(t) +JV (xt , k, t) − JV (xt , k, t)

under zero initial conditions we have (jj )

J ζ T (t)Ξku ζ(t),

where ζ(t) = x(t − τ (t)) x(t) x(t − δ1 ) . . . x(t − N δ) . . . w(t) z(t) , ⎤ ⎡˜ ˜ ψ11 ψ12 0 ψ˜14 ψ˜15 ⎢ ψ˜22 ψ˜23 ψ˜24 ψ˜25 ⎥ ⎥ ⎢ (jj ) ˜ ˜ ˜ ⎥ Ξku = ⎢ ⎢ ψ33 ψ34 ψ35 ⎥ ⎣ ψ˜44 0 ⎦ ψ˜55 T with ψ˜33 = −γ 2 ; ψ˜15 = Gdjk ; ψ˜25 = Gjk 0 · · · 0 0 · · · 0 0 ; ψ˜35 = Hwjk ; ψ˜55 = −I. (jj ) In fact J 0 if and only if Ξku < 0. This leads to z(t)2 ≤ γw(t)2 . That is

∞ ∞ z T (t)z(t) dt γ 2 wT (t)w(t) dt . E 0

0

(jj ) Ξku

< 0 is not an LMI, as it involves various nonlinear (jj ) matrix terms. Pre and post multiplying Ξku with diag Pk−1 Pk−1 I R−1 I ˜ α = P˜k Qα P˜k , R ˜ α = P˜k Rα P˜k , and the H∞ control gain and letting P˜k = Pk−1 , Q (jj ) −1 matrices be Kj k = Yj k P˜k , we get Φ˜ku which is given in (11) with exception that It is clearly seen that

(1) w1

T = Ajk P˜k + P˜k ATjk + YjT k Bjk + Bjk Yj k +

s

˜1 πkk P˜k Pk P˜k + Q˜1 − R

k =1 −1 and ψ˜ . From the property of Markovian jump transition matrix, the 44 = −R s ˜ −1 P˜k term k =1 πkk P˜k Pk P˜k can be written as P¯ −πkk P˜k and −R−1 = −P˜k R ˜ − 2P˜k . This leads to LMIs (9) and (10). Hence the proof is complete. R

44

4

S. Jeeva Sathya Theesar and P. Balasubramaniam

Numerical Simulations

In this section, we apply the above derived criterion to H∞ control of a modiﬁed computer simulated single link robot arm [6]. The time varying delayed model can be given as x˙ 1 (t) = μx2 (t) + (1 − μ)x2 (t − τ (t)) Mk gl μD(t) (1 − μ)D(t) 1 sin(x1 (t)) − x2 (t) − x2 (t − τ (t)) + u(t) + aw(t) x˙ 2 (t) = − Jk Jk Jk Jk z(t) = x1 (t) + 0.2w(t), k = 1, 2, 3, (16)

where x1 (t), x2 (t), u(t), and z(t) are the angle of the arm, the angular velocity, the control input, and the control output, respectively. w(t) is the exogenous disturbance attenuation input. Let the dynamics jumps between the three states of mass Mk and the inertia Jk such that M1 = J1 = 1, M1 = J1 = 5, and M1 = J1 = 10. The transition ⎡ rate matrix of⎤ the markovian jump op−0.3 0.25 0.05 eration modes is given as Π = ⎣ 0.1 −0.2 0.1 ⎦. The length of the arm 0.03 0.07 −0.1 l = 0.5, acceleration of gravity g = 9.81, and Damping D(t) ∈ [1.8, 2.2]. Also here μ ∈ [0, 1] is the constant retarded coeﬃcient.For TS fuzzy modelsin(x1 )−ρx1 x1 (1−ρ) , x1 = 0 ; ing of (16), we use the membership functions μ1 (x1 ) = 1 x1 = 0 x1 −sin(x1 ) x1 (1−ρ) , x1 = 0 μ2 (x) = , where ρ = 10−2 /π. The MJFS deﬁned in (6) can 0 x1 = 0 be easily obtained matices

with following x1 (t) 0 μ 0 μ 0 μ x(t) = , A12 = , A13 = , , A11 = −gl −2μ −gl −0.4μ −gl −0.2μ x (t) 2

0 μ 0 μ 0 μ , A22 = , A23 = , A21 = −ρgl −2μ −ρgl −0.4μ −ρgl −0.2μ

0 1−μ 0 1−μ Ad11 = Ad21 = , Ad12 = Ad22 = , 0 −2(1 − μ) 0 −0.4(1 − μ)

0 1−μ 0 0 , B11 = B21 = , B12 = B22 = , Ad13 = Ad23 = 1 0.2 0 −0.2(1 − μ)

0 0 B13 = B23 = , Bw1 = Bw2 = , G1 = G2 = 1 0 , and 0.1 0.1 Gw1 = Gw2 = 0.2. The main purpose of giving this model is to ﬁnd the desirable H∞ state feedback controller for time varying delayed single link robot arm model with performance level γ. Now the LMIs (9) and (10) are solved numerically for j, j = 2, k = 3, and N the number of decomposition by Matab LMI toolbox. The maximum allowable delay limit τ is given with H∞ performance level γ. For N = 3, and γ = 1, μ = 0.9, ϕ(t) = [0.5 ∗ π, −1], ∀t ∈ [0.4635, 0], and r0 = 1 the solution states of (16) is simulated and plotted in Fig. 1with Markovian switching signal that changes the mode randomly according to . For reference Fig. 2 is provided

H∞ Fuzzy Control of Delayed MJFS

45

Table 1. Performance of the H∞ control criterions derived in Threorem 1, for time varying delayed single link robot arm model (16) H∞ Performance level / τ N=2 N=3 N=5 γ = 0.3 0.2274 0.3276 0.4430 γ=1 0.2384 0.3486 0.4635

to show how system states behave in three diﬀerent modes separately. It is worth noting that the results presented in [6] is only for constant delay and cannot solve the H∞ control problem of model (16).

8 x1 6

2

x

2

4

r

t

2

state

0

1

−2 −4 −6

0

−8 −10

0

50

40

30

20

10

0

10

20

time

30

40

50

time

Fig. 1. State trajectory of model (15) with Markovian switching signal rt

8

6

4

x1

x1

x

6

3

x

4

2

2

x1 2

4

x

2

2 1 0

0

0 state

state

state

2

−2

−2

−4

−4

−6

−6

−8

−1 −2 −3 −4

−8

0

10

20

30

40

time

50

−10

−5 0

10

20

30 time

40

50

−6

0

10

20

30

40

50

time

Fig. 2. State trajectories of (15) at three diﬀerent modes

Also to cater the need of models that consists of constant delay considering 0 τ˙ (t) μ (having additional information on delay), we can extend the Theorem 1, based on [7].

5

Conclusion

In this paper, we have studied the H∞ control problem of MJFS with time varying delay. The less conservative delay dependent suﬃcient criterions are derived. Using the single link robot arm model, the eﬀectiveness of the conditions

46

S. Jeeva Sathya Theesar and P. Balasubramaniam

are tested numerically. In fact, there are cases when noise and network dropout can reduce the stability and performance of the systems, that will be our future research.

References 1. Xiong, J., Lam, J.: Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers. Automatica 42, 747–753 (2006) 2. Xu, S., Lam, J., Mao, X.: Delay-dependent H∞control and ﬁltering for uncertain Markovian jump systems with time varying delays. IEEE Trans. Circuits Syst. I, Reg. Papers 45, 2070–2077 (2007) 3. Wu, H., Cai, K.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man, Cybern, Part B 36, 509–519 (2006) 4. Nguang, S.K., Assawinchaichote, W., Shi, P., Shi, Y.: Robust H∞ control design for uncertain fuzzy systems with Markovian jumps: an LMI approach. In: Proc. of Amer. Contr. Conf., pp. 1805–1810 (2005) 5. Dong, J., Yang, G.: Fuzzy controller design for Markovian jump nonlinear systems. Int. J. Control, Automation, and Systems 5, 712–717 (2007) 6. Zhang, Y., Xu, S., Zhang, J.: Delay-dependent robust H∞ control for uncertain fuzzy Markovian jump systems. Int. J. Control, Automation, and Systems 7, 520– 529 (2009) 7. Zhang, X.M., Han, Q.: A delay decomposition approach to delay dependent stability for linear systems with time-varying delays. Int. J. of Robust Non. Cont. 19, 1922– 1930 (2009)

Assessing Morningness of a Group of People by Using Fuzzy Expert System and Adaptive Neuro Fuzzy Inference Model Animesh Biswas1,*, Debasish Majumder2, and Subhasis Sahu3 1

Department of Mathematics, University of Kalyani, Kalyani – 741235, India 2 Department of Mathematics, JIS College of Engineering, Kalyani – 741235, India 3 Department of Physiology, University of Kalyani, Kalyani – 741235, India {abiswaskln,debamath}@rediffmail.com, [email protected]

Abstract. In this paper two computational systems, one is based on fuzzy logic system and other is based on adaptive neuro fuzzy inference system (ANFIS), are developed for assessing morningness of a group of people and the result is compared for finding the best system in the assessment process. In fuzzy rule based system, the linguistic terms for assessing morningness are quantified by using fuzzy logic and a fuzzy expert system is developed. On the other side, an ANFIS model is generated to assess data for training and testing the model in accordance with a preference scale adopted to quantify the responses of subjects for a reduced version of Morningness-Eveningness Questionnaire (rMEQ). The result reflects that ANFIS is able to assess morningness better than fuzzy expert inference system. Keywords: Chronotypology, rMEQ, Fuzzy Expert System, Mamdani Fuzzy Inference System, ANFIS, Morningness.

1 Introduction Assessing morningness is now getting importance in various fields since human performance is the key to the success. Morningness, or the preference for morning or evening activities, is an individual difference in circadian rhythms with potential applications in optimizing work schedules, sports performance and academic activities. Circadian rhythms, cyclic fluctuation in many physiological functions, are thought to influence adjustment to shift-work. The researchers are interested in studying morningnes and eveningness orientation by developing several self reporting questionnaires. To measure these parameters often referred to collectively as morningness. Several self reporting questionnaires have been developed to measure diurnal type or distinction between ‘Larks’ and ‘Owls’ [1, 2, 3, 4, 5, 6]. Among the questionnaires, Morningness-Eveningness Questionnaires (MEQ) of Horne and Ostberg [1], Circadian *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 47–56, 2011. © Springer-Verlag Berlin Heidelberg 2011

48

A. Biswas, D. Majumder, and S. Sahu

Typology Questionnaires (CTQ) of Folkard et.al. [3] and Composite Morningness Questionnaires (CMQ) of Smith et.al. [6] were widely used and well accepted. Again, among the three cronotypological questionnaires the reduced version of MEQ (rMEQ) was short, easily understandable and may be considered as the best morningnesseveningness predictor for Indian people [7]. In the present study rMEQ type questionnaire is selected to assess the morningnes of a group of people. In rMEQ, the different types of questions or linguistic terms are asked to fill up the questionnaire. The linguistic terms are very often fuzzily defined. In this context fuzzy logic [8, 9] is seemed to be very much useful to handle the collected data in a proper justified way. The study of fuzzy logic began as early as the 1960s. In the 1970s, fuzzy logic was combined with expert systems to become a fuzzy logic system that was then implemented in many industrial [10, 11], medical [12] and environmental [13] applications. But the application of fuzzy logic in the field of ergonomics, especially, assessing morningness of the group of people is yet to appear in the literature. Again, no standard method is found for transforming human knowledge or experience into the rule base or data base of a fuzzy inference system. Also, there is a need of effective methods for tuning the membership functions so as to minimize the output error. In this perspective, ANFIS [14] can serve as a basis for constructing a set of fuzzy if-then rules with appropriate membership functions to generate the stipulated input-output pairs. In the present study, two types of analysis viz., Fuzzy Expert System based on Mamdani Fuzzy Inference Process and ANFIS based on Takagi-Sugeno Fuzzy Control System are developed in systematic ways and the achieved result is compared in the context of Indian Environment and Western Climatic Environment.

2 Mode of Data Collection The sample consists of 530 post-graduate students randomly taken from University of Kalyani, West Bengal, INDIA who answered the survey item on voluntary basis. Almost fifty percent of the students are resident students and fifty percent are day boarders. They all maintain diurnal schedule. All of them are acquainted with English. All students reported to maintain diurnal schedule. After describing details about the study, the English version of rMEQ was handed over to them. They filled up the questionnaire and return it. The study was conducted between 11:00 a.m. to 4:00 p.m. In the present study the parameters which are considered as fuzzy inputs are Average Rising Time for Feeling Best (X1), Feelings within First Half an Hour after waken up in the morning (X2), Sleeping Time (X3), Feeling Best Time in a Day (X4) and Self Assessment about Morningness (X5) and are measured on a scale ranging from 0 to 5, from 0 to 4, from 0 to 5, from 0 to 5 and from 0 to 6, respectively. Output parameter Morningness (Z) is a linguistic variable and is measured in points on a scale from 0 to 6.

3 Designing Fuzzy Expert System Fuzzy expert system is viewed as an inference system in which expert’s knowledge is used along with fuzzy control theory especially in time of defining fuzzy sets of each

Assessing Morningness of a Group of People

49

linguistic variable and generating rule. The process of fuzzification of the inputs, evaluation of the rules and aggregation of all required rules and making an inference is known as fuzzy control. The type of control theory which is used in the present study to measure the morningness is based on Mamdani type fuzzy controller. The inputs to the fuzzy logic models are linguistic variables which are presented by fuzzy sets. In Mamdani model, the outputs are expressed in terms of linguistic variables described by some fuzzy sets. The fuzzy sets for the inputs are related to the output fuzzy sets through IF-THEN rules, which describe the knowledge about the behavior of the system. The outline of the used model is shown in Fig. 1.

Fig. 1. Outline model of morningness fuzzy expert system

In the proposed approach, rules are generated by the combination of collected data and expert experiences for the fuzzy rule based system. In the present study, arithmetic means of evaluated points corresponding to the responses given by 530 students for each question are calculated first. The evaluated values corresponding to five different linguistic variables are found as X1 = 4.00, X2 = 2.93, X3 = 3.45, X4 = 3.24 and X5 = 3.28. These numerical values are fuzzified by using fuzzification process. For each input linguistic variable, four triangular fuzzy sets are defined as shown in Fig. 2 – Fig. 6 and for the output parameter, three triangular types of fuzzy sets are defined as shown in Fig. 7. After fuzzification of the inputs, the degree to which each part of the antecedent is satisfied for each rule is determined. The fuzzified values and linguistic variables show which rules will be selected to be fired in the fuzzy implication process. In the present study, 16 rules are generated with the help of expert’s opinion as shown in Fig. 8. If the antecedent of a given rule has more than one part, the fuzzy operator AND is applied to obtain only one membership value that represents the result of the antecedent for that rule. Then, this membership value is used for obtaining the output value. In the present study, MIN (minimum) function is used for AND operator. In the next step the aggregation process is performed to combine the derived outputs of each rule into a single fuzzy set. Aggregation only works once for the output variable Morningness, just prior to the final step, defuzzification. Here, the MAX (maximum) method is being used in the aggregation process.

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Fig. 2. Membership Functions for Input Variable Fig. 3. Membership Functions for Input Variable Average Rising Time for Feeling Best Feelings within First Half an Hour after waken up in the morning

Fig. 4. Membership Functions for Input Variable Fig. 5. Membership Functions for Input Variable Sleeping Time Feeling Best Time in a Day

Fig. 6. Membership Functions for Input Variable Fig. 7. Membership Functions for Fuzzy Output Self Assessment about Morningness Parameter Morningness

Fig. 8. Snapshot of Fuzzy rules (partial)

The implication, aggregation and defuzzification calculations have been performed by using Fuzzy Rule Viewer of software MATLAB 7.1 which is also used to define the input and output parameters. The output screen which shows implication of the antecedent, aggregation of the consequent and defuzzification process for Morningness is shown in the following Fig. 9.

Assessing Morningness of a Group of People

51

Fig. 9. The Output Screen

Now, three fuzzy sets are developed to assess the morningness in the following manner and the defuzzified value is put into the sets to find the degree of morningness of the group of students in the University.

Fig. 10. Measurement of the Degree of Morningness

The membership value of morningness corresponding to 4.54, is found as 0.51, 0.23, 0 (Fig. 10). The result shows that the group of students of University of Kalyani is morning oriented and the fuzzy membership degree corresponding to the fuzzy set ‘Morning (M)’ is 0.51 and corresponding to the fuzzy set ‘Intermediate (I)’ is 0.23.

4 The ANFIS Model Formulation The ANFIS architecture is designed for Takagi-Sugeno fuzzy inference system. A typical ANFIS consists of five layers which perform different actions in the ANFIS

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A. Biswas, D. Majumder, and S. Sahu

are described below. Every layer consists of some nodes (circular or square type). The circular type nodes represent nodes that are fixed whereas the square type nodes designate nodes that have parameters which are to be learnt (Fig. 11). For simplicity, a system that has two inputs x and y and one output z has been illustrated below and the rule base of two IF-THEN rules of first order Takagi-Sugeno model has been used for this purpose. Rule 1: if

and

then

Rule 2: if

and

then

where x and y are inputs; and ( 1,2 are appropriate fuzzy sets; , and 1,2 contribute to the output of the system. ( 1,2) are output parameters and (

Fig. 11. Architecture of ANFIS network

The outputs of each of the five layers of ANFIS architecture are described as follows: Layer 1. Every node i in this layer is an adaptive node with a node function , for

1,2;

, for

3,4.

where 1, 2 and 3, 4 are the linguistic terms associated with the 1, 2 and 3,4 can adopt inputs x and y, respectively and any membership function, e.g., triangular, trapezoidal, generalized bell-shaped, Gaussian functions or other types. In fact, 1,2,3,4) specify the degrees to which the given inputs satisfy the corresponding linguistic terms. Layer 2. Every node i in this layer is a fixed node which multiplies the signals coming from the nodes of the previous layer and sends the product out. The outputs 1, 2 of this layer are represented as: , where

1,2) represent the firing strength of the

1,2. rule.

Assessing Morningness of a Group of People

53

Layer 3. All nodes in this layer are also fixed nodes. The node of this layer calcurule’s firing strength to the sum of the firing strengths of all lates the ratio of the rules. The outputs 1, 2 of this layer are represented as: ,

∑

where

1,2.

1, 2 are known as normalized firing strengths.

Layer 4. Every node i in this layer is an adaptive node with a node function , where , 1,2 are the outputs of the layer 3 and nomials (for a first order Sugeno model). Parameters layer are referred to as consequent parameters.

1,2 1, 2 are first order polyand 1, 2 in this

,

Layer 5. The single node in this layer is a fixed layer which computes the overall output as the summation of all incoming signals, i.e., ∑

∑

∑

.

In this paper ANFIS toolbox of software MATLAB 7.1 is used for creating an ANFIS model for which the following stages are performed to assess the morningness of a group of people: 1. Loading 530 pairs sample data (450 pairs for training purpose and 80 pairs for testing purpose) as shown in Fig.12. 2. Generating a Fuzzy Inference System (FIS) model (Fig. 13) using subtractive clustering method with the following parameters as shown in Table 1. Table 1. Parameters of subtractive clustering method Parameters

Range of Influence

Squash Factor

Accept Ratio

Reject Ratio

Value

0.5

1.51

0.5

0.15

3. Training the FIS for 100 steps to tune all the modifiable parameters using hybrid learning algorithm which is a combination of back propagation algorithm and least means square method and setting error of tolerance at zero as there is a no information about how the error behave (Fig. 14). 4. Testing the FIS for validity (Fig. 15).

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A. Biswas, D. Majumder, and S. Sahu

Fig. 12. Plotted data (training and testing)

Fig. 13. The Structure of FIS

Fig. 14. Diagram of error rate in different training stages

Fig. 15. Diagram of the prediction result

Thus an ANFIS model is developed for assessing morningness of a group of people and it is found that the average testing error is negligibly small. Now, considering the values corresponding to five different linguistic variables evaluated from the preference scale values of rMEQ type questionnaire are taken as X1 = 4.00, X2 = 2.93, X3 = 3.45, X4 = 3.24 and X5 = 3.28. The assessment through the newly generated ANFIS model is found as Z = 3.38 (Fig. 16).

Fig. 16. Diagram of ANFIS rules and output

Fig. 17. Measurement of the Degree of Morningness

The membership value of morningness corresponding to 0.13,

0.81,

3.38, is found as

0 (Fig. 17).

Assessing Morningness of a Group of People

55

The result obtained by proceeding through this process is found as ‘Intermediate’ type with membership value 0.81 and morning oriented with membership value 0.13.

5 Conclusions In this study two methodologies are developed for assessing morningness in a justified way. The first approach is based on fuzzy expert system. In this method the rules which are considered in the modeling process are based on expert’s expertise on that domain. The result shows that the group of students is morning oriented. Again, in the second methodology, an ANFIS method is developed. In this process it is found that the group of people is less morning oriented as that of first approach. In view of physiological perspective, the later method is more justifiable than the former one. Also ANFIS model is used as an alternative way to measure morningness in which the use of hybrid learning procedure is advantageous as the proposed architecture can refine fuzzy if-then rules obtained from human experts to describe the input-output behavior of a complex system. Moreover, if human expertise is not available, we can still set up intuitively reasonable initial membership functions and start the learning process to generate a set of fuzzy if-then rules to approximate a desired data set. Also, the structure of ANFIS is identified by using subtractive clustering method which concerns with the selection of an appropriate input-space partition style and the number of membership functions on each input and is equally important to the successful applications of ANFIS. Effective partition of input-space can decrease the number of the rules and thus increase the speed in both learning and application phases. Further the ANFIS model can be improved by using genetic algorithm in the learning process of modifiable parameters. However, it is hoped that the proposed methodologies may open up new horizon in the way of assessing morningness through fuzzy control. Acknowledgements. The authors remain grateful to the anonymous referees for their constructive comments and valuable suggestions in improving the quality of the paper. Also the authors are thankful to all the subjects who volunteered for the study and gladly given consent.

References 1. Horne, J.A., Ostberg, O.: A Self Assessment Questionnaire to Determine Morningness, Eveningness. Int. J. Chronobio. 4, 97–110 (1976) 2. Horne, J.A., Ostberg, O.: Individual Differences in Human Circadian Rhythm. Biological Psycho. 5, 179–190 (1977) 3. Folkard, S., Monk, T.H., Lobban, M.C.: Toward a Predictive Test of Adjustment to Shift Work. Ergonomics 22, 79–91 (1979) 4. Torsvall, L., Akerstedt, T.: A Diurnal Type Scale Construction, Consistency and Validation in Shift Work. Environment and Health 6, 283–290 (1980) 5. Moog, R.: Morningness- evening Types and Shift Work a Questionnaire Study. In: Reinberg, A., Vieux, N., Andlaner, P.C. (eds.) Night and Shift Work: Biological and Social Aspects, pp. 481–488. Pergamon Press, Oxford (1981)

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6. Smith, C.S., Reilly, C., Midkiff, K.: Evaluation of Three Circadian Rhythm Questionnaires with Suggestion for Improved Measures of ‘Morningness’. J. Appl. Psycho. 74, 728–738 (1989) 7. Sahu, S.: An Ergonomic Study on Suitability of Choronotypology Questionnaires on Bengalee (Indian) Population. Indian J. Biological Sci. 15, 1–11 (2009) 8. Zadeh, L.A.: Fuzzy Sets. Inf. and Control. 8, 338–353 (1965) 9. Pal, B.B., Biswas, A.: A Fuzzy Multilevel Programming Method for Hierarchical Decision Making. In: Pal, N.R., Kasabov, N., Mudi, R.K., Pal, S., Parui, S.K. (eds.) ICONIP 2004. LNCS, vol. 3316, pp. 904–911. Springer, Heidelberg (2004) 10. Buckley, D.I.: The Nature of Structural Design and Safety. Ellis Horwood, Chichester (1980) 11. Gurcanli, G.E., Mungen, U.: An Occupational Safety Risk Analysis Method at Construction Sites using Fuzzy Sets. Int. J. Industrial Ergonomics 39, 371–387 (2009) 12. Vila, M.A., Delgado, M.: On Medical Diagnosis using Possibility Measures. Fuzzy Sets and Syst. 10, 211–222 (1983) 13. Cao, H., Chen, G.: Some Applications of Fuzzy Sets of Meteorological Forecasting. Fuzzy Sets and Syst. 9, 1–12 (1983) 14. Venugopal, C., Devi, S.P., Rao, K.S.: Predicting ERP User Satisfaction – an Adaptive Neuro Fuzzy Inference System (ANFIS) Approach. Intelligent Inform. Mgmt. 2, 422–430 (2010)

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink Kumaresan Nallasamy, Kuru Ratnavelu, and Bernardine R. Wong Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia [email protected]

Abstract. In this paper, the unsteady Navier-Stokes Takagi-Sugeno (T-S) fuzzy equations (UNSTSFEs) are represented as a differential algebraic system of strangeness index one by applying any spatial discretization. Since such differential algebraic systems have a difficulty to solve in their original form, most approaches use some kind of index reduction. While processing this index reduction, it is important to take care of the manifolds contained in the differential algebraic equation (DAE) /singular system (SS) for each fuzzy rule. The Navier-Stokes equations are investigated along the lines of the theoretically best index reduction by using several discretization schemes. Applying this technique, the UNSTSFEs can be reduced into DAE. Optimal control for NavierStokes T-S fuzzy system with quadratic performance is obtained by finding the optimal control of singular T-S fuzzy system using Simulink. To obtain the optimal control, the solution of matrix Riccati differential equation (MRDE) is found by solving differential algebraic equation (DAE) using Simulink approach. The solution of Simulink approach is equivalent or very close to the exact solution of the problem. An illustrative numerical example is presented for the proposed method. Keywords: Differential algebraic equation, Matrix Riccati differential Equation, Navier-Stokes equation, Optimal control and Simulink.

1 Introduction A fuzzy system consists of linguistic IF-THEN rules that have fuzzy antecedent and consequent parts. It is a static nonlinear mapping from the input space to the output space. The inputs and outputs are crisp real numbers and not fuzzy sets. The fuzzification block converts the crisp inputs to fuzzy sets and then the inference mechanism uses the fuzzy rules in the rule-base to produce fuzzy conclusions or fuzzy aggregations and finally the defuzzification block converts these fuzzy conclusions into the crisp outputs. The fuzzy system with singleton fuzzifier, product inference engine, center average defuzzifier and Gaussian membership functions is called as standard fuzzy system [18]. Two main advantages of fuzzy systems for the control and modelling applications are (i) uncertain or approximate reasoning, especially difficult to express a mathematical P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 57–64, 2011. © Springer-Verlag Berlin Heidelberg 2011

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N. Kumaresan, K. Ratnavelu, and B.R. Wong

model (ii) decision making problems with the estimated values under incomplete or uncertain information [20]. Stability and optimality are the most important requirements in any control system. For optimality, it seems that the field of optimal fuzzy control is totally open. Singular systems contain a mixture of algebraic and differential equations. In that sense, the algebraic equations represent the constraints to the solution of the differential part. These systems are also known as degenerate, descriptor or semi-state and generalized state-space systems. The system arises naturally as a linear approximation of system models or linear system models in many applications such as electrical networks, aircraft dynamics, neutral delay systems, chemical, thermal and diffusion processes, large-scale systems, robotics, biology, etc., see [3,4,5,11]. As the theory of optimal control of linear systems with quadratic performance criteria is well developed, the results are most complete and close to use in many practical designing problems. The theory of the quadratic cost control problem has been treated as a more interesting problem and the optimal feedback with minimum cost control has been characterized by the solution of a Riccati equation. Da Prato and Ichikawa [6] showed that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Solving the Matrix Riccati Differential Equation (MRDE) is a central issue in optimal control theory. The needs for solving such equations often arise in analysis and synthesis such as linear quadratic optimal control systems, robust control systems with H2 and H∞-control [22] performance criteria, stochastic filtering and control systems, model reduction, differential games etc. One of the most intensely studied nonlinear matrix equations arising in Mathematics and Engineering is the Riccati equation. This equation, in one form or another, has an important role in optimal control problems, multivariable and large scale systems, scattering theory, estimation, detection, transportation and radiative transfer [7]. The solution of this equation is difficult to obtain from two points of view. One is nonlinear and the other is in matrix form. Most general methods to solve MRDE with a terminal boundary condition are obtained on transforming MRDE into an equivalent linear differential Hamiltonian system [8]. By using this approach, the solution of MRDE is obtained by partitioning the transition matrix of the associated Hamiltonian system [17]. Another class of methods is based on transforming MRDE into a linear matrix differential equation and then solving MRDE analytically or computationally [12,15,16]. However, the method in [14] is restricted for cases when certain coefficients of MRDE are non-singular. In [8], an analytic procedure of solving the MRDE of the linear quadratic control problem for homing missile systems is presented. The solution K(t) of MRDE is obtained by using K(t)=p(t)/f(t), where f(t) and p(t) are solutions of certain first order ordinary linear differential equations. However, the given technique is restricted to single input. Simulink is a MATLAB add-on package that many professional engineers use to model dynamical processes in control systems. Simulink allows creating a block diagram representation of a system and running simulations very easily. Simulink is really translating block diagram into a system of ordinary differential equations. Simulink is the tool of choice for control system design, digital signal processing (DSP) design, communication system design and other simulation applications [1]. This paper focuses upon the implementation of Simulink approach to compute optimal control for linear singular fuzzy system.

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink

59

Although parallel algorithms can compute the solutions faster than sequential algorithms, there has been no report on Simulink solutions for MRDE. This paper focuses upon the implementation of Simulink approach for solving MRDE in order to get the optimal solution. This paper is organized as follows. In section 2, the statement of the problem is given. In section 3, solution of the MRDE is presented. In section 4, numerical example is discussed. The final conclusion section demonstrates the efficiency of the method.

2 Statement of the Problem In computational fluid dynamics, a typical example of the equations of gas dynamics under the assumptions of incompressibility is the Navier Stokes equation [19]. It consists of as many differential equations as the dimension of the model indicates and the condition of incompressibility, see e.g.[13]: ∂u = –u.∇+ν Δu –∇p + f ∂t 0 = ∇.u

(1)

These equations, together with the appropriate initial and boundary conditions, are yet d to be solved in Ωx [0,T], where Ω is a bounded open domain in R [d = (2 or 3) dimension of the model] and T the endpoint of the time interval. Two dimensional cases are considered here for easy computation and simplification. The results are valid for a three dimensional model as well. The domain of reference is assumed to be rectangular. This is indeed a restriction. But, at the exact places, it will be pointed out, whether some techniques may be generalized to other domains or not. After applying the method of lines (MOL), i.e. carrying out a spatial discretization by finite difference or finite element techniques, the equations in (1) can be written as a differential algebraic system. Mu′(t) = K(u)u(t) –Bp(t) +f(t) T 0 = B u(t)

(2)

See [2]. Here u(t), p(t) and f(t) are approximations to the time and space dependent quantities u, p and f of (4). The matrix M is symmetric and positive definite. Quantity B stands for discrete gradient operator, while K(u) represents linear and nonlinear velocity terms. The DAE (2) is of a higher index (i.e. non-decoupled), since pressure p does not appear in the algebraic condition. If we assume that B is a full column rank, then the differentiation index is two. Since p is only determined up to an additive constant, B has, in general, a rank deficiency which causes the undeterminedness of at least one solution component. The concept of the differentiation index can not be applied to such systems. Kunkel and Mehrmann [9] have generalized the index concept to the case of over and underdetermined DAEs. Their so called strangeness index (or s-index) μ is the number of additional block columns needed in the derivative array [10] that can filter out a strangeness free system by transformations from the left. This system then represents a DAE of differentiation index one with possibly undetermined components

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or a system of ordinary differential equations. Therefore, μ is one lower than the differentiation index if the system is a DAE of at least differentiation index one without undeterminedness. For ordinary differential equations (differentiation index zero), μ is defined as zero. The system (2) is of a higher index, namely s-index 1. Such systems have a difficulty to solve in their original form because of the differential and algebraic components and strangeness [9]. It would be appropriate to remove this strangeness before solving the DAE. In most of the Navier-Stokes solution techniques, this is done without explicitly mentioning that an index reduction is carried out. If the index reduction is omitted, the results may become unsatisfactory, especially in the unsteady case. In [21], examples are computed when a steady state is reached and it is stated that satisfactory smoothness is achieved. But, this is completely impractical, if long time computations are carried out. A characterization is also guaranteed by the concept of s-index when the DAE has an unique solution. However, this is not the main advantage of this approach over the usual concept of the differentiation index. The biggest progress seems to be that [9] provides a way to reformulate the higher index DAE as a strangeness free system with the same dimension and same solution structure as the original system. In other words, it is possible to rewrite a DAE of higher index in a so called normal form of s index zero. This form not only reflects the manifold included in the original system but also all the hidden manifolds. Thus, using strangeness free normal form, a consideration of all manifolds is ensured which makes this approach superior over other index reduction variants. Moreover, the derivative term is not transformed so that no errors in time are caused by the index reduction strategy for Navier Stokes equations. By this index reduction process, the linear differential algebraic system of arbitrary sindex μ, E x′(t)=A(t)x(t)+f(t), under suitable assumptions [10] can be transformed into strangeness free normal form by means of utmost μ.3+2 rank decisions. This procedure is described in [9]. Consider the two dimensional Navier-Stokes (T-S) fuzzy equations ∂u = –ui.∇+ν Δui –∇p + fi(t) ∂t

(3)

0 = ∇.ui The above equation can be transformed into singular T-S fuzzy system using the method described in [9]. The singular time-invariant system by taking fi(t) = 0 in (3) Ei x′(t)=Aix(t)+Biu(t), x(0)=x0, n

(4)

where the matrix Ei is singular, x(t) ∈ R is a generalized state space vector and u(t) m nxn nxm ∈ R is a control variable. Ai∈ R and Bi∈R are known coefficient matrices associated with x(t) and u(t) respectively, x0 is given initial state vector and m ≤ n. In order to minimize both state and control signals of the feedback control system, a quadratic performance index is usually minimized:

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink

61

tf J=

½

∫[xT(t)Qx(t)+uT(t)Ru(t)]dt,

T

T

x (tf)Ei SEix(tf)+ ½

0 nxn

nxn

where the superscript T denotes the transpose operator, S∈R and Q∈R are symmetric and positive definite (or semidefinite) weighting matrices for x(t), R∈Rm xm is a symmetric and positive definite weighting matrix for u(t). It will be assumed that |sEi-Ai| ≠ 0 for some s. This assumption guarantees that any input u(t) will generate one and only one state trajectory x(t). If all state variables are measurable, then a linear state feedback control law –1

u(t)= – R

T

Bi Ki(t)Eix(t),

nxn

where Ki(t) ∈R is a symmetric matrix and the solution of MRDE. The relative MRDE for the linear singular fuzzy system (4) is T

T

T

T

–1

Ei Ki′(t)Ei + Ei Ki(t)Ai + Ai Ki(t)Ei +Q – Ei Ki(t) BiR

T

Bi Ki(t)Ei = 0

(5)

T

with the terminal condition Ki(tf)= Ei SEi. In the following section, the MRDE (5) is going to be solved for Ki(t) in order to get the optimal solution.

3 Simulink Solution of MRDE Simulink is an interactive tool for modelling, simulating and analyzing dynamic systems. It enables engineers to build graphical block diagrams, evaluate system performance and refine their designs. Simulink integrates seamlessly with MATLAB and is tightly integrated with state flow for modelling event driven behavior. Simulink is built on top of MATLAB. A Simulink model for the given problem can be constructed using building blocks from the Simulink library. The solution curves can be obtained from the model without writing any codes. As soon as the model is constructed, the Simulink parameters can be changed according to the problem. The solution of the system of differential equation can be obtained in the display block by running the model. 3.1 Procedure for Simulink Solution Step 1. Select the required number of blocks from the Simulink Library. Step 2. Connect the appropriate blocks. Step 3. Make the required changes in the simulation parameters. Step 4. Run the Simulink model to obtain the solution.

4 Numerical Example Consider the optimal control problem: Minimize tf J= ½ xT(tf)EiTSEix(tf)+ ½ ∫[xT(t)Qx(t)+uT(t)Ru(t)]dt, 0

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subject to the linear singular fuzzy system Ri : If xj is Tji(mji,σji), i = 1, 2 and j = 1, 2,3, then Ei x′(t)=Aix(t)+Biu(t), x(0)=x0 , where S= 1.1517 0.1517 0.1517 1

A2=

–2 2 0 –4

Ei = ,

Bi = ,

1 0

0 0

A1= ,

0 R=1, Q = 1 ,

–1 1 0 –2

,

1 1 1 1

The numerical implementation could be adapted by taking tf =2 for solving the related MRDE of the above linear singular fuzzy system with the matrix A1 . The appropriate matrices are substituted in MRDE. The MRDE is transformed into differentia algebraic equation (DAE) in k11 and k12. The DAE can be changed into a system of differential equations by differentiating the algebraic equation. In this problem, the value of k22 of the symmetric matrix K(t) is free and let k22=0. Then the optimal control of the system can be found out by the solution of MRDE. 4.1 Solution Obtained Using Simulink The Simulink model is constructed for MRDE. The Simulink model is shown in Figure 1. The numerical solution of MRDE is calculated by Simulink and displayed in Table 1. The numerical solution curve of MRDE by Simulink is illustrated in Figure 2.

Fig. 1. Simulink Model for MRDE

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink

63

Fig. 2. Simulink Curve for MRDE Table 1. Simulink Solution of MRDE t 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

k11 0.0003 0.0007 0.0015 0.0033 0.0074 0.0164 0.0368 0.0828 0.1891 0.4467 1.1517

k12 -0.9997 -0.9993 -0.9985 -0.9967 -0.9926 -0.9836 -0.9632 -0.9172 -0.8109 -0.5533 0.1517

Similarly the solution of the above system with the matrix A2 can be found out using Simulink.

5 Conclusion The optimal control for Navier-Stokes fuzzy equations can be obtained by finding the optimal control for DAE. The optimal control is found out by solving MRDE using Simulink. To obtain the optimal control, the solution of MRDE is computed by solving Differential algebraic equation (DAE) using Simulink. The Simulink solution is equivalent to the exact solution of the problem. Accuracy of the solution computed by Simulink approach to the problem is qualitatively better. A numerical example is given to illustrate the proposed method. Acknowledgements. The funding of this work by the UMRG grant (Account No: RG099/10AFR) is gratefully acknowledged.

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N. Kumaresan, K. Ratnavelu, and B.R. Wong

References 1. Albagul, A., Khalifa, O.O., Wahyudi, M.: Matlab and Simulink in Mechatronics Education. Int. J. Eng., Edu. (IJEE) 21, 896–905 (2005) 2. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical solution of Initial Value Problems in Differential Algebraic Equations. SIAM, North Holland (1989) 3. Campbell, S.L.: Singular systems of differential equations. Pitman, Marshfield (1980) 4. Campbell, S.L.: Singular systems of differential equations II. Pitman, Marshfield (1982) 5. Dai, L.: Singular control systems. Lecture Notes in Control and Information Sciences. Springer, New York (1989) 6. Da Prato, G., Ichikawa, A.: Quadratic control for linear periodic systems. Appl. Math. Optim. 18, 39–66 (1988) 7. Jamshidi, M.: An overview on the solutions of the algebraic matrix Riccati equation and related problems. Large Scale Systems 1, 167–192 (1980) 8. Jodar, L., Navarro, E.: Closed analytical solution of Riccati type matrix differential equations. Indian J. Pure and Appl. Math. 23, 185–187 (1992) 9. Kunkel, P., Mehrmann, V.: A new class of discretization methods for the solution of linear differential algebraic equations. SIAM J. Numer. Anal. 33, 1941–1961 (1996) 10. Kunkel, P., Mehrmann, V.: Local and global invariants of linear differential algebraic equations and their relation. Electronics Trans. Numer. Anal. 4, 138–157 (1996) 11. Lewis, F.L.: A survey of linear singular systems. Circ. Syst. Sig. Proc. 5(1), 3–36 (1986) 12. Lovren, N., Tomic, M.: Analytic solution of the Riccati equation for the homing missile linear quadratic control problem. J. Guidance. Cont. Dynamics 17, 619–621 (1994) 13. Quartapelle, I.: Numerical Solution of the incompressible Navier-Stokes Equations. Birkhäuser, Basel (1993) 14. Razzaghi, M.: A Schur method for the solution of the matrix Riccati equation. Int. J. Math. and Math. Sci. 20, 335–338 (1997) 15. Razzaghi, M.: Solution of the matrix Riccati equation in optimal control. Information Sci. 16, 61–73 (1978) 16. Razzaghi, M.: A computational solution for a matrix Riccati differential equation. Numerical Math. 32, 271–279 (1979) 17. Vaughu, D.R.: A negative exponential solution for the matrix Riccati equation. IEEE Trans. Automat. Control 14, 72–75 (1969) 18. Wang, L.X.: Stable and optimal fuzzy control of linear systems. IEEE Trans. Fuzzy Systems 6(1), 137–143 (1998) 19. Weickert, J.: Navier-Stokes equations as a differenctial algebraic system. Preprint SFB 393/96-08, Technische Universität Chemnitz-Zwickau (1996) 20. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inform. Sciences, Part I: (8), 199–249, Part II:(8), 301–357, Part III:(9), 43–80 (1975) 21. Zeng, S., Wesseling, P.: Multigrid solution of the incompressible Navier-Stokes equations in general coordinates. SIAM J. Numer. Anal. 6, 1764–1784 (1994) 22. Zhou, K., Khargonekar, P.: An algebraic Riccati equation approach to H ∞ optimization. Systems Control Lett. 11, 85–91 (1988)

Multi-objective Optimization in VLSI Floorplanning P. Subbaraj, S. Saravanasankar, and S. Anand National Centre for Advance Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, India [email protected]

Abstract. This paper describes use of a multiobjective optimization method, AMOSA to the VLSI ﬂoorplanning problem. The proposed model provides for decision maker choice from among the diﬀerent tradeoﬀ solutions. The objective is to minimize the area and wirelength with the ﬁxed out line constraint. AMOSA is improved with B*tree and B*tree coded AMOSA is introduced to improve the performance of AMOSA for VLSI ﬂoorplanning problem. The MCNC benchmarks are considered as test system. The results are compared and validated. Keywords: VLSI, Floorplanning, Simulated Annealing Algorithm and AMOSA.

1

Introduction

Floorplanning is an important stage in the Very Large Scale Integrated circuit (VLSI) design process. After partitioning the relative location of the diﬀerent modules in the layout, needs to be decided before the placement (placing the cells in the exact location) stage, with the objective of minimizing the total area occupied by the circuit. Floorplanning proved to be NP hard [7]. For years many algorithms have been developed applying stochastic search methods, such as simulated annealing and genetic algorithms, PSO, however only a few of them work explicitly on multi objective optimization [6,4]. Most of the literatures used the weighted sum approach to minimize the area and wire length simultaneously. This approach always has a diﬃculty to assign the weights and may results in undesirable bias towards a particular objective. In this paper we proposes B*tree coded AMOSA for VLSI ﬂoorplanning problem. The solution is encoded as a B*tree. Proposed algorithm has the advantages of AMOSA and usage of B*tree produce less space complexity [23].

2

VLSI Floorplanning Problem

The VLSI ﬂoorplanning problem has two objectives, minimizing the area and minimizing the wirelength. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 65–72, 2011. c Springer-Verlag Berlin Heidelberg 2011

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P. Subbaraj, S. Saravanasankar, and S. Anand

Objective Function and Constraint

The objective function of the problem is formulated as follows. The Cost function Φ for a ﬂoor plan solution F is given by, A minΦ(F1 ) = (1) + K3 (arcost)2 normA W.L minΦ(F2 ) = (2) + K3 (arcost)2 normW.L Where, Φ(F1 )= Area Objective. Φ(F2 )= Wirelength Objective. A= area of the current solution F . normA = average area of 1000 random iteration. W.L = wire length of the current solution F . normW.L = average wirelength of 1000 random iteration. K3 = weight factor for aspect ratio is assigned as one. arcost = ( Current ﬂoorplan aspect ratio - Desired ﬂoorplan aspect ratio ). Desired aspect ratio = 1.0. Wire length of a net is calculated by the half perimeter of the minimal rectangle enclosing the centers of the modules. In this work Aspect Ratio (AR) is considered to be the given constraint for the modern ﬁxed outline ﬂoorplanning problem. For a collection of blocks with total area (A) and a given maximum permissible percentage of dead space (Γ ), a ﬁxed outline with an Aspect Ratio R∗ (i.e. height/width) is constructed .The height H ∗ and width W ∗ of the outline are deﬁned as in [1]; Only solutions satisfying the ﬁxed outline constraint are considered as feasible. (3) H ∗ = (1 + Γ )AR∗ , W ∗ = (1 + Γ )A/R∗ 2.2

B*tree Encoding in AMOSA

Floorplan of a VLSI circuit can be divided into two categories. 1. The slicing structure [16,22]. 2. The non-slicing structure [8,14,21]. A non slicing structure can be represented by, sequence pair [12], boundary slicing grid (BSG) [14,9,11,13,15], Otree [8] and B*tree [23]. Yun-chih chang et al., proposed a B*tree representaion based on ordered binary trees and the admissible placement [8]. Yun-chich chang proved [23] that B* tree representation runs 4.5 times faster, consumes about 60 percentage lesser memory and results in smaller silicon area than the Otree [8]. Normal AMOSA having sequence representaion. Due to these advantages of B* tree, it has been used in AMOSA to represent the solution.

3

B*tree Coded Archived Multiobjective Simulated Annealing Algorithm(AMOSA)

Classical optimization methods can at best ﬁnd one solution in one simulation run, thereby making those methods inconvenient to solve multiobjective optimization problems. Evolutionary algorithms, on the other hand, can ﬁnd multiple optimal solutions in one single run due to their population approach.

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67

The AMOSA algorithm is described in [2]. In this work B*tree coded multiobjective simulated annealing algorithm (AMOSA) is selected to solve VLSI ﬂoorplanning problem, because it has been demonstrated to be among the most eﬃcient algorithm for multiobjecitve optimization on a number of benchmark problems [2]. In addition, it has been shown that AMOSA outperforms two other contemporary multiobjective evolutionary algorithms: Pareto-archived evolution strategy (PAES) and Nondominated sorting genetic algorithm-II (NSGA-II) in terms of ﬁnding a diverse set of solutions and in converging near the true Paretooptimal set. 3.1

AMOSA Algorithm

AMOSA is a simulated annealing based multiobjective optimization algorithm that incorporates the concept of archive in order to provide a set of tradeoﬀ solutions for the VLSI ﬂoorplanning problem. Simulated Annealing algorithm has been proved [5,10] to be very simple compared to Genetic Algorithm [20] and particle swarm optimization [3] methodologies. Simulated annealing [10] is widely used for ﬂoorplanning [17,19,18,5]. It is an optimization scheme with non zero probability for accepting inferior (uphill) solutions. The probability depends on the diﬀerence of the solution quality and the temperature. The probability is typically deﬁned by (4) prob = min 1, e(−Δc/T ) Where ΔC is the diﬀerence between cost of the neighboring state and current state, and T is the current temperature. In the classical annealing schedule, the temperature is reduced by a ﬁxed ratio λ, for each iteration of the annealing process. The AMOSA algorithm incorporates the concept of an archive where the non-dominated solutions seen so far are stored. Two limits are kept on the size of the archive: a hard or strict limit denoted by HL, and a soft limit denoted by SL. The algorithm begins with the initialization of a number (γ × SL, 0 < γ < 1 ) of solutions each of which represents a state in the search space. The multiple objective functions are computed. Each solution is reﬁned by using simple hillclimbing and domination relation for a number of iterations. Thereafter the non-dominated solutions are stored in the archive until the size of the archive increases to SL. If the size of the archive exceeds HL, a single-linkage clustering scheme is used to reduce the size to HL. Then, one of the points is randomly selected from the archive. This is taken as the current-pt, or the initial solution, at temperature T=Tmax. The current-pt is perturbed to generate a new solution named new-pt, and its objective functions are computed. The domination status of the new-pt is checked with respect to the current-pt and the solutions in the archive. A quantity called amount of domination Δdoma,b between two solutions a and b is deﬁned as follows: Δdoma,b =

M i=1,fi (a)=fi (b)

fi (a) − fi (b) Ri

(5)

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where fi (a) and fi (b) are the ith objective values of the two solutions and Ri is the corresponding range of the objective function. Based on domination status diﬀerent cases may arise viz., accept the (i) new-pt, (ii) current-pt or (iii) a solution from the archive. Again, in case of overﬂow of the archive, clustering is used to reduce its size to HL. The process is repeated N times for each temperature that is annealed with a cooling rate of α(< 1) till the minimum temperature Tmin (Tmin = 1 ∗ e−4 ) is attained. The process thereafter stops, and the archive contains the ﬁnal non-dominated solutions. 3.2

Cooling Schedule and Local Search

To implement the cooling schedule of AMOSA, temperature updating function Tn is formulated as shown in equation 6. The temperature is reduced as per the cooling schedule. Tn = Tn−1 ∗ α (6) Local search is performed for every Annealing temperature. Number of local searches is decided by the equation 7. N = {k1 ∗ SCKT }

(7)

Where, SCKT =Size of the circuit. k1 =A constant assigned with a value of 400. k1 gives more exploration capability to AMOSA. The parameter SCKT gives ﬂexibility to the algorithm. When the size of the circuit increases, the local search also will increase. 3.3

Creation of Neighboring Solution

To get the neighboring solution, Yun-Chih Chang proposed a Fast-SA method, where the perturbation of B ∗ tree with the perturbation probability is done using only one of the following three methods [23]; 1) Rotate a block, 2) Move a node/block to another place and 3) Swap two nodes/blocks. The same procedure has been adopted in B*tree coded AMOSA.

4

Obtaining the True Pareto-Front

An alternative approach, weighted sum method with (single objective) FastSA, is experimented to solve the VLSI Floorplanning problem. It is used to validate the results obtained using AMOSA. A pareto optimal set is deﬁned as a set of solutions that are not dominated by any feasible member of the search space; they are optimal solutions of the multi objective optimization problem. To evaluate the performance of AMOSA, a reference Pareto-optimal front is needed. One of the classical methods is weighted sum method. Using this method only one solution may be found along the pareto-front in a single simulation run. the FastSA is used to obtain the true pareto-front. the sum of weights is always 1. The weight value for the ﬁrst objective function is w1 . The weight w1 is linearly varied from 0 to 1 produce the pareto optimal front.

Multi-objective Optimization in VLSI Floorplanning

5

69

Test Results

The proposed B*coded AMOSA is implemented in C++ programming language on Intel Pentium-D 3.20GHz IBM machine. B*tree coded AMOSA and FastSA are tested with MCNC (Microelectronics center of North Carolina) benchmarks to ﬁnd the ﬂoorplan solutions with the dual objectives of minimizing the wire length and area. Details of the benchmarks is shown in Table 1. Table 1. Details of MCNC bench marks MCNC Circuit #Modules # Nets # I/O Pad # Pins hp 11 83 45 309 ami33 33 123 42 522 ami 49 49 408 22 953

5.1

Parameters, Results and Discussion

The results are sensitive to algorithm parameters, typical of heuristic techniques. Hence, it is required to perform repeated simulations to ﬁnd suitable values for the parameters. The best parameters for the AMOSA, selected through twenty test simulation runs. The parameters are, Alpha (α)=0.84. Initial Temp (Tini )=100. Soft Limit (SL)=101. Hard Limit (HL)=51. The benchmark has diﬀerent sizes of layouts, as discussed in section 5. The extreme solution of all benchmarks is shown in table 2 and shown on the pareto-front of Fig. 1(a), 1(b) and 1(c) Identiﬁcation of pareto-front. The pareto-optimal front obtained by diﬀerent using AMOSA and FastSA for benchmark ami49, ami33, and hp is shown in Fig. 1(a), 1(b) and 1(c) respectively. It shows the trade of between the area and wirelength. In single objective optimization FastSA produces the solution in diﬀerent runs for diﬀerent weights as described in section 4. In contrast, B*tree coded AMOSA produces the pareto optimal front in a single simulation run and the execution time is less ( 200 sec for ami49 ) compare to the single objective optimization process. Table 2. Extreme solutions of FastSA and AMOSA ami49 Bench mark

Area (mm2 ) FastSA 36.65 48.22 B*tree 36.65 coded AMOSA 47.16

WL (mm) 1833 837.4 1833

ami33 Area (mm2 ) 1.19 1.408 1.19

WL (mm) 114.5 57.64 114.5

hp Area (mm2 ) 9.173 15.08 9.173

WL (mm) 275.4 178.6 275.4

725.6 1.366 51.09 10.73 157.2

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(a) ami49

(b) ami33

(c) hp

Fig. 1. Pareto-optimal front of : (a), (b) and (c)

Trade-oﬀ analysis. From the Fig. 1(a), 1(b) and 1(c) it is clear that the increase in Area gives decrease in wirelength, and this directly causes variation in white space. This observation will help the decision maker to select a single solution from the pareto-optimal front. Form the Fig. 1(c), it is very clear that Fast SA can not able to ﬁnd the optimal solutions when comparing to AMOSA.

6

Conclusion

In this paper an attempt is made to solve multiobjective VLSI ﬂoorplanning problem using B*tree coded AMOSA. Multiobjective optimization enables evolution of tradeoﬀ solutions between diﬀerent problem objectives through identiﬁcation of the Pareto-front. The B*tree coded AMOSA algorithm provides that this can be done with very high computational eﬃciency and provides more exploration capability. This approach is quite ﬂexible so that other formulations using diﬀerent objectives and/or a larger number of objectives are possible.Proposed approach has been compared with the FastSA method, from the results it is clear that, “B*tree coded AMOSA” enabling the decision maker to understand their options, ultimately resulting in lower White space in lower simulation time. Acknowledgement. The authors thank the Department of Science and Technology, New Delhi, Government of India. For the support under the project no: SR/S4/MS: 326/06.

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References 1. Adya, S.N., Markov, I.L.: Fixed-outline ﬂoorplanning through better local search. In: ICCD 2001: Proceedings of the International Conference on Computer Design: VLSI in Computers & Processors, pp. 328–334. IEEE Computer Society, Austin (2001) 2. Bandyopadhyay, S., Saha, S., Maulik, U., Deb, K.: A simulated annealing based multi-objective optimization algorithm: Amosa. IEEE Transactions on Evolutionary Computation 12, 269–283 (2008) 3. Chen, G., Guo, W., Chen, Y.: A PSO-based intelligent decision algorithm for vlsi ﬂoorplanning. In: Soft Computing - A Fusion of Foundations, Methodologies and Applications, vol. 14, pp. 1329–1337 (2010) 4. Chen, J., Chen, G., Guo, W.: A discrete PSO for multi-objective optimization in vlsi ﬂoorplanning. In: ISICA 2009: Proceedings of the 4th International Symposium on Advances in Computation and Intelligence, pp. 400–410. Springer, Heidelberg (2009) 5. Chen, T.-C., Chang, Y.-W.: Modern ﬂoorplanning based on b*-tree and fast simulated annealing. IEEE Trans on CAD 25, 510–522 (2006) 6. Fernando, P., Katkoori, S.: An elitist non-dominated sorting based genetic algorithm for simultaneous area and wirelength minimization in vlsi ﬂoorplanning. In: Proceedings of 21st International Conference on VLSI Design, pp. 337–342 (2008) 7. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979) 8. Guo, P.-N., Cheng, C.-K., Yoshimura, T.: An O-tree representation of non-slicing ﬂoorplan and its applications. In: DAC 1999: Proceedings of the 36th Annual ACM/IEEE Design Automation Conference, pp. 268–273. ACM, New York (1999) 9. Kang, M., Dai, W.: General ﬂoorplanning with L-shaped, T-shaped and soft blocks based on bounded slicing grid structure. In: Proceedings of Asia and South Paciﬁc -Design Automation Conference, pp. 265–270 (1997) 10. Kirpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983) 11. Murata, H., Fujiyoshi, K., Kaneko, M.: VLSI/PCB placement with obstacles based on sequence-pair. In: ISPD 1997: Proceedings of the 1997 International Symposium on Physical Design, pp. 26–31. ACM, New York (1997) 12. Murata, H., Fujiyoshi, K., Nakatake, S., Kajitani, Y.: Rectangle-packing-based module placement. In: ICCAD 1995: Proceedings of the 1995 IEEE/ACM International Conference on Computer-aided Design, pp. 472–479. IEEE Computer Society, Washington, DC (1995) 13. Murata, H., Kuh, E.S.: Sequence-pair based placement method for hard/soft/preplaced modules. In: ISPD 1998: Proceedings of the 1998 International Symposium on Physical Design, pp. 167–172. ACM, New York (1998) 14. Nakatake, S., Fujiyoshi, K., Murata, H., Kajitani, Y.: Module placement on BSGstructure and IC layout applications. In: ICCAD 1996: Proceedings of the 1996 IEEE/ACM International Conference on Computer-aided Design, pp. 484–491. IEEE Computer Society, Washington, DC (1996) 15. Nakatake, S., Fujiyoshi, K., Murata, H., Kajitani, Y.: Module placement on BSGstructure with Pre-Places modules and rectilinear modules. In: Proceedings of the Asia and South Paciﬁc Design Automation Conference (ASP-DAC 1998), pp. 571– 576 (1998)

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16. Otten, R.H.: Automatic ﬂoorplan design. In: DAC 1982: Proceedings of the 19th Design Automation Conference, pp. 261–267. IEEE Press, Piscataway (1982) 17. Sechen, C., Sangiovanni-Vincentelli, A.: The timberwolf placement and routing package. IEEE J.Solid State Circuits 20, 510–522 (1985) 18. Sechen, C.: Chip-planning, placement, and global routing of macro/custom cell integrated circuits using simulated annealing. In: DAC 1988: Proceedings of the 25th ACM/IEEE Design Automation Conference, pp. 73–80. IEEE Computer Society Press, Los Alamitos (1988) 19. Sechen, C., Sangiovanni-Vincentelli, A.: Timberwolf 3.2: a new standard cell placement and global routing package. In: DAC 1986: Proceedings of the 23rd ACM/IEEE Design Automation Conference, pp. 432–439. IEEE Press, Piscataway (1986) 20. Tang, M., Sebastian, A.: A genetic algorithm for vlsi ﬂoorplanning using o-tree representation. In: Rothlauf, F., Branke, J., Cagnoni, S., Corne, D.W., Drechsler, R., Jin, Y., Machado, P., Marchiori, E., Romero, J., Smith, G.D., Squillero, G. (eds.) EvoWorkshops 2005. LNCS, vol. 3449, pp. 215–224. Springer, Heidelberg (2005) 21. Wang, T.-C., Wong, D.F.: An optimal algorithm for ﬂoorplan area optimization. In: DAC 1990: Proceedings of the 27th ACM/IEEE Design Automation Conference, pp. 180–186. ACM, New York (1990) 22. Wong, D.F., Liu, C.L.: A new algorithm for ﬂoorplan design. In: DAC 1986: Proceedings of the 23rd ACM/IEEE Design Automation Conference, pp. 101–107. IEEE Press, Piscataway (1986) 23. Wu, G.-M., Wu, S.-W., Chang, Y.-W., Chang, Y.-C.: B*-trees: A new representation for non-slicing ﬂoorplans. In: Proceedings of the 37th Design Automation Conference, pp. 458–463 (2000)

Approximate Controllability of Fractional Order Semilinear Delay Systems N. Sukavanam and Surendra Kumar Indian Institute of Technology Roorkee, Rorkee 247667, Uttarakhand, India {nsukvfma,sk001dma}@iitr.ernet.in

Abstract. In this paper, we prove the approximate controllability for a class of semilinear delay control systems of fractional order of the form dα y(t) = Ay(t) dtα y0 (θ) = φ(θ),

+ v(t) + f (t, yt , v(t)), t ∈ [0, τ ]; θ ∈ [−h, 0],

where A is linear operators and f is a nonlinear operator deﬁned on appropriate Banach space. Keywords: Controllability, Fractional order delay system, Inﬁnite dimensional system.

1

Introduction

Let V be a Banach space and C([−h, 0], V ) be the Banach space of all continuous functions from [−h, 0] to V . The norm on C([−h, 0], V ) is deﬁned as xt C =

sup xt (θ)V , for t ∈ (0, τ ].

−h≤θ≤0

Consider the following fractional order semilinear delay equation dα y(t) dtα = Ay(t) + v(t) + f (t, yt , v(t)), t ∈ (0, τ ]; θ ∈ [−h, 0], y0 (θ) = φ(θ),

(1)

where 0 < α < 1; A : D(A) ⊆ V → V is a closed linear operator with dense domain D(A) generating a C0 -Semigroup T (t); the state y(·) and the control function u(·) take values in V ; the operator f : [0, τ ] × C([−h, 0], V ) × V → V is nonlinear. If y : [−h, τ ] → V is a continuous function then yt : [−h, 0] → V is deﬁned as yt (θ) = y(t + θ) for θ ∈ [−h, 0] and φ ∈ C([−h, 0], V ). For the above system (1) controllability results are not available in the literature. But several authors have proved controllability of fractional order system of the form (1), when the nonlinear term is independent of the control v [1,2,3,4]. In [5,6] the approximate controllability of ﬁrst order delay control systems of the form (1) (when α = 1) has been proved when f is a function of both yt and v. In this paper, we prove the approximate controllability of the system (1), when 0 < α < 1. The paper is organized as follows: In Section 2, we present some necessary preliminaries. The approximate controllability of the system (1) is proved in Section 3. In Section 4 an example is given to illustrate our result. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 73–78, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Preliminaries

In this section some basic deﬁnitions which are useful for further development are given. Definition 1. A real function f (t) is said to be in the space Cα , α ∈ R if there exists a real number p > α, such that f (t) = tp g(t), where g ∈ C[0, ∞) and it is said to be in the space Cαm if and only if f (m) ∈ Cα , m ∈ N . Definition 2. The Riemann-Liouville fractional integral operator of order β > 0 of function f ∈ Cα , α ≥ −1 is defined as t 1 (t − s)β−1 f (s)ds. I β f (t) = Γ (β) 0 where Γ is the Euler gamma function. Note that if f ∈ Cα the above integral exists only under the constants β > 0 and p > α ≥ −1. m Definition 3. If the function f ∈ C−1 and m is a positive integer then we can define the fractional derivative of f (t) in the Caputo sense as

1 dα f (t) = α dt Γ (m − α)

t

(t − s)m−α−1 f m (s)ds, m − 1 ≤ α < m.

0

For more details about the theory of fractional diﬀerential equations we refer the books [7,8,9]. The state function is deﬁned as yt (0), t ∈ (0, τ ]; yt (θ) = y0 (θ), θ ∈ [−h, 0]. Definition 4. A function y(·, φ, v) ∈ C([−h, τ ], V ) is said to be a mild solution of (1) if it satisfies t 1 T (t)φ(0) + Γ (α) (t − s)α−1 T (t − s)[v(s) + f (s, ys , v(s))]ds, t ∈ (0, τ ]; 0 yt (θ) = φ(θ),

θ ∈ [−h, 0],

on [−h, τ ]. Let y(τ, φ(0), v) be the state value of system (1) at time τ corresponding to the control v and the initial value φ(0). The system (1) is said to be approximate controllable in time interval [0, τ ], if for every desired ﬁnal state x1 and > 0 there exists a control function v ∈ V such that the solution x(t, φ(0), v) of (1) satisﬁes x(τ, φ(0), v) − x1 <

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75

Definition 5. The set Kτ (f ) = {x(τ, φ(0), v) ∈ V : x(·, φ, v) ∈ C([−h, τ ], V ) is a mild solution of (1)} is called the Reachable set of the system (1). Definition 6. The system (1) is said to be approximate controllable on [0, τ ] if Kτ (f ) = V where Kτ (f ) denote the closure of Kτ (f ). When f = 0, then system (1) is called corresponding linear system denoted by (1)∗ . In this case, Kτ (0) denotes the reachable set of the linear system (1)∗ . It is clear that (1)∗ is approximately controllable if Kτ (0) = V.

3

Controllability Results

In this section, we prove the approximate controllability of the semilinear system (1) under suitable conditions. Consider the linear system with a control u as dα x(t) dtα = Ax(t) + u(t), t ∈ (0, τ ]; (2) x0 (θ) = φ(θ), θ ∈ [−h, 0], Assumptions[H]. Consider the following assumptions: (H1) The linear system (2) is approximate controllable. (H2) The nonlinear operator f (t, xt , u(t)) satisﬁes the Lipschitz condition i.e. there exists a positive constant l > 0 such that f (t, xt , u(t)) − f (t, yt , v(t))V ≤ l[xt − yt C + u(t) − v(t)V ] for all xt , yt ∈ C([−h, 0], V ); u, v ∈ V and t ∈ (0, τ ]. Theorem 1. Under assumptions (H1)-(H2) the semilinear control system (1) is approximate controllable if l < 1. Proof. Let xt (θ) be a mild solution of (2) corresponding to a control u and consider the following semilinear system dα y(t) dtα = Ay(t) + f (t, yt , v(t)) + u(t) − f (t, xt , v(t)), t ∈ (0, τ ]; (3) θ ∈ [−h, 0], y0 (θ) = φ(θ), On comparing (1) and (3), it can be seen that the control function v satisﬁes the following condition v(t) = u(t) − f (t, xt , v(t)). Later in the proof we show that there exists a unique v(t) satisfying the above equation. The mild solution of (2) is t 1 T (t)φ(0) + Γ (α) (t − s)α−1 T (t − s)u(s)ds, t ∈ (0, τ ]; 0 xt (θ) = (4) φ(θ), θ ∈ [−h, 0],

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and the mild solution of (3) is ⎧ t 1 (t − s)α−1 T (t − s) ⎨ T (t)φ(0) + Γ (α) 0 yt (θ) = ×[f (s, ys , v(s)) + u(s) − f (s, xs , v(s))]ds, t ∈ (0, τ ]; ⎩ φ(θ), θ ∈ [−h, 0].

(5)

If θ ∈ [−h, 0], then from (4) and (5), we get x0 (θ) − y0 (θ) = 0

(6)

and if t ∈ (0, τ ], then we have t 1 xt (0) − yt (0) = (t − s)α−1 T (t − s)[f (s, xs , v(s)) − f (s, ys , v(s))]ds Γ (α) 0 Taking norm on both sides, we get xt (0) − yt (0)V ≤

1 Γ (α)

≤

M Γ (α)

t 0 t 0

(t − s)α−1 T (t − s)f (s, xs , v(s)) − f (s, ys , v(s))V ds (t − s)α−1 f (s, xs , v(s)) − f (s, ys , v(s))V ds

where M is a constant such that T (t) ≤ M , for all t ∈ [0, τ ]. Using assumption (H2), we have t Ml xt (0) − yt (0)V ≤ (t − s)α−1 xs − ys C ds (7) Γ (α) 0 From (6) and (7), we get xt (θ) − yt (θ)V ≤

Ml Γ (α)

t

(t − s)α−1 xs − ys C ds, ∀ − h ≤ θ ≤ 0

0

This implies that xt − yt C ≤

Ml Γ (α)

t

(t − s)α−1 xs − ys C ds

0

Hence Gronwall’s inequality implies that x(t) = y(t) for all t ∈ [−h, τ ]. From this we infer that every solution of the linear system with control u is also a solution of the semilinear system with control v. Hence, Kτ (f ) ⊃ Kτ (0). Now since Kτ (0) is dense in V (by assumption (H1)) it follows that Kτ (f ) is dense in V . Hence the system (1) is approximate controllable. The proof will be complete if we show that there exists a v(t) ∈ V such that v(t) = u(t) − f (t, xt , v(t)) Let v0 ∈ V and vn+1 = u − f (t, xt , vn ), for n = 1, 2, · · ·. Then, we have vn+1 − vn V ≤ f (t, xt , vn−1 ) − f (t, xt , vn )V ≤ lvn − vn−1 V ≤ ln v1 − v0 V

Approximate Controllability of Fractional Order Semilinear Delay Systems

77

Since l < 1, R.H.S. of above inequality tends to zero as n → ∞. Therefore, the sequence vn is Cauchy in V . Since V is a complete normed linear space, vn converges to an element v ∈ V . Now (u − vn+1 ) − f (t, xt , v)V = f (t, xt , vn ) − f (t, xt , v)V ≤ lvn − vV Since R.H.S. of above inequality tends to zero as n → ∞, we get f (t, xt , v) = lim (u − vn+1 ) = u − v n→∞

Hence v = u − f (t, xt , v). Uniqueness of v can be proved easily. This completes the proof.

4

Example

In this section we present an example to illustrate our results. Let V = L2 (0, π) d2 y d2 and A = dx 2 with D(A) consisting of all y ∈ V with dx2 and y(0) = 0 = y(π). Put φn (x) = ( π2 )1/2 sin(nx); 0 ≤ x ≤ π, n = 1, 2, · · ·, then {φn , n = 1.2, · · ·} is an orthonormal base for V and φn is the eigenfunction corresponding to the eigenvalue λn = −n2 of the operator A, n = 1, 2, ···. Then the C0 -semigroup T (t) generated by A has exp(λn t) as the eigenvalues and φn as their corresponding eigenfunctions [10]. Let us consider the following semilinear control system of the form ∂α ∂2 y(t, x) = y(t, x) + u(t, x) + f (t, y(t − h, x), u(t, x)); t ∈ (0, τ ], 0 < x < π ∂tα ∂t2 y(t, 0) = y(t, π) = 0; t > 0 y(t, x) = φ(t, x); t ∈ [−h, 0],

(8)

where φ(t, x) is continuous. If the conditions (H1) and (H2) are satisﬁed, then the approximate controllability follows from Theorem 3.1. For example, if we consider the function f as f (t, z, v) = l[zφ3 + vφ4 ], where l ∈ (0, 1), then f satisﬁes (H2) with Lipschitz constant l < 1. From Theorem 3.1 the approximate controllably of (8) follows.

References 1. Abd El-Ghaﬀar, A., Moubarak, M.R.A., Shamardan, A.B.: Controllability of Factional Nonlinear Control System. Journal of Fractional Calculus 17, 59–69 (2000) 2. Balachandran, K., Park, J.Y.: Controllability of Fractional Integrodiﬀerential Systems in Banach Spaces. Nonlinear Analysis: Hybrid Systems 3, 363–367 (2009) 3. Bragdi, M., Hazi, M.: Existence and Controllability Result for an Evolution Fractional Integrodiﬀerential Systems. Int. J. Contemp. Math. Sciences 5(19), 901–910 (2010)

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4. Tai, Z., Wang, X.: Controllability of Fractional-Order Impulsive Neutral Functional Inﬁnite Delay Integrodiﬀerential Systems in Banach Spaces. Applied Mathematics Letters 22, 1760–1765 (2009) 5. Dauer, J.P., Mahmudov, N.I.: Approximate Controllability of Semilinear Functional Equations in Hilbert Spaces. J. Math. Anal. Appl. 273, 310–327 (2002) 6. Sukavanam, N., Tomar, N.K.: Approximate Controllability of Semilinear Delay Control Systems. Nonlinear Functional Analysis and Applications 12(1), 53–59 (2007) 7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Diﬀerential Equations. Elsevier, Amsterdam (2006) 8. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations. Wiley, New York (1993) 9. Podlubny, I.: Fractional Diﬀerential Equations. Academic Press, New York (1999) 10. Naito, K.: Controllability of Semilinear Control Systems Dominated by the Linear Part. SIAM J. Control and Optimization 25(3), 715–722 (1987)

Using Genetic Algorithm for Solving Linear Multilevel Programming Problems via Fuzzy Goal Programming Bijay Baran Pal1,*, Debjani Chakraborti2, and Papun Biswas3 1

Department of Mathematics, University of Kalyani, Kalyani-741235, W.B., India Tel.: +91-33-25825439 [email protected] 2 Department of Mathematics, Narula Institute of Technology, Kolkata-700109, W.B., India [email protected] 3 Department of Electrical Engineering, JIS College of Engineering, Kalyani-741235, W.B., India [email protected]

Abstract. This article presents a fuzzy goal programming (FGP) procedure for modeling and solving multilevel programming (MLP) problems by using genetic algorithm (GA) in a large hierarchical decision making system. In the proposed approach, an GA scheme is introduced first for searching of solutions at different stages and thereby solving the problem and making decision in the order of hierarchy of execution of decision powers of the decision makers (DMs) located at different hierarchical levels. In the proposed GA scheme, Roulette-wheel selection scheme, single point crossover and random mutation are adopted to search a satisfactory solution in the hierarchical decision system. To illustrate the potential use of the approach, a numerical example is solved. Keywords: Multilevel Programming, Fuzzy Programming, Fuzzy Goal Programming, Goal Programming, Genetic Algorithm, Membership Function.

1 Introduction The MLP is a special field of study in the area of mathematical programming for solving decentralized planning problems with multiple DMs involving different objectives in a large hierarchical decision organization. In MLP problems (MLPPs), one DM is located at each of the different hierarchical levels and each control a decision vector and an objective function separately in the decision making context. The execution of the decision power is sequential from an upper-level to a lower-level, and each DM tries to optimize his/her own benefit, where the objectives often conflict each other in the decision making situation. In a practical decision situation, although the execution of the decision powers is sequential, the decisions of the upper-level DMs are often affected by the reactions of the lower-level DMs due to their dissatisfactions with the decisions of the upper-level DMs. *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 79–88, 2011. © Springer-Verlag Berlin Heidelberg 2011

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As a consequence, most of the hierarchical decision organizations are frequently faced with the problems of distribution of decision powers to the DMs for overall benefit of the organization. During the last four decades, MLPPs as well as bilevel programming problems (BLPPs) as a special case of MLPPs have been studied deeply in [1], [2], [3], [4], [5], [6] from the view point of their potential use to different decision problems. But the use of the classical approaches often lead to the paradox that the decision power of a lower-level DM dominates that of a higher-level DM. To overcome this situation, an ideal point dependent solution approach based on the preferences of the DMs for achievement of their objectives has been introduced by Wen and Hsu [7]. But their method does not always provide a satisfactory decision in a highly conflicting hierarchical decision situation. In order to overcome the shortcomings of the classical approaches, fuzzy programming (FP) [8] approach to hierarchical decision problems has been introduced by Lai [9] in 1996 to solve BLPPs as well as MLPPs. The solution concept of Lai has been further extended by Shih et al. [10] and Shih and Lee [11] with a view to make a balance of decision powers to the DMs. In these approaches, the elicited membership functions for the fuzzy goals are redefined again and again to reach a satisfactory decision in the solution search process. To avoid such a computational difficulty, a goal satisficing method in goal programming GP [12] for minimizing the regrets of the DMs in case of a BLPP has been studied by Pal and Moitra [13]. However, the extensive study on fuzzy MLPPs is at an early stage. Now, in a decision making environment, GAs initially introduced by Holland [14] which based on natural selection and population genetics have appeared as a flexible and robust tools for solving multiobjective decision problems [15]. The GA based solution approaches to decision problems in the framework of GP in [12], [16] have been studied [17] in the past. The use of GAs to FP formulations of BLPPs as well as MLPPs has been studied in [18], [19] by the pioneer researchers in the field. The potential use of GA based on GP method to Interval-valued MLPPs has been studied [20] in the recent past. However, the GA based solution method to the FGP formulations of BLPPs as well as MLPPs is yet to circulate in the literature. In this paper, the efficient use of GA to MLPPs for distribution of proper decision powers to the DMs within a hierarchical decision structure is presented. In the model formulation, the individual decision of each of the DMs is determined first by employing the proposed GA scheme to define the fuzzy goals of the problem. Then, the optimal decision of the problem under the framework of minsum FGP is determined through the proposed GA solution search process. A numerical illustration is provided to expound the approach. Now, the formulation of the MLPP is presented in the Section 2.

2 Formulation of MLPP In the hierarchical decision situation, let X = ( x1 , x 2 ,⋅ ⋅ ⋅, x n ) be the vector of decision variables involved with different levels of the decision system. Then, let Z i be the objective function and X i be the control vector of the decision variables of the i -th level DM, i = 1,2,..., I; I ≤ n where ∪ {X i = 1,2,..., I } = X i

i

Using Genetic Algorithm for Solving Linear MLPPs via FGP

81

Then, the general MLPP within a hierarchical nested decision structure can be presented as: Find X ( X1 , X 2 ,... . , X I ) so as to: max Z 1 ( X) = X1

I

∑

i =1

p1i X i

(top-level)

where, for given X1 ; X 2 , X 3 ,..., X I solve max Z 2 ( X) = X2

I

∑

i =1

p 2i X i

(second-level)

where, for given X1 and X 2 ; X 3 , X 4 ,..., X I solve … … … … … … where, for given X1 , X 2 ,..., X I −1 ; X I solves max Z I ( X) = XI

I

∑

i =1

p Ii Xi

(I -th level) I

subject to ( X1 , X 2 ,..., X I ) ∈ S = { ∑ A i X i ≤ b} , X i ≥ 0, i = 1,2,..., I i =1

(1)

where p ji ( j = 1,2,..., I ) and b are constant vectors, A i (i = 1,2,..., I ) are constant matrices. It is also assumed that S (≠ Φ) is bounded. Now, it is to be observed that the problem (1) is multiobjective in nature and the execution of decision power of the DMs is sequential from the top-level to the bottom level. Now, to develop the fuzzy goals of the problem and then to solve the problem under the framework of the FGP, a GA scheme is introduced in the Section 3.

3 GA Scheme for MLPP In the literature of the GAs, there are a number of schemes [14] for generation of new populations with the use of the different operators: selection, crossover and mutation. Here, the binary coded representation of a candidate solution called chromosome is considered to perform genetic operations in the solution search process. The conventional Roulette wheel selection scheme in [15], single-point crossover [15] and bit-bybit mutation operations are adopted to generate offspring in new population in the search domain defined in the decision making environment. The fitness score of a chromosome v (say) in evaluating a function, say, eval( Ev ) , is based on maximization or minimization of an objective function called the fitness function defined on the basis of DM’s needs and desires in the decision making context. The fitness function is defined as eval( E v ) = ( Z i ) v , i = 1,2,..., I ; v = 1,2,..., pop_size.

(2)

where Z i represents the objective function of the i -th level DM given in (1), and where the subscript v is used to indicate the fitness value of the v -th chromosome, v = 1,2,..., pop_size. The best chromosome with largest fitness value at each generation is determined as:

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E * = max{eval( E v ) v = 1,2,..., pop_size}, or, E * = min{eval( E v ) v = 1,2,..., pop_size} , depending on searching of the maximum or minimum value of an objective. Now, formulation of FGP of the problem (1) by defining the fuzzy goals is presented in the Section 4.

4 FGP Problem Formulation To formulate the FGP model of the given problem, the imprecise aspiration levels of the objectives of the DMs at all the levels and the decision vectors of the upper-level DMs as well as the tolerance limits for achieving the respective aspired levels are determined first. Then, they are characterized by their membership functions for measuring the degree of achievement of the aspired goal levels in the decision situation. In the present decision situation, the individual best decisions of the DMs are taken into consideration and they are evaluated by using the proposed GA scheme. Let ( X1i , X i2 ,..., X iI ; Z *i ) be the best decision of the i-th level DM, where

Z i* = max Z i ( X) . X∈S

Then the fuzzy goal objectives can be presented as Z i ) Z i* ,

i = 1,2,..., I .

Similarly, the fuzzy goals for the control vectors of the upper-level DMs appear as X i ) X ii ,

i = 1,2,..., I − 1.

Now, from the view point of considering the relaxation made by an upper-level DM for the benefit of a lower-level DM, the lower-tolerance limit for the i-th fuzzy goal Z i can be determined as: Z ii +1[= Z i ( X1i +1 , X i2+1 ,..., X iI+1 )] < Z i* , i = 1,2,..., I − 1.

Again, since the I-th level DM is at the bottom level, the lower-tolerance limit of the objective Z I can be determined as Z Im [= min( Z i ( X1i , X i2 ,..., X iI ); i = 1,2,...I − 1)] < Z *I

Now, since the DMs are motivated to co-operate each other and a certain relaxation on the decision of each of the upper-level DMs is made for the benefit of a lower level DM, the lower tolerance limit of the decision X i can be determined as X in ( X ii +1 < X in < X ii ), i = 1,2,...I − 1. Then, the construction of the membership functions of the defined fuzzy goals is presented in the Section 4.1.

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83

4.1 Construction of Membership Functions

For the defined lower-tolerance limits of the fuzzy goals, the membership functions can be constructed as follows: ⎧ 1 ⎪ ⎪⎪ Z (X) − Z ii +1 μ Zi [Z i (X)] = ⎨ i * i +1 ⎪ Zi − Zi ⎪ ⎪⎩ 0

if Z i (X) ≥ Z *i , if Z ii +1 ≤ Z i (X) < Z i* , if Z i (X) < Z ii +1, i = 1,2,...,I − 1.

⎧ 1 ⎪ ⎪⎪ Z (X) − Z Im μ Z I [Z I (X)] = ⎨ I * m ⎪ ZI − ZI ⎪ ⎪⎩ 0 ⎧ ⎪ ⎪⎪ X μ Xi [ X i ] = ⎨ i i ⎪ Xi ⎪ ⎪⎩

(3)

if Z I (X) ≥ Z *I , if Z Im ≤ Z I (X) < Z *I ,

(4)

if Z I (X) < Z Im , if X i ≥ X ii ,

1 − X in

if X in ≤ X i < X ii ,

− X in

if X i < X in ,

0

(5)

i = 1, 2, . . . , I − 1.

Now, the FGP model formulation is presented in the following Section 4.2. 4.2 FGP Model Formulation

In the FGP model formulation, the membership functions are transformed into goals by assigning the aspiration level unity and introducing under- and over-deviational variables to each of them. Then, achievement of the aspired goal levels to the extend possible by minimizing the sum of the weighted under-deviational variables in the goal achievement function on the basis of weights of importance of achieving the goals is taken into account. The minsum FGP formulation of the problem can be presented as: Find ( X1 , X 2 ,..., X I ) so as to: I

Minimize Z =

∑ i

and satisfy μ Z i :

=1

wi− d i− +

Z i ( X) − Z ii +1 Z i*

−

Z ii +1

2 I −1 ∑

l = I +1

w l− d l− ,

+ d i− − d i+ = 1, i = 1,2,..., I − 1.

m μ Z I : Z I ( X) − Z I + d − − d + = 1 I I Z *I − Z Im

84

B.B. Pal, D. Chakraborti, and P. Biswas n μ Xi : X i − X i + d − − d + = I , l l l X ii − X in

l = I + i, where i = 1,2,..., I − 1.

d i+ , d i− ≥ 0, d l+ , d l− ≥ 0,

(6)

subject to the given system constraints in (1), where d i+ , d i− represent the over- and under-deviational variables, respectively, and d l+ , d l− ≥ 0, are the vectors of over- and under-deviational variables associated with the respective goals. Z represents the goal achievement function consisting of the weighted under-deviational variables and vectors of weighted under-deviational vari− ables, where the numerical weights wi (> 0), i = 1,2,..., I , and the vector of the nu− merical weights w l (> 0), l = I + 1, I + 2,...,2 I − 1 , represent the relative weights of importance of achieving the goals to their aspired levels, and they are determined as:

wi− = wI− =

1 Z i*

− Z ii +1 1

Z *I

−

Z Im

, i = 1,2,..., I − 1 , and w l− =

1 X ii

− X in

, l = I + i, where i = 1,2,..., I − 1.

Now, since MLPP is a large hierarchical decision structured problem and the objectives at the various decision levels often conflict each other in the decision making situation, the use of conventional FGP solution approach may not always provide the satisfactory decision from the view point of distribution of proper decision power to the DMs in the decision making environment. Here, the GA method as a decision satisficer [14] rather than optimizer can be employed in the solution search process. The fitness function under the GA scheme appears as: eval( E v ) = ( Z i ) v =

I

∑

i =1

wi− d i− +

2 I −1 ∑

l = I +1

w l− d l− , v = 1,2,..., pop_size.

(7)

In, the decision search process, the best chromosome E* with the highest score at a generation is determined as: E * = min{eval( E v ) v = 1,2,..., pop_size} in the genetic search process. The efficient use of the proposed approach is illustrated by a numerical example.

5 An Illustrative Example The following trilevel problem, studied by Anandalingam [1] is considered. Find ( x1 , x 2 , x3 ) so as to

max Z 1 = 7 x1 + 3x 2 − 4 x3 x1

(top-level)

Using Genetic Algorithm for Solving Linear MLPPs via FGP

85

where, for given x1 , x 2 and x3 solve max Z 2 = x 2

(middle-level)

x2

where, for given x1 and x 2 , x3 solves max Z 3 = x3

(bottom-level)

x3

subject to x1 + x2 + x3 ≤ 3, x1 + x2 − x3 ≤ 1, x1 + x2 + x3 ≥ 1, − x1 + x2 + x3 ≤ 1, x3 ≤ 0.5, x1, x2 , x3 ≥ 0

(8)

Now, to solve the problem by employing the proposed GA scheme, the following genetic parameter values are incorporated in the decision search process. • • • •

population size = 100 probability of crossover = 0.8 probability of mutation = 0.08 length of the chromosome = 30.

The GA is implemented using the Programming Language C. The execution is performed in an Intel Pentium IV PC with 2.66 GHz. Clock-pulse and 1 GB RAM. Now, following the procedure, the individual optimal solutions of the three successive levels are obtained as: 1 1 1 (i) ( x1 , x 2 , x 3 ; Z 1 ) = (1.4997, 0, 0.4998 ; 8.4990) (ii) ( x12 , x 22 , x 32 ; Z 2 ) = (0.3567, 1.000, 0.3567 ; 1.000)

(top-level) (middle-level)

3 3 3 (iii) ( x1 , x 2 , x 3 ; Z 3 ) = (0.5522, 0.4914, 0.50 ; 0.50)

(bottom-level)

Then, the fuzzy goals are obtained as: Z 1 )8.4990, Z 2 )1, Z 3 )0.5 and

x1 )1.4997, x 2 )1.

Following the procedure, the lower-tolerance limits of the objective goals are determined as:

Z12 = 4.1, Z 23 = 0.4914, Z 3m = 0.3567 . Then, the top-level and middle-level DMs feel that their control variables x1 and x 2 should be relaxed up to 1.35 and 0.8, respectively, for the benefit of the respective lower-level DMs, and not beyond of them. So, x1n = 1.35 ( x12 < 1.35 < x11 ) and

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x 2n = 0.8 ( x 23 < 0.8 < x 22 ) act as lower tolerance limits of the decisions x1 and x 2 , respectively. Now, following the procedure and using the above numerical values, the membership functions of the defined fuzzy goals can be constructed by (3), (4) and (5). Then, the resultant FGP model appears as: Find ( x1 , x 2 , x3 ) so as to Minimize Z =

1 1 1 1 1 − d 1− + d 2− + d 3− + d 4− + d5 4.399 0.5086 0.1433 0.1497 0 .2

and satisfy: 7 x1 + 3 x 2 − 4 x 3 − 4.1 + d 1− − d 1+ = 1 4.399 x − 0.4914 μZ2 : 2 + d 2− − d 2+ = 1 0.5086 x − 0.3567 + μ Z3 : 3 + d 3− − d 3 = 1 0.1433 x1 − 1.35 μ x1 : + d 4− − d 4+ = 1 0.1497 x − 0.8 μ x2 : 2 + d 5− − d 5+ = 1 0 .2 μ Z1 :

d q+ , d q− ≥ 0, q = 1,2,.....,5. subject to the given system constraints in (8) Now, employing the GA scheme with the evaluation function in (7), the resultant solution of the problem is obtained as ( x1 , x 2 , x 3 ) = (1.201, 0.398, 0.5) with ( Z 1 , Z 2 , Z 3 ) = (7.319, 0.398, 0.5) The achieved membership values of the objective goals are μ Z1 = 0.7 , μ Z 2 = 0, μ Z 3 = 1 The solution achieved here is the most satisfactory one from the view point of distributing the proper decision powers to the DMs in the decision making environment. Now, the solutions of the problem obtained previously by using the different approaches are presented as follows: (a) The solution of the problem obtained by using the conventional mathematical programming approach [1] is ( x1 , x 2 , x3 ) = (0.5 ,1, 0.5 ) with ( Z1 , Z 2 , Z 3 ) = (4.5 , 1, 0.5) . (b) The solution of the problem obtained by using the conventional FGP approach [21] is ( x1 , x 2 , x3 ) = (1.5, 0, 0.5) with ( Z 1 , Z 2 , Z 3 ) = (8.5, 0, 0.5) .

Using Genetic Algorithm for Solving Linear MLPPs via FGP

87

(c) The solution of the problem obtained by using fuzzy compensatory γ-operator [11] in fuzzy sets is: ( x1 , x 2 , x3 ) = (0.9, 0.6, 0.5) with ( Z 1 , Z 2 , Z 3 ) = (6.1, 0.6, 0.5) A graphical representation of the goal values achieved with the use of different approaches is displayed in the Fig. 1.

Fig. 1. Graphical representation of goal achievement under different approaches

A comparison of the results shows that that the proposed GA based FGP solution is the most satisfactory one from the viewpoint of arriving at a compromise decision by maintaining the hierarchical order of the DMs in the decision making environment.

Note: It is to be noted that the decision of the middle-level DM is improved here under the proposed approach with a least sacrifice of the decision of the top-level DM than the use of conventional FGP approach [21]. As a matter of fact, it may be claimed that, as GA is a volume-oriented (global one) search method and a goal satisficer [15] rather than objective optimizer, the incorporation of the GA solution search process to the FGP formulation of the problem makes the model more effective to make a reasonable balance of execution of decision powers of the DMs in the decision environment.

6 Conclusion The main advantage of the GA solution approach to the FGP model of the problem is that the higher degree of satisfaction with the solution can be searched on the basis of the needs and desires of the DMs in the decision making environment. The proposed approach can easily be extended to solve fuzzy MLPPs having the characteristics of fractional criteria. An extension of the approach to fuzzy multilevel decentralized planning problems is one of the current research problems. However, it is hoped that the proposed approach can contribute to future study in the field of real life large-scale hierarchical decision making problems.

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Acknowledgements. The authors are grateful to the anonymous Reviewers and Prof. P. Balasubramaniam, Organizing Chair ICLICC2011, for their helpful suggestions and comments which have led to improve the quality and clarity of presentation of the paper.

References 1. Anandalingam, G.: A mathematical programming model of decentralized multi-level system. Journal of Operational Research Society 39(11), 1021–1033 (1988) 2. Bard, J.F., Falk, J.E.: An explicit solution to multi-level programming problem. Computers and Operations Research 9, 77–100 (1982) 3. Bialas, W.F., Karwan, M.H.: On two level optimization. IEEE Transactions on Automatic Control 27, 211–214 (1982) 4. Bialas, W.F., Karwan, M.H.: Two level linear programming. Management Sciences 30, 1004–1020 (1984) 5. Burton, R.M.: The multilevel approach to organizational issues of the firm-Critical review. Omega 5, 446–457 (1977) 6. Candler, W.A.: A linear bilevel programming algorithm, a comment. Computers and Operations Research 15, 297–298 (1988) 7. Wen, U.P., Hsu, S.T.: Efficient solution for linear bilevel programming problem. European Journal of Operational Research 62, 354–362 (1991) 8. Liu, B.: Theory and Practice of Uncertain Programming, 1st edn. Physica-Verlag, Heidelberg (2002) 9. Lai, Y.J.: Hierarchical optimization. A satisfactory solution. Fuzzy Sets and Systems 77, 321–335 (1996) 10. Shih, H.S., Lai, Y.J., Lee, E.S.: Fuzzy approaches for multi-level programming problems. Computers and Operations Research 23, 73–91 (1996) 11. Shih, H.S., Lee, S.: Compensatory fuzzy multiple level decision making. Fuzzy Sets and Systems 14, 71–87 (2000) 12. Ignizio, J.P.: Goal Programming and Extensions. D.C. Health, Lexington (1976) 13. Pal, B.B., Moitra, B.N.: A fuzzy goal programming approach for solving bilevel programming problems. In: Pal, N.R., Sugeno, M. (eds.) AFSS 2002. LNCS (LNAI), vol. 2275, pp. 91–98. Springer, Heidelberg (2002) 14. Holland, J.H.: Genetic Algorithms and Optimal Allocations of Trials. SIAM Journal of Computing 2(2), 88–105 (1973) 15. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading (1989) 16. Charnes, A., Cooper, W.W.: Management models and industrial applications of linear programming. John Wiley & Sons, New York (1961) 17. Gen, M., Ida, K., Lee, J., Kim, J.: Fuzzy nonlinear goal programming using genetic algorithm. Computers and Industrial Engineering 33(1-2), 39–42 (1997) 18. Hejazi, S.R., Memariani, A., Jahanshahloo, G., Sepehri, M.M.: Linear bilevel programming solution by genetic algorithm. Computers and Operations Research 29, 1913–1925 (2002) 19. Sakawa, M., Nishizaki, I., Hitaka, M.: Interactive fuzzy programming for multi-level 0-1 programming problems with fuzzy parameters through genetic algorithms. Fuzzy Sets and Systems 117, 95–111 (2001) 20. Pal, B.B., Chakraborti, D., Biswas, P.: A Genetic Algorithm Based Goal Programming Approach to Interval-Valued Multilevel Programming Problems. In: 2nd International Conference on Computing, Communication and Networking Technologies, pp. 1–7 (2010) 21. Pal, B.B., Pal, R.: A linear multilevel programming method based on fuzzy programming. In: Proceedings of APORS 2003, vol. 1, pp. 58–65 (2003)

Intuitionistic Fuzzy Fractals on Complete and Compact Spaces D. Easwaramoorthy and R. Uthayakumar Department of Mathematics, The Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Dindigul, Tamil Nadu, India Tel.: +91 451 2452371; Fax: +91 451 2453071 [email protected], [email protected]

Abstract. We develop the Hutchinson-Barnsley theory for generating the intuitionistic fuzzy fractals through iterated function system, which consists of ﬁnite number of intuitionistic fuzzy contractive mappings on intuitionistic fuzzy metric space. For that, we prove the existence and uniqueness of fractals in a complete intuitionistic fuzzy metric space and a compact intuitionistic fuzzy metric space by using the intuitionistic fuzzy Banach contraction theorem and the intuitionistic fuzzy Edelstein contraction theorem respectively. Keywords: Fractal Analysis; Iterated Function System; Intuitionistic Fuzzy Metric Space; Contraction; Complete; Compact; Precompact.

1

Introduction

Mandelbrot deﬁned Fractal as a subset of a complete metric space for which the fractal dimension strictly exceeds the topological dimension in 1975 and the theory was popularized by various mathematicians [2,3,7]. The theory of fuzzy sets was introduced by Zadeh in 1965 [15]. Many authors have introduced and discussed several notions of fuzzy metric space [4,8] and ﬁxed point theorems with interesting consequent results in fuzzy metric spaces [4,5,12]. Then the concept of intuitionistic fuzzy metric space was given by Park [10] recently and the subsequent ﬁxed point results in the intuitionistic fuzzy metric spaces were investigated by Alaca and et al. [1]. Hutchinson [7] and Barnsley [2] initiated and developed the theory, called as Hutchinson-Barnsley theory (HB theory), in order to deﬁne and construct the fractal as a compact invariant subset of a complete metric space generated by the IFS of contractions, by using the Banach ﬁxed point theorem. Here, we initiate the fractal theory in the intuitionistic fuzzy metric spaces. In this paper, we propose a generalization of Hutchinson-Barnsley theory in order to deﬁne and construct the intuitionistic fuzzy IFS fractals through iterated function system. For that, we prove the existence and uniqueness of fractals in

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 89–96, 2011. c Springer-Verlag Berlin Heidelberg 2011

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a complete intuitionistic fuzzy metric space and a compact intuitionistic fuzzy metric space by using the intuitionistic fuzzy Banach contraction theorem and the intuitionistic fuzzy Edelstein contraction theorem, respectively.

2

Metric Fractals

Now we recall the Hutchinson-Barnsley theory (HB theory) to deﬁne and construct the IFS fractals in the complete metric space. Deﬁnition 1 ([2]). Let (X, d) be a metric space and Ko (X) be the collection of all non-empty compact subsets of X. Deﬁne, d(x, B) := inf y∈B d(x, y) and d(A, B) := supx∈A d(x, B) for all x ∈ X and A, B ∈ Ko (X). The Hausdorﬀ metric or Hausdorﬀ distance (Hd ) is a function Hd : Ko (X) × Ko (X) −→ Ê deﬁned by Hd (A, B) = max {d(A, B), d(B, A)} . Then Hd is a metric on the hyperspace of compact sets Ko (X) and hence (Ko (X), Hd ) is called a Hausdorﬀ metric space. Deﬁnition 2 ([2,7]). Let (X, d) be a metric space and fn : X −→ X, n = 1, 2, 3, ..., No (No ∈ Æ) be No - contraction mappings with the corresponding contractivity ratios kn , n = 1, 2, 3, ..., No. The system {X; fn , n = 1, 2, 3, ..., No} is called an Iterated Function System (IFS) or Hyperbolic Iterated Function Syso tem with the ratio k = maxN n=1 kn . Then the Hutchinson-Barnsley operator (HB operator) of the IFS is a function F : Ko (X) −→ Ko (X) deﬁned by F (B) =

No

fn (B).

n=1

Further, the HB operator (F ) is a contraction mapping on (Ko (X), Hd ). Theorem 1 (HB Theorem [2,7]). Let (X, d) be a complete metric space and {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} be an IFS. Then, there exists only one compact invariant set A∞ ∈ Ko (X) of the HB operator (F ) or, equivalently, F has a unique ﬁxed point namely A∞ ∈ Ko (X). Deﬁnition 3 ([2]). The ﬁxed point A∞ ∈ Ko (X) of the HB operator F described in the Theorem 1 is called the Attractor (Fractal) of the IFS. Sometimes A∞ ∈ Ko (X) is called as Metric Fractal generated by the IFS and so called as IFS Metric Fractal.

3

Intuitionistic Fuzzy Metric Space

Deﬁnition 4 ([1,10]). A 5-tuple (X, M, N, ∗, ♦) is said to be an intuitionistic fuzzy metric space if X is an arbitrary (non-empty) set, ∗ is a continuous t-norm [14], ♦ is a continuous t-conorm [14] and M, N are fuzzy sets on X 2 × [0, ∞) satisfying the following conditions:

Intuitionistic Fuzzy Fractals on Complete and Compact Spaces

(a) (b) (c) (d) (e) (f ) (g) (h) (i) (j) (k) (l) (m)

91

M (x, y, t) + N (x, y, t) ≤ 1; M (x, y, 0) = 0; M (x, y, t) = 1 if and only if x = y; M (x, y, t) = M (y, x, t); M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s); M (x, y, ·) : [0, ∞) −→ [0, 1] is left continuous; limt→∞ M (x, y, t) = 1 N (x, y, 0) = 1; N (x, y, t) = 0 if and only if x = y; N (x, y, t) = N (y, x, t); N (x, y, t)♦N (y, z, s) ≥ N (x, z, t + s); N (x, y, ·) : [0, ∞) −→ [0, 1] is right continuous; limt→∞ N (x, y, t) = 0

for all x, y, z ∈ X and t, s > 0. Then (M, N, ∗, ♦) or simply (M, N ) is called an intuitionistic fuzzy metric on X. The functions M (x, y, t) and N (x, y, t) represents the degree of nearness and the degree of non-nearness between x and y in X with respect to t, respectively. Deﬁnition 5 ([10]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. The open ball B(x, r, t) with center x ∈ X and radius r, 0 < r < 1, with respect to t > 0, is deﬁned as B(x, r, t) = {y ∈ X : M (x, y, t) > 1 − r, N (x, y, t) < r} . Deﬁne τ(M,N ) = {A ⊂ X : f or each x ∈ A, there exists t > 0 and r ∈ (0, 1) such that B(x, r, t) ⊂ A}. Then τ(M,N ) is a topology on X induced by an intuitionistic fuzzy metric (M, N ). Deﬁnition 6 ([1,11,13]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. We say that X is complete if every Cauchy sequence is convergent. It is called compact if every sequence contains a convergent subsequence. We say that X is precompact if for each 0 < r < 1 and t > 0 there exists a ﬁnite subset A of X such that X = a∈A B(a, r, t). Deﬁnition 7 ([1]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. We say that the mapping f : X −→ X is intuitionistic fuzzy B-contraction (intuitionistic fuzzy Sehgal contraction) if there exists k ∈ (0, 1) such that M (f (x), f (y), kt) ≥ M (x, y, t) and N (f (x), f (y), kt) ≤ N (x, y, t) for all x, y ∈ X and t > 0. Here, k is called the intuitionistic fuzzy B-contractivity ratio of f . We say that the mapping f : X −→ X is intuitionistic fuzzy Edelstein contraction if M (f (x), f (y), ·) ≥ M (x, y, ·) and N (f (x), f (y), ·) ≤ N (x, y, ·) for all x, y ∈ X such that x = y. Here, k is called the intuitionistic fuzzy Edelstein contractivity ratio of f .

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Hausdorﬀ Intuitionistic Fuzzy Metric Space

In [6], Gregori et al. deﬁned the Hausdorﬀ intuitionistic fuzzy metric on intuitionistic fuzzy hyperspace Ko (X) and constructed the Hausdorﬀ intuitionistic fuzzy metric space as follows. Deﬁnition 8 ([6]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. We shall denote by Ko (X), the set of all non-empty compact subsets of X. Deﬁne, M (x, B, t) := supy∈B M (x, y, t), M (A, B, t) := inf x∈A M (x, B, t) and N (x, B, t) := inf y∈B N (x, y, t), N (A, B, t) := supx∈A N (x, B, t) for all x ∈ X and A, B ∈ Ko (X). Then we deﬁne the Hausdorﬀ intuitionistic fuzzy metric (HM , HN , ∗, ♦) as follows: HM (A, B, t) = min {M (A, B, t), M (B, A, t)} and HN (A, B, t) = max {N (A, B, t), N (B, A, t)} . Here (HM , HN ) is an intuitionistic fuzzy metric on the hyperspace of compact sets, Ko (X), and hence (Ko (X), HM , HN , ∗, ♦) is called a Hausdorﬀ intuitionistic fuzzy metric space. Now we recall that if (X, U ) is a uniform space, then the Hausdorﬀ-Bourbaki uniformity HU (of U ) on Ko (X), has as a base the family of sets of the form HU = {(A, B) ∈ Ko (X) × Ko (X) : B ⊂ U (A) and A ⊂ U (B)}, where U ∈ U . This deﬁnition is valid even upto the set of all non-empty subsets of X instead of Ko (X). Further, if (X, M, N, ∗, ♦) is an intuitionistic fuzzy metric space, then {Un : n ∈ Æ} is a (countable) base for a uniformity U(M,N ) on X compatible with τ(M,N ) , where Un = {(x, y) ∈ X × X : M (x, y, 1/n) > 1 − 1/n, N (x, y, 1/n) < 1/n}, for all n ∈ Æ (See the Lemma 2.6 in [13]). U(M,N ) is called the uniformity induced by (M, N ). Especially, U(HM ,HN ) is exactly the uniformity induced by the Hausdorﬀ fuzzy metric (HM , HN ). In this connection, we frame the following results. Theorem 2. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space . Then, the Hausdorﬀ-Bourbaki uniformity HU(M,N ) , coincides with the uniformity U(HM ,HN ) on Ko (X).

Proof. For each n ∈ Æ, it is enough to prove the following:

{(A, B) ∈ Ko (X) × Ko (X) : B ⊂ Un+1 (A) and A ⊂ Un+1 (B)} ⊂ {(A, B) ∈ Ko (X) × Ko (X) : HM (A, B, 1/(n + 1)) ≥ 1 − 1/(n + 1), HN (A, B, 1/(n + 1)) ≤ 1/(n + 1)} ⊂ {(A, B) ∈ Ko (X) × Ko (X) : HM (A, B, 1/n) ≥ 1 − 1/n, HN (A, B, 1/n) ≤ 1/n} ⊂ {(A, B) ∈ Ko (X) × Ko (X) : B ⊂ Un (A) and A ⊂ Un (B)}. This concludes that, HU(M,N ) = U(HM ,HN ) on Ko (X).

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Theorem 3. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then 1. A sequence {x}n∈ is a Cauchy sequence in the intuitionistic fuzzy metric space (X, M, N, ∗, ♦) if and only if it is a Cauchy sequence in the uniform space (X, U(M,N ) ). 2. An intuitionistic fuzzy metric space (X, M, N, ∗, ♦) is complete if only if the uniform space (X, U(M,N ) ) is complete. Proof. It is proved in ([11], Theorem 3.4) that if {x}n∈ is a Cauchy sequence in (X, U(M,N ) ), then {x}n∈ is a Cauchy sequence in the intuitionistic fuzzy metric space (X, M, N, ∗, ♦), so the uniform space (X, U(M,N ) ) is complete whenever (X, M, N, ∗, ♦) is complete intuitionistic fuzzy metric space. It is easy to prove that the converse of these two results. Theorem 4. An intuitionistic fuzzy metric space (X, M, N, ∗, ♦) is precompact if only if the uniform space (X, U(M,N ) ) is precompact. Proof. Since an intuitionistic fuzzy metric space (X, M, N, ∗, ♦) is precompact if and only if every sequence in X has a Cauchy subsequence ([13], Theorem 2.4) and a sequence in (X, M, N, ∗, ♦) is Cauchy if and only if it is Cauchy in the uniform space (X, U(M,N ) ) (by Theorem 3), we realize that (X, M, N, ∗, ♦) is precompact if only if (X, U(M,N ) ) is precompact. Remark 1. It is well known that, if (X, U ) is a uniform space, then (Ko (X), HU ) is complete (respectively precompact) if and only if (X, U ) is complete (respectively precompact) (See [9]). Theorem 5. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then (Ko (X), HM , HN , ∗, ♦) is complete if and only if (X, M, N, ∗, ♦) is complete. Proof. By Theorem 3, (Ko (X), HM , HN , ∗, ♦) is complete if and only if (Ko (X), U(HM ,HN ) ) is a complete uniform space. Since by Theorem 2, HU(M,N ) = U(HM ,HN ) on (Ko (X), it follows from Remark 1 that (Ko (X), UHM ,HN ) is complete if and only if (X, U(M,N ) ) is complete. Therefore, we conclude that (Ko (X), HM , HN , ∗, ♦) is complete if and only if (X, M, N, ∗, ♦) is complete. Now we can easily prove the following result as in similar arguements in the above Theorem 5, by using Theorem 4 and Remark 1. Theorem 6. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then (Ko (X), HM , HN , ∗, ♦) is precompact if and only if (X, M, N, ∗, ♦) is precompact. Theorem 7. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then (Ko (X), HM , HN , ∗, ♦) is compact if and only if (X, M, N, ∗, ♦) is compact. Proof. Since an intuitionistic fuzzy metric space is compact if and only if it is precompact and complete (See Corollary 2.7 in [13]), and by Theorems 5 and 6, we deduce that (Ko (X), HM , HN , ∗, ♦) is compact if and only if (X, M, N, ∗, ♦) is compact.

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Fractals in the Intuitionistic Fuzzy Metric Space

In this paper, our primer interests are to deﬁne intuitionistic fuzzy fractals on complete and compact spaces. Deﬁnition 9. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space and fn : X −→ X, n = 1, 2, 3, ..., No (No ∈ Æ) be No - intuitionistic fuzzy Bcontractions (respectively intuitionistic fuzzy Edelstein contractions) with the corresponding contractivity ratios kn , n = 1, 2, 3, ..., No. The system {X; fn , n = 1, 2, 3, ..., No} is called an Intuitionistic Fuzzy Iterated Function System (IF-IFS) in the intuitionistic fuzzy metric space (X, M, N, ∗, ♦). Then the Intuitionistic Fuzzy Hutchinson-Barnsley operator (IF-HB operator) of the IF-IFS is a function F : Ko (X) −→ Ko (X) deﬁned by F (B) =

No

fn (B).

n=1

Theorem 8. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorﬀ intuitionistic fuzzy metric space. Suppose f : X −→ X is an intuitionistic fuzzy B-contraction with the corresponding contractivity ratio k ∈ (0, 1). Then, HM (f (A), f (B), kt) ≥ HM (A, B, t)

and

HN (f (A), f (B), kt) ≤ HN (A, B, t)

for all A, B ∈ Ko (X) and t > 0. Theorem 9. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorﬀ intuitionistic fuzzy metric space. Suppose f : X −→ X is an intuitionistic fuzzy Edelstein contraction. Then, HM (f (A), f (B), .) ≥ HM (A, B, .)

and

HN (f (A), f (B), .) ≤ HN (A, B, .)

for all A, B ∈ Ko (X) such that A ∩ B = ∅. Lemma 1. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorﬀ intuitionistic fuzzy metric space. If A, B, C, D ∈ Ko (X), then HM (A ∪ B, C ∪ D, t) ≥ min{HM (A, C, t), HM (B, D, t)} and HN (A ∪ B, C ∪ D, t) ≤ max{HN (A, C, t), HN (B, D, t)} for all t > 0. The proofs of the above Theorems and Lemmas are straightforward and are omitted.

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Theorem 10. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorﬀ intuitionistic fuzzy metric space. Suppose {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} is an IF-IFS of intuitionistic fuzzy B-contractions with the same contractivity ratio k ∈ (0, 1) for each n ∈ {1, 2, ..., No} . Then the HB operator F is an intuitionistic fuzzy B-contraction with the ratio k. Proof. Lemma 1 and Theorem 8 completes the proof. Similarly we can prove the following result. Theorem 11. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorﬀ intuitionistic fuzzy metric space. Suppose {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} is an IF-IFS of intuitionistic fuzzy Edelstein contractions. Then the HB operator F is an intuitionistic fuzzy Edelstein contraction. Theorem 12 (HB Theorem for Complete Space). Let (X, M, N, ∗, ♦) be a complete intuitionistic fuzzy metric space. Let {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} be an IF-IFS of intuitionistic fuzzy B-contractions and F be the IF-HB operator of the IF-IFS. Then, there exists only one compact invariant set A∞ ∈ Ko (X) of the HB operator (F ) or, equivalently, F has a unique ﬁxed point namely A∞ ∈ Ko (X). Proof. Since (X, M, N, ∗, ♦) is a complete intuitionistic fuzzy metric space and by Theorem 5, we have (Ko (X), HM , HN , ∗, ♦) is also complete Hausdorﬀ intuitionistic fuzzy metric space. By Theorem 10 shows that that the HB operator F is an intuitionistic fuzzy B-contraction. Then by the Intuitionistic Fuzzy Banach Contraction Theorem (Theorem 7 in [1]), we conclude that F has a unique ﬁxed point, namely A∞ ∈ Ko (X). Theorem 13 (HB Theorem for Compact Space). Let (X, M, N, ∗, ♦) be a compact intuitionistic fuzzy metric space. Let {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} be an IF-IFS of intuitionistic fuzzy Edelstein contractions and F be the IFHB operator of the IF-IFS. Then, there exists only one compact invariant set A∞ ∈ Ko (X) of the HB operator (F ) or, equivalently, F has a unique ﬁxed point namely A∞ ∈ Ko (X). Proof. Similar arguements of the Theorem 12, Theorems 7 and 11 completes the proof. Now we deﬁne the Intuitionistic Fuzzy Attractor or Fractal on complete space (respectively compact space) as the set A∞ ∈ Ko (X), which is described in the HB Theorem 12 for complete space (respectively HB Theorem 13 for compact space). Such A∞ ∈ Ko (X) is also called as Fractal generated by the IF-IFS of intuitionistic fuzzy B-contractions (respectively intuitionistic fuzzy Edelstein contractions) and so called as Intuitionistic Fuzzy Fractal on complete space (compact space). This study will leads to develop more interesting concepts and properties of fractals in the fuzzy spaces.

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Conclusion

In this study, we proposed a generalization of Hutchinson-Barnsley theory in order to deﬁne and construct the intuitionistic fuzzy IFS fractals with respect to Hausdorﬀ intuitionistic fuzzy metric through iterated function system of intuitionistic fuzzy contractions by using the intuitionistic fuzzy version of two ﬁxed point theorems. Acknowledgments. The research work has been supported by University Grants Commission (UGC - MRP & SAP), New Delhi, India.

References 1. Alaca, C., Turkoglu, D., Yildiz, C.: Fixed points in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 29, 1073–1078 (2006) 2. Barnsley, M.: Fractals Everywhere, 2nd edn. Academic Press, USA (1993) 3. Barnsley, M.: SuperFractals. Cambridge University Press, New York (2006) 4. George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64, 395–399 (1994) 5. Gregori, V., Sapena, A.: On ﬁxed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems 125, 245–252 (2002) 6. Gregori, V., Romaguera, S., Veeramani, P.: A note on intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 28, 902–905 (2006) 7. Hutchinson, J.E.: Fractals and self similarity. Indiana University Mathematics Journal 30, 713–747 (1981) 8. Kramosil, I., Michalek, J.: Fuzzy metrics and statistical metric Spaces. Kybernetika 11, 336–344 (1975) 9. Morita, K.: Completion of hyperspaces of compact subsets and topological completion of open-closed maps. General Topology and its Applications 4, 217–233 (1974) 10. Park, J.H.: Intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 22, 1039–1046 (2004) 11. Park, J.S., Kwun, Y.C., Park, J.H.: Some properties and Baire space on intuitionistic fuzzy metric space. In: Sixth International Conference on Fuzzy Systems and Knowledge Discovery, pp. 418–422. IEEE Press, Los Alamitos (2009) 12. Rodriguez-Lopez, J., Romaguera, S.: The Hausdorﬀ fuzzy metric on compact sets. Fuzzy Sets and Systems 147, 273–283 (2004) 13. Saadati, R., Park, J.H.: On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 27, 331–344 (2006) 14. Schweizer, B., Sklar, A.: Statistical metric spaces. Paciﬁc Journal of Mathematics 10, 314–334 (1960) 15. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)

Research on Multi-evidence Combination Based on Mahalanobis Distance Weight Coefficients Si Su and Runping Xu No. 3 Wanshou Road, Haidian District, 100036 Beijing, China [email protected]

Abstract. To solve the contradiction in using Dempster’s method for the combination of highly conflicting evidences, an improved evidence combination method is presented based on Mahalanobis distance weight coefficients. First of all, the similarities between evidences are used as an approach to judge whether conflict exists. If there are more than 3 evidences consisting of conflicts, the Mahalanobis Distance algorithm can be used to calculate the distance between each evidence and the others to obtain the evidences’ weight coefficients, which could be transformed into BPA functions by means of the coefficients, and finally the Dempster’s method is used for the combination. Stimulation results show that this method can deal with conflicting evidences effectively, calculate faster than traditional algorithms, reduce more uncertainty in recognizing results, and retain the advantages of Dempster’s method in tackling non-conflicting evidences. Keywords: conflicting evidence; D-S evidence theory; information fusion.

1 Introduction Dempster’s evidence combination theory aims for uncertainty reasoning promoted by Dempster and Shafer at late 60s and early 70s in the 20th century. Oriented towards basic hypothesis set within the frame of discernment, this method is the further expansion of the probability theory applied to sensor measurements fusion in different levels. This theory is an improved method for uncertainty reasoning adapted to multisensor information fusion [1]. Nevertheless, shortcomings also exist in Dempster’s method, that is ineffectiveness in handling highly conflicting evidences. Results of evidence combination are basically right when there is no or little conflicts; whereas results contrary to common sense are arrived at in time of serious conflicts. To solve this problem, retaining the advantages of traditional algorithms and adopting the concept of the Mahalanobis Distance [2] from statistical theory, this paper presents an improved evidence combination method based on evidence weight coefficients. More accurate results of multi-sensor information fusion could be obtained by use of this method.

2 An Overview of Dempster Theory and Some Improvement Let m1 and m2 be functions of Basic Probability Assignment (BPA) of two evidences in the frame of discernment. The Dempster combination formula could be presented as P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 97–104, 2011. © Springer-Verlag Berlin Heidelberg 2011

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m(C ) =

∑ m ( A)m ( B)

A∩ B = C

∈ , A∈Θ, B∈Θ,

In formula 1, C Θ

1

2

(1)

1− k k=

∑ m ( A)m ( B)

A ∩ B =φ

1

2

indicating the

conflicting degree of two evidences. When larger conflict appears, k is close to 1, and the combination result would always be unreasonable [3]. Many scholars have brought improved suggestions to solve this problem which could be generally categorized into two classes: one is improvement of combination rules, and the other is perfection of evidence models [3-9]. Yager first improved D-S evidence theory by classifying the conflicting evidences into set Θ [4]. Though conflicts are removed, some useful information is also lost. When the conflict level is high, recognition results would fall under the set Θ, that is, many fields are unknown, which is useless for decision making. Besides, Yager’s solution could not get rid of “one-vote veto” problem existing in Dempster’s method. When one evidence is totally inconsistent with many, the focal element supported by them in the combination would be fully vetoed [5]. Reference [6] raises weighted additive strategy to process the existing evidence combination. Adding each evidence together in derangement to achieve conformity, conflicts could be removed consequently; then Dempster’s method could be used to synthesize to get the final result. However, this kind of correctness lies on utilizing repeatedly the existing evidences. In Reference [7], Murphy presents when there are many evidences in the system, basic assignment probability of these evidences should be equalized and combined for many times with Dempster’s method. But this is just the simple average of multi-source information without considering the relevance of these evidences. While Reference [8] unfolds an improvement of Murphy’s way by Deng Yong in taking the relevance into account. Using the formula of Euclidean Distance algorithm to calculate the distance between evidences, this method ensures the credibility of evidences greatly. Weighted average of evidences and combination making by Murphy’s method could process conflicting evidences powerfully except for low recognition speed. Combining the advantages of traditional algorithms, this paper overcomes “one vote veto” problem in Dempster and Yager’s method. Judgment should be made first on whether conflicting evidences exist based on degree of similarity between the evidences. Supposing there are over three conflicting evidence combinations, the Mahalanobis Distance algorithm can be used later to calculate the distance between one evidence and other groups of evidences so as to obtain the evidence weight coefficients. Then combination is to be made using the transformed results of BPA functions. This way could ensure the axiom BPA functions should add to 1 and also meet the requirement of exchangeability in combination order. Compared with traditional algorithm, this method possesses faster recognition speed, reduces the uncertainty largely, as well as retains the merit of Dempster’s method in processing non-conflicting evidence combinations.

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3 Improved Combination Method Based on Mahalanobis Distance Weight Coefficients Let the frame of discernment in fusion system include many completed incompatible hypothetical propositions {A1, A2, A3,…, Aw}, the BPA functions of evidence e1, e2, e3,…, en are respectively called as m1, m2, m3,…, mn. Then synthetic procedure could be carried out as follows: 1) Test whether there is conflict between evidences. Evidence ei and ej could find their BPA functions mi and mj, with a maximum of max[ mi ( Ap )] and 1≤ p ≤ w

max[mi ( Aq )] respectively. If Ap and Aq correspond to the same recognition result,

1≤ q ≤ w

there is no evidence conflict; otherwise, mi and mj are treated as two vectors, similarity coefficient between them would be calculated as follows:

δ (mi , m j ) =

miT m j

(2)

mi m j

≤p

Let p be threshold value. If σ(mi,mj) , that means no conflict between ei and ej. Test BPA functions m1, m2, m3,…, mn in pairs to find out conflicting evidences, then go to Formula 2); if no conflicting evidence is discovered after the whole test, then go to Formula 5); 2) Confirm the total number of evidences n. If n=2, go to Formula 3); if n>2, go to Formula 4); 3) Apply the Additive Strategy to process BPA functions m1, m2, and then go to Formula 5); 4) Calculate the distance between one evidence and the other groups of evidences. Except for evidence ei, all remaining evidences could be viewed as a unity G, the average is vector for w dimension, and covariance ∑ is matrix of w w order. Using the Mahalanobis Distance algorithm to calculate the distance between ei and the unity G:

μ

×

d (i ) = (mi − μ ) ∑ −1 (mi − μ )T , i=1,2,…,n

(3)

Here, d(i) indicates the proximity between evidence ei and others. Weight coefficients vector of n dimension is defined as: 2

n ⎧ ⎫ ⎪⎪ ∑ d (i ) − d (i ) ⎪⎪ , i=1,2,…,n β (i ) = ⎨ i =1 n ⎬ ⎪ max[∑ d (i ) − d (i )] ⎪ ⎪⎩ 1≤i ≤n i =1 ⎪⎭

(4)

Here β(i) indicates the importance of evidence ei among the list e1, e2, e3,…, en. By use of weight coefficients, BPA functions m1, m2, m3,…, mn could be converted as following, and then turn to Formula 5).

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∈

mi' ( A) = β (i ) mi ( A) , ∀ A Θ, A≠Θ, i=1,2,…,n

(5)

mi' (Θ) = β (i )mi (Θ) + [1 − β (i )] , i=1,2,…,n

(6)

5) Dempster’s method is used to synthesize the evidences e1, e2, e3,…, en, and the final result could be arrived at. In conclusion, the overall procedure of combination put forward in this paper could be reduced as Figure 1. Evidences

Test whether conflicts exist yes

no

Confirm the number of evidences n

n>2 Calculate the distance d between evidences

Calculate the weight coefficients

β

Calculate the transformed BPA functions

n=2 Process BPA functions in Additive Strategy

Combination in Dempster method

Recognition results

Fig. 1. Total process of combination

4 Simulation Analysis Assuming a maneuvering target is detected in a sea battlefield and identified as the hostile force, five different sensors would provide information for this target. The frame of discernment could be: { A = fighter, B = helicopter, C = UAV }, and the BPA functions of evidences transformed by calculated data from these five sensors could be indicated as below: e1: m1(A) = 0.5, m1(B) = 0.2, m1(C) = 0.3 e2: m2(A) = 0.5, m2(B) = 0.2, m2(C) = 0.3 e3: m3(A) = 0.6, m3(B) = 0.1, m3(C) = 0.3 e4: m4(A) = 0.6, m4(B) = 0.1, m4(C) = 0.3 e5: m5(A) = 0.6, m5(B) = 0.1, m5(C) = 0.3 4.1 If Data Provided Regular Let threshold value for conflict test p=0.5. Six methods, such as Dempster’s, Yager’s, Additive Strategy, Murphy’s in Reference [7], Deng Yong’s in Reference [8] and improved method proposed here are applied respectively to synthesize the combination results. Results are compared in Table 1.

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Table 1. Comparison of different evidence combination methods on condition of no conflict Evidence Combination

e1 + e2 + e3

e1 + e2 + e3+ e4

e1 + e2 + e3+ e4 + e5

Combination Method Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper

m(A)

Recognition Results A

m(B)

m(C)

m(Θ)

0.9417 0.2100

0.0179 0.0040

0.0404 0.0090

0 0.7770

0.9286

0.0195

0.0519

0

A

0.9257 0.9240 0.9417 0.9760 0.1260

0.0198 0.0196 0.0179 0.0031 0.0004

0.0544 0.0564 0.0404 0.0209 0.0027

0 0 0 0 0.8709

A A A A

Θ

0.9695

0.0034

0.0271

0

A

0.9671 0.9654 0.9760 0.9889 0.0756

0.0038 0.0036 0.0031 0.0005 0.0000

0.0291 0.0310 0.0209 0.0106 0.0008

0 0 0 0 0.9235

A A A A

Θ

0.9856

0.0006

0.0138

0

A

0.9843 0.9831 0.9889

0.0007 0.0006 0.0005

0.0150 0.0162 0.0106

0 0 0

A A A

Θ

Many facts are indicated by this table. When sensors are working normally, recognition results using Yager’s method are all unknown term. That is because it takes out conflicting factors in combination, and most BPA functions of conflicting evidences are classified into uncertainty set Θ. This method is applicable in handling conflicting evidences, but as to no or little confliction it shall not apply. By comparison, the other methods could ensure the accuracy of results, and recognition result focalizes as the increasing of evidences supported. Despite this, recognition results applying Dempster’s or the method proposed in this paper possesses less uncertainty compared with others, which indicates the merit of Dempster is maintained in the improved method of this paper. 4.2 If Interference Occurs in the Second Sensor It works out of order because of electromagnetic interference from the rivaling force, and then BPA functions are changed to: e2: m2(A) = 0, m2(B) = 0.2, m2(C) = 0.8 So conflicts between the evidences are obvious. Combination results using these methods are compared in Table 2. From Table 2, the problem “one vote veto” appears in both Dempster’s and Yager’s method when big conflict exists between the evidences. That is when there is only one evidence completely vetos A while most don’t, the final combination results would completely veto A, which is surely unreasonable. Using Yager’s method, most

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Table 2. Comparison of different evidence combination methods on condition that one conflicting evidence exists Evidence Combination

e1 + e2 + e3

e1 + e2 + e3 + e4

e1 + e2 + e3 + e4 + e5

Combination Method Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper

Recognition Results C

m(A)

m(B)

m(C)

m(Θ)

0 0

0.0526 0.0040

0.9474 0.0720

0 0.9240

0.3022

0.0330

0.6648

0

C

0.3169 0.4517 0.6648

0.0298 0.0290 0.0851

0.6533 0.5193 0.2502

0 0 0

C C A

0 0

0.0182 0.0004

0.9818 0.0216

0 0.9780

Θ

0.4721

0.0086

0.5193

0

C

0.4962 0.6959 0.7477 0 0

0.0077 0.0060 0.0490 0.0061 0.0000

0.4962 0.2981 0.2034 0.9939 0.0065

0 0 0 0 0.9935

A, C A A C

0.6439

0.0020

0.3541

0

A

0.6668 0.8434 0.8671

0.0017 0.0011 0.0125

0.3315 0.1556 0.1204

0 0 0

A A A

Θ

C

Θ

BPA functions of conflicting evidences are classified into the uncertainty set Θ. With the increase of evidences supporting A, the uncertainty of recognition results also keeps on increasing. With regard to Additive Strategy, Murphy’s method and Deng Yong’s method, the recognition results are C when there are three evidences, and the correct results could only be arrived at when more evidences appear. It shows low recognition speed of these methods; in practical application, more evidences need to be collected to ensure the accuracy of recognition results. When using Murphy’s method, that is just a simple average of weight coefficients of evidences without considering the degree of practical importance, which makes evidences of little importance affecting overall combination result. Uncertainty factors in recognition results need to be further reduced. Although Deng Yong’s method makes up for the shortcomings of Murphy’s, it can’t compare with the method proposed here in recognition speed and competence in reducing the uncertainty of recognition results. 4.3 If Interferences Occur in Both the Second and Fifth Sensor They work out of order because of strong electromagnetic interference from the rivaling force, and then BPA functions are changed to: e2: m2(A) = 0, m2(B) = 0.2, m2(C) = 0.8 e5: m5(A) = 0.1, m5(B) = 0.8, m5(C) = 0.1

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Table 3. Comparison of different evidence combination methods on condition that two conflicting evidences exist Evidence Combination

e1 + e2 + e3 + e4 + e5

Combination Method Dempster Yager Additive Strategy Murphy DengYong This Paper

Recognition Results C

m(A)

m(B)

m(C)

m(Θ)

0 0

0.1290 0.0003

0.8710 0.0022

0 0.9975

0.5233

0.0748

0.4020

0

A

0.4377 0.6861 0.8334

0.1246 0.0382 0.0218

0.4377 0.2757 0.1448

0 0 0

A, C A A

Θ

In the case of interference coming from multiple sensors simultaneously, combination results using these methods are compared in Table 3. From Table 3, the question “one vote veto” still exists in both Dempster’s and Yager’s method when two sensors are interfered simultaneously. More supporting evidences are needed to arrive at the correct recognition results in Murphy’s method. Although the Additive Strategy and Deng Yong’s method could come at the accurate result, the way raised in this paper could reduce the uncertainty of recognition results better. Besides, summing up the results getting from Tables 1, 2 and 3, this improved method could make sure all BPA functions should add to 1. Moreover, by making use of Dempster’s method to synthesize the transformed BPA functions, it ensures the exchangeability of evidence combination order.

5 Conclusions Aiming at the shortcomings of Dempster’s method in processing conflicting evidences, an improved evidence combination method is presented in this paper based on Mahalanobis Distance evidence weight coefficients. Simulation results indicate that this method could efficiently process conflicting evidence combination, tackle the problem of “one vote veto”, satisfy the need that the sum of BPA functions should be 1, and ensure the exchangeability of evidence combination order. Compared with other traditional methods, this one is possessed of many merits, such as achieving high speed of recognition, efficiently reducing the uncertainty of recognition results and retaining good features of Dempster’s method in coping with non-conflicting evidence combination.

References 1. Chongzhao, H., Hongyan, Z., Zhansheng, D.: Multi-source Information Fusion. Tsinghua University Press, Beijing (2006) 2. Yan, W., Silian, S., Aiqing, W.: Mathematic Statistics And MATLAB Engineering Data Analysis. Tsinghua University Press, Beijing (2007)

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3. Haiyan, L., Zonggui, Z., Xi, L.: Combination of Conflict Evidences in D-S Theory D-S. J. Journal of University of Electronic Science and Technology of China 37, 701–704 (2008) 4. Yager, R.R.: On the Dempster–Shafer framework and new combination rules. J. Information Science 41, 93–137 (1989) 5. Fenghua, L., Lize, G., Yixian, Y.: An Efficient Approach for Conflict Evidence Combination. J. Journal of Beijing University of Posts and Telecommunications 31, 28–32 (2008) 6. Bin, Q., Xiaoming, S.: A Data Fusion Method for Single Sensor under Interfere environment. J. Journal of Projectiles, Rockets, Missiles and Guidance 27, 305–307 (2006) 7. Murphy, C.K.: Combining belief functions when evidence conflicts. J. Decision Support Systems 29, 1–9 (2000) 8. Yong, D., Wenkang, S., Zhenfu, Z.: Efficient Combination Approach of Conflict Evidence. J. Journal Infrared Millimeter and Waves 23, 27–32 (2004) 9. Quan, P., Shanying, Z., Yongmei, C.: Some Research on Robustness of Evidence Theory. J. ACTA Automatic Si 27, 798–805 (2001)

Mode Based K-Means Algorithm with Residual Vector Quantization for Compressing Images K. Somasundaram and M. Mary Shanthi Rani Image Processing lab Department of Computer Science and Applications Gandhigram Rural Institute- Deemed University Gandhigram

Abstract. Image compression plays a vital role in many online applications like Video Conferencing, High Definition Television, Satellite Communication and other applications that demand fast and massive transmission of images. In this paper, we propose a Mode based K-means method that combines K-Means and Residual Vector Quantization(RVQ) for compressing images. Three processes are involved in this approach; Partitioning and Clustering, Pruning and Construction of Master codebook and Residual vector Quantization. Extensive experiments show that this method obtains a fast solution with better compression rate and comparable PSNR than conventional K-Means algorithm. Keywords: K-Means, Mode, Median, Residual Vector Quantization, Break Even Point, Master Codebook, Residual Codebook.

1 Introduction Vector quantization (VQ) is an effective means for lossy compression of speech and digital images. K-Means [1] is a widely used Vector Quantization technique known for its computational efficiency and speed. It has a great number of applications in the fields of image and video compression, Pattern Recognition, and Watermarking, However, it has some pitfalls like apriori fixation of number of clusters and random selection of initial seeds. Wrong choice of initial seeds may yield poor results and results in increase in time for convergence. We begin with an overview of existing methods based on K-Means algorithm. A pioneering work of seeds initialization was proposed by Ball and Hall(BH) [2]. A similar approach named as Simple Cluster Seeking(SCS) was proposed by Tou and Gonzales [3]. The SCS method chooses the first input vector as the first seed and the rest of the seeds are selected if they are ‘d’ distance apart from all selected seeds. The SCS and BH method are sensitive to the parameter ‘d’ and the order of the inputs. Astrahan [4] suggested a method using two distance parameters, d1 and d2 which first computes the density of each point in the dataset. The highest density point is chosen as the first seed and subsequent seed points are chosen in the order of decreasing density subject to the condition that each new seed point be at least at a distance of ‘d2’ from all other previously chosen seed points. The drawback in this approach is that it is very sensitive to the values of d1 and d2 and requires hierarchical clustering. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 105–112, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Kaufman and Rousseau [5] introduced a method that estimates the density through pair wise distance comparisons and initializes the seed clusters using the input samples from the areas with high local density. A significant drawback of the method is its computational complexity which may sometimes be higher than k-means when ‘n’ is large. Katsavounidis et al (KKZ) [6] suggested a parameterless approach, which chooses the first seed near the “edge” of the data, by choosing the vector with the highest norm. Then, it chooses the next seed to be the point that is farthest from the nearest seed in the set chosen so far. This method is very inexpensive (O (kn)) and is easy to implement but is sensitive to outliers. Bradley and Fayyad (BF) [7] proposed an initialization method that is suitable for large datasets. The main idea of their algorithm is to select ‘m’ subsamples from the data set, apply the k-means on each subsample independently, keep the final k centers from each subsample and produce a set that contains mk points. They apply the k-means on this set ‘m’ times: at the first time, the first k points as initial centers and at the second time, the second k points and so on. The algorithm returns the best k centers from this set. They use 10 subsamples from the data set, each of size 1% of the full dataset size. Finally, a last round of k-means is performed on this dataset and the cluster centers of this round are returned as the initial seeds for the entire dataset. This method generally performs better than k-means and converges to the local optimum faster. However, it still depends on the random choice of the subsamples and hence, may obtain poor clusters in an unlucky session. Fahim et al. [8] proposed a method to minimize the number of distance calculations required for convergence. Arthur and Vassilvitskii [9] proposed the k-means++ approach, which is similar to the KKZ method. In k-means++, the next point chosen will be the one with the probability proportional to the minimum distance of this point from already chosen seeds. Note that due to the random selection of first seed and probabilistic selection of remaining seeds, different runs have to be performed to obtain a good clustering. Single Pass Seed Selection (SPSS) [10] algorithm initializes first seed and the minimum distance that separates the centroids is based on the point which is close to more number of other points in the data set. An efficient fast algorithm to generate VQ codebook was proposed by Kekre et al[11] which uses sorting method to generate codebook and the code vectors are obtained using median approach. Recently, Residual VQ has captured the attention of researchers as they reduce memory and computation complexity [12]-[14]. A low bit rate still image compression scheme by compressing the indices of Vector Quantization (VQ) and generating residual codebook was proposed by Somasundaram and Dominic[15]. The proposed method is an innovative approach that combines the merits of mode based seed selection and residual vector quantization The motivation of this approach is multi-fold. Firstly, the use of mode parameter exploits correlation among similar blocks for clustering resulting in low bit rate and preserves image quality as well. Second, the use of divide and conquer strategy results in significant reduction of computation time. Third, residual vector quantization phase encodes the residual image, further enhancing the reconstructed image quality.

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The rest of the paper is organized as follows: In Section 2, we briefly discuss Mode based K-means with Residual Vector Quantization (RVQ) approach. Experimental results are presented in Section 3. Finally, Section 4 concludes by outlining the merits and demerits of the proposed method.

2 Mode Based K-Means with RVQ Mode based K-means is a simple and fast clustering algorithm. It adopts the divide and conquer strategy by dividing the input data space into classes and apply K-means clustering. Mode based K-means works in three phases. In the first phase, the input vectors are divided into “N” classes based on the mean value of each vector. The distribution of image data determines the number of classes ‘N’. As the range ‘R’ of the value of a pixel in image data is 0-255, the range of the domain of each class is set based on the ratio of ‘R’ to the number of classes(N). The range of the domain of each class Di is calculated as follows; Di = (max_pixel_value – min_pixel_value) / N * I

(1)

where i is the Class_index between 1 and N, max_pixel_value and min_pixel_value are the maximum and minimum pixel values in the input image data respectively. This mean based grouping of input data speeds up convergence subsequently reducing computation time. K-Means algorithm is performed on each class of vectors with initial seeds. The number of initial seeds ‘M[i]’ in each class Ci is proportional to its size and is calculated as M[i] = Class-size[i] / T

(2)

Where ‘T’ is the ratio of total number of training vectors and number of initial seeds ‘K’ and ∑M[i] = K , i = 1…N. The novelty of our approach is using mode for choosing the initial seeds. The initial seeds are the high mode vectors among the class population. High mode vectors are the vectors whose values are mode values of each dimension of the class vectors, in descending order. If mode_vector[i,j] is the ith mode vector, then mode_vector(i,j)= pixel value that occupies ith position in descending order of frequency of occurrence(mode) in jth dimension of all the vectors of a class. (3) In the second phase, the code vectors generated from each class are subjected to pruning based on a minimum (Euclidean) distance threshold “D” with the code vectors of the partially generated code book. A new code vector v1i of a class Ci, will be added to the master codebook only if its Euclidean distance with all of the code vectors of classes C1 ..Ci-1 is greater than a predefined threshold “D”. The value of ‘D’ can be used to tune rate-distortion performance. Increase in the value of ‘D’ results in reduction in bit-rate but at the cost of image quality. Experiments have revealed that the value of ‘D’ should lie in the range 2-4 to achieve better rate-distortion

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performance. Also, pruning of code vectors done at the construction phase significantly reduces the code book size and computation time as well. The third phase constructs the residual codebook. It computes the residual image by computing the difference between the input vector and the code vector. Next, the residual vectors with the same code index are grouped together which results in ‘K’ groups. One median vector (16-element) is constructed from each group by taking the median value of each column in the group. Next, each median vector (16-element) is converted to 4-element by dividing the 16-element vector into four 4-element blocks and representing each block by its mean value. Thus, the values of 4-element vector are the mean values of the four sub blocks. The resulting ‘K’ 4-element vectors form the residual code book. Grouping vectors by their code index provides two advantages. First, input vectors quantized to same code vector are highly correlated which may result in highly correlated residual vector group .This correlation can be utilized to reduce the size of the residual codebook. Second, we don’t require any additional index for residual code book. One master index is used for both the master code book and residual codebook as well. This significantly reduces the index overhead, despite the use of two code books. 2.1 Algorithm 2.1.1 Seed Generation Phase 1. Divide the 512x512 input image data into 4x4 image blocks and convert the blocks into training vectors(16-vecctor) of dimension 16. 2.

Classify the training vectors into “N” classes based on the mean value of its elements of each vector.

3.

Calculate the range of the domain Di of each class using equation (1).

4.

Calculate ‘T’ which is equal to the ratio of total number of training vectors and number of initial seeds ‘K’.

5.

Determine the number of initial seeds ‘M[i]’ for each class using equation (2).

6.

Construct mode vectors whose values are the mode value (value with highest frequency of occurrence) of its respective dimension in descending order until we get the desired number of initial seeds ‘M[i]’ using equation (3).

7.

Perform K-means algorithm on each class with ‘M’ initial seeds.

2.1.2 Pruning and Construction Phase 8. Calculate the Euclidean distance between each of the newly generated code vectors of the current class Ci and the code vectors in the partial codebook constructed from classes C1..Ci-1 so far. 9.

Add only those code vectors to the code book whose distance with all of the partial codebook vectors is greater than a predefined threshold ‘D’.

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109

10. Repeat this process for all the classes Ci, i=1 .. N 11. The final code book resulting after step 10 is the initial codebook for the entire data set. 12. Perform a final run of K-means on the the entire data set with this final codebook as initial seeds. 13. The final code book resulting after step 12 is the master codebook for the entire data set. 2.1.3 Residual Vector Quantization (RVQ) Phase 14. Compute the residual vector which is the difference between the input vector and its corresponding code vector. 15. Group all residual vectors having the same master code index. 16. Construct the median vector for each group by calculating the median value of each column of the residual vectors belonging to the same group. 17. Convert each median vector (16-element) to 4-element vector by dividing it into four 2x2 blocks and representing each sub block by its mean value. 18. Construct the residual codebook with the median vectors resulting out of step 17. The master code book, residual code book and the code indices form the compressed stream of data. 2.1.4 Decoding Phase 1 Determine the code vector from the master code book based on the Master code index of each input vector. 2

Retrieve the residual code vector (4-element) from the residual code book using Master code index.

3

Convert the residual 4-element vector to 16-element vector by replacing the pixel values of the 2x2 sub blocks with their respective mean value.

4

Construct the final code vector by adding or subtracting the corresponding residual values of residual vector (16-vector) from the master code vector.

3 Results and Discussion We carried out experiments using the proposed method on several standard images of size 512 x 512 on a machine with core2 duo processor at 2 GHZ using MATLAB 6.5. The time taken for code book generation, bit rate in terms of BPP (Bits Per Pixel) and PSNR values computed for standard images Lena, Barbara, Boat, Peppers and Couple using our method and K-Means algorithm are given in Table 1. We note from

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the Table 1 that the time taken for code book generation with the proposed method reduced by 2 times in the case of Boat image to about 17 times for Peppers when compared to K-Means algorithm. It is further observed that the bit rate has also improved without significant reduction in the image quality measured in terms of PSNR value. Experimental results demonstrate that Mode based k-means shows optimal performance in terms of Computation time, Bit rate and PSNR for a specific value of ‘N’ which can be considered as the Break-Even Point (BEP). Further analysis of results reveals that BEP is image-specific and is equal to c(log2K) for some constant ‘c’ which lies in the range between 0.5 and 1 where K is the number of initial seeds. Moreover, the results show that the value of c depends on the distribution of frequently occurring pixels in input data (hit-no). The value of c is equal to 0.5 for images whose hit-no is less than 200 and 1 for images with hit_no >200. Besides, the size of the final codebook is reduced further in the pruning phase and is found to be less than K thereby enhancing compression rate. It is worth noting that the initial value of K is chosen to be less than 256 to increase bit rate without compromising picture quality by coding the residuals in Residual Vector Quantization phase. In terms of memory requirements, the proposed method requires storage for the master and the residual codebooks, as well as for storing code indices. As the master codebook consists of a total of K 16-element vectors, the size P in bits of the master codebook is given by P = K × 16 × 8 bits. However, the residual codebook requires storage only for K 4-element vectors and as the value of each vector element does not exceed 8, the size Q in bits of the residual codebook is given by Q = K × 4 × 3 bits. Despite using two codebooks, the memory requirements of the proposed method (P + Q) is still less than conventional K-Means. For visual comparison, the original and reconstructed images of Lena by the proposed method are shown in Fig.1 and Fig. 2.

Fig. 1. Original Lena image

Fig. 2. Reconstructed Lena image by Mode based K-means

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Table 1. Perfomance analysis of Mode based K-Means on test images Image

Method

BEP (N)

K-Means Lena

Barbara

Boat

Peppers

Couple

Bridge

Computation time 1500

Bit rate

K

PSNR

0.125

256

33.01

350

0.113

208

32.812

2075

0.125

256

28.2

Mode Based K-means K_Means

4

Mode based K-means K-Means

8

160

0.115

210

27.972

-

362

0.125

256

31.4

Mode based K-Means K-Means

8

150

0.119

214

31.319

-

3160

0.125

256

31.4

Mode based K-Means Kmeans

8

189

0.119

214

31.176

-

1196

0.125

256

28.898

Mode based K-Means K-Means

4

188

0.118

212

28.699

-

1368

0.125

256

25.723

Mode Based K-Means

8

249

0.116

211

25.454

4 Conclusion Mode Based K-Means proposes a three phase method in which the use of mode parameter for initial seed selection has been investigated. Mode based Kmeans shows superior performance than traditional kmeans with random choice of initial seeds significantly in terms of computation time with comparable PSNR and Bit rate. This would be an ideal choice for applications of image compression where time complexity plays a crucial role like video conferencing, online search engines of image and Multimedia databases.

References [1] Lloyd, S.P.: Least square quantization in PCM. IEEE Trans. Inform. Theory 28, 129–137 (1982) [2] Ball, G.H., Hall, D.J.: PROMENADE-an on-line pattern recognition system. Stanford Research Inst. Menlo Park, Stanford University (1967) [3] Tou, J., Gonzalez, R.: Pattern Recognition Principles, p. 377. Addision-Wesley, Reading (1977) [4] Astrahan, M.M.: Speech analysis by clustering (1970)

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[5] Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis, p. 342. Wiley, New York (1990) [6] Katsavounidis, I., Kuo, C.C.J., Zhen, Z.: A new initialization technique for generalized Lloyd iteration. IEEE. Sig. Process. Lett. 1, 144–146 (1994) [7] Bradley, P.S., Fayyad, U.M.: Refining initial points for K-means clustering. In: Proceeding of the 15th International Conference on Machine Learning (ICML 1998), July 24-27, pp. 91–99. ACM Press, Morgan Kaufmann, San Francisco (1998) [8] Fahim, A.M., Salem, A.M., Torkey, F.A., Ramadan, M.: An efficient enhanced k- means clustering algorithm. J. Zhejiang Univ. Sci. A. 7, 1626–1633 (2006) [9] Arthur, D., Vassilvitskii, S.: k-means++: The advantages of careful seeding. In: Proceeding of the 18th Annual ACM-SIAM Symposium of Discrete Analysis, January 7-9, pp. 1027–1035. ACM Press, New Orleans (2007) [10] Karteeka Pavan, K., Rao, A.A., Dattatreya Rao, A.V., Sridhar, G.R.: Single Pass Seed Selection Algorithm for k-Means. Journal of Computer Science 6(1), 60–66 [11] Kekre, H.B., Sarode, T.K.: An Efficient Fast Algorithm to Generate Codebook for Vector Quantization. In: Proceedings of International Conference on Emerging Trends in Engineering and Technology, Thadomal Shahani, pp. 62–67 [12] Juang, B.H., Gray, A.H.: Multiple stage vector quantization for speech coding. In: Proc. IEEE Int. Conf. Acoust. Speech,Signal Process., vol. 1, pp. 597–600 (1982) [13] Kossentini, F., Smith, M.J.T., Barnes, C.F.: Image Coding Using Entropy- Constrained Residual Vector Quantization. IEEE Trans. Image Processing 4, 1349–1357 (1995) [14] Wu, F., Sun, X.: Image Compression by Visual Pattern Vector Quantization (VPVQ). In: Data Compression Conference, 1068-0314/08 © (2008) [15] Somasundaram, K., Dominic, S.: Lossless VQ Index Coding Scheme for Image Compression. In: IEEE International Conference on Signal and Image Processing, vol. 1, pp. 89–94 (2006)

Approximation Studies Using Fuzzy Logic in Image Denoising Process Sasi Gopalan1, Madhu S. Nair2, Souriar Sebastian3, and C. Sheela4 1 Department of Mathematics, School of Engineering, Cochin University of Science and Technology, Kochi – 682022 2 Department of Computer Science, University of Kerala, Kariavattom, Thiruvananthapuram – 695581 3 Department of Mathematics, St. Albert’s College, Ernakulam, Kerala 4 Department of Mathematics, Federal Institute of Science and Technology, Angamali, Kerala

Abstract. In this work Lossy methods are used for the approximation studies, since it is suitable for natural images such as photographs. When comparing the compression codecs, it used as an approximation to human perception of reconstruction quality. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation. Fuzzy Logic Systems are blending functions and it has been used to develop a Modified Fuzzy Basis Function (MFBF) and also its approximation capability has been proved. An iterative formula (Lagrange Equation) for the fuzzy optimization has been adopted in this paper. This formula closely relates with the membership function in the RGB colour space. By maximizing the weight in the objective function the noise in the image reduced, so that the filtered image approximates the original image. Keywords: Approximation, Fuzzy, Denoising, color images.

1 Introduction In fuzzy image processing the function approximation is possible only by the modification in the membership functions. The concept of fuzzy basis functions is related with the membership functions [1, 2, 3, 8]. The designed Modified Fuzzy Basis Function (MFBF) facilitates faster approximation when compared to other available filters. The centroid defuzzification method is used with weight as the scaling function and the maximization problem is discussed. That is, by the maximization of weight (for small distances) the function is optimized. An experimental study in connection with our work shows that the designed MFBF can be used more effectively to reduce noise from an image signal. Colours in RGB space are represented by a 3-D vector with first element being red, second being green and third being blue respectively. A colour image C represented by a 2-D array of vectors where (i, j ) = xi , j defines a position in C called pixel and

Ci , j ,1 = xi , j ,1 , Ci , j , 2 = xi , j , 2 and Ci , j ,3 = xi , j ,3 denotes the red,

green and blue components respectively of C. The distance measure of pixels used in MFBF for the approximation studies. In fuzzy image processing the function P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 113–120, 2011. © Springer-Verlag Berlin Heidelberg 2011

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approximation is possible only by the modification in the membership functions. The concept of fuzzy basis functions is related with the membership functions [1, 2, 3, 5, 7, 8]. The designed Modified Fuzzy Basis Function (MFBF) facilitates faster approximation when compared to other available filters. The centroid defuzzification method is used with weight as the scaling function and the maximization problem is discussed. That is, by the maximization of weight (for small distances) the function is optimized. An experimental study in connection with our work shows that the designed MFBF can be used more effectively to reduce noise from an image signal. Colours in RGB space are represented by a 3-D vector with first element being red, second being green and third being blue respectively. A colour image C represented by a 2-D array of vectors where (i, j ) = xi , j defines a position in C called pixel and

Ci , j ,1 = xi , j ,1 , Ci , j , 2 = xi , j , 2 and Ci , j ,3 = xi , j ,3 denotes the red, green and blue components respectively of C. The distance measure of pixels used in MFBF for the approximation studies.

2 Interpolation Using Fuzzy Systems The concept of Fuzzy Basis Function [FBF] is clearly defined in [8]. Hence the entire static mapping for multiple rule antecedent is of the form p

p

p

n

∑ ∑ ⋅ ⋅ ⋅ ⋅∑ a ∏ A

ki i

j

F (a, x ) = g a , x =

k n =1 k n −1 =1 p

k1 =1

p

∑∑

k n =1 k n =1 =1

( xi )

i =1

p

n

⋅ ⋅ ⋅ ⋅∑∏ Aiki ( xi ) k1 =1 i =1

where i = 1, 2,...., n , j = j ( k 1 , k 2 ,...., k n ) The sub index

i of ki indicates the variable xi that is being considered and ki

indicates the fuzzy sets and order of the variable

xi from left to right.

According to [3, 4, 10, 11]

F ( a, x) = g a , x =

m= pn

∑a Φ j

j

( x) = aT Φ ( x)

j =1

That is, F ( a, x ) is presented as a convex linear combination of the basis functions and hence

F ( a, x) = g a , x =

m= pn

∑a Φ j

j

( x) = a j , where Φ j (x) are called fuzzy basis

j =1

functions (FBF). Thus the interpolation capability is satisfied.

3 Proposed Algorithm The basics of noise reduction is clearly defined in [9,10,12].

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3.1 Definition (Distance Measure in Colour Space) Let d be the Euclidean distance, RG-red green, NRG- Neighbour Red green. Let

RG = ( xi , j ,1 , xi , j , 2 ) and NRG = ( xi+k , j +l ,1 , xi+ k , j +l , 2 ) . Then

d ( RG, NRG ) =

(x

− xi , j ,1 ) + ( xi + k , j + l ,2 − xi , j ,2 ) 2

i + k , j + l ,1

2

3.2 Definition (Weight in Fuzzy Noise Reduction) [4, 9] The weight function is defined as the product of the membership functions. That is, Weight = Ai , j ( Dist ( RG , NRG )) × Ai , j ( Dist (RB, NRB )) =

min[ Dist ( RG, NRG )), Dist (RB, NRB )]

3.3 Modified Fuzzy Basis Function (MFBF) Here the modified fuzzy filter is defined as n

g (xi , j ) =

m

∑∑x

i − k , j −l i =1 j =1 n m

∑∑ Φ i =1 j =1

where

Φ i−k , j −l ( xi , j )

i − k , j −l

( xi , j )

Φ i −k , j −l is the product of the membership function (weight function) and the

distance function. The maximum weight is equal to unity if the distance between colour pixel and its neighboring pixel is small. This can be brought to triangular form. The Modified Fuzzy Basis Function is

Φ( xi , j ) =

Φ i − k , j − l ( xi , j )

n

m

∑∑ Φ i =1 j =1

i − k , j −l

.

( xi , j )

Therefore by section 2,

g (xi , j ) = ∑∑ xi − k , j − l Φ ( xi , j ) m

n

i =1 j =1

By adjusting the weight means the membership function is modified. This variation in distance between the pixels directly affects the standard deviation of the Gaussian noise. i.e., for small distances its PSNR value will be high. So the best approximation is obtained with the Modified Fuzzy Membership Function (MFBF).

4 Weights in RGB Colour Space According to [10], the non linear function

Φ k ,l can be approximated as a step like

function. The following figure shows that for small distances the weight is maximum.

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Fig. 1. Approximation of weight function

In this case the output of the fuzzy filter is represented as n

gi , j =

n

∑ ∑Φ

k ,l k =−n l =−n n n

∑ ∑Φ

k =−n l =−n

If

( xi , j ) xi − k , j −l , k ,l

( xi , j )

xi. j − xi − k , j − l and xi+k . j +l − xi , j is small then xi − k , j − l and xi + k , j + l are assumed

in the same flat area (here Φ k , l

= 1 ). If not they are assumed in different flat area

( Φ k , l = 0) . The concept of gradient method is discussed in [6]. In the gradient method, a parameter is updated by subtracting a value which is proportional to the gradient of the cost function. 4.1 Fuzzy Optimization The iterative formula is

S k (n + 1) = S k (n) − α

∂E ( n) ∂Φ k

Sk (n) is the value Sk ( Sk is the height of the k th step of the step like function) at time point n, α is a small positive coefficient and E is the mean where

square error. Using this iterative formula we start with an algorithm and by the fine tuning we can choose the suitable rule for the function approximation. Therefore by Banach fixed point theorem we can find fixed points corresponding to different rules. 4.2 Optimization Problem This is a maximization problem. For the maximum weight the desired image approximates with the filtered image. Therefore we define the following Legrange equation as the Objective function

E = Wk ,l − α ( f i , j − g i , j )

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E − The error W − The weight factor α − Small positive constant f i. j − Desired output g i , j − Filtered output

(a)

(b) PSNR = 22.10

(c) PSNR = 26.57

(d) PSNR = 27.66

(e) PSNR =28.38

(f) PSNR = 29.33

(a)

(b) PSNR = 18.59

(c) PSNR = 24.63

(d) PSNR = 24.73

(e) PSNR =25.16

(f) PSNR = 25.79

Fig. 2. (a) Original Peppers image (256×256) (b) Noisy image (Gaussian noise, σ = 30) (c) After applying Mean filter (3×3 window) (d) After applying Median filter (5×5 window) (e) After applying FBF with K=3 (7×7 window) and L=2 (5×5 window) (f) MFBF with K=3 (7×7 window) and L=2 (5×5 window)

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4.3 Peak Signal–to-Noise Ratio (PSNR) [9, 12] PSNR is the most commonly used measure of quality of reconstruction of lossy compression (e.g. for image compression). This metric is used to compare results of codec type. Typical values for the PSNR in lossy image and video comparison are between 30 and 50, where higher is better. Therefore,

PSNR[ g (i, j , c) − f (i, j , c)] = 10 log10 [ S 2 / MSE ( g (i, j , c) − f (i, j , c)] where MSE is the mean square error and

∑∑∑ [g 3

MSE[ g i , j ,C − f i , j ,C ] =

N

M

c =1 i =1 j =1

i , j ,C

− f i , j ,C

]

2

3.N .M

f (i, j , c) is the desired colour image and g (i, j , c) is the filtered colour image of size N .M (c = 1,2,3 corresponding to the colours red, green and blue Here

respectively).

(a)

(d) PSNR = 21.52

(b) PSNR = 16.10

(c) PSNR = 22.03

(e) PSNR =22.75

(f) PSNR = 23.61

Fig. 3. (a) Original Gantry crane image (400×264) (b) Noisy image (Gaussian noise, σ = 40) (c) After applying Mean filter (3×3 window) (d) After applying Median filter (3×3 window) (e) After applying FBF with K=3 (7×7 window) and L=2 (5×5 window) (f) After applying Proposed Fuzzy filter with K=3 (7×7 window) and L=2 (5×5 window)

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Table 1. Corresponding to different distances and for different filters the PSNR value is calculated and shown PSNR (dB) σ=5

σ = 10

σ = 20

σ = 30

σ = 40

Noisy

34.13

28.12

22.10

18.57

16.08

Mean

28.05

27.70

26.57

25.13

23.72

Median

32.31

30.81

27.66

25.02

22.95

FLF Method

34.12

31.79

28.38

25.85

23.76

MFBF Method

34.22

32.77

29.33

26.51

24.18

5 Experimental Studies The PSNR values for the following figures are verified for small distances. The MFBF is applied in three natural photographs and its approximation is also satisfied. The results are shown in fig. 2 and 3. The PSNR values are shown in Table 1. The PSNR values obtained at different noise levels are shown in fig. 4.

Fig. 4. For restored image approximating noise image for distance (σ = 5 )

6 Conclusion The main claim of this work is the formation of Modified Fuzzy Basis Function (MFBF) in RGB colour space. It is shown that MFBF approximates the noisy image with minimum error and it is better than any other filters.

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References 1. De Luca, A., Termini, S.: A definition of non probabilistic entropy on the setting of fuzzy sets theory. Inform and Control 20, 301–312 (1972) 2. Alimi, A.M., Hassine, R., Selmi, M.: Beta fuzzy logic systems: Approximation Properties in the MISO case. Int. J. Appl. Math. Comput. Sci. 13(2), 225–238 (2003) 3. Kosko, B.: Fuzzy Associative Memories in fuzzy systems. In: Kandel., A. (ed.), AdditionWesley Pub.co., New York (1987) 4. Moser, B.: Sugeno Controllers with a bounded number of rules are nowhere dense. Fuzzy Sets and System 104(2), 269–277 (1999) 5. Kosko, B.: Neural Networks and Fuzzy systems: A Dynamical Approach To Machine Intelligence. Prentice Hall, Inc., Upper Saddle River (1992) 6. Simon, D.: Innovatia Software, Gradient Descent Optimization of fuzzy functions, May 29 (2007) 7. Kerre, E.E.: Mike Nachtegael-2000, Fuzzy Techniques in Image Processing (Studies in Fuzziness and Soft Computing), 1st edn. Physica-Verlag, Heidelberg (2000) 8. Kim, H.M., Mendel, J.M.: Fuzzy Basis Functions: Comparisons with other Basis Functions. IEEE Transactions on Fuzzy systems 3(2), 158–168 (1995) 9. Nair, M.S., Wilscy, M.: Modified Method for Denoising Color Images using Fuzzy Approach, SNPD. In: 2008 Ninth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, pp. 507–512 (2008) 10. Lu, R., Deng, L.: An Image Noise Reduction Technique Based on Fuzzy Rules. In: IEEE International conference on Audio, Language and Image Processing, pp. 1600–1605 (2008) 11. Gopalan, S., Sebastian, S., Sheela, C.: Modified Fuzzy Basis Function and Function Approximation. Journal of Computer & Mathematical Sciences 1(2), 263–273 (2010) 12. Wilscy, M., Nair, M.S.: A New Fuzzy-Based Additive Noise Removal Filter. In: Advances in Electrical Engineering and Computational Science. Lecture Notes in Electrical Engineering, vol. 39, pp. 87–97 (2009), doi:10.1007/978-90-481-2311-7_8

An Accelerated Approach of Template Matching for Rotation, Scale and Illumination Invariance Tanmoy Mondal1 and Gajendra Kumar Mourya2 1 Graphics and Intelligence Based Script Technology Group Centre for Development of Advanced Computing, Pune, India 2 Computational Instrumentation Unit Central Scientific Instruments Organisation, Chandigarh, India [email protected], [email protected]

Abstract. Template matching is the process for determining the presence and location of a certain reference in the scene sub image. The conventional approach like spatial correlation, cross correlation is computationally expensive and error prone when the object in the image is rotated or translated. The conventional approach gives erroneous result in the presence of variation in illumination of the scene sub image. In this paper, an algorithm for a rotation, scale and illumination invariant template matching approach is proposed, which is based on the combination of multi-resolution wavelet technique, projection method and Zernike moments. It is ideally suited for highlighting local feature points in the decomposed sub images, and the result is computationally saving in the matching process. As demonstrated in the result, the wavelet decomposition along with Zernike moments makes the template matching scheme feasible and efficient for detecting objects in arbitrary orientation and rotations. Keywords: Wavelet Decomposition; Zernike Moments; Template matching; Rotation, scale and illumination invariance.

1 Introduction Template matching is the process of determining the maximum match between template of interest and the scene sub image according to the same arbitrary measure, the scene sub image might be taken at different condition. It has a great application in different domains like remote sensing, medical imaging, computer vision etc. In the last few decades template matching approach have been widely applied in image registration and recognition process as illustrated in literatures [1-3]. Various rotation invariant template matching approaches have been proposed. A couple of novel set of template matching algorithm was proposed by Choi et al. [4]. In the first approach, the matching candidates, called as vector sum of circular projections of the sub image are selected using a computationally low cost feature, later frequency domain calculation was adopted to reduce the computational cost of spatial domain. In the second approach, rotation-invariant properties of Zernike moments are applied to perform template matching only on the candidate pixels. Orientation code-based matching is robust for searching objects in cluttered environments even in the cases of illumination fluctuations resulting from shadowing or highlighting etc. Ullah et al. [5] P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 121–128, 2011. © Springer-Verlag Berlin Heidelberg 2011

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applied orientation codes for rotation invariant template matching in grey scale images. A new approach of template matching, invariant to image translation, rotation, scaling, proposed by Lin et al. [6]. An amalgamation of “ring-projection transform” to convert the 2D template in a circular region into 1D grey level signal and “parametric template vector” for template matching is achieved in their work. Tsai et al. [7] addressed a wavelet decomposition approach for rotation-invariant template matching. In the matching process, only the pixels with high wavelet coefficients based on norm in the decomposed detail sub image at a lower resolution level to compute the normalized correlation between two compared patterns is used. To perform rotation invariant template matching, they further use the ring-projection transform (RPT), which is invariant to object orientation, to represent an object pattern in the detailed sub image. The proposed method significantly reduces the computational burden of the traditional pixel-by-pixel matching. Cole-Rhodes et al. [8] introduced a registration algorithm that combines a simple yet powerful search strategy based on a stochastic gradient with two similarity measures, correlation and mutual information, together with a wavelet-based multi-resolution pyramid. Mutual information are shown to optimize with one-third the number of iterations required by correlation. Ideally, template matching should be invariant to translation, rotation and scaling in the input scene. However, the matching results of these schemes are sensitive to scale changes. A new template matching approach proposed by Lin et al. [9], in this approach, ring projection transform process is used to convert 2D template in a circular region into a 1D grey level signal as a function of radius. Then, the template matching is performed by constructing a parametric template vector (described by Tanaka et al. [10]) of the 1D grey level signal with differently scaled template of the object. However, the matching scheme becomes computational time expensive, for large searching regions. Template matching based on moments is a popular matching scheme [11, 12]. Moment invariants are properties of connected regions in binary images that are invariant to translation, rotation and scaling. They are useful because they define a simply calculated set of region properties that can be used for shape classification and object recognition. However, they require too much computation because of many integration and multiplication operations. Reddy and Chatterji [13] proposed an FFT-based matching, which is an extension to phase correlation technique. The specialty of this algorithm is its insensitivity to translation, rotation, scaling and noise. To find scale and rotational variation Fourier scaling properties and Fourier rotational properties are used; the translational movement is determined by the phase correlation techniques, but the matching method requires computation of three FFT’s and three IFFT’s, so the computation is apparently too computationally intensive. Therefore the template matching algorithm faces great challenges of computational efficiency, accuracy, and the input scene with the changes of translation, rotation and scale. The rotation invariant moment based method can be helpful but it is computationally expensive and is sensitive to noise. Araujo et al. [14] introduced the Color-Ciratefi that takes into account the color information for color template matching, which is a cumbersome problem due to color constancy. Recently they had proposed [15] a new grey scale template matching algorithm named Ciratefi, invariant to rotation, scale, translation, brightness and contrast. In their recent work for color template matching, a new similarity metric in the CIELAB space is used to obtain invariance to brightness and contrast changes. They demonstrated that Color-Ciratefi is

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123

more accurate than C-color-SIFT.Kim [16] developed another RST-invariant template matching based on Fourier transform of the radial projections named Forapro. Forapro is faster than Ciratefi in a conventional computer but the later seems to be fit to be implemented in FPGA than the former.

2 Proposed Methodology In order to deal with template matching in large size of images a multi-resolution based wavelet decomposition [17] approach is proposed, which drastically reduces the computationally expensiveness of template matching. 2.1 Multi-resolution Based Wavelet Decomposition In case of 2-D image the wavelet transform can be defined as a two dimensional scaling function , , and three two dimensional wavelets , , , and , are required in defining wavelet transform in case of discrete images. Each of this produces results of two dimensional functions. Excluding products that produce one dimensional results like (x), the four remaining products produce the separable scaling function. The pyramid structure wavelet decomposition approach consists of filtering and down sampling horizontally using 1D low pass filter L (with impulse responses ) and high pass filter H (with impulse responses ) to each row in the image , , and produces the coefficient matrix , and , and produces four sub-images x, y , , , , , , for one level decomposition. , is the coarse representation of the image and , , , , Here , , , are detail sub-images, which represents horizontal, vertical and diagonal direction of the image. The detailed 2-D pyramid decomposition algorithm, with periodic boundary conditions applied, can be expressed as follows: Let

h

Then,

= original size of the image , = low pass coefficient of specific wave let basis, 0,1,2,3, … … . 1, where is the support length of the specific filter L. = high pass coefficient of specific wave let basis, j 0,1,2,3, … … . 1, where is the support length of the specific filter H. ∑

,

.

0,1,2, … … .

for ,

1

0,1,2, … … .

,

;

1

,

∑ 0

1

.

2

,

1 and y = 0, 1, 2, 3,……N-1. . 1

for

2

;

, 2 .

,

, 2

1 and y = 0, 1, 2, 3,…… -1.

1

. ,

1

;

, 2 .

, 2

,

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A wavelet basis with long supports such as the Harr wavelet with compact support of 2 and a 4-tap Daubechies wavelet are the best choice for the investigated application, as with long support length may results in poor illumination, and can be affected by dominating boundary effect but it’s advantageous factors are, it is less sensitive to boundary effect and computationally efficient than those with long support lengths. The next decomposition level can be obtained by iterating 2-D pyramidal algorithm on the smooth image , which is the coarse representation of the original image. The multi-resolution based wavelet decomposition results in drastic reduction in the size of the image, in effect lessen computational time. Image size reduction level is denoted by 4 , where J denotes decomposition levels. If the decomposed sub-image has an over down-sampling size, it may results in false matching accordingly. As aforementioned three sum images containing, separately horizontal, vertical, and diagonal edge information of object patterns are obtained in decomposition level. These three detailed sub image can be combined in as a single composite detail, sub-image that can simultaneously contain the characteristics of horizontal, vertical, and diagonal edge information in a composite detail sum-image. The composite detail sub-image is given by: ,

,

=

,

,

+

,

+

Where,

;

,

,

,

,

are horizontal, vertical and diagonal detail sub images . Given an image of

, the size of the composite detail sub image

size

,

at resolution level

J is reduced to . Now matching process can be carried out on the composite detail sub image. The pixels with high energy values are considered for matching, these pixels are selected by comparing with threshold value. The threshold for selecting high energy value pixels is determined by Otsu’s method. Depending on the complexity of the image, the high energy valued pixels generally dominate 2030% of the entire composite detail sub image. So matching process become more robust and faster. 2.2 Zernike Moments Zernike moments are based on a set of complex polynomials that form a complete orthogonal set over the interior of the unit circle [18]. Zernike moments are defined as the orthogonal image function on these orthogonal basis functions. The basic , , is given by: , , ; where n is functions , , , | | is even and (+ve) integer, m is the non-zero integer subject to the constraints | | , is the length of the vector from the origin (centre of the image) to x, y and is the angle between the vector and axis in a counter clockwise direction and is the Zernike radial polynomial. The Zernike radial polynomial , is , defined as here

∑

,

,

! | |,

!

,

!

∑

| |,

, ,

. The basis function in equitation 1 is orthogonal thus satisfy

, ,

!

,

,

,

,

; where,

,

1; when a=b and

,

= 0;

An Accelerated Approach of Template Matching

125

otherwise the Zernike moments of order n with repetition m for a diagonal image , ∑∑ , , ; where , is defined function is given by: , , . Zernike moments extracted from a pattern are as the complex conjugate of , perfectly rotation, scale, and translation invariant. For more detailed discussion see [19,20]. 2.3 Improving Speed of the Zernike Moments Calculation By substituting equations 2 in 1 and 5 and reorganizing the term the Zernike moments can be calculated in the following form. ∑∑

,

∑

| |

,

,

=

∑

=

∑

| |

∑∑

, ,

| |

, ,

,

(1)

,

As mentioned earlier, defined in equation (1) , is noted to be the common term in the computation of Zernike moments with the same repetition as shown in figure (1) for the case of repetition m=0 and order n=0. In general to compute Zernike moments up to order N, , need to be computed for each repetition as shown in Table 1. The table demonstrates that , to be computed for each repetition up to order 10. The second row of the table corresponds to the , shown (same colour indicates repetition of , in the calculation of , ; n=0, 1...10) in figure 1. Once all the entries in the table 1 are computed, Zernike moments of any order and repetition can be calculated as a linear combination of , as shown in equation (1). It is worth to be noted that , , does not depend upon the image coordinates; therefore, they are stored on the small lookup table to save further computation. Numerical precision is another issue in the computation of high order Zernike moments. Depending on the image size and the maximum order, double precision arithmetic does not provide enough precision. 2.4 Matching Based on Zernike Moments In our application Zernike moments from order 0 up 10 (say l) is used. Therefore, for this case 36 (say k). Zernike moments are calculated as feature vector of the ,

, ,

,

, ,

,

, ,

,

, ,

, ,

,

, ,

+ + , + , +

, ,

, ,

+

, ,

,

,

, ,

,

,

, ,

,

, ,

,

, ,

,

, ,

,

, ,

, , , ,

, ,

, ,

,

, , , ,

, ,

, , , ,

, ,

, ,

,

Fig. 1. The common terms to compute Zernike moments up to 10 orders and zero repetition

the image. The repetition parameter m is automatically calculated for each order (n), as, n is even, then m will be number of even numbers present between 0 to n and vice

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versa. Let be the Zernike moments of the template , , and be the moments of , in the region which coincides with current location of . Then the correlation coefficient at point where

0,1,2, … … . .

1,

, and

Table 1. , is needed to compute Zernike moments to 10 order and m repetition repetition m

0 1 2 3 4 5 6 7 8 9 10

given by:

,

are means of

∑ ∑

∑

and

, respectively.

Table 2. List of Zernike moments and their corresponding mber of features up to order 10 Order

0 1 2 3 4 5 6 7 8 9 10

Zernike Moments

Number of Zernike Moments

1 1 2 2 3 3 4 4 5 5 6

3 Experimental Results Experiments were conducted to measure the performance of the algorithm in terms of speed and matching accuracy. As proposed in this paper, that the combination of multi-resolution based wavelet decomposition approach with accelerated Zernike moment methodology is very useful for computational time inexpensive template matching. The hierarchical Zernike moment approach utilizes the pyramid structure of the image and yielded results comparable in terms of accuracy. 3.1 Test Images The proposed algorithm is tested with a series of experiment conducted under various translations, rotation, scales and also with variation in brightness, contrast change, and added Gaussian noise. As observed from the experiment, the speed of the test merely depends on the size of the template. The following figure (2) reports the results obtained by performing experimentation in all of the images with different settings, and the proposed algorithm succesfully locates the position of each template precisely.

An Accelerated Approach of Template Matching

(a)

(c)

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(d)

(b)

(e)

(f )

Fig. 2. (a) Sample test image (PCB board), (b) reference template, (c) tested image rotated in an arbitrary angle, (d) image rotated in arbitrary angle with additive Gaussian noise of zero mean variance of 0.025 (e) image with brightness variation, (f) image with contrast variation

4 Conclusion In this paper a robust and computationally inexpensive template matching approach are proposed. The proposed method significantly reduces the number of search pixels by using the composite detail sub image at a low resolution level. The computation of correlation coefficients is only carried out for those high energy-valued pixels in the composite detail sub image. The accelerated Zernike moment representation not only reduces the computational complexity of the normalized correlation but also makes the detection invariant to different transformations. As can be observed from the experiment that the performance is rather good when SNR is more than 20 dB.

References [1] Brown, L.G.: A survey of image registration techniques. ACM Comput. Surv. 24(4), 326–376 (1992) [2] Zitová, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21(10), 997–1000 (2003) [3] Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Med. Image Anal. 2, 1–36 (1998) [4] Choi, M.S., Kim, W.Y.: A novel two stage two template matching method for rotation and illumination invariance. Pattern Recognition 35, 119–129 (2002)

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[5] Ullah, F., Kanekoi, S.: Using orientation codes for rotation invariant template matching. Pattern Recognition 37(2), 201–209 (2004) [6] Yi-Hsein, L., Chin-Hsing, C.: Template matching using parametric template vector with translation, rotation and scale invariance. Pattern Recognition 41, 2413–2421 (2008) [7] Tsai, D.M., Chiang, C.H.: Rotation-invariant pattern matching using wavelet decomposition. Pattern Recognition Lett. 23, 191–201 (2002) [8] Cole-Rhodes, A.A., Johnson, K.L., Le Moigne, J., Zavorin, I.: Multi-resolution registration of remote sensing imagery by optimization of mutual information using a stochastic gradient. IEEE Trans. Image Process. 12(11), 1495–1511 (2003) [9] Yi, L., Chen, C.: Template matching using parametric template vector with translation, rotation, scale invariance: Elsevier. Pattern Recognition 41, 2413–2421 (2008) [10] Tanaka, K., Sano, M., Ohara, S., Okudaira, M.: A parametric template method and its application to robust matching. In: IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 620–627 (June 2000) [11] Liao, S.X., Pawlak, M.: On image analysis by moments. IEEE Trans. Pattern Anal. Mach. Intell. 18(3), 254–266 (1996) [12] Flusser, J., Suk, T.: Affine moment invariants: a new tool for character recognition. Pattern Recognition Lett. 15, 433–436 (1994) [13] Reddy, B.S., Chatterji, B.N.: An FFT-based technique for translation, rotation, and scaleinvariant image registration. IEEE Trans. Image Process 5(8), 1266–1271 (1996) [14] de Araújo, S.A., Yong, K.H.: Color-Ciratefi: A color- based RST-invariant template matching algorithm. In: IWSSIP - 17th International Conference on Systems, Signals and Image Processing (2010) [15] Kim, H.Y., de Araújo, S.A.: Grayscale template-matching invariant to rotation, scale, translation, brightness and contrast. In: Mery, D., Rueda, L. (eds.) PSIVT 2007. LNCS, vol. 4872, pp. 100–113. Springer, Heidelberg (2007) [16] Kim, H.Y.: Rotation-Discriminating Template Matching Based on Fourier Coefficients of Radial Projections with Robustness to Scaling and Partial Occlusion. Pattern Recognition 43(3), 859–872 (2010) [17] Tsai, D.M., Chen, C.H.: Rotation invariant pattern matching using wavelet decomposition, http://citeseerx.ist.psu.edu/viewdoc/ download?doi=10.1.1.108.2705&rep=rep1&type=pdf [18] Amayeh, G.R., Erol, A.: Accurate and Efficient computation of High Order Zernike Moments [19] More, V.N., Rege, P.P.: Devanagari handwritten numeral identification based on Zernike moments. IEEE Xplore, pp. 1–6 (2009) [20] Qader, H.A., Ramli, A.R.: Fingerprint recognition using Zernike moments. The International Arab Journal of Information Tech. 4(4) (October 2007)

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained Histogram Equalization P. Shanmugavadivu1 and K. Balasubramanian2 1 Department of Computer Science & Applications, Gandhigram Rural Institute – Deemed University, Gandhigram [email protected] 2 Department of Computer Applications, PSNA College of Engineering & Technology, Dindigul [email protected]

Abstract. Histogram Equalization (HE) based methods are the commonly used image enhancement techniques to improve the contrast of an input image by changing the intensity level of the pixels, based on its intensity distribution. This paper presents a novel image contrast and edge enhancement technique using constrained histogram equalization. A new transformation function for histogram equalization is devised and a set of constraints are applied over the histogram of an image prior to the equalization process and then, unsharp masking is applied on the resultant image. The proposed method ensures better contrast and edge enhancement, which is measured in terms of discrete entropy. Keywords: Histogram equalization, image sharpening, edge enhancement, unsharp masking.

1 Introduction Contrast enhancement plays a major role in image processing applications such as digital photography, medical imaging, remote sensing etc. Histogram Equalization (HE) is one of the well-known methods for enhancing the contrast of an image [1]. HE uniformly distributes the intensity of the input image by stretching its dynamic intensity range using cumulative density function. The major problem of HE is the mean shift that attributes to a drastic deviation in the mean brightness of input and output images. HE based techniques are categorized into two as: (i) The Adaptive (or local) HE methods (AHE) (ii) The improved global methods based on histogram equalization or histogram specification (matching) In the AHE [1] methods, equalization is based on the statistics obtained from the histograms of neighborhood pixels. AHE methods usually provide stronger enhancement effects than global methods. However, due to their high computational load, AHE methods are not best suited for real-time applications. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 129–136, 2011. © Springer-Verlag Berlin Heidelberg 2011

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The following are the improved global HE methods: • Brightness Preserving Bi-Histogram Equalization (BBHE) BBHE [2] divides the input image into two sub-images based on its mean and the equalization is performed independently for the two sub-images. Unlike HE, BBHE preserves the original brightness of an input image. However, it creates unwanted artifacts. • Dualistic Sub-Image Histogram Equalization (DSIHE) DSIHE [3] is a simple variation of BBHE. It selectively separates the image histogram based on the gray level (threshold level) with cumulative probability density equivalent to 0.5. • Recursive Mean Separate HE (RMSHE) Chen and Ramli proposed a novel method, called as RMSHE [4] which partitions the input image’s histogram recursively based on mean. Here, some portions of histogram among the partitions cannot be expanded much, while the outside regions are expanded significantly that creates the unwanted artifacts. This is a common drawback of most of the existing histogram partitioning techniques since they keep the partitioning point fixed throughout the entire process of equalization. • Minimum Mean Brightness Error Bi-Histogram Equalization (MMBEBHE) In MMBEBHE [5], a threshold level is calculated based on Absolute Mean Brightness Error (AMBE). Using this level, the input image histogram is separated into two and the HE process is then applied to them independently. But it is recorded that the performance of RMSHE is higher than that of MMBEBHE. • Recursive Sub-Image Histogram Equalization (RSIHE) RSIHE [6] is another recursive method which resembles RMSHE. In RSIHE, the histogram is recursively partitioned, based on the median instead of mean. This method produces 2r pieces of sub-histograms. It is very difficult to find the optimal value of ‘r’. • Weighted Thresholded Histogram Equalization (WTHE) WTHE [7] was proposed by Wang and Ward in which the histogram of an input image is modified by weighting and thresholding prior to the histogram equalization. But this technique fails to sharpen and to preserve the original details of the input image, while enhancing the contrast. • Recursively Separated and Weighted Histogram Equalization (RSWHE) In RSWHE [8], the input image histogram is segmented into two or more subhistograms recursively. A weighing process is then applied on those subhistograms independently. HE process is finally applied on the weighted subhistograms. This method is computationally complex since it is recursive and the formulation of weighing module for all sub-histograms is not an easy task to be performed at a faster rate. • Sub-Regions Histogram Equalization (SRHE) Ibrahim and Kong developed a new HE technique called SRHE [9]. This method partitions the image based on the smoothed intensity values, which are obtained by convolving the input image with Gaussian filter.

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In this paper, a new edge enhancement technique based on the modified HE transformation function of SRHE and the weighing principle used in WTHE is developed. Additionally, the edge and contrast of the images are improved by the application of unsharp masking filter with better scalability.

2 Basic Histogram Equalization Let X={X(i, j)} denote a digital image, where X(i, j) denotes the gray level of a pixel at (i, j). The total number of pixels in the image is N and the image intensity is digitized into L levels that are {X0, X1, ..., XL-1}. So it is obvious that ∀X (i, j ) ∈{ X 0 , X 1 ,...., X L −1 } . Suppose nk denotes the total number of pixels with gray level of Xk in the image, then the probability density of Xk is given as: p( X k ) =

nk , k = 0, 1, ...., L − 1 N

(1)

The plot between p(Xk) and Xk is defined as the histogram of an image. The cumulative density function (CDF) based on the image's PDF is defined as: k

c( X k ) = ∑ p( X i )

(2)

i =0

where k = 0, 1 ,..., L-1 and it is known that c(XL-1)=1. The transformation function of HE is defined as: f ( X k ) = X 0 + ( X L −1 − X 0 ) × c( X k ), k = 0, 1, ...., L − 1

3

(3)

The Proposed Method for Edge and Contrast Enhancement

The proposed technique accomplishes edge and contrast enhancement at three levels through i) Development of modified HE transformation function ii) Classifying input image histograms based on weighing and thresholding and the application of modified HE transformation function and iii) Unsharp masking over the resultant image. 3.1 Modified HE Transformation Function It is evident that there is significant intensity variation between the HEed image and the image obtained after inversing the input image, equalizing and then re-inversing it [9]. It is illustrated in Fig. 1(a) – 1(e). It is prominently visible from Fig. 1(b) and Fig. 1(e) that there exists a qualitative difference due to intensity variations between them. This intensity variation is due to the fact that the low probable intensity values of the input image gain importance after inversion. In the normal HE process, they are almost ignored and hence the contrast and edges of the image are not enhanced significantly.

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(a)

(b)

(c)

(d)

(e)

Fig. 1. (a) Original Truck Image (b) HEed Truck Image (c) Negative of Truck Image (d) HEed of Fig.1(c) (e) Negative of Fig.1(d)

In order to exploit the advantages of these two approaches, it is proposed to take the intensity average of Fig. 1(b) and Fig. 1(e). The transformation function f1 for the image with inversion process is given in equation (4). ⎛ X L −1 ⎞ f 1 ( X k ) = X 0 + ( X L −1 − X 0 ) × ⎜⎜1 − ∑ p( X i )⎟⎟ ⎝ i= X k ⎠

The average of the output image fnew is obtained from f and f1 as:

f new ( X k ) =

1 ( f ( X k ) + f 1 ( X k )) 2

Xk ⎞ ⎛ ⎟ ⎜ X 0 + ( X L −1 − X 0 ) × ∑ p ( X i ) + i= X 0 ⎟ 1⎜ = ⎜ ⎟ X L −1 2⎜ ⎞⎟ ⎛ X + ( X L −1 − X 0 ) × ⎜⎜1 − ∑ p ( X i )⎟⎟ ⎟ ⎜ 0 ⎠⎠ ⎝ i= X k ⎝

(4)

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133

Xk ⎛ ⎞ ⎜ X 0 + ( X L−1 − X 0 ) × ∑ p ( X i ) + ⎟ ⎟ i= X 0 1⎜ = ⎜ ⎟ X 2⎜ ⎞⎟ ⎛ k −1 X + ( X L−1 − X 0 ) × ⎜⎜ ∑ p ( X i )⎟⎟ ⎜ 0 ⎟ ⎠⎠ ⎝ i= X 0 ⎝ ⎛ ⎛ X k −1 ⎞ ⎞ ⎜ X 0 + ( X L −1 − X 0 ) × ⎜ ∑ p( X i ) + p( X k )⎟ + ⎟ ⎜ ⎟ ⎟ 1⎜ ⎝ i= X 0 ⎠ = ⎜ ⎟ X 2⎜ ⎛ k −1 ⎞ ⎟ ⎜ X 0 + ( X L −1 − X 0 ) × ⎜⎜ ∑ p ( X i )⎟⎟ ⎟ ⎝ i= X 0 ⎠ ⎝ ⎠ X k −1 ⎤ 2 X 0 ( X L−1 − X 0 ) ⎡ = + ⎢ p( X k ) + 2 ∑ p( X i )⎥ 2 2 i= X 0 ⎣ ⎦ X k −1 ⎡ ⎤ f new ( X k ) = X 0 + ( X L −1 − X 0 ) ⎢0.5 p ( X k ) + ∑ p ( X i )⎥ i= X0 ⎣ ⎦

(5)

The HE process of the proposed method uses this transformation function (equation (5)). 3.2 Weighing and Thresholding

Once the probability densities are found using equation (1) for the individual pixels of an input image, the probability density values are modified [7] using the weighing and thresholding method as follows: p c ( X k ) = Τ( p ( X k )) ⎧ pu ⎪ ⎪⎛ p ( X k ) − p l = ⎨⎜⎜ ⎪⎝ p u − p l ⎪0 ⎩

if p ( X k ) > p u

r

⎞ ⎟⎟ * p u ⎠

⎫ ⎪ ⎪ if p l < p ( X k ) ≤ p u ⎬ ⎪ ⎪ if p ( X k ) ≤ p l ⎭

(6)

where pc(Xk) is the weighted and thresholded probability density for the pixels with gray level Xk; Pu is the upper constraint which is computed as Pu = v * max(PDF) where 0.1 < v < 1.0; ‘r’ is the power factor which accepts values as 0.1 < r < 1.0 and Pl is the lower constraint whose value is arbitrarily selected as low as possible (for e.g. Pl=0.0001). The transformation function T(p(Xk)) fixes the original probability density value at an upper threshold Pu and at a lower threshold Pl and transforms all values between the upper and lower thresholds using a normalized power law function with index r>0. This clamping procedure is mainly used for enhancing the contrast of an image. The CDF is obtained using the modified probability density values as in equation (7).

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k

cc ( X k ) = ∑ pc ( X i )

(7)

i =0

Then, the HE procedure is applied using the modified HE transformation function given in equation (5). 3.3 Unsharp Masking

The high frequency domain of an image plays a vital role in enhancing its visual appearance, with respect to the edges and contrast. The classic linear unsharp masking (UM) is one of the ideal techniques which is often employed for this purpose [1]. UM produces an edge image g(x,y) from an input image f(x,y) as follows: g ( x, y ) = f ( x, y ) − f smooth ( x, y )

(8)

where fsmooth(x,y) is the smoothed version of f(x,y). This edge image can be used for sharpening by adding it back to the original image as: f sharp ( x, y ) = f ( x, y ) + k * g ( x, y )

(9)

where, ‘k’ is the scaling constant which decides the degree of sharpness of an image. ‘k’ usually accepts the values in the range 0.1 to 1.0. After the input image’s probability density values are weighed, thresholded and equalized, the resultant image is treated with unsharp masking filter which sharpens the image by means of enhancing the edges.

4

Results and Discussion

The proposed image enhancement algorithm is developed in MATLAB 7.0. In this proposed technique, three major parameters which control the enhancement of contrast as well as edges are identified. They are: • the value of ‘v’ for the calculation of Pu • the value of normalized power law function index ‘r’ and • the value of UM scaling constant ‘k’

The optimal values for ‘v’, ‘r’ and ‘k’ are identified through iterative process. The objective function, discrete entropy E(X) is used to measure the richness of details in an image after enhancement [10]. It is defined as: 255

E ( X ) = − ∑ p ( X k ) log 2 ( p ( X k ))

(10)

k =0

In order to evaluate the performance of the proposed method, standard images such as Bottle, Truck, Village, Woman and Putrajaya are chosen. The performance of the proposed method is quantitatively measured in terms of discrete entropy. The entropy values are computed for those test images after the application of our proposed method. The same are calculated for contemporary HE methods such as GHE, BBHE,

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Table 1. Comparison of Discrete Entropy Values Image

Original

Bottle Truck Village Woman Putrajaya

7.4472 6.5461 7.4505 7.2421 6.8116

(a)

(d)

Proposed Method 7.3717 6.4082 7.3006 7.0981 6.7068

GHE

BBHE

DSIHE

RMSHE

5.9776 5.8041 5.9769 5.9594 5.7758

7.2826 6.3796 7.2455 6.9889 6.6223

7.2477 6.3781 7.2404 6.9824 6.5952

7.2924 6.4163 7.2982 7.0451 6.6436

(b)

(c)

(e)

(f)

Fig. 2. (a) Original Putrajaya image and the resultant images of (b) Proposed Method (c) GHE (d) BBHE (e) DSIHE (f) RMSHE

DSIHE and RMSHE. The comparative entropy values are detailed in the Table 1. It is evident from this table that the proposed method has higher entropy values which are very closer to the original images. This signifies that the proposed method is found to preserve the details of the original image in the enhanced image. The qualitative performance of the proposed and contemporary methods is illustrated using Putrajaya image which is shown in Fig. 2(a) – 2(f). Fig. 2(c) – 2(f) are the enhanced images using the standard methods such as GHE, BBHE, DSIHE and RMSHE respectively. It can be noted from these images that the standard methods failed to enhance the edges of the input image. Fig. 2(b) is the result of the proposed method in which the edges as well as the contrast of the original image are enhanced and the optimal values of ‘v’, ‘r’ and ‘k’ are found to be 0.2, 0.5 and 0.9 respectively. Moreover, Fig. 2(b) clearly shows that the visual perception of the proposed

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method is better than those of the other HE techniques and is free from overenhancement. The proposed method is found to produce comparatively better results for other test images too.

5 Conclusion In general, the mean gray level of the input image is different from that of the histogram equalized output image. The proposed method is proved to solve this problem effectively. This method accomplishes two major desired objectives of edge and contrast enhancement of any given input image. It is computationally simple and is easy to implement. It provides subjective scalability in edge as well as contrast enhancement, by appropriately adjusting the control parameters namely ‘v’, ‘r’ and ‘k’. This method suitably finds application in the field of consumer electronics, medical imaging, etc.

References 1. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Prentice Hall, Englewood Cliffs (2002) 2. Kim, Y.: Contrast enhancement using brightness preserving bi-histogram equalization. IEEE Transactions on Consumer Electronics 43, 1–8 (1997) 3. Wan, Y., Chen, Q., Zhang, B.: Image enhancement based on equal area dualistic subimage histogram equalization method. IEEE Transactions on Consumer Electronics 45, 68–75 (1999) 4. Chen, S., Ramli, A.R.: Contrast enhancement using recursive mean-separate histogram equalization for scalable brightness preservation. IEEE Transactions on Consumer Electronics 49, 1301–1309 (2003) 5. Chen, S., Ramli, A.R.: Minimum mean brightness error bi-histogram equalization in contrast enhancement. IEEE Transactions on Consumer Electronics 49, 1310–1319 (2003) 6. Sim, K.S., Tso, C.P., Tan, Y.Y.: Recursive sub-image histogram equalization applied to gray-scale images. Pattern Recognition Letters 28, 1209–1221 (2007) 7. Wang, Q., Ward, R.K.: Fast image/video contrast enhancement based on weighted thresholded histogram equalization. IEEE Transactions on Consumer Electronics 53, 757–764 (2007) 8. Kim, M., Chung, M.: Recursively separated and weighted histogram equalization for brightness preservation and contrast enhancement. IEEE Transactions on Consumer Electronics 54, 1389–1397 (2008) 9. Ibrahim, H., Kong, N.S.P.: Image sharpening using sub-regions histogram equalization. IEEE Transactions on Consumer Electronics 55, 891–895 (2009) 10. Wang, C., Ye, Z.: Brightness preserving histogram equalization with maximum entropy: A variational perspective. IEEE Transactions on Consumer Electronics 51, 1326–1334 (2005)

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs S.T. Jaya Christa1 and P. Venkatesh2 1

Department of Electrical and Electronics Engineering, Mepco Schlenk Engineering College, Sivakasi – 626 005, Tamil Nadu, India Mobile: 91-9486288111 [email protected] 2 Department of Electrical and Electronics Engineering, Thiagarajar College of Engineering, Madurai – 625 015, Tamil Nadu, India [email protected]

Abstract. The optimal placement of Thyristor Controlled Series Compensators (TCSCs) in transmission systems is formulated as a multi-objective optimization problem with the objective of maximizing transmission system loadability, minimizing transmission loss and minimizing the cost of TCSCs. The thermal limits of transmission lines and voltage limits of load buses are considered as security constraints. Multi-Objective Particle Swarm Optimization (MOPSO) and Multi-Objective Comprehensive Learning Particle Swarm Optimizer (MOCLPSO) are applied to solve this problem. The proposed approach has been successfully tested on IEEE 14 bus system and IEEE 118 bus system. From the results it is inferred that MOCLPSO can obtain an evenly distributed pareto front. A multi-criterion decision-making tool, known as TOPSIS, has been employed to arrive at the best trade-off solution. Keywords: MOCLPSO, System loadability, Thyristor Controlled Series Compensator, Transmission loss, Multi-objective optimization, TOPSIS.

1 Introduction Many real-world problems generally involve simultaneous optimization of multiple, often conflicting objective functions. In recent years, several evolutionary algorithms have been successfully applied to solve multi- objective optimization problems. An algorithm which finds an ideal set of tradeoff solutions which are close to and uniformly distributed across the optimal tradeoff surface can be considered to be well suited for a particular problem. One of the problems faced by Power System Engineers is to increase the power transfer capability of existing transmission systems. This problem can be overcome by installing Flexible AC Transmission Systems (FACTS) devices which has recently gained much attention in the electrical power world [1]. Finding the optimal location of a given number of FACTS devices and their parameters is a combinatorial optimization problem. To solve such a problem, heuristic methods can be used [2]. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 137–144, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Many researches were carried out on the optimal allocation of FACTS devices considering a single objective of loadability maximization and several techniques have been applied for finding the optimal location of different types of FACTS devices in power systems [3],[4],[5],[6],[7]. Several methods have been proposed to solve multi-objective optimization problems [8],[9]. Recently, multi-objective particle swarm optimization techniques have received added attention for their application in power system problems for FACTS devices placement [10],[11]. Decision making is an important task in multi-objective optimization. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was proposed in [12] to solve classical Multi-Criteria Decision Making (MCDM) problems. In this paper, one of the FACTS devices namely Thyristor Controlled Series Compensator (TCSC) is employed to analyse the increase in power transfer capabilities of transmission lines and reduction in transmission loss considered along with the minimization of the installation cost of TCSCs. In this paper, Multi-Objective Particle Swarm optimization (MOPSO) and Multi-Objective Comprehensive Learning Particle Swarm optimizer (MOCLPSO) have been applied for the optimal allocation of single and multiple TCSCs. The best compromise solution from the set of pareto-optimal solutions is selected based on TOPSIS method. The IEEE 14 bus system and IEEE 118 bus system are taken as test systems to investigate the performance of the MOPSO and MOCLPSO techniques.

2

Problem Formulation

The objective of this paper is to find out the optimal location and parameters of TCSCs which simultaneously maximize the loadability of transmission lines and minimize the transmission loss in the system and minimize the cost of TCSCs, while satisfying the line flow limits and bus voltage limits. The objectives for placing the TCSCs are described below. The constraints considered in this problem are the power balance constraints, line flow constraints, bus voltage constraints and TCSC parameter operating range. 2.1 Objectives for the Problem 2.1.1 Loadability Maximization The objective function for loadability maximization is given as, Maximize where

λ

f1 = λ

(1)

is a scalar parameter representing the loading factor.

2.1.2 Loss Minimization The objective function for loss minimization is given as, Minimize

f2 =

NL

∑ (P k =1

LK

)

(2)

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs

139

where,

PLK - real power loss of line k NL - number of transmission lines 2.1.3 Minimization of Installation Cost of TCSCs The objective function for minimization of installation cost of a TCSC is given as, Minimize

f3 = CTCSC

x

S

x

1000

(3)

Where,

CTCSC - cost of TCSC in US$ / kVAR S - operating range of the TCSC in MVAR The cost of TCSC is given as,

CTCSC = 0.0015S 2 − 0.713 S + 153.75

(4)

3 Application of MOPSO and MOCLPSO for the Multi-Objective Optimization Problem In this paper, two multi-objective optimization techniques namely Multi-Objective Particle Swarm Optimization (MOPSO) algorithm and Multi-Objective Comprehensive Learning Particle Swarm Optimizer (MOCLPSO) are employed for solving the multi-objective loadability maximization, loss minimization and minimization of installation cost of TCSCs problem. MOPSO is an extension of Particle Swarm Optimization (PSO) algorithm to handle multi-objective optimization problems [13]. MOPSO algorithm uses an external archive to record the non-dominated solutions obtained during the search process. MOCLPSO is an extension of Comprehensive Learning Particle Swarm Optimizer (CLPSO) to handle multiple objective optimization problems [14]. In CLPSO, the particle swarm learns from the global best (gbest) solution of the swarm, the particle’s personal best (pbest) solution, and the pbests of all other particles [15]. In this paper, pbest is selected randomly as in [16]. In this paper, gbest is selected by randomly choosing a particle from the non-dominated solutions. The crowding distance method of density estimation is employed in MOCLPSO to maintain the archive size when the archive reaches the maximum allowed capacity.

4 TOPSIS Method of Decision Making After solving the multi objective problem, the researcher is required to select a solution from the finite set by making compromises. In this paper, TOPSIS method of decision making is adopted to select the best solution. The basic principle of TOPSIS method is that the chosen points should have the shortest distance from the ideal solution and the farthest distance from the negative ideal solution. The researcher has to express the importance of criteria to determine the weights of the criteria.

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5 Numerical Results and Discussion The IEEE 14-bus power system and the IEEE 118-bus power system are used as the test systems. The numerical data of IEEE 14-bus system are taken from [17] and the numerical data of IEEE 118-bus system are taken from [7]. To successfully implement MOPSO and MOCLPSO, the values for the different parameters are to be determined correctly. In this paper, the inertia weight is linearly decreased from 0.9 to 0.1 over the iterations and the acceleration coefficients are taken as 2. For MOCLPSO, learning probability, Pc is set as 0.1 and elitism probability, Pm is chosen as 0.4. The archive size is maintained as 100 for both MOPSO and MOCLPSO. 5.1 IEEE 14 Bus System Generally, when the loadability increases, the total loss also increases. By optimally placing the TCSCs, the loadability can be increased and the loss can be minimized. Objective values obtained by MOPSO and MOCLPSO for the optimal placement of one TCSC are shown in Fig. 1, and Fig.2 respectively. The two extreme solutions of the pareto front obtained using MOPSO are f1 =1.000064, f 2 = 35.9167,

f 3 = 3.6991x105 and f1 =1.147815, f 2 = 49.996, f 3 = 6.6477x105. The two extreme solutions of the pareto front obtained using MOCLPSO are f1 =1.000064, f 2 = 35.7381, f 3 = 1.276x108 and f1 =1.147815, f 2 = 49.996, f 3 = 6.6477x105. From TOPSIS method, the best compromise solutions for the placement of one TCSC are selected among the pareto-optimal solutions obtained using MOPSO and MOCLPSO and are shown in Table 1. In this paper, the weight assigned to the loadability objective, w1 is 0.6, the weight assigned to the loss objective, w2 is 0.1 and the weight assigned to the cost objective,

w3 is 0.3.

Table 1. The best compromise solutions based on TOPSIS for placing 1 TCSC and 2 TCSCs

% Cost of Number of Optimization Loading Loss reduction TCSCs TCSCs Method Factor (MW) in loss (US$) 5 5.9434x10 MOPSO 1.1287 47.431 2.738 1 TCSC 5 MOCLPSO 1.1281 47.3769 2.738 5.935x10 MOPSO

1.1478 49.6745 1.894

2 TCSCs MOCLPSO 1.1576 50.9394 1.298

Lines Parameter of with TCSC (p.u.) TCSC 7-8 -0.14092 7-8 -0.14092 -0.14092 7 7-8 9.5296x10 3-4 -0.047153 7-8 -0.14092 1.4506x107 9-14 -0.048668

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs

Fig. 1. Objective values obtained with MOPSO for the placement of one TCSC

Fig. 2. Objective values obtained using MOCLPSO for the placement of one TCSC

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The comparison of the number of fitness evaluations and computation time for MOPSO and MOCLPSO for the placement of one number of TCSC and two numbers of TCSCs are given in Table 2. The performance metric, spread is calculated to judge the performance of the algorithms and it is also presented in Table 2. The results prove that, MOCLPSO is able to obtain a more evenly distributed pareto front when compared to MOPSO. Table 2. Comparison of computation time and performance metrics for IEEE 14 bus system

Particulars No of fitness evaluations Average computation time for one trial (secs) Spread

MOPSO MOCLPSO 1 TCSC 2 TCSC 1 TCSC 2 TCSC 12,000 18,000 12,000 18,000 455 834 465 840 0.0701 0.0938 0.0075 0.0723

5.2 IEEE 118 Bus System A minimum of 5 TCSCs is considered for IEEE 118 bus system. The best compromise solutions selected using TOPSIS are presented in Table 3. Lower value of spread is obtained for MOCLPSO for this case also. The results obtained for the test systems prove that MOCLPSO is efficient in finding the optimal placement of TCSCs for this multi-objective optimization problem. Table 3. The best compromise solutions based on TOPSIS for placing five TCSCs in IEEE 118 bus system

Optimization Loading Method Factor

Loss % reduction (MW) in loss

Cost of TCSCs (US$)

MOPSO

1.188

158.183

0.0722

7.6132x108

MOCLPSO

1.188

158.057

0.1518

4.1563x108

Lines with TCSCs 75-118 106-107 104-105 76-118 105-106 75-118 106-107 104-105 76-118 105-106

Parameters of TCSCs (p.u.) -0.03848 -0.1464 -0.03024 -0.04352 -0.04376 -0.03848 -0.00183 -0.03024 -0.04352 -0.04376

6 Conclusion In this paper, the optimal placement of TCSCs for transmission system loadability maximization, transmission loss minimization and minimization of installation cost of TCSCs has been formulated as a multi-objective optimization problem. The thermal

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limits of transmission lines and voltage limits of load buses are considered as security constraints. This problem is solved by using MOPSO and MOCLPSO. The proposed approach has been successfully tested on two IEEE test systems namely, IEEE 14 bus system and IEEE 118 bus system. The pareto optimal solutions obtained by MOPSO are compared with those obtained by MOCLPSO. From the performance metric evaluated for the pareto optimal solutions obtained by MOPSO and MOCLPSO, it is inferred that MOCLPSO performs well with a good spread of solutions along the pareto front. TOPSIS method has been adopted to ease the decision making process. Acknowledgement. The authors thank the managements of Thiagarajar College of Engineering and Mepco Schlenk Engineering College for their support. Simulation results are obtained by modifying the MOCLPSO matlab codes made available on the Web at http://www.ntu.edu.sg/home/EPNSugan. The authors express their sincere thanks to Prof.P.N. Suganthan, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore for providing the MOCLPSO matlab codes.

References 1. Hingorani, N.G.: High power electronics and Flexible AC Transmission System. IEEE Power Engineering Review 8(7), 3–4 (1988) 2. Sait, S.M., Youssef, H.: Iterative computer algorithms with applications in engineering: solving combinatorial optimization problems, 1st edn. IEEE Computer Society Press, California (1999) 3. Gerbex, S., Cherkaoui, R., Germond, A.J.: Optimal location of multi-type FACTS devices in a power system by means of genetic algorithms. IEEE Transactions on Power systems 16(3), 537–544 (2001) 4. Hao, J., Shi, L.B., Chen, C.: Optimising location of Unified Power Flow Controllers by means of improved evolutionary programming. IEE Proceedings on Generation, Transmission and Distribution 151(6), 705–712 (2004) 5. Mori, H., Maeda, Y.: Application of two-layered tabu search to optimal allocation of UPFC for maximizing transmission capability. In: IEEE International Symposium on Circuits and Systems (ISCAS), Island of Kos, Greece, pp. 1699–1702 (2006) 6. Rashed, G.I., Shaheen, H.I., Cheng, S.J.: Optimal Location and Parameter Settings of Multiple TCSCs for Increasing Power System Loadability Based on GA and PSO Techniques. In: Third International Conference on Natural Computation (ICNC), Haikou, Hainan, China, vol. 4, pp. 335–344 (2007) 7. Saravanan, M., Mary Raja Slochanal, S., Venkatesh, P., Prince Stephen Abraham, J.: Application of particle swarm optimization technique for optimal location of FACTS devices considering cost of installation and system loadability. Electric Power Systems Research 77(3-4), 276–283 (2007) 8. Deb, K.: Multi-objective optimization using evolutionary algorithms, 1st edn. John Wiley and Sons, New York (2001) 9. Deb, K.: Current trends in evolutionary multi-objective optimization. Int. J. Simul. Multidisci. Des. Optim. 1, 1–8 (2007) 10. Eghbal, M., Yorino, N., Yoshifumi Zoka El-Araby, E.E.: Application of evolutionary multi objective optimization algorithm to optimal VAr expansion and ATC enhancement problems. International Journal of Innovations in Energy Systems and Power 3(2), 6–11 (2008)

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11. Benabid, R., Boudour, M., Abido, M.A.: Optimal location and setting of SVC and TCSC devices using non-dominated sorting particle swarm optimization. Electric Power Systems Research 79(12), 1668–1677 (2009) 12. Hwang, C.L., Yoon, K.: Multiple attribute decision making: methods and applications. Springer, Berlin (1981) 13. Coello, C.A.C., Lechuga, M.S.: MOPSO: A proposal for multiple objective particle swarm optimization. In: IEEE World Congress on Computational Intelligence, pp. 1051–1056 (2003) 14. Huang, V.L., Suganthan, P.N., Liang, J.J.: Comprehensive learning particle swarm optimizer for solving multiobjective optimization problems. International Journal of Intelligent Systems 21, 209–226 (2006) 15. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Evaluation of comprehensive learning particle swarm optimizer. In: Pal, N.R., Kasabov, N., Mudi, R.K., Pal, S., Parui, S.K. (eds.) ICONIP 2004. LNCS, vol. 3316, pp. 230–235. Springer, Heidelberg (2004) 16. Coello, C.A.C., Pulido, G.T., Lechuga, M.S.: Handling multiple objectives with particle swarm optimization. IEEE Trans. Evol. Comput. 8, 256–279 (2004) 17. The University of Washington: College of Engineering. Power systems test case archive, http://www.ee.washington.edu/research/pstca

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space P.C. Pal and Dinbandhu Mandal Department of Applied Mathematics Indian School of Mines, Dhanbad-826004 [email protected], [email protected]

Abstract. An analysis is presented to consider the problem of torsional body forces within a Kelvin-Voigt type visco- elastic material half space. Hankel & Fourier transform are used to calculate the disturbance for a body force located at a constant depth from the free surface. Numerical results are obtained and the natures are shown graphically. Keywords: Torsional body force, Visco elastic medium, Kelvin-Voigt material, Hankel Transform, Fourier cosine Transform.

1 Introduction The studies of propagation of torsional surface waves in anisotropic visco-elastic medium are of great importance, as they help in investigation to the structure of earth and in exploration of natural resources inside the earth as well. Such studies are also important in elastic medium due to their wide application in geophysics and to understand the cause of damage due to earthquake. In many industrial applications, the engineers are faced with stress analysis of three dimensional bodies in which geometry and loading involved are axisymmetric. For axisymmetric problems the point loads are generalized to ring loads. The determination of stresses and displacement in an isotropic, homogeneous elastic medium due to torsional oscillation has been a subject of considerable interest in solid mechanics and applied mathematics. In particular, torsional vibration of anisotropic visco-elastic material is largely used in measuring shear constants, shear velocities and shear elasticity of liquid. The problem of torsional body force in visco-elastic material is solved by many authors. Bhattacharya (1986) has studied the wave propagation in Random MagnetoThermo Visco-elastic Medium. Mukherjee and Sengupta (1991) have discussed the surface waves in thermo visco-elastic media of higher order. Manolis and Shaw (1996) have discussed the harmonic wave propagation through visco elastic heterogeneous media. Dey et (1996) have considered the propagation of torsional surface waves in visco-elastic medium. Drozdov (1997) has studied on a constitutive model for nonlinear visco elastic media. Lee and Wineman (1999) have discussed a model for non linear visco-elastic torsional response of an elastomeric bushing. Pal (2000, 2001) have discussed the effect of dynamic visco-elasticity on vertical and torsional vibrations of a half-space and the torsional body forces. Romeo (2001) has studied the P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 145–151, 2011. © Springer-Verlag Berlin Heidelberg 2011

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properties of wave propagator and reflectivity in linear visco-elastic media. Chen and Wang (2002) have discussed the torsional vibrations of elastic foundation of saturated media. Dey etl. (2003) have discussed the propagation of torsional surface waves in an elastic layer with void pores over an Elastic half space with void pores. Kristek and Moczo (2003) have discussed the Seismic wave propagation in visco elastic media with material discontinuities. Narain and Srivastava (2004) have discussed the magneto elastic torsional vibration of non-homogeneous Aeolotropic cylindrical shell of visco-elastic solid. Cerveny and Psencik (2006) have discussed the particle motion of plane waves in visco elastic anisotropic media. Borcherdt and Bielak (2010) have discussed the visco elastic waves in layered media. In this paper, we discuss the problem of torsional disturbance due to a decaying body forces within a visco-elastic medium of Kelvin-Voigt type material. Using the Hankel and Fourier cosine transforms, the solution is obtained for type of body forces located over a constant depth from the plane face. Numerical results are obtained for Kelvin-Voigt material and compared with elastic solid. Variations of disturbance are exhibited graphically for different value of α for elastic material and a constant value of depth from the surface.

2 Formulation of Problem Let (r,θ,z) be cylindrical polar coordinates. The origin is taken to any point of the boundary of the half space and z-axis is taken in the vertically downward directions. The material of the medium is taken to be visco elastic of Kelvin Voigt type material. Medium contains a body force which is torsional in nature and as a function r, z and t. For torsional disturbances the stress-strain relations as

τ rr =τ θθ =τ zz =τ rz =0

⎛ ⎞ ⎛ ⎞ ⎜ μ ⎟ ⎜ μ ⎟ ⎛ ∂v v ⎞ τ =2 ⎜ ⎟e = ⎜ ⎟⎜ - ⎟ 1 1 ⎝ ∂r r ⎠ ⎜ 1+ ⎟ ⎜ 1+ ⎟ ⎝ iωδ ⎠ ⎝ iωδ ⎠ rθ

rθ

⎛ ⎞ ⎛ ⎞ ⎜ μ ⎟ ⎜ μ ⎟ ∂v and ⎜ ⎟τ =⎜ ⎟ 1 1 ⎜ 1+ ⎟ ⎜ 1+ ⎟ ∂z ⎝ iωδ ⎠ ⎝ iωδ ⎠ θz

where μ = Lame constant,

ω =frequency parameter and δ =relaxation parameter.

τrθ and τθz are shear stresses in the solid. Here the motion is symmetrical about z-axis so v r = v z = 0 and v(r, z, t) (say), and is independent of θ and is circumferential component of displacement. The only non-zero equation of motion in terms of stresses is ∂τ rθ ∂r

where, Fθ is the body force.

+

2 r

τ rθ +

∂τ θz ∂z

∂ v 2

+ Fθ =ρ

∂t

2

.

(1)

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space

We take body force

147

Fθ in the form Fθ = F(r,z)e ,ω>0 -ωt

(2)

.

The equation of motion (1) in terms of displacement component v may be written as

⎡∂

k⎢

2

⎣ ∂r

v 2

+

1 ∂v

-

r ∂r

v r

2

∂ v⎤ 2

+

∂z

2

-ωt

⎦

1+

Let us take v (r, z, t ) = f (r, z )e ∂f 2

∂r

2

+

1 ∂f

2

, where,

1 iωδ

, ω > 0 and so final equation m ay be w ritten as +

r ∂r

∂t

μ

k=

− ωt

∂ v 2

⎥ + F(r,z)e =ρ

∂f ∂z

⎧1

−⎨

⎩r

+

2

ρw

2

k

⎫ F ⎬f + ⎭ k

=0

.

(3)

3 Method of Solution Eq. (3) will be solved using Hankel transform and Fourier cosine transform. The pair of transformation is defined as ∞

( f , F ) = ∫ ( f , F )rJ ( ξr ) dr 1

1

0

(f

2

2

, F2 ) =

π

(4)

∞

∫ ( f , F ) cos ( ηz ) dz . 1

1

0

where J1 ( ξr ) is Bessel unction of first kind and order one. Now applying Hankel transform to eq. (3), we have ∂ f1 2

∂z

2

⎡

− ⎢ξ + 2

ρω

⎣

⎤ ⎥f ⎦

2

1

k

+

F1 k

=0

.

(5)

The initial and boundary conditions of the problem require that: v(r, z, t) =

and the plane boundary is stress free so at

∂v ∂t

(r, z, t) at t = 0 .

∂F1

=0

∂z

on z=0. We may assume that

z → ∞.

Now applying Fourier cosine transform to (5), we have f 2 ( ξ, η ) =

F2

⎡ ⎣

k ⎢ξ + η + 2

2

ρω k

2

⎤ ⎥ ⎦

(6) ∂f1 ∂z

→0

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P.C. Pal and D. Mandal

Hence, by inversion theorem on Hankel and Fourier transform, we have finally v(r, z, t) which is given by v(r, z, t) = f(r,z)e =

e

− ωt

k

− ωt

2

∞ ∞

∫∫ π

ξF2 ( ξ, η ) J 1 ( ξr ) cos ( ηz ) dξdη ρω ⎤ ⎡ ⎢ξ + η + k ⎥ ⎣ ⎦

.

(7)

2

2

0 0

2

4 Decaying Torsional Body Force To determine F2 in (7), we assume that the body force acts inside the half space at a depth z=h(>0). The body force is defined as Fθ = f ( r, z ) e

= p (r) δ (z − h ) e

− ωt

− ωt

.

(8)

i.e. body force is too large at z=h and occurs as p(r) behaves. Hence ∞

f1 ( ξ, z ) = p ( r ) δ ( z − h ) rJ1 ( ξr ) dr

∫ 0

and F2 ( ξ, η ) =

2

∞

∫ p ( ξ )δ ( z − h ) cos ( ηz ) dz π 1

0

2

=

where,

P1 ( ξ ) =

π

p1 ( ξ ) cos ( ηh )

∞

∫ rp ( r ) J ( ξr ) dr 1

1

0

Hence finally v ( r, z, t ) =

where, n = ξ + α 2

2

and α =

ρω

2

1 e

− ωt

π k

⎡π ∫ p ( ξ ) ⎢⎣ 2n e ∞

−n(z+h)

1

0

+

π

e

−n(z −h)

2n

⎤ ξJ ( ξr ) dξ . ⎥⎦ 1

2

2

.

k

5 Special Case Case 1. Let us assume that p(r) the radial dependence of applied force be defined as p(r) =

⎧Qr, r ≤ a ⎨ ⎩0, r > a

(9)

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space

149

Q and a are constants then a

p1 ( ξ ) = Qr J 1 ( ξr )dr

∫

2

(10)

0

=

Q ξ

⎡ ⎢⎣ −a

2

2

J 0 ( ξa ) +

aJ 1

ξ

( ξa ) ⎤⎥ ⎦

.

Hence from (9), the displacement v(r,z,t) is given as v ( r, z, t ) =

Qe

− ωt

∞

∫

2k

e

− n( z + h )

+e

−n(z −h)

⎡ ⎢⎣ −a

n

0

2

J 0 ( ξa ) +

2 ξ

⎤ ⎥⎦

aJ1 ( ξa ) J1 ( ξr ) dξ .

(11)

Case2. Let ⎧⎪Qr/ ( a -r 2

2

p(r)= ⎨

)

1/2

, r
⎪⎩0

,r>a

Q again a constant. This shows that the body force is in the form of angular displacement over a circular area of radius a and zero elsewhere. p1 ( ξ ) =Q

a

∫ 0

r J 1 ( ξr ) dr 2

(a

2

-r

2

)

1/2

If a=1, then according to Gradshteyn and Ryzhik [1996, pp. 977], we get p1 ( ξ ) =

Q ⎡ sinξ

⎤ ⎢⎣ ξ -cosξ ⎥⎦

ξ

Hence from (9), the displacement v(r,z,t) is given as v ( r, z, t ) =

Qe

-ωt

2k

∞

∫

e

-n ( z+h )

+e

-n ( z-h )

(12)

1

n

0

⎡ sinξ ⎤ ⎢⎣ ξ -cosξ ⎥⎦J ( ξr ) dξ .

6 Numerical Results and Discussions For Kelvin-Voigt material, we assume the visco elastic parameter α 2 = hence the torsional disturbances is given by

ρω2 and a=1 k

vke ωt = I (say) (on thesurface z = 0) Q

where, ∞

I=

∫

(ξ

cosh

0

(ξ

2

2

)

+α h

+α

2

2

)

2 ⎡ ⎤ ⎢⎣ − J ( ξ ) + ξ J ( ξ ) ⎥⎦ J ( ξr ) dξ . 0

1

1

(13)

for special case (1) and ∞

I=

∫ 0

for special case (2).

cosh

(ξ

2

2

( ξ +α ) 2

)

+α h ⎡ sinξ 2

⎤ ⎢⎣ ξ -cosξ ⎥⎦ J ( ξr ) dξ . 1

(14)

150

P.C. Pal and D. Mandal -3

2

x 10

r=0.9 r=0.7 r=0.5

1.8 1.6 1.4 1.2 [I]

1 0.8 0.6 0.4 0.2 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[α ]

Fig. 1. Variation of torsional disturbance with visco elastic parameter (case1)

-3

3

x 10

r=0.9 r=0.7 r=0.5

2.5

[I]

2

1.5

1

0.5

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[α ]

Fig. 2. Variation of torsional disturbance with visco elastic parameter (case2)

7 Conclusion The value of I for different values of α (visco elastic parameter) are evaluated numerically from the Eqs. (13) and (14) for a particular value of h=3.75(depth) and radial distance taken as r<1. The integrals (13) and (14) are evaluated using the known

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space

151

result of Gradshteyn and Ryzhik(1996). From fig.1 and fig.2 it is concluded that the torsional disturbance are almost same at initial instant, then decrease with the increases of α(visco elastic parameter). As the torsional disturbance depends on applied body force, it is clear that, the disturbance would decrease with the increase of r and deviates too much for certain value of α.It is also inferred that the values of I for the both cases decreases exponentially and magnitude of I are showing same nature for various value of r.

References 1. Drozdov, A.D.: A Constitutive Model for Nonlinear Visco-elastic Media. Int. Jn. of sol. and str. 34, 2685–2707 (1997) 2. Borcherdt, R.D., Bielak, J.: Visco-elastic Waves in Layered Media. Earthquake Spectra 26, 901–903 (2010) 3. Cerveny, V., Psencik, I.: Particle Motion of Plane Waves in Visco-elastic Anisotropic Media. Russian Geo. and Geoph. 47, 551–562 (2006) 4. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic press, London (1996) 5. Kristek, J., Moczo, P.: Seismic Wave Propagation in Visco-elastic Media with Material Discontinuities: A 3D Fourth-order Staggered Grid Finite-Difference Modeling. Bull. of the Seism. Soc. of Am. 93, 2273–2280 (2003) 6. Chen, L., Wang, G.: Torsional Vibrations of Elastic Foundation of Saturated Media. Soil Dynamics and Earthquake Eng. 22, 223–227 (2002) 7. Mukherjee, A., Sengupta, P.R.: Surface Waves in Thermo Visco-elastic Media of Higher Order. Ind. J. of Pure Appl. Math. 22, 159–167 (1991) 8. Romeo, M.: Properties of Wave Propagator and Reflectivity in Linear Visco-elastic Media. Q. J. of Mech. and App. Maths. 54, 213–226 (2001) 9. Manolis, G.D., Shaw, R.P.: Harmonic Wave Propagation Through Visco-elastic Heterogeneous Media Exhibiting Mild Stochasticity-II Application. Soil Dyn. Earthq. Eng. 15, 129–139 (1996) 10. Bhattacharyya, R.K.: On Wave Propagation in Random Magneto-Thermo Viscoelastic Medium. Ind. J. Pure Appl. Math. 17(5), 705–772 (1986) 11. Pal, P.C.: A Note on the Torsional Body Force in a Visco-elastic Half Space. Ind. J. Pure Appl. Math. 31, 207–213 (2000) 12. Pal, P.C.: Effect of Dynamic Visco-elasticity on Vertical and Torsional Vibrations of a Half-Space. Sadhana 26, 371–377 (2001) 13. Narain, S., Srivastava, H.K.: Magneto Elastic Torsional Vibration of Non-Homogeneous Aeolotropic Cylindrical Shell of Viscoelastic Solid. Defence Science Journal 54, 443–454 (2004) 14. Lee, S.B., Wineman, A.: A Model for Non Linear Visco-elastic Torsional Response of an Elastomeric Bushing. Acta Mechanica 135, 199–218 (1999) 15. Dey, S., Gupta, A.K., Gupta, S.: Propagation of Torsional Surface Waves in Visco-elastic Medium. Int. J. for Num. and Analytical Methods in Geomechanics 20, 209–213 (1996) 16. Dey, S., Gupta, S., Gupta, A.K., Kar, S.K., De, P.K.: Propagation of Torsional Surface Waves in an Elastic Layer with Void Pores over an Elastic Half Space with Void Pores. Tamkang Jn. of Science and Eng. 6, 241–249 (2003)

Medical Image Binarization Using Square Wave Representation K. Somasundaram and P. Kalavathi Image Processing Lab Department of Computer Science and Applications Gandhigram Rural Institute – Deemed University Gandhigram – 624 302, Tamil Nadu, India [email protected], [email protected]

Abstract. This paper describes a new approach for medical image binarization based on square wave representation. A square wave is a type of wave form, where the input signal has two levels +1 (foreground) and -1 (background). The signal switches between these levels based on the threshold value computed at that level with the specified time interval. In this method, a local threshold value is calculated at every interval using the current intensity value. Then, the image pixel is assigned with a value +1 or -1 using this local threshold value. The experimental results show that the proposed method reduces the complexity and increases the seperability factor in medical image segmentation. The result obtained by our method is comparable to or better than Otsu’s thresholding method. Keywords: Medical image, binarization, thresholding, segmentation, square wave binarization.

1 Introduction Almost all the medical image processing techniques need to produce a binary form of original image and it plays an essential role in many medical image segmentation techniques [1-4]. Advantages of obtaining first a binary image is that it reduces the complexity of the data and simplifies the process of image segmentation. Threshold is one of the most powerful techniques for image segmentation. The medical image is converted to binary image using the threshold value. Thresholding creates a binary image from gray scale images by turning all pixels below the threshold to zero representing the background and all pixels about that threshold to one for foreground representation. Finding the correct threshold value is a challenging process. Because of its efficiency in performance and its simplicity, thresholding techniques have been studied extensively and a large number of thresholding methods have been developed [5]. Automatic thresholding techniques are classified into two main groups: global and local. In global methods, a fixed threshold value is used for the whole image, whereas P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 152–158, 2011. © Springer-Verlag Berlin Heidelberg 2011

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in local methods the threshold changes dynamically. Global thresholding techniques [6] failed to separate the objects when the background is uneven due to poor illumination condition. A local threshold value is computed for each pixel on the basis of information contained in a local neighborhood of the pixel. All these techniques optimize a criterion function based on information obtained from either histogram generation or spatial distribution. One of the well known and widely used methods, Otsu’s thresholding method [7], is based on discriminant analysis to find the maximum separability of classes. For every possibility of threshold values, Otsu evaluated the goodness of this value if used as a threshold. This evaluation includes the heterogeneity of both class and the homogeneity of every class. In this paper, a new technique for automatic thresholding of medical image is proposed. The intensity value of the images is represented as a square wave form (+1 or -1) by computing a local threshold for each pixel. The proposed method does not require image histogram or any complex function to compute threshold value. The performance of our method is evaluated with MRI and CT scan images and is found to give satisfactory results. The remaining part of the paper is organized as follows: In section 2, we briefly explained the methods and materials that are used in the proposed approach. In section 3, the experimental results and discussion are given. In section 4, the conclusion is given.

2 Methods and Materials A square wave is a periodic waveform consisting of instantaneous transition between two levels [8]. An ideal square wave alternates regularly and instantaneously between two levels. The square wave illustrated in Fig.1 has period 2 and levels -1/2 and 1/2. Other common levels for square waves include (-1, 1) and (0, 1) (digital signals).

Fig. 1. Square wave with period 2 and levels -1/2 and 1/2

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Square wave has many definitions; it can be defined as simply the sin function of a sinusoid.

x(t ) = sgn(sin(t ))

(1)

where, x(t) will be 1 when the sinusoid is positive, -1 when the sinusoid is negative, and 0 at the discontinuities. Representing the pixel intensity into square wave form, first we need to simplify the image intensity values for easy manipulation. Logarithm is a method to represent numbers lying in a wide range of magnitude. We use of base 2 logarithms for simplifying the intensity values and it is obtained from:

if f ( x, y ) ≠ 0

⎧log ( f ( x, y )) f t ( x, y ) = ⎨ 2 ⎩0

otherwise

(2)

where f(x, y) is the intensity of the input image and we avoided the ∞ (infinitive) value by assigning 0 for log2 (0). Now the simplified values in ft are used for square wave representation. Then the square wave is generated with the period 2π for every element of ft. Switching between the levels in the square wave is decided by the local threshold value Ti. The local threshold value Ti is computed using the current ft(x, y) at the coordinate (x, y) as given below:

⎛ d ⎞ Ti = 2 × f t ( x, y ) × ⎜ ⎟ ⎝ 100 ⎠

(3)

where, d is the duty cycle and represent the percentage of period ft is positive. By default the duty cycle of the square wave is 50%. It is up half the time and down half the time. In our method d value is calculated by

d=

S × 100 N( f )

(4)

where, N(f) represent the total number of pixels in f, S denotes the number of foreground pixel and is calculated by counting the number of pixel in f with intensity value greater than the pixel intensity of the first peak of its histogram. Since, in the medical images, the background is black. Therefore we assumed that the first peak represent the total number of background pixels. Then using the threshold value Ti the binary image g is obtained as:

⎧1 g ( x, y ) = ⎨ ⎩0

if f t ( x, y ) ≤ Ti 1

Otherwise

where, f1t(x, y) is the normalized ft(x ,y) value between 0 and 2π. The steps involved in the proposed method are summarized as follows: Step-1: Let f(x, y) is the intensity values of the input medical image Step-2: Compute ft(x, y) by Eq. (2)

(5)

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Step-3: Repeat steps (i) to (iii) for every ft(x, y) value in ft. (i) Compute the local threshold value Ti using Eq.(3) and (4). (ii) Let t1 be the normalized value of ft(x, y) between 0 and 2π in f1t (iii) Compare t1 and Ti and construct a binary image g by Eq.(5). 2.1 Thresholding Performance Evaluation The performance of the thresholding method is evaluated by comparing the output image with the ground-truth image. If the ground-truth image is not available, the performance is evaluated using region non-uniformity (NU) measure. The NU measure [9], is not based on the ground-truth data, but judges the intrinsic quality of the segmented value and is defined as:

NU = σo2 / σT2

(6)

where, σT2 represent the variance of the whole image and σo2 represent the foreground variance. It is expected that a well-segmented image will have NU measure close to 0, while the worst case NU =1 corresponds to an image for which background and foreground are indistinguishable up to second order moments. 2.2 Materials We have used 20 volumes of MR brain images obtained from the Internet Brain Segmentation Repository (IBSR) [9] of the Centre for Morphometric Analysis (CMA) at the Massachusetts General Hospital and from the website ‘The Whole Brain Atlas’ (WBA) [10] maintained by the Department of Radiology and Neurology of Brigham and Women’s Hospital, Harvard Medical School, the Library of Medicine, and the American academy of neurology. We have also used 10 CT scan images collected from the internet for our experiments.

3 Results and Discussion We carried out experiments by applying the proposed method on the collected images and compared the performance of the new technique with the Otsu’s method. The selected sample images and their corresponding thresholded image by the proposed and Otsu’s methods are shown in Fig.2. Each row in Fig.2 represents images numbered from 1 to 10. Images 1 to 5 are CT scan images and 6 to 10 are MR brain images. Images 6 and 7 are T1-weighted images. PD-weighted images are shown in Images 8 and 9. A T2-weighted image is shown in image 10. To compare the performance of the proposed method, we calculated the non-uniformity (NU) measure applying Eq.(6) on both the methods for the images shown in Fig. 2 and are given in Table 1. The drawback of this method is it does not produce good result for noisy images and is illustrated in Image 1 of Fig.2 and Table 1. This can be avoided by adapting any noise removal techniques prior to the segmentation.

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Fig. 2. Binary image obtained by thresholding. Column 1 original image, column 2 binary image by the proposed method and column 3 by Otsu’s method.

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Fig. 2. (continued) Table 1. Computed NU value for the selected images of Fig.1 by proposed and Otsu’s methods

S.No Image 1 Image 2 Image 3 Image 4 Image 5 Image 6 Image 7 Image 8 Image 9 Image 10

Proposed 0.3531 0.2785 0.0890 0.2668 0.2075 0.0586 0.0626 0.1590 0.2073 0.1656

Otsu 0.2518 0.2795 0.0891 0.2689 0.2275 0.0686 0.0757 0.1593 0.2222 0.1626

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4 Conclusion In this method, we have developed a new automatic thresholding technique for the medical images by square wave representation. The proposed method computes a local threshold at every interval using the current intensity value. The quality of the binary image obtained by this method is evaluated using non-uniformity measure. The results are comparable to or better than Otsu’s method. Our method requires lower computational complexity. It is straightforward to extend this method to threshold general gray-scale images.

References 1. Pal, N.R., Pal, S.K.: A Review on Image Segmentation Techniques. Pattern Recognition 26, 1277–1294 (1993) 2. Sahoo, P.K., Soltani, S., Wong, A.K.C.: A Survey of Thresholding Techniques. Computer Vision Graphics and Image Processing 41, 233–260 (1988) 3. Trier, O.D., Jain, A.K.: Goal-Directed Evaluation of Binarization Methods. IEEE Transaction on Pattern. Analysis and Machine Intelligence 17, 1191–1201 (1995) 4. Trier, O.D., Taxt, T.: Evaluation of Binarization Methods for Document Images. IEEE Transaction on Pattern. Analysis and Machine Intelligence 17, 312–315 (1995) 5. Sezgin, M., Sankur, B.: Survey over Image Thresholding Techniques and Quantitative Performance Evaluation. Journal of Electronic Imaging 13(1), 146–165 (2004) 6. Lee, S.U., Chung, S.Y., Park, R.H.: A Comparative Performance Study of Several Global Thresholding Techniques for Segmentation. Computer Vision, Graphics and Image Processing 52, 171–190 (1990) 7. Otsu, N.: A Threshold Selection Method from Gray-level Histograms. IEEE Transaction on Systems, Man, and Cybernetics 9(1), 62–66 (1979) 8. Maths Reference available online, http://www.mathreference.com 9. IBSR data set available online, http://www.cma.mgh.harvard.edu/ibsr/index.html 10. WBA (Whole Brain Atlas) MRI brain images available online, http://www.med.harvard.edu/AANLIB/home.html

Solving Two Stage Transportation Problems P. Pandian and G. Natarajan Department of Mathematics, School of Advanced Sciences, V I T University, Vellore-632 014, Tamil Nadu, India [email protected], [email protected]

Abstract. A new method is proposed for solving a two-stage transportation problem, which is based on zero point method [5]. This method is very simple, easy to understand and apply and also, provides more than one solution to the two-stage transportation problem. The proposed method helps the decision makers in the logistics related issues by aiding them in the decision making process and providing an optimal solution in a simple and effective manner. Keywords: Transportation problem, Optimal solution, Zero point method, Two-stage transportation problem.

1 Introduction The transportation problem (TP) deals with shipping commodities from different sources to destinations. The objective of the TP is to determine the shipping schedule that minimizes that total shipping cost while satisfying supply and demand limits. The TP finds application in industry, planning, communication network, scheduling, transportation and allotment etc. In literature [1, 2, 3, 9], a good amount of research is available to obtain an optimal solution for TPs with equality constraints. In some circumstances due to storage constraints, destinations are unable to receive the quantity in excess of their minimum demand. In this case, a single shipment is not possible. Therefore, items are shipped to destinations from the origins in two stages. Initially, the minimum demands of the destinations are shipped from the origins to the destinations. After consuming part of whole of this initial shipment, they are prepared to receive the excess quantity in the second stage. Such type of transportation problems are known as two stage transportation problems. The main objective of the two-stage transportation problem is to transport the items from the origins to the destinations in two stages such way that the total transportation costs in the two stages are minimum. In both the stages, the transportation of the product from sources to the destinations is done in parallel. Sonia and Rita Malhotra [7] proposed a polynomial bound algorithm for a two stage time minimization problem to obtain optimal schedules for stage I and stage II such that the sum of the corresponding transportation times is the least. Sonia, Rita P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 159–165, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Malhotra and Puri [8] introduced a polynomial time algorithm for two-stage time minimizing problem to determine a global optimal solution. Pandian and Natarajan [5] proposed a new method namely, zero point method for finding an optimal solution to a transportation problem without using a basic feasible solution and optimality test procedure. Nagoor Gani and Abdul Razak [4] solved a two-stage cost minimizing fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers using a parametric approach. Ritha and Merline Vinotha [6] proposed a method for finding a best compromise solution to a multiobjective two-stage fuzzy transportation problem using geometric programming approach. In this paper, we propose a new method for solving two stage cost transportation problems, which is based on zero point method [5]. This method is very simple, easy to understand and apply and provides more than one optimal solutions to the transportation problem. The proposed method helps the decision makers to choose an appropriate solution according to their current situation. With help of the numerical example, the proposed method is illustrated.

2 Preliminaries Consider the following two-stage cost minimization transportation problem: (P) Minimize z = z1 + z 2 Subject to m

n

∑ ∑ cij xij ≤ z1

i =1 j=1 m

n

∑ ∑ cij yij ≤ z 2

i =1 j=1 n

∑ xij ≤ a i , i = 1,2,…,m

j=1 m

∑ xij = k j , j = 1,2,…,n

i =1 n

n

j =1

j =1

∑ y ij = a i − ∑ xij , i = 1,2,…,m m

∑ y ij = b j − k j , j = 1,2,…,n

i =1

xij , yij ≥ 0 ,

i = 1,2,..., m and

j = 1,2,..., n

and integers

where cij is the cost of shipping one unit from supply point i to the demand point

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j; ai is the supply at supply point i; b j is the demand at demand point j; k j is the maximum storage capacity of the jth destination, xij is the number of units shipped from supply point i to demand point j in the stage I and y ij is the number of units shipped from supply point i to demand point j in the stage II. Consider the following cost transportation problem: (G)

m

n

Minimize w = ∑ ∑ cijtij i =1 j =1

Subject to n

∑ tij = ai , i = 1,2,…,m

(1)

j=1 m

∑ t ij = b j , j = 1,2,…,n

(2)

i =1

tij ≥ 0 , for all i and j and integers,

where cij is the cost of shipping one unit from supply point i to the demand point j; t ij is the number of units shipped from supply point i to demand point j; ai is the

supply at i and b j is the demand at j . We, now prove the following theorem, which establishes a relation between the optimal solutions of the two transportation problems (G) and (P). Theorem 1: The optimal solution of the problem (G) yields an optimal solution of the problem (P). Further, the optimal objective value of (G) is same as the optimal objective value of (P). Proof: Let {tijD , i = 1,2,..., m and j = 1,2,..., n} be an optimal solution of the problem

(G) with objective value wD . Let {xijD , i = 1,2,..., m and j = 1,2,..., n} and { yijD , i = 1,2,..., m and j = 1,2,..., n} be two sets of positive real numbers which are generated from the set {tijD , i = 1,2,..., m and j = 1,2,..., n} such that n

D ∑ xij ≤ a i , i = 1,2,…,m

j=1 m

D ∑ xij = k j , j = 1,2,…,n.

i =1

yijD = tijD − xijD , i = 1,2,..., m and j = 1,2,..., n .

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m n m n Let z1D = ∑ ∑ cij xijD and z 2D = ∑ ∑ cij yijD . i =1 j=1

i =1 j=1

Then, z1D + z2D = wD . Clearly, {z1D , z2D , xijD and yijD , i = 1,2,..., m and j = 1,2,..., n} is a feasible solution to the problem (P). Suppose that

{z1D , z2D , xijD and yijD , i = 1,2,..., m and j = 1,2,..., n} is not an optimal

solution to the problem (P). Then, there exists a feasible

{z1 , z2 , xij and yij , i = 1,2,..., m and j = 1,2,..., n}

of (P) such that z = z1 + z2 < wD . Since {z1 , z 2 , xij and yij , i = 1,2,..., m and j = 1,2,..., n} is a feasible solution of (P),

then {tij = xij + yij , i = 1,2,..., m and j = 1,2,..., n} is a feasible solution of (G). Now,

m

w= ∑

n

m

∑ cijt ij = ∑

i =1 j =1

n

∑ cij ( xij + y ij )

i =1 j =1 m n

m

n

D = ∑ ∑ cij xij + ∑ ∑ cij y ij ≤ z1 + z 2 < w

i =1 j =1

i =1 j=1

which contradicts that {tijD , i = 1,2,..., m and j = 1,2,..., n} is an optimal solution to (G). Thus, {z1D , z2D , xijD and yijD , i = 1,2,..., m and j = 1,2,..., n} is an optimal solution to the problem (P). Hence the theorem. 2.1 A Procedure for Finding an Optimal Solution to Two Stage Transportation Problem

We, now introduce a new method for finding various optimal solutions for two stage TPs. The proposed method proceeds as follows. Step 1. Obtain an optimal solution to the given TP without considering the minimum capacity of the destinations by zero point method [5]. Step 2. Obtain the various possible solutions to the Stage I and Stage II problems of the given two stages TP using the optimal solution obtained in the Step.1. Step 3. The combined solution(s) of the various optimal solutions obtained from the Step 2. yields various possible optimal solutions to the given two-stage transportation problem.

The proposed method is illustrated by the following example.

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Example 1: Consider the following transportation problem:

O1

D1 5

D2 8

D3 3

Supply 10

O2

7

4

5

4

O3

2

6

9

4

O4

4

6

6

12

Demand

12

14

14

with maximum capacity of the destinations are b1 = 6 ; b2 = 7 and b3 = 7 respectively. Now, by the zero point method , we obtain the optimal solution to the given transportation problem without considering the minimum demand. D1

D2

D3 10

O1 O2

Supply 10

4

O3

4

O4

8

4

Demand

12

14

4 4 12 14

in addition, the minimum transportation cost is 110. Now, since maximum capacity of destinations are b1 = 6 ; b2 = 7 and b3 = 7 , we can arrange the optimal solution of the TP in the following ten ways. D1

D2

O1 O2

D3 10 (7)

4 (4+3)

O3

4 (0+1+2+3+4)

O4

8 (6+5+4+3+2)

4 (3+4)

Demand

12 (6)

14 (7)

Supply 10 4 4 12

14 (7)

Now, the given two stage transportation problem has the following ten optimal solutions with minimum total transportation costs is Rs. 110.

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Stage-I optimal solutions x31 = 0 ; x41 = 6 ; x22 = 4 ; x42 = 3 ; x13 = 7 with total transportation cost is Rs.79 x31 = 0 ; x41 = 6 ; x22 = 3 ; x42 = 4 ; x13 = 7 with total transportation cost is Rs.81 x31 = 1 ; x41 = 5 ; x22 = 4 ; x42 = 3 ; x13 = 7 with total transportation cost is Rs.77 x31 = 1 ; x41 = 5 ; x22 = 3 ; x42 = 4 ; x13 = 7 with total transportation cost is Rs.79

transportation cost is Rs.31 x31 = 4 ; x41 = 2 ; x22 = 1 ; x42 = 0 ; x13 = 3 with total transportation cost is Rs.29 x31 = 3 ; x41 = 3 ; x22 = 0 ; x42 = 1 ; x13 = 3 with total transportation cost is Rs.33 x31 = 3 ; x41 = 3 ; x22 = 1 ; x42 = 0 ; x13 = 3 with total transportation cost is Rs.31

x31 = 2 ; x41 = 4 ; x22 = 4 ; x42 = 3 ; x13 = 7 with total transportation cost is Rs.75 x31 = 2 ; x41 = 4 ; x22 = 3 ; x42 = 4 ; x13 = 7 with total transportation cost is Rs.77 x31 = 3 ; x41 = 3 ; x22 = 4 ; x42 = 3 ; x13 = 7 with total transportation cost is Rs.73 x31 = 3 ; x41 = 3 ; x22 = 3 ; x42 = 4 ; x13 = 7 with total transportation cost is Rs.75

x31 = 2 ; x41 = 4 ; x22 = 0 ; x42 = 1 ; x13 = 3 with total transportation cost is Rs.35 x31 = 2 ; x41 = 4 ; x22 = 1 ; x42 = 0 ; x13 = 3 with total transportation cost is Rs.33 x31 = 1 ; x41 = 5 ; x22 = 0 ; x42 = 1 ; x13 = 3 with total transportation cost is Rs.37 x31 = 1 ; x41 = 5 ; x22 = 1 ; x42 = 0 ; x13 = 3 with total transportation cost is Rs.35

9

x31 = 4 ; x41 = 2 ; x22 = 4 ; x42 = 3 ; x13 = 7 with total transportation cost is Rs.71

x31 = 0 ; x41 = 6 ; x22 = 0 ; x42 = 1 ; x13 = 3 with total transportation cost is Rs.39

10

x31 = 4 ; x41 = 2 ; x22 = 3 ; x42 = 4 ; x13 = 7 with total transportation cost is Rs.73

x31 = 0 ; x41 = 6 ; x22 = 1 ; x42 = 0 ; x13 = 3 with total transportation cost is Rs.37

2

3

4

5

6

7

8

Stage-II optimal solutions x31 = 4 ; x41 = 2 ; x22 = 0 ;

x42 = 1 ; x13 = 3 with total

Remark 1: These types of various optimal solutions are much useful to decision makers when they are handling various types of the transportation problems in two stages.

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Remark 2: We can extend the procedure to fuzzy two stage transportation problems. Remark 3: The number of optimal solutions of the considering example of the two stage cost minimizing transportation problem in [4] by the proposed method is five with the total minimum transportation cost is 156, but Nagoor Gani and Abdul Razak [4] obtained only one solution with the total minimum transportation cost is 168.

3 Conclusion In real life situations, the two-stage cost minimization transportation problem occurs due to minimum capacity of destinations. The main aim of this problem is to find the two schedules for shipping the minimum requirements to the destinations from the sources in stage-I and the left out quantity is supplied to the destinations in stage-II, from the sources which have surplus quantity left after the completion of stage-I such that the total transportation cost is minimum. The proposed method based on the zero point method provides more than one optimal solutions for a two-stage transportation problem. When the decision makers are handling various types of logistic problems having destination capacity parameters, the proposed method provides more options and can be served an important tool for them to choose an appropriate solution according to their current situation.

References 1. Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963) 2. Gupta, S.K.: Linear Programming and Network Model. Affiliated East-West Press Private Limited, New Delhi (1989) 3. Kasana, H.S., Kumar, K.D.: Introductory Operations Research Theory and Applications. Springer International Edition, New Delhi (2005) 4. Nagoor Gani, A., Abdul Razak, K.: Two Stage Fuzzy Transportation Problem. Journal of Physical Sciences 10, 63–69 (2006) 5. Pandian, P., Natarajan, G.: A New Method for Finding an Optimal Solution for Transportation Problems. International J. of Math. Sci. & Engg. Appls. 4, 59–65 (2010) 6. Ritha, W., Merline Vinotha, J.: Multiobjective Two Stage Fuzzy Transportation Problem. Journal of Physical Sciences 13, 107–120 (2009) 7. Malhotra, S., Malhotra, R.: A polynomial Algorithm for a Two – Stage Time Minimizing Transportation Problem. OPSEARCH 39, 251–266 (2002) 8. Malhotra, S., Malhotra, R., Puri, M.C.: Two Stage Interval Time Minimizing Transportation Problem. ASOR Bulletin. 23, 2–14 (2004) 9. Winston, W.L.: Introduction to Mathematical Programming Applications and Algorithms. Duxbury Press, California (1991)

Tumor Growth in the Fractal Space-Time with Temporal Density P. Paramanathan1 and R. Uthayakumar2 1 2

Department of Mathematics, AMRITA Vishwa Vidyapeetham - University, Coimbatore 641105, TamilNadu, India Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram 624302, TamilNadu, India [email protected], [email protected] Abstract. The fractal properties and temporal chracteristics of tumors reveal interesting results. In this paper, we describe a mathematical model for tumor growth in the space-time domain using fractal based density. This model is an alternative for the fractal dimension approach. Determination of the temporal fractal density Dt (t) and the scaling factor at (t) for the tumor growth in the fractal space-time domain is presented. Keywords: Gompertzian growth, Fractal space-time, Density, Tumor growth.

1

Introduction

During last three decades, a variety of mathematical models for tumor growth have been developed by using diﬀerent concepts from diﬀerential equation, stochastic processes and fractal geometry. Eventhough the models based on fractal geometry are suitable for tumor growth due to its multiscale nature. Waliszewski described the fractal properties of tumor growth through Gompertzian curve [5-8]. Molski improved the work of Waliszewski and determine the time depenent temporal fractal dimension for tumor grwoth in the space-time [10]. Bru et al., studied the ﬂuctuations in the behaviour of tumor growth using fractal analysis [11]. Xu and Feng analyzed a mathematical model for tumor growth under indirect eﬀect of inhibitors with time delay in proliferation using fractal geometry [12]. Based on the growth of tumor, Gompertzian curve is a fractal i.e, fractal dimension exists, it changes along the time scale, the Gompertzian function is a contractable mapping of the Banach space R of the real numbers, holds the Banach theorem about the ﬁxed point, and its derivative is ≤ 1. It denotes that local, intrasystemic time in which tumor growth occurs, possesses a fractal structure [4]. The computation of temporal fractal dimension reveals only the structural properties of the tumor growth. It will not analyze the variations in the growth depending on space occupied by the interacting cells. So we need a tool to analyze the density properties of the tumor growth. In this paper, we study a mathematical modeling of tumor growth with its fractal density properties. The idea of this model studied in this paper was initiated by Molski, and P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 166–173, 2011. c Springer-Verlag Berlin Heidelberg 2011

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recently this study has drawn attentions in the ﬁeld of mathematical modeling of tumor growth. The model we study in this paper is established by modifying the model of Molski by replacing fractal dimension with density. Here Gomprtzian curve is used for reﬂecting the changes in the tumor growth with respect to fractal density. The Gompertz curve is deﬁned by the following function, d

−at

G(t) = G0 e a (1−e

)

(1)

widely applied to ﬁt the demographic, biological and medical data, in particular the growth of organisms, organs, tissues and tumors, has been taken into considerations [9]. Here, G(t) stands for a number of cells or tumor weight after time t, G0 stands for the inital mass, volume, diameter or number of proliferating cells and a is retardation constant whereas d denotes the initial growth or regression rate constant. Let F be a subset of plane. The density of F at x is lim

a→0

area(F ∩ B(x, a)) area(F ∩ B(x, a)) = lim a→0 area(B(x, a)) πa2

where B(x, a) is the closed disc of radius a and centre x [1]. The classical Lebsegue density theorem tells us that, for a Borel set F, this limit exists and equals 1 when x ∈ F and 0 when x ∈ / F , except for a set of x of area 0. If F is smooth curve in the plane and x is a point of F (other than end point), then F ∩ B(x, a) is close to a diameter of B(x, a) for small a and length(F ∩ B(x, a)) = 1. a→0 2a lim

If x ∈ / F then this limit is clearly 0. The lower and upper densities of the biological, physical or mathematical object under consideration are as follows. H s (F ∩ B(x, a)) , (2a)s a→0

ds (F, xi ) = lim and s

H s (F ∩ B(x, a)) a→0 (2a)s

d (F, xi ) = lim

i = 1, 2, 3, ....

i = 1, 2, 3, ....

(2)

(3)

Here H 2 is essentially area, so if s = 2, we are in the Lebesgue density situation. The morphometric computer-aided image analysis reveals that tumorigenesis occurs in the space-time with the spatial fractal density (involving Hausdorﬀ measure) [2-3]. The fractal density can be deﬁned also by the self-similar fractal scaling function (4) y(t) = at (2t)dt in which y(t) characterizes the time-evolution of the system, dt is its temporal fractal density whereas at is a scaling factor. In the case of biological systems the fractal structure of space in which cells interact and diﬀerentiate is essential for their self-organization and emergence of

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the hierarchical network of multiple cross-interacting cells, sensitive to external and internal conditions. Hence, the biological phenomena take place in the space whose densities are not represented only by integer numbers of Euclidean space. The mean values dt and at can be obtained by the ﬁt of the fractal density function (4) to experimental Gompertz curves. To this aim the three intervals are taken into considerations (i) from zero to the inﬂection time ti , in which a number of cells reaches a 37% of the ﬁnal cell number, (ii) from the inﬂection time to the time-point at which the process of diﬀerentiation reaches a plateau and (iii) at which the number of cells remains constant [10]. A much more eﬀective approach is deriving time-dependent analytical expressions for dt (t) and at (t), as they would permit a calculation of temporal fractal density and scaling factor at an arbitrary moment of time or their mean values at an arbitrary interval. The ﬁrst attempt in this respect was unsuccessful, as both dt (t) and at (t) were derived using the relationship −at

d

at (2t)dt = e a (1−e

)

(5)

Reconstructing (3) using s = 2 we get −at

d

4t dt a2t = e a (1−e

)

, t>0

(6)

which did not take into account proper boundary conditions for t → 0 lim 4t dt a2t → 0,

t→0

−at

d

lim e a (1−e

)

t→0

→1

(7)

Hence the derived formula dt (t) = 4t dt a2t ,

lim dt (t) → 0,

lim dt (t) → 0

t→∞

t→0

(8)

To ﬁnd the explicit form of dt (t) and at (t) the relation d

−at

4t dt a2t = e a (1−e

)

−1

(9)

and its ﬁrst derivative d

−at

4 dt a2t = d e−at e a (1−e

)

(10)

should be employed. The ﬁrst of them satisﬁes the boundary conditions d −at lim 4t dt a2t = lim e a (1−e ) − 1 = 0 t→0

t→0

d −at lim 4t dt a2t = lim e a (1−e ) − 1 = at ⇔ lim dt = 0

t→∞

t→∞

t→∞

Combining (9) and (10), we arrive at the analytical expressions d −at d 2t e a (1−e ) − 1 −at d dt (t) = te−at e a (1−e ) , at (t) = d −at 8 at d e−at e a (1−e )

(11)

(12)

Tumor Growth in the Fractal Space-Time with Temporal Density

169

which satisﬁes the boundary conditions lim dt (t) = 1,

t→0

lim dt (t) = 0

t→∞

(13)

hence, the model proposed predicts in an early stage of the Gompertzian growth is a linear growth according to the equation y(t → 0) = 4t a2t ,

dt = 1

(14)

Similar boundary relations can be speciﬁed for the scaling factor at (t). They take the forms 1 dG(t) G(t) − G0 d lim = e a −1 = lim at (t) (15) = d = lim at (t), lim t→∞ t→∞ t→0 G0 t→0 dt G0 The limit at (t → 0) is undeﬁned due to the fundamental requirement t > 0 in (4). From (12) we can calculate the mean temporal fractal density and the scaling factor for evolving Gompertzian systems at an arbitrary period of time by making use of the equations ft (t) =

te 1 ft (t), N t=t d

N=

te − td p

(16)

Here td and te denote the beginning and the end of period under consideration; N is the number of the time-points in the period (td , te ), whereas p is an interval (step) between two neighbouring time-points used in calculations. The main objective of the present study is experimental veriﬁcation of the model proposed. To this aim, we determine the time-dependent temporal fractal density dt (t) and the scaling factor at (t) for the Flexner-Jobling rat’s tumor. We shall also be concerned with calculation of the mean values of dt (t) and at (t) for other kinds of rat’s tumors. 1.1

Methods

The proposed model is performed by making use of the formulae (12), the boundary conditions (15) and the experimental Gompertz curve. First, we employ the curve evaluated by Laird for Flexner-Jobling rat’s tumor whose growth is characterized by the parameters: a = 0.0490 ± 0.0063 day, b = 0.394 ± 0.066 day and G0 = 0.015g [10]. The generated plots of dt (t) and at (t) presented in Fig.1, and calculated the values of Gompertz function (1) and its derivative to determine the limits according to Eq.(15). All calculations were performed using MATLAB 7.0 software for symbolic calculations. The results are reported in Table 1. From the determined analytical expressions (12), we can calculate their mean values by taking advantage of the formula (15) and Gompertzian parameters evaluated by Laird [9] for diﬀerent kinds of rat’s tumors: Flexner-Jobling, R3a7, R4c4 and a7R3. The values of those parameters, the calculated mean values of the tempoW −K ral fractal density dt (t) and scaling factor at (t) and the values of dt (t) W −K and at (t) obtained are presented in Table 2.

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Fig. 1. Plots of the temporal Fractal desnity dt (t) and the scaling factor at (t) for Flexner-Jobling rat’s tumor whose growth is characterized by the parameters: a=0.0490 day and d=0.394 day and G0 =0.015g. Plots of the scaling factor at (t) are presented for short and long periods.

Tumor Growth in the Fractal Space-Time with Temporal Density

171

Table 1. The limiting values of the quantities used in the experimental test of the model proposed Time [day]

0

0.0001

∞

20

[G(t) − G0 ]/G0 0 0 - 2886.14 dG(t)/dt 0.284 0.38403 0 G−1 0 undeﬁned 0.43201 0.08 2886.14 at (t) 1 1.000001 1.96 0 dt (t)

Table 2. The limiting values of the quantities used in the experimental test of the model proposed Tumor

2

dt (t) dt (t)W −K

at (t) at (t)W −K

td

te

p

N

F −J 3.6732 D=0.0490(63) 0.653 r=0.394(66) 0.088

2.381 0.756 0.100

0.0357 200.24 2000.96

0.154 99.87 2035.18

0 50.5 107

50.5 107 157.5

0.1 0.1 0.1

505 565 505

a7R3 D=0.063(22) r=737(162)

3.492 0.755 0.535

2.381 1.283 0.756 187.52 0.100 20,435.06

0.154 99.87 2035.18

0 43.5 58.7

43.5 58.7 102.2

0.1 0.1 0.1

435 132 435

R4C4 D=0.078(12) r=540(20)

3.768 0.525 0.039

2.381 0.756 0.100

1.339 225.32 560.18

0.154 99.87 2035.18

0 35.2 76.0

35.2 76.0 111.2

0.1 0.1 0.1

352 408 352

R3a7 D=0.124(11) r=1.28(25)

3.616 0.622 0.019

2.381 0.756 0.100

1.392 253.2 642.35

0.154 99.87 2035.18

0 21.8 28.4

21.8 28.4 50.2

0.1 0.1 0.1

218 66 218

Results and Discussions

The performance of our calculations proved that formulae (12) applied to the Flexner-Jobling rat’s tumor growth satisfactorily describe their time-dependence in the period (0, ∞). The temporal fractal density of the Flexner-Jobling’s tumor increases from 1 for t = 0 to a maximal value 2.32 for t=20 and then decreases to zero (see Fig.1). As to the scaling factor at (t), its value for t = 0 is equal to Gompertzian parameter d = 0.758 and increases to value of 0.95 for t = 1, then it decreases to a minimal value of 0.5 for t = 20 and increases again to a maximal constant value of 0.9903. We conclude that the analytical formulae (12) describing the time-dependence of the temporal fractal density and scaling factor very well reproduce the growth of the Flexner-Jobling rat’s tumor in the spacetime with temporal fractal density. The possibility of mapping the Gompertz function, describing the tumor growth, onto the fractal density function conﬁrms that Gompertzian growth is self-similar. These results indicate that at the ﬁrst stage of the Flexner-Jobling rat’s tumor growth, which includes the period from zero to the inﬂection time

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1 ti = ln a

b = 42.54 day a

(17)

diﬀerent complex processes at microscopic level take place, which result in the speciﬁc time-change of the temporal fractal density and the scaling factor. If we assume that time is a continuous variable, it is clear that initiation of tumorigenesis cannot take place in the space-time with temporal fractal density dt = 0 as at this time-point an extrasystemic time characterizes dt = 1. An analysis of the mean fractal density presented in Table 2 reveals that they are consistent with those obtained by ﬁtting procedure whereas the mean values of the scaling factor diﬀer from those derived previously. The reasons for this discrepancy are unknown at present. The proposed method permits calculation of dt (t) and at (t) for an arbitrary period of time. It should be pointed out here that the results obtained for the latter seem to be unreliable - probably due to a low quality data used in Laird’s analysis [9].

3

Conclusion

The discussed results shows that the model proposed here is the most consistent and easy method for computing Fractal Density of tumor growth in space-time. We have proved that the analytical formulae describing the time-dependence of the temporal fractal density and scaling factor very well reproduce the growth of the Flexner-Jobling rat’s tumor in particular and growth of other rat’s tumors in general and the calculated mean values of temporal fractal density coincide very well with those obtained by ﬁtting procedure.

References 1. Falconer, J.: Fractal Geometry-Mathematical Foundations and Applications. John Wiley and Sons, Chichester (2003) 2. Losa, G.A., Nonnenmacher, T.F.: Self-similarity and fractal irregularity in pathologic tissues. Modelling Pathology 9, 174–182 (1996) 3. Sedive, R.: Fractal tumors: their real and virtual images. Wien Klin Wochenschr 108, 547–551 (1996) 4. Peitgen, H.O., Jurgens, H.H., Saupe, D.: Chaos and Fractals. Springer, Heidelberg (1992) 5. Waliszewski, P., Molski, M., Konarski, J.: Collectivity and evolution of fractal dynamics during retinoid-induced diﬀerentiation of cancer cell population. Fractals 7, 139–149 (1999) 6. Waliszewski, P., Molski, M., Konarski, J.: On the modiﬁcation of fractal self-space during cell diﬀerentiation or tumor progression. Fractals 8, 195–203 (2000) 7. Waliszewski, P., Konarski, J.: Neuronal diﬀerentiation and synapse formation in space-time with fractal dimension. Synapse 43, 252–258 (2002) 8. Waliszewski, P., Konarski, J.: The Gompertzian curve reveals fractal properties of tumor growth. Chaos Solitons & Fractals 16, 665–674 (2003) 9. Laird, A.K.: Dynamics of tumor growth. British Journal of Cancer 18, 490–502 (1964)

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10. Molski, M., Konarski, J.: Tumor growth in the space-time with temporal fractal dimension. Chaos Solitons & Fractals 36(4), 811–818 (2008) 11. Bru, A., Casero, D., de Franciscis, S., Herero, M.A.: Fractal analysis of tumor growth. Math. and Comp. Model. 47, 546–559 (2008) 12. Xu, S., Feng, Z.: Analysis of a mathematical model for tumor growth under indirect eﬀect of inhibitors with time delay in proliferation. Jour. of Math. Anal. and Appl. 374, 178–186 (2011)

The Use of Chance Constrained Fuzzy Goal Programming for Long-Range Land Allocation Planning in Agricultural System Bijay Baran Pal1,*, Durga Banerjee2, and Shyamal Sen3 1

Department of Mathematics, University of Kalyani Kalyani-741235, West Bengal, India Tel.: +91-33-25825439 [email protected] 2 Department of Mathematics, University of Kalyani Kalyani-741235, West Bengal, India [email protected] 3 Department of Mathematics, B.K.C. College Kolkata-700108, West Bengal, India [email protected] Abstract. This paper describes how the fuzzy goal programming (FGP) can be efficiently used for modelling and solving land allocation problems having chance constraints for optimal production of seasonal crops in agricultural system. In the proposed model, utilization of total cultivable land, different farming resources, achievement of the aspiration levels of production of seasonal crops are fuzzily described. The water supply as a productive resource and certain socio-economic constraints are described probabilistically in the decision making environment. In the solution process, achievement of the highest membership value (unity) of the membership goals defined for the fuzzy goals of the problem to the extent possible on the basis of the needs and desires of the decision maker (DM) is taken into account in the decision making horizon. The potential use of the approach is demonstrated by a case example of the Nadia District, West Bengal (W. B.), INDIA. Keywords: Agricultural Planning, Chance Constrained Programming, Fuzzy Programming, Fuzzy Goal Programming, Goal Programming.

1 Introduction The mathematical programming models to agricultural production planning have been widely used since Heady [1] in 1954 demonstrated the use of linear programming (LP) for land allocation to cropping plan in agricultural system. From the mid-’60s to ’80s of the last century, different LP models studied [2, 3] for farm planning has been surveyed by Glen in [4] in 1987. Since most of the farm planning problems are multiobjective in nature, the goal programming (GP) methodology in [5] as a prominent *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 174–186, 2011. © Springer-Verlag Berlin Heidelberg 2011

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tool for multiobjective decision analysis has been efficiently used to land-use planning problems [6] in the past. Although, GP has been widely accepted as a promising tool for multiobjective decision making (MODM), the main weakness of the conventional GP methodology is that the aspiration levels of the goals need be stated precisely. To overcome the above difficulty of imprecise in nature of them, fuzzy programming (FP) approach in [7] to farm planning problems has been deeply studied [8] in the past. The FGP approach [9] as an extension of conventional GP to agricultural production planning problems has also been studied by Pal et al. [6,10] in the past. Now, in most of the real-world decision situations, the DMs are often faced with the problem of inexact data due to inherent uncertain in nature of the resource parameters involved with the problems. To deal with the probabilistically uncertain data, the field of stochastic programming (SP) has been studied [11] extensively and applied to various real-life problems [12, 13] in the past. The use of chance constrained programming (CCP) to fuzzy MODM problems has also been studied by Pal et al. [14] in the recent past. However, consideration of both the aspects of FP and SP for modeling and solving real-life decision problems has been realized in the recent years from the view point of occurrence of both the fuzzy and probabilistic data in the decision making environment. Although, fuzzy stochastic programming (FSP) approaches to chance constrained MODM problems have been investigated [15] by active researchers in the field, the extensive study in this area is at an early stage. Now, in the agricultural production planning context, it is worthy to mention that the sustainable supply of water depends solely on the amount of rainfall in all the seasons throughout a year. As such, water supply to meet various needs is very much stochastic in nature. Although, several modeling aspects of water supply system have been investigated [12] in the past, consideration of probabilistic parameters to the agricultural systems in fuzzy decision environment is yet to be circulated in the literature. In this article, utilization of total cultivable land, different farming resources, achievement of the aspiration levels of production of seasonal crops are fuzzily described. The water supply as a productive resource and certain socio-economic constraints are described probabilistically in the decision making environment. In the solution process, achievement of the highest membership value (unity) of the membership goals defined for the fuzzy goals of the problem to the extent possible on the basis of the needs and desires of the decision maker (DM) is taken into account in the decision making horizon. The potential use of the approach is demonstrated by a case example of the Nadia District, West Bengal (W. B.), INDIA. Now, the general chance constrained FGP formulation is presented in the Section 2.

2 Problem Formulation The general form of a fuzzy MODM problem with chance constrained can be stated as:

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Find X (x1,x2,…,xn) so as to satisfy

⎛> ⎞ ⎜~⎟ Z k (X) ⎜ ⎟ ⎜< ⎟ ⎝~⎠

gk ,

k = 1, 2,….,K.

(1)

subject to X ∈ S{X ∈ R n | Pr[H( X) ≥ b] ≥ p, X ≥ 0 , H, b ∈ R m },

(2)

where X is the vector of decision variables in the bounded feasible region S (≠Φ), and where ) and 1indicate the fuzziness of ≥ and ≤ restrictions, respectively, in the sense of Zimmermann [7], and where gk be the imprecise aspiration level of the k-th objective. Pr stands for probabilistically defined (linear / nonlinear ) constraints set H(X), b is a resource vector, and p (0< p <1) is the vector of satisficing probability levels of the defined constraints. Now, to formulate the FGP model of the problem, the fuzzy goals in (1) are first characterized by their membership functions in [7] to measure the degree of achievement of the goals. Then, they are transformed into membership goals [16] by assigning the highest membership value (unity) as the aspiration level for goal achievement and introducing under- and over- deviational variables to each of them. Again, the probabilistic constraints are converted into their deterministic equivalent to employ the proposed approach in the process of solving the problem. 2.1 Construction of Membership Goals of Fuzzy Goals

The membership goal expression of the membership function μk (X) defined for the fuzzy goal Zk ( X) ) g k appears as [16]: μ k ( X) :

Z k (X ) − g lk + d k− − d k+ = 1, k ∈ K 1 g k − g lk

(3)

where, g lk and (g k − g lk ) represent the lower tolerance limit and tolerance range , re−

+

spectively, for achievement of the associated k-th fuzzy goal. Also, dk ≥ 0 and dk ≥ 0 are the under- and over- deviational variables, respectively, of the k-th membership goal μk (X) . Similarly, the membership goal expression for the fuzzy goal Zk ( X) 1 g k takes the form: g − Z k (X) μ k (X) : uk (4) + d k− − d k+ = 1, k ∈ K 2 g uk − g k where, g uk and (g uk − g k ) represent the upper tolerance limit and tolerance range, respectively, for achievement of the associated k-th fuzzy goal, and where K1 ∪ K2 = {1,2,..., K} with K1 ∩ K2 = ĭ.. 2.2 Deterministic Equivalent of Chance Constraints

The deterministic equivalent of a chance constraint depends on randomness and probability distribution of the parameters involved with the constraints. In the present

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decision situation, the resource vector b in (2) is taken as normally distributed random parameter. Then, the deterministic equivalent of the i-th constraint takes the form [15]: Hi (X) ≥ E(bi ) + Fi−1(pi ) {var (bi ) }

, i= 1,2, …,m

(5)

where, E(bi) and var(bi) represent mean and variance of bi and F −1 (.) represents the inverse of the probability distribution function F(.) of standard normal variate. Now, the general FGP model of the problem is presented in the following Section 2.3. 2.3 FGP Model Formulation

In a fuzzy decision making situation, the aim of the decision maker ( DM) is to achieve the highest membership value of each of the defined membership goals to the extent possible by minimizing the under deviational variables of each of them, and that depends on the needs and desires of the DM. The FGP model formulation under a pre-emptive priority structure can be presented as [10]: Find X(x1,x2,…,xn) so as to Minimize Z = [P1(d–), P2(d–), ..., Pr(d–), ..., PR(d–)] and satisfy the membership goals in (3) and (4) ,subject to the system constraints set in (5); where Z represents the vector of R priority achievement function. Pr(d–) is a linear function of the weighted under-deviational variables, where Pr(d–) is of the form K

Pr (d–) = ∑ w rk d rk , −

−

k = 1, 2, ..., K ; (R ≤ K),

k =1

where d −rk is renamed for d −k to represent it at the r-th priority level, w −rk (>0) is the numerical weight associated with d −rk and it designates the weight of importance of achieving the aspired level of the k-th goal relative to other which are grouped at the r-th priority level and where w rk values are determined as [16]: − w rk

⎧ ( g −1g ) , for the defined μ k ( X) in (3) ⎪ k kl r =⎨ 1 ⎪⎩ ( g ku −g k ) r , for the defined μ k ( X) in (4)

where (g k − g kl ) r and (g ku − g k ) r are used to present g k − g kl and g ku − g k respectively, at the r-th priority level. Now, the FGP model formulation of the proposed problem is presented in the Section 3.

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3 FGP Model Formulation of the Problem The decision variables and different types of parameters involved with the problem are defined first in the following Section 3.1. 3.1 Definition of Decision Variables and Parameters (a) Decision Variable: lcs = Allocation of land for cultivating the crop c during the season s, c = 1,2, ..., C; s = 1, 2, ..., S. (b) Productive resource parameters: • Fuzzy resources: LAs = Total farming land (hectares (ha)) currently in use for cultivating the crops in the season s. MHs = Estimated total machine hours (in hrs.) required during the season s. MDs = Estimated total man-days (in days) required during the season s. = Estimated total amount of the fertilizer f (f = 1,2,…,F) (in quintals (qtls.)) Ff required during the planning year. RS = Estimated total amount of cash (in Rupees) required per annum for supply of the productive resources. • Probabilistic resource:

WSs = Total supply of water (in inch / ha) required during the season s. (c) Fuzzy aspiration levels: = Annual production level (in qtls.) of the crop c. Pc MP = Estimated total market value (in Rupees) of all the crops yield during the planning year. (d) Probabilistic aspiration levels: = Ratio of annual production of the i-th and j-th crop (i, j = 1, 2, ...,C; i ≠ j). Rij = Ratio of annual profits obtained from the i-th and the j-th crops (i, j=1,2,...,C; rij i ≠ j). (e) Crisp coefficients: MHcs = Average machine hours (in hrs.) required for tillage per ha of land for cultivating the crop c during the season s. MDcs = Man days (in days) required per ha of land for cultivating the crop c during the season s. Ffcs = Amount of the fertilizer f required per ha of land for cultivating the crop c during the season s. Pcs Acs MPcs

= Estimated production of the crop c per ha of land cultivated during the season s. = Average cost for purchasing seeds and different farm assisting materials per ha of land cultivated for the crop c during the season s. = Market price (Rupees / qtl.) at the time of harvest of the crop c cultivated during the season s.

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(f) Random coefficients: Wcs = Estimated amount of water consumption (in inch) per ha of land for cultivating the crop c during the season s. 3.2 Description of Fuzzy Goals and Chance Constraints (a) Land utilization goal: The land utilization goal for cultivating the seasonal crops appears as: C

∑l

cs

c =1

< LA s , ~

s = 1, 2, . . . , S.

(b) Productive resource goals: • Machine-hour goals: An estimated number of machine hours is to be provided for cultivating the land in different seasons of the plan period. The fuzzy goals take the form: C

∑ MH

, ~> MH S

cs .l cs

c =1

s = 1, 2,..., S.

• Man-power requirement goals: A number of labors are to be employed through out the planning period to avoid the trouble with hiring of extra labors at the peak times. The fuzzy goals take the form: C

∑ MD

~> MD s . s = 1, 2,..., S.

cs .l cs

c =1

• Fertilizer requirement goals: To maintain the fertility of the soil, different types of fertilizer are to be used in different seasons in the plan period. The fuzzy goals take the form: C

∑F

fcs

c =1

.l cs ~ > Ff ,

f = 1, 2, ..., F; s = 1, 2,..., S.

(c) Cash expenditure goal: An estimated amount of money (in Rs.) is involved for the purpose of purchasing the seeds, fertilizers and other productive resources. The fuzzy goals take the form: S

C

s =1

c =1

∑ ∑A

cs

.l cs ~ > RS

(d) Production achievement goals: To meet the demand of agricultural products in society, a minimum achievement level of production of each type of the crops is needed. The fuzzy goals appear as: S

∑P

cs .l cs

s =1

~> Pc

,

c = 1, 2,..., C.

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(e) Profit goal: A certain level of profit from the farm is highly expected by the farm decision maker.

The fuzzy profit goal appears as:

S

C

s =1

c =1

∑ ∑ ( MP

cs

. P cs − A cs ) . l cs ~ > MP .

3.3 Description of Chance Constraints

The different chance constraints of the problem are presented in the following Sections. (a) Water-supply constraints: An estimated amount of water need be supplied to the soil for sustainable growth of the crop c cultivated during the season s. But, water-supply resources solely depends on rainfall and so probabilistic in nature. The water-supply constraints appear as: C

∑W l

Pr[

cs cs

≥ WSs ] ≥ ps , s =1,2,…,S.

c=1

where ps (0< ps <1) denotes the satisficing level of probability for the supply of water. (b) Production-ratio constraints: To meet the demand of the primary food products in society, allocation of land for the crops production in different seasons should be made in such a way that certain ratios of total production of major crops can be maintained. The production-ratio constraints appear as: S ⎡ S ⎤ Pr⎢( Pis .lis ) /( P js .l js ) ≥ R ij ⎥ ≥ p ij , i, j = 1, 2, ..., C, and i ≠ j. ⎢⎣ s =1 ⎥⎦ s =1

∑

∑

where pij (0< pij <1) denotes the satisficing level of probability for the ratios of i-th and j-th crops. (c) Profit-ratio constraints:

Here, similar to the case in production-ratio constraints, the profit-ratio constraints are random in nature. The profit-ratio constraints take the form: S ⎡ S ⎤ Pr ⎢ ( ( MP is .Pis − A cs ) l is ) /( ( MP js .P js − A cs ) l js ) ≥ rij ⎥ ≥ q ij , ⎢⎣ s =1 ⎥⎦ s =1

∑

∑

i, j = 1, 2, ..., C, and i ≠ j.

where, qij (0< qij <1) denotes the satisficing level of probability for the i-th and j-th profit-ratio.

4 An Illustrative Example: A Case Study The land-use planning problem for production of the principal crops of the District Nadia of West Bengal, India is considered to illustrate the FGP model. The data of the planning years: 2003-2004, 2004-2005 and 2005-2006 were collected from different

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181

agricultural planning units. The sources are: District Statistical Hand Book, Nadia, 2006-2007 [17]; Action Plan Records (2005-2006, 2004-2005 and 2003-2004) [18]; Soil Testing and Fertilizer Recommendation [19]; The Nadia Gramin Bank; Department of Agri-Irrigation [20]. The three seasonal crop-cycles: Pre-kharif, Kharif and Rabi successively appear in West Bengal during a planning year. The decision variables and different types of data involved with the problem are summarized in the following Tables 1–4. Table 1. Summary of the seasonal crops and decision variables

Season (s)

Pre-kharif (1)

Crop (c)

Jute (1)

Sugarcane (2)

Variable (lcs)

l11

l21

Kharif(2) BoroAusWheat Amanpaddy paddy (6) paddy (4) (5) (3) l31

l42

l53

l63

Rabi (3) Mustard (7)

Potato (8)

Pulses (9)

l73

l83

l93

Table 2. Data description of the aspired goal levels and tolerance limits

Goal 1. Land utilization (’000 hectares) : (i) Pre-kharif season (ii) Kharif season (iii) Rabi season 2. a) Machine-hours (in hrs.) : (i) Pre-kharif season (ii) Kharif season (iii) Rabi season b) Man-days (days) : (i) Pre-kharif season (ii) Kharif season (iii) Rabi season c) Fertilizer requirement (metric ton) : (i) Nitrogen (ii) Phosphate (iii) Potash 3. Production (’000 metric ton) : (a) Jute (b) Sugarcane (c) Rice (d) Wheat (e) Mustard (f) Potato (g) Rabi pulse 4. Cash expenditure (Rupees Lac.) 5. Profit (Rupees Lac.)

Aspiration Level

Tolerance Limit Lower Upper

272.14 272.14 272.14

----------

309.33 309.33 309.33

14522.30 7253.57 35373.01

13070.07 6528.22 31835.71

14274.1 6513.1 11312.2

12846.69 5861.79 10180.98

----------------------

51.5 19.7 19.7

46.36 17.73 17.73

----------

339.660 78.798 829.00 126.28 94.71 526.18 38.6 82636.4175 81470.8825

325.152 70.919 800 111.78 86.10 478.34 28.3 ----73323.794

---------------------90900.0592 ----

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Table 3. Data description of productive resource utilization, cash expenditure and market price

Crops

MHs

MDs

Ff N

P

K

PA

CE

MP

Jute 66.72 90 40 20 20 2603 17297.00 1050 Sugarcane 166.76 123 200 100 100 70364 30887.50 1200 Aus 138.99 60 40 20 20 2256.039 14331.80 1350 Aman 66.72 60 40 20 20 2153.989 12849.20 1600 Boro 266.87 60 100 50 50 3482.690 23721.60 1200 Wheat 66.72 39 100 50 50 2187.633 11119.50 1100 Mustard 33.36 30 80 40 40 869.182 8401.40 1700 Potato 111.19 70 150 75 75 21087.719 37312.10 500 Pulses 49.06 15 20 50 20 725.641 4942.00 2200 Note: MHs = machine hours (in hrs/ha), MDs = man-days (days/ha), Ff = fertilizer (kg/ha): N=Nitrogen, P = Phosphate, K = Potash; PA = production achievement (kg/ha), CE = cash expenditure (Rs / ha), MP = market price (Rs / qtl). Table 4. Data description of water-supply, water-utilization, production-ratio and profit-ratio

WU (i)

Year 2003-2004

2004-2005

2005-2006

2006-2007

1 2 3 4 5 6 7 8 9

20 60 34 50 70 15 10 18 10 (116.93, 159.85, 264.62)

20 60 34 50 70 15 10 18 10 (100.44, 147.77, 243.49) 6

20 60 34 50 70 15 10 18 10

WS(PKS,KS,RS)

20 60 34 50 70 15 10 18 10 (119.42, 147.76, 335.92) 7.39

(100.96, 133.19, 224.10)

PDR(Rice and Wheat) 6.22 6.6 PR(Jute and Aus1.17 2.27 2 5.5 paddy) Note: WU(i)= Water-utilization (inch/ha) for the i-th crop (i=1,2,…,9), WS(.)= Water-supply (inch), PKS= Pre-kharif season, KS= Kharif season, RS= Rabi season, PDR= Production-ratio, PR= Profit-ratio.

Now, using the data Tables 1-3, the membership functions of the defined fuzzy goals can be constructed by using the expressions in (3) and (4). The fuzzy goals appear as follows: Land utilization goals: The membership goals for land utilization in the three consecutive seasons appear as

The Use of Chance Constrained FGP for Long-Range Land Allocation Planning μ 1 : 8.3 2 − 0.027 ( l11 + l 21 + l 31 ) + d 1− − d 1+ = 1

(Pre-kharif)

μ 2 : 8.3 2 − 0.027 ( l 21 + l 42 ) + d 2− − d 2+ = 1

(Kharif)

μ 3 : 8.32 − 0.027 (l21 + l53 + l63 + l73 + l83 + l93 ) + d 3− − d 3+ = 1

183

(6)

(Rabi)

Productive resource goals: Machine-hour goal μ 4 : 0. 0459 l 11 + 0. 0383 l 21 + 0. 0957 l 31 − 9 + d −4 − d +4 = 1

μ 5 : 0 . 0919 l 21 + 0 . 0766 l 42 − 8 . 999 +

d 5−

−

d 5+

(Prekharif)

=1

(Kharif)

μ 6 : 0. 0157 l 21 + 0 .0754 l 53 + 0.0 189 l 63 + 0.0 094 l 73 + 0. 0314 l 83

(7)

(Rabi)

+ 0. 0139 l 93 − 9 .012 + d 6− − d 6+ = 1

Man-power goals: μ 7 : 0.0630 l11 + 0.0287 l 21 + 0.0420 l 31 − 9.03 + d 7− − d 7+ = 1

(Prekharif)

μ 8 : 0.0629l 21 + 0.0921l 42 − 8.98 + d 8− − d 8+ = 1

(Kharif)

μ 9 : 0.0362 l 21 + 0.0530 l 53 + 0.0345 l 63 + 0.0265 l 73 + 0.0619 l 83 + 0.0133 l 93 − 9 +

d 9−

− d 9+ = 1

(8)

(Rabi)

Fertilizer requirement goals: μ 10 : 0.0097 l 11 + 0 . 0388 l 21 + 0 . 0116 l 31 + 0 . 0116 l 42 + 0 . 0252 l 53 + 0 . 0233 l 63 + 0 . 0194 l 73 − + + 0 . 0388 l 83 + 0 . 0039 l 93 − 9 . 001 + d 10 − d 10 =1

(N)

μ 11 : 0.0127 l 11 + 0 . 0508 l 21 + 0 . 0152 l 31 + 0 . 0152 l 42 + 0 . 0329 l 53 + 0 . 00202 l 63 − + + 0 . 0254 l 73 + 0 . 0761 l 83 + 0 . 0203 l 93 − 8 . 89 + d 11 − d 11 =1

(P)

μ 12 : 0.0253 l11 + 0 . 0501 l 21 + 0 . 0152 l 31 + 0 . 0152 l 42 + 0 . 0329 l 53 + 0 . 0304 l 63 − + + 0 . 0254 l 73 + 0 . 0761 l 83 + 0 . 0203 l 93 − 9 + d 12 − d 12 =1

(K)

(9)

(10) Production achievement goals: − + μ 14 : 0 . 0778 l 31 + 0 . 0742 l 42 + 0 . 01201 l 53 − 27 . 5862 + d 14 − d 14 = 1 (Rice) − + μ 15 : 0 . 200 l11 − 22 . 4119 + d 15 − d 15 =1

(Jute)

− + μ 16 : 0 . 1509 l 63 − 7 . 7089 + d 16 − d 16 =1

(Wheat)

− + μ17 : 3.93l21 − 5.9 + d17 − d17 =1

(Sugercane )

− + μ 18 : 0 . 1009 l 73 − 10 + d 18 − d 18 =1

(Mustard)

− + μ 19 : 0 . 4408 l 83 − 9 . 9987 + d 19 − d 19 =1

μ

20

: 0 . 0704 l 93 − 2 . 7476 + d

− 20

− d

+ 20

= 1

(Poteto) (Pulses)

(11)

Profit achievement goal: μ 21 : 0.0123 l11 + 0.0916 l 21 + 0 .0198 l 31 + 0.0265 l 42 + 0 .0222 l 53 + 0 .0159 l 63 + 0 .0078 l 73 + 0.08364 l 83 + 0 .0135 l 93 − 8.9999 + d −21 − d +21 = 1

(12)

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Now, using the data in the Table 4 and following the procedure, the deterministic equivalent of the defined chance constraints can be obtained by using the expression (5). Water-supply constraints:

20 l11 + 60 l 21 + 34 l 31 ≥ 4576 . 99 ,

50 l 42 ≥ 6318 . 53

70l53 + 15l 63 + 10l 73 + 18l83 + 10l93 ≥ 12891.72,

(13)

where the satisfaction of probability levels are taken 0.70, 0.80, 0.90, respectively. Production-ratio constraint: The ratio of the two crops rice and wheat are considered here as the major agricultural products. The production ratio constraint appears as:

( 2 . 256 l 31 + 2 . 154 l 42 + 3 . 483 l 53 ) /( 2 . 188 l 63 ) ≥ 7 . 851 ,

(14)

where the satisfaction of the probability level is considered 0.90. Profit-ratio constraint: The profit ratio for Jute and Aus-paddy in the pre-kharif season is taken into account here. The profit-ratio constraint takes the form:

(100.32l11 ) /(969,45l 21 + 161.25l31 ) ≥ 3 . 448 ,

(15)

where the probability of satisfaction of the profit ratio constraint is taken as 0.70. Now, the executable FGP model under the four assigned priorities appears as: Find {l cs | c = 1,2,...,9; s = 1,2,3} so as to: MinimizeZ= − − − − − [P1(0.014 d −14 +0.06 d15 +0.088 d16 +0.048 d17 +0.14 d18 +0.028 d19 +0.26 d −20 ), P2(0.027 d1− +0.027 d −2 +0.027 d 3− ),P3(0.012 d −4 +0.025 d 5− +0.014 d 6− +0.2 d 7− +0.09 d 8− − − − − +1.2 d11 +0.65 d12 ), P4(0.00012 d13 +0.0001 d −21 )] +0.06 d 9− +3.7 d10

(16)

and satisfy the membership goals in (6)-(12), subject to the systen constraints in (13) – (15). The model solution for goal achievement is presented in the Table 5. Table 5. Land allocation and crops - production under the proposed model

Crop(c) Land allocation Production

Jute 117.06

Sugarcane 1.76

Rice 398.24

Wheat 57.71

Mustard 109.02

Potato 24.95

Pulses 53.23

304.68

123.54

824.51

126.26

94.76

526.26

38.63

The existing and allocation and production plan is presented in the Table 6.

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185

Table 6. Existing land allocation and crops - production of the year 2006-07

Crop(c) Land allocation Production

Jute 130.5

Sugarcane 1.1

Rice 250.3

Wheat 46.9

Mustard 79.6

Potato 5.7

Pulses 39.0

339.66

77.4

677.7

102.6

69.1

120.2

28.3

A comparison shows that a better cropping plan is obtained here from the view point of achieving the goal levels of crops-production on the basis of the needs and desires of the DM in the decision making environment.

5 Conclusion In the framework of proposed approach, the other different parameters (fuzzy / probabilistic) can easily be incorporated without involving any computational difficulty. In future studies, the proposed approach can be extended to cropping plan problems having the fuzzy satisficing probability levels of the chance constraints in the decision situation. Finally, it is hoped that the solution concept presented here can contribute to future studies in farming and other stochastic MODM problems in the current uncertain decision making arena. Acknowledgements. The authors are thankful to the Organizing Chair ICLICC 2011, and the anonymous Referees for their valuable comments and suggestions to improve the clarity and quality of presentation of the paper.

References 1. Heady, E.O.: Simplified Presentation and Logical Aspects of Linear Programming Technique. Journal of Farm Economics 36, 1035–1054 (1954) 2. Wheeler, B.M., Russell, J.R.M.: Goal Programming and Agricultural Planning. Operational Research Quarterly 28, 21–32 (1977) 3. Nix, J.S.: Farm management. The State-of-the-Art (or Science). Journal of Agricultural Economics 30, 277–292 (1979) 4. Glen, J.: Mathematical Models in Farm Planning: A Survey. Operations Research 35, 641– 666 (1987) 5. Ignizio, J.P.: Goal Programming and Extensions. D. C. Health, Lexington (1976) 6. Pal, B.B., Moitra, B.N.: Using Fuzzy Goal Programming for Long Range Production Planning in Agricultural Systems. Indian Journal of Agricultural Economics 59(1), 75–90 (2004) 7. Zimmermann, H.-J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1, 45–55 (1978) 8. Slowinski, R.: A Multicriteria Fuzzy Linear Programming Method for Water Supply System Development Planning. Fuzzy Sets and Systems 19, 217–237 (1986) 9. Tiwari, R.N., Dharmar, S., Rao, J.R.: Fuzzy Goal Programming – an Additive Model. Fuzzy Sets and Systems 24, 27–34 (1987) 10. Biswas, A., Pal, B.B.: Application of Fuzzy Goal Programming Technique to Land Use Planning in Agricultural System. Omega 33, 391–398 (2005)

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11. Charnes, A., Cooper, W.W.: Chance-constrained Programming. Management Science 6, 73–79 (1959) 12. Feiring, B.R., Sastri, T., Sim, I.S.M.: A Stochastic Programming Model for Water Resource Planning. Mathematical Computer Modeling 27(3), 1–7 (1998) 13. Bravo, M., Ganzalez, I.: Applying Stochastic Goal Programming: A case study on water use planning. European Journal of Operational Research 196, 1123–1129 (2009) 14. Pal, B.B., Sen, S.: A Fuzzy Goal Programming Approach to Chance Constrained Multiobjective Decision Making Problems. In: Proceedings of International Conference on Rough Sets, Fuzzy Sets and Soft Computing (ICRFSC 2009), pp. 529–539. Serials Publications (2009) 15. Hulsurkar, S., Biswal, M.P., Sinha, S.B.: Fuzzy Programming Approach to Multi-objective Stochastic Linear Programming Problems. Fuzzy Sets and Systems 88, 173–181 (1997) 16. Pal, B.B., Moitra, B.N., Maulik, U.: Goal Programming Procedure for Fuzzy Multiobjective Linear Fractional Programming Problem. Fuzzy Sets and Systems 139, 395–405 (2003) 17. District Statistical Hand Book, Nadia, Department of Bureau of Applied Economics and Statistics, Govt. of W. B., India (2006) 18. Action Plan for the year 2004–2005 and 2005–2006, Office of the Principal Agricultural Officer, Nadia, Govt. of W. B., India 19. Basak, R.K.: Soil testing and fertilizer recommendation. Kalyani Publishers, New Delhi (2000) 20. Department of Agri-irrigation, Office of the Executive Engineer, Krishnanagar, Nadia, Govt. of W. B., India (2009)

A Fuzzy Goal Programming Method for Solving Chance Constrained Programming with Fuzzy Parameters Animesh Biswas* and Nilkanta Modak Department of Mathematics, University of Kalyani Kalyani – 741235, India [email protected], [email protected]

Abstract. This paper develops a fuzzy goal programming methodology for solving chance constrained programming problem involving fuzzy numbers and fuzzy random variables which follow standard normal distribution. In the model formulation process, the problem is converted into an equivalent fuzzy programming problem by applying chance constrained programming technique. Then the problem is divided into equivalent sub problems by considering the tolerance limit of the fuzzy numbers relating to the system constraint having different forms of membership functions in the decision making context. Afterwards the objective and the constraints resulting from chance constraints are converted into fuzzy goals by assigning some imprecise aspiration levels. In the decision process, a fuzzy goal programming methodology is introduced to find the most satisfactory solution in the decision making environment. To demonstrate the potentiality of the proposed approach, an illustrative example, studied previously, is solved and is compared with the existing methodologies. Keywords: Chance constrained programming, normal distribution, goal programming, membership function, fuzzy programming, fuzzy goal programming.

1 Introduction As a special field of mathematical programming, stochastic programming deals with the situation where some or all of the parameters described by stochastic variables rather than deterministic quantities. In 1959, Charnes and Cooper [1] first introduced chance constrained programming (CCP) technique for solving stochastic programming problem. During 1960s to 1980s, CCP and its different aspects was investigated by Kataoka [2], Miller and Wagner [3], Panne and Pop [4], Pre/kopa [5] and other researchers. Stanch-Minasian & Wefs [6] and Vlerk [7] discussed research bibliographical surveys in the field of stochastic programming. The basic idea of all stochastic programming models is to convert the probabilistic nature of the problem into an equivalent deterministic situation according to the specified confidence levels and the nature of the distribution followed by the random variables. In real life decision making environment, in spite of having a vast decision making experience, the decision makers (DMs) can not always articulate the goals precisely *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 187–196, 2011. © Springer-Verlag Berlin Heidelberg 2011

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due to not only with the stochastic or uncertain behavior of the problem, but also the imprecision involved with the goals in the model formulation process of the problem. As a result the methodologies developed only for solving stochastic programming problem can not perform well and also fail to provide satisfactory answers. To overcome such a situation, Ludhandjula [8] introduced a fuzzy stochastic programming methodology based on fuzzy set theory [9]. Chakraborty [10] studied CCP in fuzzy environment by considering aspiration level and utility function for objective function and fuzzy probability involved with the system constraints. Nanda et al. [11] presented a methodology for solving chance constrained fuzzy linear programming. A fuzzy weighted additive approach for stochastic fuzzy programming (FP) has been discussed by Iskander [12]. The methodologies developed so far are based on only the fuzzy inequalities or fuzzy goals involved with the problem. But in real world decision making context it is found that the fuzziness is involved inherently with the DMs’ ambiguous understanding of the nature of parameters associated with the problem. As a result the coefficients involved with the problem are very often fuzzily described. Also the developed approaches suffer from the fact of computational difficulty with re-evaluation of the problem repetitively for adjustment of the satisfaction of the DMs. In the present paper, a fuzzy goal programming (FGP) technique is developed to solve CCP problems having random fuzzy coefficients with the system constraints. The formulation process is presented in the next section.

2 Fuzzy CCP (FCCP) Model Formulation An FCCP problem can be presented as , ,…, ∑

Find so as to Max subject to, ∑ ∑

; ;

1, 2, … , 1, 2, … , ;

0;

1, 2, … ,

(1)

1, 2, … , ; 1, 2, … , denote the normally distributed independent where fuzzy random variables; 1, 2, … , represent right sided fuzzy numbers; , 1, 2, … , ; 1, 2, … , represent fuzzy numbers with triangular membership function and , (for 1, 2, … , ; 1, 2, … , represent real numbers. and var be the mean and variance of the normally distributed Let for 1, 2, … , ; 1, 2, … , . fuzzy random variable Now by applying CCP methodology the above problem (1) is converted to the equivalent FP problem as ∑

Max ∑

subject to, ∑

∑

Φ ;

1,

2, … , ;

; 0;

1, 2, … , ; 1, 2, … , . (2)

A Fuzzy Goal Programming Method for Solving CCP with Fuzzy Parameters

189

Here Φ represents the cumulative density function of the standard normal fuzzy random variable.

3 Formulation of FP Model To formulate the FP model of the problem, the problem is divided into two sub prob1, 2, … , of (1). lems by considering the right sided fuzzy goal values Let be the tolerance limit of the fuzzy goal values 1, 2, … , . A right sided fuzzy number can be represented by the following figure: 1

0

bi + δ i

bi

Fig. 1. Right sided fuzzy number

Now considering the aspiration level of the fuzzy goal values of the system constraints relating to the chance constraints in (1), problem (2) can be written as ∑

Max ∑

subject to, ∑ Max

and

;

1,

2, … , ;

;

Φ 1,

∑ 2, … , ;

;

1, 2, … , 1, 2, … , .

0;

(3)

∑ ∑

subject to,

/

∑

Φ

∑

/

; 1, 2, … , 1, 2, … , . (4)

0;

1, 2, … , ; 1, 2, … , are normally distributed fuzzy random Again, since variables and the decision variables , , … . , are unknowns, some fuzzy random ∑ variables, 1, 2, … , , can be introduced as ; 1, 2, … , , which is normally distributed with the respective mean and variance given by ∑

1, 2, … ,

;

∑

and

;

1, 2, … ,

.

Applying mean and variance of the fuzzy random variable 1, 2, … , , the problem (3) and (4) are converted into the following respective forms (5) and (6) as: Max

∑ Φ

Subject to ∑ and Max Subject to ∑

; ;

1,

1, 2, … , 2, … , ;

0;

1, 2, … , .

0;

1, 2, … , . (6)

(5)

∑ Φ ;

; 1,

1, 2, … , 2, … , ;

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, ,

A triangular fuzzy number can be presented as the following membership function 0

and is expressed by

or

where and denote respectively the left and right tolerance values from the real number a. Now the triangular fuzzy mean and variance of the newly introduced fuzzy random 1, 2, … , , is considered as , , ; variable , , , respectively. Also the coefficients, 1, 2, … , ; 1, 2, … , and requirement vectors 1, 2, … , of the fuzzily described system constraints are described by the triangular fuzzy numbers as: , , ; , , for 1, 2, … , ; 1, 2, … , ; 1, 2, … , . Then the problem (5) and (6) can be decomposed into the following sub problems. 3.1 Lower Decomposed Sub Problem (LDSP) The lower decomposed sub problems can be found as Max

∑ Φ

subject to, ∑ and Max subject to, ∑ Let and and (8).

; ;

1, 2, … , 2, … , ;

0;

1, 2, … , .

(7)

1, 2, … , 1, 2, … , ;

0;

1, 2, … , .

(8)

1,

∑ Φ

; ;

be the respective objective values obtained by solving the models (7)

3.2 Upper Decomposed Sub Problem (UDSP) Again, the upper decomposed sub problems can be found as Max

∑ Φ

subject to, ∑ and Max subject to, ∑

;

; 1,

1, 2, … , 2, … , ;

0;

1, 2, … , . (9)

;

1, 2, … , 2, … , ;

0;

1, 2, … , . (10)

∑ Φ ;

1,

A Fuzzy Goal Programming Method for Solving CCP with Fuzzy Parameters

191

Here, let and be the respective objective values obtained by solving (9) and (10). Then the membership functions can be derived as explained in the next section.

4 Construction of Membership Functions Regarding the satisfying region of the two sub problems in 3.1 it is easy to verify [13] that the following relations hold for the system constraints involved with the LDSP as ; and for

for ;

.

Thus the relation regarding the objective values can be expressed as . is absolutely Now, it is to be understood that for the LDSP the value higher than acceptable to the DMs and the value lower than will be totally ignored by the DMs. . Hence the fuzzy objective goal for the LDSP can be expressed as: Thus the membership function for the LDSP can be written as 0

if if

1

.

(11)

if

Similarly, proceeding through the same arguments, regarding the UDSP in 3.2, the membership function for the UDSP can be described as 0

if if

1

.

(12)

if

Further it can be proved [13] that corresponding to the four decomposed system constraints resulting from the chance constraints, two membership functions can be drawn on the basis of the fuzziness involved with the parameters. This can be expressed in the following form for 1, 2, … , as 1

if Φ

if

0

Φ if

1

Φ if

Φ 0

Φ

Φ

if

Φ if

Φ

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Now the aim of the decision maker is to achieve highest degree to the extent possible of each of the membership functions define in the decision making context.

5 FGP Model Formulation Considering the highest degree as the aspiration level of each of the defined membership functions, the flexible membership goals can be presented as Min D = 1;

so as to

1; 1; 1 ,

, 0;

,

,

subject to ∑ , , , 1, 2, … , ;

0; with 1,

;∑ . 2, … , ;

; .

. 1, 2, … ,

. ;

0; (13)

where D represent the fuzzy goal achievement function consisting of the weighted under deviational variables, and where 0 , 1, 2, … , are the numerical weights representing the relative importance of achieving the aspired levels of the goals. To asses the relative importance of the fuzzy goals properly, the values ( 1, 2, … , ) are determined as [14]: ;

;

.

The minsum goal programming [15] technique is used to solve the problem (13).

6 An Illustrative Example To explore the potential use of the proposed approach, the modified version of the problem studied by Nanda et al. [11] is considered. The fuzzy CCP problem is presented as Find so as to Max subject to

, 2

3 ;

,

0.6 ; 0.

(14)

where , are normally distributed independent fuzzy random variables with respective mean and variance represented by the following fuzzy numbers: E var

9.95, 10, 10.05 , var 3.9, 4, 4.1 .

5.9, 6, 6.1 ; E

4.95, 5, 5.05 ,

A Fuzzy Goal Programming Method for Solving CCP with Fuzzy Parameters

193

represents a right sided fuzzy number with the following membership function: 1

if

50

if 50

μ

60.

0 if 60 , , are triangular fuzzy numbers associated with the fuzzily described system constraints having the form: 2.95, 3, 3.05 ,

1.95, 2, 2.05 ,

29.5, 30, 30.5 .

By applying general CCP technique, problem (14) can be described as 2

Max

3 Φ 0

subject to ;

,

0.6 (15)

Now considering the right sided fuzzy goal value straint, the above problem can be divided as Max subject to 50

2

3 ;

and Max subject to 60

associated with the chance con-

2

Φ 0

0.6

,

0.6

,

Φ 0

(16)

3 ;

(17)

Again, on the basis of the fuzzily defined mean and variance of the fuzzy random variable and also considering the fuzzy number associated with the other system constraints, the LDSP can be formulated as Max 2 3 4.95 0.253 5.9 subject to 50 9.95 2.95 1.95 29.5; , 0 and

2

Max

3.9

; (18)

3

subject to 50 10.05 3.05 2.05

5.05 0.253 6.1 30.5; , 0

4.1

; (19) 27.52,

On solving (18) and (19) the solutions are achieved as respectively. Similarly, the UDSP can be written as Max

2

26.97,

3

subject to 60 9.95 4.95 0.253 5.9 1.95 29.5; , 0 2.95

3.9

; (20)

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2

and

Max

subject to

60 10.05 3.05 2.05

3 5.05 0.253 6.1 30.5; , 0

4.1

; (21)

On solving (20) and (21) the achieved solutions are 33.03, 32.36, respectively. Now the fuzzy goals of the LDSP and UDSP are obtained as 27.52 and 33.03, respectively. Also the respective lower tolerance values of Z are considered as 26.97 and 32.36. Now the membership functions can be constructed by following the procedure in (11), (12). Again, for the decomposed system constraints relating to the chance constraint the following membership function can be drawn 1,

if 9.95

60

9.95

4.95 4.95

Φ Φ

0.6 5.9 0.6 5.9

3.9 3.9

50 ,

9.95

4.95

10 Φ

0,

if 9.95

4.95

Φ

0.6 5.9

3.9

60

1,

if 10.05

5.05

Φ

0.6 6.1

4.1

50

60

10.05

if 50

if 50 0,

5.05

Φ

10.05

5.05

10 Φ

if 10.05

5.05

Φ

0.6 5.9

3.9

60

0.6 6.1

4.1

,

0.6 6.1

4.1

60

0.6 6.1

4.1

60

Hence the FGP model for the defined membership functions appear as Find so as to Min D = subject to

; ; with

The software LINGO (6.0) is used to solve the problem.

; (22)

A Fuzzy Goal Programming Method for Solving CCP with Fuzzy Parameters

195

The optimal solution of the problem (22) is obtained as 2.08 and 9.62. The achieved objective value of the given problem is found as 33.02. The solution obtained by using the methodology developed by Nanda, et al. [11] is 0, 9.67 and the corresponding objective value is 29.01. This solution reflects that the decision obtained by proceeding the proposed approach is most satisfactory in the fuzzy chance constrained decision making arena.

7 Conclusions In this paper, an FGP model is developed for solving fuzzy CCP problems efficiently, where achievement of each decomposed fuzzy goals to the highest membership value to the extent possible is taken into account. The proposed approach does not involve any computational load with evaluation of the problem in a repetitive manner unlike the previous FP approaches. Also, the proposed model is flexible enough to introduce the tolerance limits to the fuzzy goals initially to arriving at the satisfactory decisions on the basis of the needs and desires of the DM. Also this paper captures the fuzziness involved not only with the fuzzy goals, but also the fuzziness involved inherently with the parameters to define the resource constraints. An extension of the proposed approach to fully fuzzified CCP may be one of the current research problems. In future studies the proposed approach may be extended to multiobjective fuzzy CCP and also in a hierarchical decision making arena involving fuzzy CCP problems. Acknowledgements. The authors are thankful to the anonymous referees for their valuable comments and suggestions in improving the quality of the paper.

References 1. Charnes, A., Cooper, W.W.: Chance Constrained Programming. Mgmt. Sci. 6, 73–79 (1959) 2. Kataoka, S.: A Stochastic Programming Model. Econometric 31, 181–196 (1963) 3. Miller, L.B., Wagner, H.: Chance-Constrained Programming with Joint Constraints. Ops. Res. 13, 930–945 (1965) 4. Panne, V.D., Popp, W.: Minimum Cost Cattle Feed under Probabilistic Protein Constraints. Mgmt. Sci. 9, 405–430 (1963) 5. Prekopa, A.: Contribution to the Theory of Stochastic Programming. Mgmt. Sci. 4, 202– 221 (1973) 6. Stancu-Minasian, I.M., Wets, M.J.: A Research Bibliography in Stochastic Programming. Ops. Res. 24, 1078–1119 (1976) 7. Maarten, V., Van Der, H.: Stochastic programming bibliography(2003), http://mally.oco.rug.nl/spbib.html/ 8. Luhandjula, M.K.: Linear Programming under Randomness and Fuzziness. Fuzzy Sets Sys. 10, 45–55 (1983) 9. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965) 10. Chakraborty, D.: Redefining Chance Constrained Programming in Fuzzy Environment. Fuzzy Sets Sys. 125, 327–333 (2002)

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11. Nanda, S., Panda, G., Dash, J.K.: Chance Constrained Programming with Fuzzy Inequality Constraints. Opsearch 45, 34–48 (2008) 12. Iskander, M.G.: A Fuzzy Weighted Additive Approach for Stochastic Fuzzy Goal Programming. Appl. Math. Compn. 154, 543–553 (2004) 13. Lu, J., Zhang, G., Ruan, D., Wu, F.: Multiobjective Group Decision Making. Imperial College Press, London (2007) 14. Biswas, A., Pal, B.B.: Application of Fuzzy Goal Programming Technique to Land Use Planning in Agricultural System. Omega 33, 391–398 (2005) 15. Ignizio, J.P.: Goal Programming and Extensions. Lexington D. C. Health. MA, USA (1976)

Fractals via Ishikawa Iteration Bhagwati Prasad and Kuldip Katiyar Department of Mathematics Jaypee Institute of Information Technology A-10, Sector-62, Noida, UP-201307 India [email protected], [email protected]

Abstract. Fractal geometry is an exciting area of interest with diverse applications in various disciplines of engineering and applied sciences. There is a plethora of papers on its versatility in the literature. The basic aim of this paper is to study the pattern of attractors of the iterated function systems (IFS) through Ishikawa iterative scheme. Several recent results are also obtained as special cases. The results obtained are illustrated through figures generated by Matlab programs. Keywords: Fractals, Attractor, Iterated function system, Ishikawa iterate.

1 Introduction The word fractal was coined by Mandelbrot [12] in 1975. It is however, remarked that many of the fractals and their descriptions go back to the mathematician of past like George Cantor (1872), Helge von Koch (1904), Waclaw Sierpinski (1916), Gaston Julia (1916), Felix Hausdorff (1919) and others [14]. Mandelbrot [12] demonstrated that these early mathematical fractals have many features in common with the shapes found in nature (also see, for instance, [2], [4], [8], [9], [14], [15], and several references thereof). Mandelbrot founded his insights in the idea of self-similarity, requiring that a true fractal “fracture” or break apart into smaller pieces that resemble the whole. Hutchinson [9] and Barnsley [4] initiated the way to define and construct fractals as compact invariant subsets of a complete metric space with respect to the union of contractions. The idea of iterated function systems (IFS) for finite number of Banach contractions presented by Hutchinson and Barnsley is exciting as it helps to define a fractal as a fixed point of an operator based on the contractions (see Kigami [11]). Fractal theory has emerged as a new frontier of science with significant roles in the study of applicable areas of sciences, engineering, medicine, business, textile industries and several other areas of human activity. For a detailed analysis in this respect, one may refer to Falconer [8], Peitgen et al [14]-[15] and several references thereof. The theory of IFS was developed by Hutchinson [9] in 1981 and popularized by Michael Barnsley [4]. The existence, convergence and uniqueness results are based on contraction properties of the IFS with respect to the Hausdorff metric for sets. Iterative schemes are used to generate fractals through IFS. A detailed discussion on various iterative schemes is given in Berinde [6]. Recently, Singh et al. [16,17] obtained fractals using Mann iterative scheme. In this paper, we study the fractal pattern through Ishikawa iterative scheme. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 197–203, 2011. © Springer-Verlag Berlin Heidelberg 2011

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2 Preliminaries In this section we present the basic definitions and concepts required for our results: Definition 2.1 [02]. Let (X, d) be a metric space. A transformation w: X → X is said to R iff d(w(x), w(y)) ≤ s d(x, y) for all x, y be Lipschitz with Lipschitz constant s X. A transformation w: X → X is called contractive iff it is Lipschitz with Lipschitz [0, 1). A Lipschitz constant s [0, 1) is also called a contraction factor. constant s

∈

∈

∈

∈

Definition 2.2 [01]. A hyperbolic iterated function system (IFS) consists of a complete metric space (X, d) together with a finite set of contraction mappings wi: X → X, with respective contractivity factors si for i = 1, 2, ..., n. This IFS is represented by {X; wi: i = 1, 2, ..., n} with contractivity factor s = max{si: i = 1, 2, ..., n}. Definition 2.3 [02]. Let (X, d) be a metric space and H(X) denote the nonempty compact subsets of X. Then the Hausdorff metric h in H(X) is defined as h(A, B) = max {d(A, B), d(B, A)} for all A, B

∈

∈

∈ H(X),

B): a A). where d(A, B) = max(min(d(a, b): b The following theorem of Kigami [11, p. 10] ensures the existence of a self similar set in a complete metric space: Theorem 2.4 [11]. Let (X, d) be a complete metric space and wi: X → X be a Banach contraction with respect to the metric d for i = 1, 2, …, n. Then there exists a unique nonempty compact subset A of X that satisfies A = w1(A) ∪ w2(A) ∪ . . . ∪ wn(A).

(2.1)

A is called the self similar set with respect to {w1, w2, . . . , wn}. We now present a lemma as given in Barsnley [2] which guarantees a contraction map in {H(X), h} out of a contraction map on (X, d). Lemma 2.5 [02]. Let w: X → X be a contraction on a metric space (X, d) with contractivity factor s. Then w: H(X) → H(X) defined by

w( B ) = {w( x) : x ∈ B} ∀B ∈ H ( X ) is a contraction on {H ( X ), h} with contractivity factor s. The following theorem ensuring the existence of a unique fixed point (also called an attractor) of the IFS is fundamental to our results. (see [1] - [2]). Theorem 2.6 [02]. Let {X; wi, i = 1, 2, ..., n} be a hyperbolic iterated function system with contractivity factor s. Then the transformation W: H(X) → H(X) defined by n

∈

W ( B) = ∪ wi ( B) i =1

H(X) is a contraction mapping on the complete metric space (H(X), h) for all B with contractivity factor s. That is, h(W(B), W(C)) ≤ s h(B, C) for all B, C H(X). Its

∈

Fractals via Ishikawa Iteration

unique fixed point, A

∈

n

H(X) obeys

A = limi → ∞ W ( B ) for any B i

199

A = W ( A) = ∪ wi ( A) and is given by

∈ H(X).

i =1

We are now in a position to present our main results.

3 Experiments and Results We first present an overview of the popular iterative procedures. Let X be a non empty set and w: X → X. Then for any point u0 in X, the iterative scheme un+1 = w(un) n = 0, 1, 2, … is called Picard iterate and a Picard orbit is defined as O(w, u0) := {un: un = w(un-1), n = 0, 1, 2, …}

(3.1)

If the sequence {un} is constructed in the following manner un+1 = αn w(un) + (1- αn) un

(3.2)

for n = 0, 1, 2, 3, …, where 0 < αn ≤ 1. This procedure is called Mann iteration procedure [13]. Further, the sequence {un} may also be constructed as follows. un+1 = αn w(vn) + (1- αn) un, vn = βn w(un) + (1- βn) un

(3.3)

for n = 0, 1, 2, 3, …, where 0 < αn ≤ 1, 0 ≤ βn ≤ 1. Then sequence {un} constructed above is called Ishikawa iteration [10]. We take u0=(x0, y0) as our initial choice and construct the sequence of iterates following scheme (3.3) with αn = α, βn = β and 0 < αn ≤ 1, 0 ≤ βn ≤ 1. Such sequence {(xn, yn)} is essentially studied by Ishikawa [10] in nonlinear analysis, called Ishikawa sequence of iterates (also see Berinde [6]). The Ishikawa iterates with β = 0 are the Mann iterates and for β = 0, α = 1 are Picard iterates. It is also noticed that the function iteration is one step feedback process while Mann iteration is two step feedback procedures, since it requires two inputs. Fractals have been generated in the literature using Picard and Mann iterates by a number of authors (see Barnsley [2], Singh [16] and some references thereof). We use (3.3) which is a three step feedback procedure. We plot the attractors of IFS {X; w1, w2} where X = [0, 1] × [0, 1]

⊂ R and

1 ⎛ x 3y 5 x 3y 3 ⎞ w1 ( x, y ) = ⎜ − + , + + ⎟; p1 = 2 ⎝ 2 8 16 2 8 16 ⎠ 1 x 3 y 11 ⎞ ⎛ x 3y 3 + ,− + + ⎟; p1 = w2 ( x, y ) = ⎜ + 2 ⎝ 2 8 16 2 8 16 ⎠

2

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Here p1 and p2 are the probabilities of using maps w1 and w2 respectively (see Barnsley [3]). For different choices of the parameters α and β, the fractal pattern (attractor) obtained is illustrated in the figure given below.

α = 1, β = 0

α = 1, β = 0.1

α = 1, β = 0.2

α = 1, β = 1

Fig. 1. Attractor of given IFS using Ishikawa iterates of sequence for α = 1 and β = 0, 0.1, 0.2, 1

α = 0.8, β = 0

α = 0.8, β = 0.4

Α = 0.8, β = 0.7

α = 0.8, β = 1

Fig. 2. Attractor of given IFS using Ishikawa iterates of sequence for α = 0.8 and β = 0, 0.4, 0.7, 1

Fractals via Ishikawa Iteration

α = 0.6, β = 0

α = 0.6, β = 0.7

α = 0.6, β = 0.9

α = 0.6, β = 1

201

Fig. 3. Attractor of given IFS using Ishikawa iterates of sequence for α = 0.6 and β = 0, 0.7, 0.9, 1

α = 0.5, β = 0

α =0.5, β = 1

Fig. 4. Attractor of given IFS using Ishikawa iterates of sequence for α = 0.5 and β = 0, 1

α = 0.2, β = 0

α = 0.2, β = 1

Fig. 5. Attractor of given IFS using Ishikawa iterates of sequence for α = 0.2 and β = 0, 1

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α = 0.1, β = 0

α = 0.1, β = 1

Fig. 6. Attractor of given IFS using Ishikawa iterates of sequence for α = 0.1 and β = 0, 1

4 Conclusion While observing figures given in section 3, it is noticed that there are remarkable changes found in the fractal pattern corresponding to the values of parameters α and β (see figure 1-6.). When we take αn = α and βn = β = 0 following the scheme (3.3) the fractal pattern for α =1, α = 0.8, α =0.5, α =0.1 while β = 0 is same as that is obtained by Singh et al [16]. At any stage of iterates (more than 1000) the emerging patterns of the attractor for values of α and β are analyzed in the following manner: 1.

2.

For α = 1 and β ≈ 0.2, we get a disconnected attractor of iterated function system. When α decreases from 1 to 0.6 the disconnectivity in attractor of iterated function system is achieved at increasing values of β from 0.2 to 0.8, i.e., when α = 1, attractor of IFS disconnected at β ≈ 0.2 and when α decreases to 0.6, attractor remains overlapping up to β ≈ 0.7 and it becomes disconnected at β ≈ 0.8. For 0.2 ≤ α ≤ 0.5 and changing 0 ≤ β ≤ 1, attractors of the IFS change but remain overlapping.

For α = 0.1, 0 ≤ β ≤ 1 the dusty patterns of attractors of the IFS seem to remain unchanged.

References 1. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Springer, Heidelberg (2009) 2. Barnsley, M.F.: Fractals Everywhere, 2nd edn. Revised with the assistance of and a foreword by Hawley Rising, III. Academic Press Professional, Boston MA (1993) 3. Barnsley, M.F.: Superfractals. Cambridge University Press, New York (2006) 4. Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. Roy. Soc. Lond. A399, 243–275 (1985) 5. Barnsley, M.F., Hutchinson, J.E., Stenflo, O.: Fractals with partial self similarity. Adv. Math. 218(6), 2051–2088 (2008) 6. Berinde, V.: Iterative Approximation of Fixed Points. Springer, Heidelberg (2007) 7. Edgar Gerald, A.: First course in chaotic dynamical systems. Benjamin/Cumming publishing Co. Inc., Menlo Park (1986) 8. Falconer, K.: Techniques in fractal geometry. Wiley, England (1997)

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9. Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981) 10. Ishikawa, S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44(1), 147– 150 (1974) 11. Kigami, J.: Analysis on Fractals. Cambridge University Press, England (2001) 12. Mandelbrot, B.B.: The Fractal Geometry of Nature, Updated and Augmented, International Business Machines, Thomas J. Watson Research Centerw. H. Freeman And Company, New York (1977) 13. Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 44, 506–510 (1953) 14. Peitgen, H.O., Jürgens, H., Saupe, D., Maletsky, E., Perciante, T., Yunker, L.: Fractals for The Class room: Strategic activities, vol. 1. Springer, Heidelberg (1991) 15. Peitgen, H.O., Jürgens, H., Saupe, D.: Chaos and Fractals: New Frontiers of Science. Springer, Heidelberg (1992) 16. Singh, S.L., Jain, S., Mishra, S.N.: A new approach to superfractals. Chaos, Solitons and Fractals 42, 3110–3120 (2009) 17. Singh, S.L., Prasad, B., Kumar, A.: Fractals via iterated functions and multifunctions. Chaos, Solitons and Fractals 39, 1224–1231 (2009)

Numerical Solution of Linear and Non-linear Singular Systems Using Single Term Haar Wavelet Series Method K. Prabakaran1 and S. Sekar2 1

Research and Development Centre, Bharathiar University, Coimbatore – 641 046, Tamil Nadu, India [email protected] 2 Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, Mayiladuthurai – 609 305, Tamil Nadu, India

Abstract. In this paper, a new method known as Single Term Haar Wavelet Series (STHWS) has been presented to obtain the solution for linear and nonlinear singular systems. This new approach provides a better effectiveness to find discrete solutions of linear and non-linear singular systems for any length of time t. This is a direct method and can be easily implemented in a digital computer. Keywords: linear singular systems, non-linear singular systems, Haar wavelet, single term Haar wavelet series, operational matrix.

1 Introduction The Haar wavelet was initiated and independently developed by Haar. Recently, the Haar theory has been innovated and applied to various fields in sciences and engineering applications. In particular, Haar Wavelets have been applied extensively for image processing, signal processing in communications and proved to be a useful mathematical tool. Palanisamy and Balachandran [1] introduced a new method for the analysis of linear singular systems through single-term Walsh series (STWS). Balachandran and Murugesan [2] applied STWS method for stable, unstable, singular and stiff systems. They have observed that the method is suitable for singular systems. Hsiao [3] examined the state analysis of liner time delayed systems via Haar wavelets. He has shown that Single Term Haar Wavelets method provides better accuracy on solving linear systems. Hsiao and Wang [4] studied state analysis of time varying singular non linear and bilinear systems via Haar wavelets. They have proved that Single Term Haar Wavelets method is convenient and accurate when it is applied to solve time varying singular, nonlinear and bilinear systems. Murugesan et al. [5] made an analysis on second order multivariable linear system using STWS technique and Runge Kutta method. On comparing these two methods, they have observed that Runge Kutta method is better than STWS technique. Murugesan et al. [6] presented a study of second order time invariant and time varying state space systems using the STWS technique. They have concluded that P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 204–212, 2011. © Springer-Verlag Berlin Heidelberg 2011

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205

−9

STWS method is highly stable and gives the accuracy up to 10 with the true solutions. Sepehrian and Razzaghi [7] proposed a method for finding the solution of time varying singular bilinear systems using STWS. They have demonstrated the validity and applicability of the technique by an example. Lepik [8] discussed about the numerical solution of ODE and PDE by using Haar wavelet techniques. He has applied the new technique called the segmentation method and checked the applicability and the efficiency of the method. Bujurke et al. [9] investigated the numerical solutions of stiff systems from nonlinear dynamics using single term Haar wavelet series. It has been proved that the efficiency and the desired accuracy of solution can be achieved using STHWS. The main aim of this paper is to determine the effectiveness of the method STHWS and to show the accuracy by applying it to some linear and non-linear singular differential equations. The present paper has been organized as follows: In section 2, we describe the basic properties of Haar and the procedures of STHWS technique. In section 3, we discuss the application of STHWS technique to linear and non-linear singular systems. Finally, in section 4, we present some illustrative examples to demonstrate the accuracy and brief comparison with exact solutions.

2 STHWS Technique 2.1 Single Term Haar Wavelet The orthogonal set of Haar wavelets

hi (t ) is a group of square waves with magnitude

of ± 1 in some intervals and zeros elsewhere. In general,

(

)

hn (t ) = h1 2 j t − k , where n = 2 j + k ,

j ≥ 0, 0 ≤ k < 2 j , n, j , k ∈ Z 1 ⎧ ⎪⎪1, 0 ≤ t < 2 h1 (t ) = ⎨ ⎪− 1 , 1 ≤ t < 1 ⎪⎩ 2

(1)

(2)

Namely, each Haar wavelet contains one and just one square wave, and is zero elsewhere .Just these zeros make Haar wavelets to be local and very useful in solving linear and non-linear singular systems. 2.2 Single Term Haar Wavelet Series Any function y (t), which is square integrable in the interval [0, 1) can be expanded in a Haar series with an infinite number of terms

206

K. Prabakaran and S. Sekar ∞

y (t ) = ∑ ci hi (t ), where i = 2 j + k ,

(3)

i =0

j ≥ 0, 0 ≤ k < 2 j , t ∈ [0,1) where the Haar coefficients 1

ci = 2 j ∫ y (t )hi (t )dt

(4)

0

are determined such that the following integral square error

ε

is minimized

2

m −1 ⎡ ⎤ ε = ∫ ⎢ y (t ) − ∑ ci hi (t )⎥ dt , where m = 2 j , j ∈ {0} ∪ N i =0 ⎦ 0 ⎣ 1

(5)

Furthermore,

⎧2 − j , i = l = 2 j + k , j ≥ 0,0 ≤ k < 2 j ⎫ −j = = h ( t ) h ( t ) dt 2 δ ⎨ ⎬ il ∫0 i 1 ⎩0 , i ≠ l ⎭ 1

(6)

Usually, the series expansion Eq. (2) contains an infinite number of terms for a smooth y (t). If y (t) is a piecewise constant or may be approximated as a piecewise constant, then the sum in Eq. (2) will be terminated after m terms, that is m =1

t ∈ [0,1)

y (t ) ≈ ∑ ci hi (t ) = c(m ) h(m ) (t ), T

i =0

c(m ) (t ) = [c0 c1 ...c m−1 ] , T

(7)

hm (t ) = [h0 (t ), h0 (t ),........, hm−1 (t )]

T

where ‘T’ indicates transposition, the subscript m in the parentheses denotes their T

dimensions, C (m ) h( m ) (t ) denotes the truncated sum. Since the differentiation of Haar wavelets results in generalized functions, which in any case should be avoided, the integration of Haar wavelets are preferred. Integration of Haar wavelets should be expandable in Haar series t

∞

0

i =0

∫ hm (τ )dτ = ∑ Ci hi (t )

(8)

If we truncate to m = 2 terms and use the above vector notation, then integration is performed by matrix vector multiplication and expandable formula into Haar series with Haar coefficient matrix defined by [4]. n

1

∫ h( ) (τ )dτ ≈ M ( m

0

m× m )

h(m ) (t ),

t ∈ [0,1) (9)

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207

h(m ) (t )h(Tm ) (t ) ≈ M (m × m ) (t ) , and M (1×1) (t ) = h0 (t ) and satisfying

M (mxm ) (t )c(m ) = C (mxm ) h( m ) (t ) and C (1x1) = c0

.

2.3 Operational Matrix of the Haar Wavelet

The first four haar function vectors can be expressed as follows

h⎛ 1 ⎞ = [1,1,1, o ] ,

h⎛ 3 ⎞ = [1,1,−1, o ]

T

T

⎜ ⎟ ⎝8⎠

⎜ ⎟ ⎝8⎠

h⎛ 5 ⎞ = [1,−1, o,1] , T

⎜ ⎟ ⎝8⎠

h⎛ 7 ⎞ = [1,−1, o,−1]

T

⎜ ⎟ ⎝8⎠

This can be written in matrix form as 1 1⎤ ⎡1 1 ⎢1 1 − 1 − 1⎥ ⎥ H4 = ⎢ ⎢1 − 1 0 0⎥ ⎢ ⎥ ⎣0 0 1 − 1⎦

⎡h(m ) (1 / 2m), h(m ) (3 / 2m), h(m ) (3 / 2m),⎤ ⎥ ⎣⎢..........................h(m ) ((2m − 1) / 2m). ⎦⎥

In general, we have H ( m ) = ⎢

⎡1 1 ⎤ H (1) = [1], H (2 ) = ⎢ ⎥ ⎣1 − 1⎦ where P( m ) is the m x m operational matrix and is given by

− H ( m / 2) ⎤ 0 ⎥⎦

1 ⎡2mP P(m ) = ⎢ −1 ( m / 2 ) 2 ⎣ H (m / 2 ) If P(1 x1) =

⎡2 − 1⎤ 1 , so P( 2 ) = ⎢ ⎥ , and P( 4 ) 2 ⎣1 0 ⎦

⎡8 − 4 − 2 − 2 ⎤ ⎢ ⎥ 1 4 0 −2 2 ⎥ . = ⎢ 0 0⎥ 16 ⎢1 1 ⎢ ⎥ 0⎦ ⎣1 − 1 0

3 Analysis of STHWS Technique 3.1 Solution of Linear Singular Systems via STHWS

Consider the linear singular system •

K (t ) x(t ) = A(t ) x(t ) + B(t )u (t )

(10)

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K. Prabakaran and S. Sekar

with the initial condition

x(0) = x 0 , where the n × n matrix K (t ) is singular for

all t, A(t) is an n × n matrix, B(t) is an n × r matrix, x(t) is an n state vector and u(t) is an r input vector. With the STHWS approach, the given function is expanded into Single term Haar

⎡ 1⎞ ⎟ by ⎣ m⎠

wavelet series in the normalized interval [0, 1), which corresponds to t ∈ ⎢0, defining mt

= τ , m being any integer.

The Eq. (10) becomes, in the normalized interval •

K (τ ) x (τ ) = where τ

B (τ ) A(τ ) x(τ ) + u (τ ) , m m

x ( 0) = x 0

(11)

∈ [0,1) .

•

Let x(τ ), x(τ ), and u (τ ) be expanded by STHWS in the nth interval as •

x(τ ) = C n h0 (τ )

x(τ ) = Bn h0 (τ )

u (τ ) = H n h0 (τ )

(12)

The following set of recursive relations has been obtained for the Linear singular systems with P = 1/2. −1

M M ⎛ ⎞ C n h0 (τ ) = ⎜ M 0 − 1 − 22 − ...... ⎟ Gn 2m 4 m ⎝ ⎠ 1 C n + x(n − 1) 2 x(n) = C n + x(n − 1)

(13)

Bn =

where G n =

(14)

A B x ( n − 1) + H n and n = 1, 2, 3,…, the interval number, m m

substituting Eq.(12) and (14) into Eq. (11). We obtain linear algebraic equation and they may be solved to obtain x(n ) .The Single Term Haar wavelet series with piecewise constant orthogonal functions is an extension of single term Haar wavelet series algorithm Eq.(10), that avoid the inverse of the big matrix induced by the kronecker product. This approach is applicable for any transform with piecewise constant basis. Therefore, in this paper the above algorithm is used in solving the linear singular problems to avail the advantages of its fast to any length of time ‘t’. The value m can be selected as large enough to increase the accuracy of the results.

Numerical Solution of Linear and Non-linear Singular Systems

209

3.2 Solution of Non-linear Singular Systems via STHWS

Consider the non-linear singular system •

K (t ) x(t ) = A(t ) x(t ) + f ( x(t )) with the initial condition x(0)

(15)

= x 0 , where the n × n matrix K (t ) is singular for all

t, A (t) is an n × n matrix, x(t) is an n state vector. With the STHWS approach, the given function is expanded into single term Haar wavelet series in the normalized

⎡ 1⎞ ⎟ by defining mt = τ , m being any ⎣ m⎠

interval [0, 1), which corresponds to t ∈ ⎢0, integer.

The Eq. (15) becomes, in the normalized interval •

K (τ ) x (τ ) =

A(τ ) f ( x(t )) x (τ ) + , m m

x (0) = x0

(16)

[ )

where τ ∈ 0,1 . With similar ideas as in the case of linear systems, we obtain the same set of recursive relations given by Eq. (13) for non-linear singular systems with P = where G n =

1 , 2

A x ( n − 1) + f (C n , x ( n − 1)) and n = 1, 2, 3…, the interval number. m

4 Numerical Examples In this section, we will illustrate two numerical examples to show the power of single term Haar wavelet series method. Example 1. Let us consider the following linear singular system. •

K (t ) x(t ) = A(t ) x(t ) + B(t )u (t ) , x(0) = x 0 −1 − 2 ⎤ ⎡ 1 0 − 2⎤ ⎡ 0 ⎢ ⎥ ⎢ where K (t ) = − 1 0 2 ⎥ , A(t ) = ⎢ 27 22 17 ⎥⎥ ⎢ ⎢⎣ 2 3 2 ⎥⎦ ⎢⎣− 18 − 14 − 10⎥⎦ 1⎤ ⎡1 2 ⎡ 0.4123 ⎤ ⎡1 ⎤ ⎢ ⎥ ⎢ ⎥ B(t ) = ⎢2 − 1 − 3⎥ , x 0 = ⎢ 0.0769 ⎥ , u = ⎢⎢0⎥⎥ ⎢⎣0 1 ⎢⎣− 1.2500⎥⎦ ⎢⎣0⎥⎦ 1 ⎥⎦

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The exact solution is

14 7 ( 2t / 3 ) 5 e −t + x 2 (t ) = e (2t / 3) + 2t − 1 , , 13 52 18 7 (2 t / 3 ) 1 x 3 (t ) = − e −t + 4 2 x1 (t ) =

(17)

The discrete solution is calculated using the recursive formula given by Eq. (14) and the exact solution is calculated using Eq. (17). The results of exact and discrete solutions are shown in table 1. Table 1.

Time

x1 (t )

Exact Solution

x 2 (t )

x1 (t )

x3 (t )

STHWS method

x 2 (t )

x3 (t )

0

0.4123931

0.0769230

-1.2500000

0.4123931

0.0769230

-1.2500000

0.5

-0.0343513

1.5029672

-2.4423217

-0.0343513

1.5029672

-2.4423217

1.0

-0.4600272

3.0975597

-3.9085345

-0.4600272

3.0975597

-3.9085345

1.5

-0.8562996

4.9273804

-5.7569932

-0.8562997

4.9273804

-5.7569932

2.0

-1.2115361

7.0854885

-8.1389188

-1.2115361

7.0854885

-8.1389188

2.5

-1.5095024

9.7017585

-11.2653575

-1.5095024

9.7017585

-11.2653576

3.0

-1.7275415

12.9574450

-15.4308481

-1.7275416

12.9574450

-15.4308482

3.5

-1.8340335

17.1055091

-21.0464523

-1.8340336

17.1055092

-21.0464524

4.0

-1.7848489

22.4989865

-28.6858531

-1.7848489

22.4989866

-28.6858532

4.5

-1.5183999

29.6305782

-39.1496896

-1.5183999

29.6305782

-39.1496896

5

-0.9487342

39.1879037

-53.5553435

-0.9487342

39.1879037

-53.5553436

Example 2. Let us consider the following non-linear singular system. •

K (t ) x(t ) = A(t ) x(t ) + f ( x(t )) , x(0) = x 0 ∗ ⎡0 1⎤ ⎡ x1 ⎤ ⎡− 1 0⎤ ⎡ x1 ⎤ ⎡ 0 ⎤ ⎢0 0 ⎥ ⎢ • ⎥ = ⎢ 0 2 ⎥ ⎢ x ⎥ + ⎢ − x 2 ⎥ , ⎣ ⎦ ⎢⎣ x 2 ⎥⎦ ⎣ ⎦ ⎦⎣ 2 ⎦ ⎣

The exact Solution is x1 (t ) = − t , x 2 (t ) = t / 2. 2

with

⎡0 ⎤ x(0) = ⎢ ⎥ ⎣0 ⎦ (18)

The discrete solution is calculated using the recursive formula given by Eq. (14) and the exact solution is calculated using Eq. (18). The results of exact and discrete solutions are shown in table 2.

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Table 2.

Time 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0 2.25 2.50 2.75 3.00

Exact Solution

x1 (t )

0 -0.25000000 -0.50000000 -0.75000000 -1.00000000 -1.25000000 -1.50000000 -1.75000000 -2.00000000 -2.25000000 -2.50000000 -2.75000000 -3.00000000

x 2 (t ) 0 0.03125000 0.12500000 0.28125000 0.50000000 0.78125000 1.12500000 1.53125000 2.00000000 2.53125000 3.12500000 3.78125000 4.50000000

STHWS method

x1 (t )

x 2 (t )

0 -0.25000000 -0.50000000 -0.75000000 -1.00000000 -1.25000000 -1.50000000 -1.75000000 -2.00000000 -2.25000000 -2.50000000 -2.75000000 -3.00000000

0 0.03125000 0.12500000 0.28127000 0.50000000 0.78128000 1.12500000 1.53128000 2.00000000 2.53128000 3.12500000 3.78127000 4.50000000

5 Conclusions A simple and easy method is introduced in this paper to obtain discrete solutions of linear and non-linear singular systems using STHWS. The efficiency and the accuracy of the STHWS method have been illustrated by suitable examples. The solutions obtained are compared well with the exact solutions. It has been observed that the solutions by our method show good agreement with the exact solutions. The tables of results show a negligible error with a range of 10-7-10-8 between exact and discrete solutions. The present method is very convenient as it requires only simple computing systems, less computing time and less memory. The STHWS method is very simple and direct which provides the solutions for any length of time t. Acknowledgment. The authors gratefully acknowledge the Dr.K.Balachandran, Head of the Mathematics Department, Bharathiar University, Coimbatore 641 046, Tamil Nadu, India, for his kind help and encouragement.

References 1. Palanisamy, K.R., Balachandran, K.: Single term Walsh series approach to singular systems. Int. J. Control 46, 1931–1934 (1987) 2. Balachandran, K., Murugesan, K.: Analysis of different systems via single term Walsh Series method. Int. J. Comput. Math. 33, 171–179 (1990)

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3. Hsiao, C.H.: State analysis of liner time delayed systems via Haar wavelets. Math. Comput. Simul. 44, 457–470 (1997) 4. Hsiao, C.H., Wang, W.j.: State analysis of time varying singular nonlinear systems via Haar Wavelets. Math. Comput. Simul. 51, 91–100 (1999) 5. Murugesan, K., Dhayabaran, D.P., Evans, D.J.: Analysis of second order multivariable linear system using the STWS technique and Runge Kutta method. Int. J. Comput. Math. 72, 367–374 (1999) 6. Murugesan, K., Dhayabaran, D.P., Amirtharaj, E.C.H.: A study of second order state space systems of time invariant and time varying circuits using the STWS technique. Int. J. Electron. 89, 305–315 (2002) 7. Sepehrian, B., Razzaghi, M.: State analysis of time varying singular bilinear systems by single term Walsh series. Int. J. Comput. Math. 80(4), 413–418 (2003) 8. Lepik, U.: Numerical solution of differential equation using Haar wavelets. Math. Comput. Simul. 68, 127–143 (2005) 9. Bujurke, N.M., Salimath, C.S., Shiralashetti, S.C.: Numerical solution of stiff systems from nonlinear dynamics using single term Haar wavelet series. Non linear. Dyn. 51, 595–605 (2008)

Cryptographic Image Fusion for Personal ID Image Authentication K. Somasundaram1 and N. Palaniappan2 1 Image Processing Lab, Department of Computer Science and Applications, The Gandhigram Rural Institute – Deemed University, Gandhigram – 624 302, India [email protected] 2 Computer Centre, The Gandhigram Rural Institute – Deemed University, Gandhigram – 624 302, India [email protected]

Abstract. Personnel identification by ID images is the main authentication source of the security system. For multiple authentication unique biometric authentication features like finger print, signature and personal identification marks can also be used. Instead of sending these features additionally, they can be embedded in the personal ID images itself. In this paper we propose an algorithm for image fusion for personal ID image using blind and zero watermarking. By using the secret key the embedded information can be extracted. Experimental result shows that the algorithm is robust against various attacks like JPEG compression and few image processing methods. Keywords: Personal ID Image, Blind Watermarking, Zero Watermarking.

1 Introduction Identification of personnel in an organization and authenticate them to access the vital resource are part of the security system. Authentication systems make use of some features of the person like personal ID image, fingerprint, iris print etc. One of the most commonly used and reliable feature is the personal ID image [1]. Along with the personal ID image other unique features like fingerprint, iris print, hand geometry, retinal patterns, signature and personal identification marks can also be used for multiple authentication. The additional authentication data need to be sent along with the personal ID image in a secured manner. Cryptography is a good solution for embedding this additional information. Confidentiality, integrity and authentication are the basic objectives of cryptography. The user can access the required data only with the authentication key. Watermarking is one of the cryptographic methods. Watermarking is a technique for labeling digital pictures by hiding secret information in the images. The key point is that the embedded information in watermarked image can neither be removed nor be decoded without the required secret keys [2]. Watermarking method is used to embed any one of the additional authentication data in the personnel ID image. When the end user receives the transmitted cover image and the key, the embedded additional data can be extracted. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 213–220, 2011. © Springer-Verlag Berlin Heidelberg 2011

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In this paper we propose a variant of the watermarking algorithm proposed by Hsien-Wen Tseng and Chi-Pin Hsieh [3] which gives better results than the original algorithm. The proposed algorithm is used to embed the additional biometric authentication features in the personnel ID image. The remaining part of the paper is organized as follows. Section 2 gives an overview of the previous works on watermarking, Section 3 explains the proposed algorithm, Section 4 gives the results and discussion and Section5 gives the conclusion.

2 Previous Works on Watermarking There are a number of algorithms available for watermarking. They are mainly classified into spatial domain and frequency domain algorithms. Simple watermarks could be embedded in the spatial domain of images by modifying the pixel values or the least significant bit (LSB) values. However, more robust watermarks could be embedded in the transform domain of images by modifying the transform domain coefficients [4]. The basic types of transform domain are Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT). Among them DWT gives better performance. The only disadvantage is its computational complexity. LL1

HL1

LH1

HH1

Fig. 1. First level Decomposition of an image using DWT

The DWT method decomposes the given image into several sub-bands using wavelet filters. Some of the commonly used wavelet filters are Haar and Daubechies. The LL sub-band contains the coarse detail and the HH, HL and LH contains the diagonal, horizontal and vertical detail of the image where n is the level of decomposition. Fig. 1 shows the first level DWT decomposition of an image. For further decomposition the LL1 sub-band is decomposed to LL2, HL2, LH2 and HH2. This process is repeated for n times for n level decomposition. Some methods embed the watermark in the LL sub-band and most of the methods embed the watermark in any of the HH, HL and LH sub-bands. Normally LL sub-band embedded watermark degrades the cover image quality. Afzel Noore et al [5] uses the high frequency bands HH1, HL1 and LH1 and mid-frequency bands HH2, HL2 and LH2 for embedding data. In MingShing Hsieh et al [2] algorithm, Qualified Significant Wavelet Tree (QSWT) based on the definition of Embedded Zero tree Wavelet coding (EZW) is used to select the significant wavelet coefficients for embedding watermark. These methods require more calculation and degrade the quality of the cover image to some extent. Methods which require the original image as secret key at decoding stage are named as non-blind watermarking methods. Methods which do not require the

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original image at decoding stage are called as blind watermarking methods. Traditional watermarking techniques mostly modify a selected portion of the cover image to embed the watermark. Watermarking method which does not alter the cover image is called as zero-watermark scheme. Jiang-Lung Liu et al [6] propose a blind watermarking method by modifying the original image in transform domain and embedding the watermark in the difference values between the original image and its reference image. There will be a small decrease or increase in the original pixel values according to the watermark values {-1, +1}. This method also degrades the quality of the cover image. Hsien-Wen Tseng and Chi-Pin Hsieh [3] proposed a blind and zero watermark algorithm for watermarking. This algorithm uses single level discrete wavelet transform and variance to embed and extract the watermark. Watermark Embedding. The watermarking image W is a binary sequence of length n, where W(i) ∈ {0,1} 1 < i < n. The values {0,1} of W are converted to {-1,1}. For security purpose the watermark sequence W is rearranged using permutation. The cover image X of size is transformed by single level DWT using CohenDaubechies-Feauveau wavelet (CDF 9/7) wavelet filter. All the co-efficients in the sub-bands LH1, HH1 and HL1 are replaced by zeros. Inverse DWT is applied to get the reference image X’ of the original image. The reference image X’ is divided into non-overlaping blocks of size 3 x 3 where 1< k < x . Variance σ2 (the average of the squared differences from the mean) of each block is calculated. The polarity of the kth block is calculated by 1 1

σ σ

.

(1)

where the is the location index of each block, is the mean variance of all the blocks and is the threshold selected. For every bit in W a block location is selected randomly, where if W(i) = +1, select a block having = +1 and W(i) = -1, select = -1, otherwise the block location is skipped. The selected block locations are stored and transmitted as secured key. The value decides the polarity of each block. As the blocks are selected randomly, there is a possibility of selecting the blocks having variance close to the mean variance . When the cover image is attacked, the pixel values may differ resulting in change of mean variance during watermark extraction. For the blocks having minimum difference from mean variance, the polarity value may change from -1 to +1 or +1 to -1. Hence a threshold value is added to the mean variance to avoid this problem. Watermark Extraction. The transmitted cover image Xw is transformed by single level DWT. All the co-efficients in the three sub-bands LH1, HH1 and HL1 are replaced by zeros. Inverse DWT is applied to get the reference image Xw’ of the original image. The reference image Xw’ is divided into non-overlapping blocks of size 3 x 3

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where 1< k < x . Variance σ2 of each block is calculated. The mean variance is also calculated. The transmitted key sequence is used to extract the watermark. Using the polarity of every block the value of watermark is extracted as follows. 1 1

W

σ σ

′ ′

.

(2)

where, belongs to the secured key sequence. Extracted watermark is re-permuted to the original version. The values in the range {-1, +1} are converted to {0, 1} to give the original watermark.

3 Proposed Algorithm We propose a variant of the previous algorithm by eliminating the threshold value t. When experimenting with different types of images, it will be an incessant problem for selecting the threshold value. An improper selection of the threshold value will affect the efficiency of the algorithm. Table 1 shows the range of variance of 3x3 non-overlapping blocks of the baboon image of size 256 x 256 pixels. Among the total 7225 blocks, the polarity values of 5123 blocks are positive and 2102 blocks are negative. The threshold value t can be selected only after a number of trials. Table 1. Range of variance values and number of blocks for the test image baboon Label -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Range of Variance Value < 10.0 10.1 – 29.9 30.0 – 49.9 50.0 – 69.9 70.0 – 89.9 90.0 – 91.42 91.43 91.44 – 99.9 100.0 -109.9 110.0 – 129.9 130.0 – 149.9 150.0 – 169.9 170.0 >

No. of blocks 1518 1585 903 623 460 34 0 156 140 254 239 191 1122

To overcome this, after calculating the polarity value , the block locations having positive and negative polarity values are separated and stored in two different arrays. Positive polarity locations are sorted in descending order and negative polarity locations are sorted in ascending order based on their variance σ2. During embedding, the required block locations are selected from the sorted arrays from the starting. Since the blocks having highest difference are used for embedding, even after attacks they can maintain the polarity value. This increases the efficiency of the algorithm.

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Attacks. Different types of attacks are used to test the robustness of the algorithm. They are JPEG image compression and few image processing methods. JPEG compression is experimented with quality factors of 100, 75, 50, 25 and 10. Image processing attacks like sharpening, Gaussian blurring at a radius of 3.0 pixels and median filtering of window size 5 x 5 are performed. Error Metrics. Correlation co-efficient and PSNR value are used as error metrics to assess the quality of the image and performance of the algorithms. Validation of the Proposed Algorithm. To evaluate the performance of the proposed algorithm, we carried out experiments by taking baboon image of size 256 x 256 pixels and a watermark data of 1000 bits. The robustness of the proposed algorithm is verified by subjecting the watermarked image to several attacks. The computed values of correlation coefficients and PSNR values for the proposed algorithm along with the existing algorithm are given in Table 2. Table 2. Correlation co-efficient and PSNR values of the extracted watermark for various attacks

Type of attack JPEG Quality Factor 100 JPEG Quality Factor 75 JPEG Quality Factor 50 JPEG Quality Factor 25 JPEG Quality Factor 10 Sharpening Gaussian Blur Median Filtering

Existing Algorithm Correlation PSNR Co-efficient (db) 0.991 50.45 0.896 27.74 0.819 22.16 0.716 17.26 0.545 12.11 0.927 31.05 0.389 2.34 0.362 2.12

Proposed Algorithm Correlation PSNR Co-efficient (db) 1.000 ∞ 0.996 ∞ 0.990 ∞ 0.986 46.02 0.837 23.22 1.000 ∞ 0.337 7.47 0.311 6.68

From Table-2, we note that the proposed algorithm performs superior than the existing algorithm. While considering the processing time, the proposed algorithm requires 0.734 seconds and existing algorithm requires 0.562 seconds. For sorting the arrays the proposed algorithm requires more time. When comparing the results this additional time is negligible.

4 Results and Discussion The proposed algorithm is used to embed the additional authentication data in the personal ID images as watermark. Grayscale Personal ID images of standard size 640x480 [7] from Indian face database [8] are used to test the above algorithm. The ID images are pre-processed with Photoshop. The unique biometric features like finger print, signature and personnel identification marks are used as additional authentication data. Fig. 2 shows few of the personal ID images and biometric authentication eatures used. The proposed algorithm is developed with Matlab version 6.5. The jpeg

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A mole near the left nose; A thin scar on the fore head; Height 132; Weight 45; (a)

(b)

(c)

(d)

(e)

Fig. 2. Personal ID images and biometric authentication features (a) Personal ID image–1 (b)Personal ID image– 2 (c) Finger print image (d) Signature image (e) Personnel identification marks

quality factors are maintained with Matlab commands itself. The image processing attacks are experimented using Photoshop. To normalize the image, a median filter of window size 3 x 3 is applied initially. 4.1 Finger Print Watermarking Finger print images are scanned using finger print reading devices. The standard size of the captured finger print image is 225 x 313. The image to be watermarked is resized to one fourth of its size and the number of bits to be embedded counts to 4368. These bits are embedded in the test images and are extracted. After extracting the watermark, the correlation co-efficients and PSNR values are computed and given in Table 3. Table 3. Correlation co-efficients and PSNR values obtained after several types of attacks Type of attack Jpeg Quality factor 100 Jpeg Quality factor 75 Jpeg Quality factor 50 Jpeg Quality factor 25 Jpeg Quality factor 10 Sharpening Gaussian Blur Median Filtering

Corr. Co-efficient 0.842 0.837 0.808 0.770 0.683 0.849 0.655 0.513

PSNR (db) 22.89 22.56 20.87 18.96 15.67 23.15 14.45 10.07

The fingerprint embedding withstands the JPEG image compression even at the quality factor of 10. It gives very poor values when attacked with median filtering. 4.2 Signature Watermarking Signature images are scanned using HP PSC1210 scanner. The standard size of the scanned signature image is 200 x 100. The image to be watermarked is resized to one fourth of its size and the number of bits to be embedded counts to 1250. These bits are embedded in the test images and extracted. The results of various types of attacks are shown Table 4.

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Table 4. Correlation co-efficients and PSNR values obtained after several types of attacks Type of attack Jpeg Quality factor 100 Jpeg Quality factor 75 Jpeg Quality factor 50 Jpeg Quality factor 25 Jpeg Quality factor 10 Sharpening Gaussian Blur Median Filtering

Corr. Co-efficient 1 1 1 1 0.99 1 0.79 0.65

PSNR (db) ∞ ∞ ∞ ∞ 54.47 ∞ 25.53 18.48

The signature embedding gives excellent results for JPEG compression. Because of the small number of bits embedded it withstands even at the quality factor of 10. It gives very poor results when attacked with median filtering. 4.3 Personal Identification Marks Watermarking Personal identification marks are combination of alphabets and numbers. The letters a-z, A-Z, 0-9, ’;’ and space are used for this purpose. The required letters are limited to 64. Their ASCII values are encoded to 1 to 64. 6 bits are used to represent every letter. The average number of bits counts to 3500. These bits are embedded in the test images and extracted. The results of various types of attacks are shown Table 5. Table 5. Correlation co-efficient values obtained after several types of attacks Type of attack Jpeg Quality factor 100 Jpeg Quality factor 75 Jpeg Quality factor 50 Jpeg Quality factor 25 Jpeg Quality factor 10 Sharpening Gaussian Blur Median Filtering

Corr. Co-efficient 1 1 0.999 0.997 0.952 1 0.89 0.81

The watermark is resilient to JPEG image compression. Since the watermark is composed of alphabets and numbers, even a single bit alteration will result in change of the letters and the result is affected.

5 Conclusion In this work we have proposed a blind and zero watermarking algorithm. This cryptographic algorithm embeds additional biometric features in the personnel ID images for better authentication methods. Experimental results on test images show that the proposed algorithm gives better results than the existing algorithm. But it failed when attacked with median filter. This algorithm may find its application in various biometric authentication systems.

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References 1. Somasundaram, K., Palaniappan, N.: Adaptive Low Bit Rate Facial Feature Enhanced Residual Image Coding Method Using SPIHT for Compressing Personal ID Images. International Journal of Electronics and Communications (2010) (in Press) 2. Hsieh, M., Tseng, D., Huang, Y.: Hiding Digital Watermarks Using Multiresolution Wavelet Transform. IEEE Transactions on Industrial Electronics 48-5, 875–882 (2001) 3. Tseng, H., Hsieh, C.: A Robust Watermarking Scheme for Digital Images Using Self Reference. In: Huang, X., Chen, Y., Ao, S. (eds.) Advances in Communication Systems and Electrical Engineering. LNEE, vol. 4, pp. 479–495. Springer, Heidelberg (2008) 4. Potdar, V.M., Han, S., Chang, E.: A Survey of Digital Image Watermarking Techniques. In: 3rd IEEE International Conference on Industrial Informatics, pp. 709–716 (2005) 5. Noore, A., Singh, R., Vatsa, M., Houck, M.M.: Enhancing Security of Fingerprints through contextual biometric watermarking. Elsevier – Forensic Science International 169-2, 188– 194 (2007) 6. Liu, J., Lou, D., Chang, M., Tso, H.: A Robust Watermarking Scheme Using Self-reference Image. Elsevier Computer Standards & Interfaces 28-3, 356–367 (2006) 7. ISO/IEC 19794-5, Information Technology – Biometric Data Interchange Formats, Part 5: Face Image data 8. Jain, V., Mukherjee, A.: The Indian Face Database (2002), http://vis-www.cs.umass.edu/~vidit/IndianFaceDatabase/

Artificial Neural Network for Assessment of Grain Losses for Paddy Combine Harvester a Novel Approach Sharanakumar Hiregoudar1,2, R. Udhaykumar3, K.T. Ramappa3, Bijay Shreshta2, Venkatesh Meda2, and M. Anantachar1 1

College of Agricultural Engineering, UAS, Raichur, Karnataka, India 2 University of Saskatchewan, Saskatoon, Canada 3 Gandhigram Rural University, Gandhigram, T.N, India [email protected]

Abstract. Paddy is a staple food for more than 93 countries and it will stay of life for future generations. Harvesting is one of the vital operations in crop production and timely harvesting is essential for getting maximum yield. Moisture content and forward speed are the two factors to overcome the post harvest losses and minimise the quantitative losses. In this paper, an artificial neural network is introduced to assess the grain losses in the field condition. The simulation result shows that the ANN method is appropriate and feasible to assess the grain losses. However, model results that, an error of (RMSE) 0.1582 for cutter bar loss, for threshing loss was 0.1299 and for separation loss was 0.1321. Hence, the ANN model help the operator / farmer to decide the time of harvest. It also minimizes the post harvest grain losses by considering crop and machine parameters. Keywords: Paddy; Combine harvester; Grain losses; ANN.

1 Introduction Rice (Oryza sativa L.) is the second most important cereal crop and a staple food [1]. It is a member of the graminae family along with wheat and corn. Rice is one of the three crops on which human being largely depend on daily food requirements. Today, the consumption of rice is greater than before because of increase in population, migration of peasants to cities and improvement of living standards [2]. This might be due to distribution of people and food habit and also environmental determinations. Bridging the gap between the demand and production of food grains is an important task before the scientists. The solutions to bridge the gap are: improve post harvest management practices, put more land under cultivation, introduce high yielding varieties and improve husbandry practices. Minimizing the post harvest losses is practically viable. “Post harvest losses” implies a measurable quantitative and qualitative loss of a given product that occurs during different phases of processing right from harvest to till bagging [3]. Harvesting is one of the vital operations in crop production and timely harvesting is essential for getting optimum yields. The percentage of ripe grains in the panicles determines the harvesting time. The crop is ready for harvest when 80 percent of the panicles turn straw-coloured and the grains in the lower portions of the panicle reach the hard-dough stage. In recent days, there has been an acute shortage of agricultural P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 221–231, 2011. © Springer-Verlag Berlin Heidelberg 2011

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labourers during harvesting season due to increased employment opportunities in urban areas for rural youth. Due to non-availability of labour and increased demand in work efficiency, introduction of paddy combine harvester becomes essential. Its introduction will balance these aspects simultaneously without any delay, as delay in harvesting would result in increased grain losses. One of the important reasons for harvesting delay could be the labour shortage [4,5]. However, the efficiency of combine harvester is affected by crop conditions at the time of harvest, operation speed and width of cutter bar. During the field operations the operator has to adjust the machine systems to suit the crop condition in the field. Many time, it is not done due to urgency to harvest bothering less about grain losses. Farmers are not aware of the amount of loss incurred during the operation and no agency is guiding them properly to reduce the post harvest losses in the field. Ultimately the grain production per unit area is not improved. Further, the nonavailability of the precise information from the operator or user may affect the necessary modifications required to improve the working of the machine, to suit the local crop conditions. In the present experiment, an attempt was made to develop an artificial neural network (ANN) for time taken, width of cut, moisture content of grain and percent grain losses by the combine harvester. The multiple regression models might be able to predict the dependent variables. But the above parameters are independent and determined as a function of input variable. Hence, a powerful tool ANNs is used in different applications in great success. Based on the previous work in different application in agricultural machinery ANNs performance a better solution to assess the required parameter [6,7,8,9,10,11]. To ascertain the above problems, ANN model was used for judging the different parameters by reducing the post harvest grain losses in the field while using paddy combine harvester.

2 Materials and Methods 2.1 Materials The experiments were conducted in the province of Karnataka, Raichur district, India to ascertain the post harvest grain losses by using combine harvester. A Class combine (Escorts make – track type, 2.1 m cutter bad width) were selected to assess the grain looses. The different operational parameters were taken during the field evaluation. The preharvest losses were tested based on the test procedure of FMO [12]. 2.2 Field Test Parameters The parameters are considered prior to harvesting operation are: area of the field, crop variety, plant height, mode of planting, grain moisture content, grain straw ratio, type of soil and soil moisture content. During operation the field parameters such as pre harvest loss, cutting loss, bundling loss, conveying loss, threshing loss, winnowing loss, time taken for harvesting, bundling, conveying, threshing, winnowing and cost of operation were recorded [12].

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2.3 Machine Operational Parameters The operational parameters are recorded when the machine was operating in the field. These include forward speed of the machine, effective working width, cutting height, time elements (as used in computing field capacity and efficiency), estimated grain yield and quality of paddy. The field evaluation trials were carried out according to BIS and RNAM test codes. The basic parameters to estimate the harvesting loses in the field conditions are cutter loses, threshing and separation loss and were calculated as follows. The cutter bar losses are considered for the action of cutter bar during the harvesting the parameters are width of cut, time taken to harvest and moisture content of straw. The threshing losses were width of cut, time taken to harvest and moisture content of grain (in this case 100 g of sample were taken and analysed damages or un-threshed), separation loss were estimated in the sieve or straw walker losses based on the previous parameters. All the losses were expressed in per cent basis. 2.4 Artificial Neural Network Multi-layer back propagation (BP) artificial neural networks with three inputs, the time taken to harvest, width of cut, moisture content of grain / moisture content of straw, and a single output, the cutter bar loss / threshing loss/ separation loss were implemented. The tan-sigmoid and purelin differential transfer functions were respectively used for the nodes in the hidden and output layer as it was a powerful configuration for complex non-linear function approximation [13]. A network model is characterized with an error space representing the error of the system for every possible combination of the weights and biases. Details on BP ANN, ALR and MT can be found in [14,15]. The performance function used was Mean Sum-squared Error (MSE), and the batch training was used in which the weights and biases are updated in the direction of the negative gradient of the performance function only after the entire training samples (epoch) has been applied to the network. The gradients calculated at each training sample are added together to determine the change in the weights and biases. Figure 1 represents a three-layered BP ANN with input column vector x, weight matrices u, v, and w, and the intermediate outputs y and z, and the output o column vectors. The elements of the first row of u are the weights connecting the first node (neuron) of the first hidden layer (jth) to I inputs, and the elements of the second row of u are the weights connecting the second node of the first hidden layer to I inputs, and so on. 2.4.1 Improving Generalization (Robust Network) One of the problems that occurs during training of ANN is overfitting, in which case the network memorizes the training samples, and produces outputs close to the targets. But, it fails to learn to generalize to new situations (samples), and produces outputs not even close to the targets. If the training samples are larger in size than the network parameters, overfitting does not occur. On the other hand, larger networks are required to model complex non-linear functions. Among two techniques, Early Stopping (ES) and Bayesian Regularization (BR), to prevent overfitting, the later is more effective especially for the smaller training samples [13]. In BR, there is no need to separate samples out of the training samples to form a validation samples as

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required in ES, and therefore, all of the samples is available for training, which is crucial for smaller data sets. When using Bayesian regularization, it is important to train the network until it reaches convergence. The sum squared error (SSE), the sum squared weights (SSW), and the effective number of parameters (EP) should reach constant values when the network has converged. The EP should remain approximately the same, no matter how large the total number of parameters in the network becomes provided network has been trained for a sufficient number of iterations to ensure convergence. y1

x1

uj1

xi

yj

uji

z1

vk1 vkj

zk

o1

wlk

vkJ

ujI

yJ

wl1

ol wlK

zK

xI Input (Layer I)

oL Layer J

Layer K

Layer L

Fig. 1. A three-layered feed forward network

3 Results and Discussion An experiment was carried out in Raichur District of Karnataka, India to assess the grain losses by using escorts make track type combine harvester. The primary objective of the study is to evolve a suitable model, which could provide optimum levels of moisture content, width of cut and speed to reduce the losses. In order to achieve this, the scatter diagram of each one of the independent variables with the loss was observed. The scatter diagram expressed a ‘U’ shaped variation, which implies that there exists an optimum level for each one of the parameters, which can minimize the loss. Apart from this the ‘ANOVA’ carried out between the blocks and also between the seasons did not show any significant difference in any of the mean value of any of these parameters. This helped to collate all the data available for combine harvester in both the blocks for two seasons. Hence all the 100 observations were taken together. Even in the combined data, the scatter diagram exhibited same type of variations with respect to each one of the variable. 3.1 Statistical Characterization of Data The descriptive statistics of the different parameters causing for the grain losses are given Table 1. All the parameters were normally distributed as confirmed by the

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Kolmogorov_Smirnov test (SYSTAT 13) and the values on skewness. The statistical analysis of the individual values showed that the losses had no significant effects on the accuracy of the models. Table 1. Descriptive statistics of the different attributes (n=100)

Variable Time Width of cut Moisture content of grain Moisture content of straw Area harvested Estimated yield Effective field capacity Field efficiency Purity of grain

Unit sec m % % ha t/ha ha/h % %

Minimum 19.95 1.68 15.05 65.05 0.15 1.68 0.44 89.40 88.48

Maximum 24.55 2.08 19.68 79.68 0.35 3.60 0.74 99.19 94.28

Mean 21.89 2.01 17.25 71.83 0.26 2.85 0.57 95.65 92.54

Skewness 0.36 -2.85 0.22 0.42 -0.33 -.075 0.56 -1.02 -0.76

The Table 2 refers to the summary of the grain losses and time parameters for the selected attributes. The variation of standard deviation clearly represents, as the mean increases the grain losses increase in respective attributes. Table 2. Summary statistics of the grain losses parameters for the selected attributes (n=100)

Variable

Unit

Minimum

Maximum

Mean

Pre harvest loss Cutter bar loss Threshing loss Straw walker loss Sieve loss

% % % %

0.13 0.48 0.47 0.17

0.26 0.99 2.21 0.49

0.17 0.75 1.36 0.33

Standard Deviation 0.03 0.10 0.51 0.08

%

0.15

0.55

0.35

0.09

3.2 Optimal Artificial Neural Network Selection The data were divided into 80:20 out of 100 samples, where 80 were used for training and 20 for testing of the model. Three different losses were selected for the experiments. The best ANN structures for all the losses considered in this work have been provided in Table 3. Each of the trained models had 4 input nodes and one output. Table 3. Summary of the best structure and optimum parameters of the ANN modeling

Components

ANN structure

Cutterbar loss Threshing loss Separation loss

3-5-1 3-5-1 3-5-5-1

Transfer function Transig Transig Transig

Epoch

R2

201 1000 233

0.84 0.92 0.98

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Fig. 2a. Cutter bar losses

Fig. 2b. Threshing losses

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Fig. 2c. Separation losses

After evaluating the different combination of the ANNs, the optimal ANN architecture has two layers with 5 neurons in the first, second cases, whereas the third case three layers with neurons of 5 gives the best fit as compared to other combination. The scaling, and the activation for the tan-sigmoid transfer function were within -1, and 1. The number of epochs required for training the ANN associated with modeling the cutter bar, separation and threshing losses were respectively 201, 233 and 1000. This indicates that the epoch size directly affected the learning ability of the networks. The 3-5-1 ANN models resulted in R2 and RMSE values of 0.84, and 1.82 (cutter bar loss), 0.92, and 2.02 (threshing loss), and the 3-5-5-1 ANN produced those values of 0.98, and 0.84 for separation loss. The commonly used network for avoiding this is to divide the training data set into two sets, one training and one for validating the model [16]. The observed data versus predicted data for testing were plotted (Fig. 2a, 2b and 2c) for all the ANN models. The developed models were tested by using the test sets each consisting 20 data points. The performance of the ANN models in predicting different losses are shown in figure 3a, 3b and 3c. The figures show predicted vs. the measured data. It was

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observed that, the predictive values were well augmented and good fits with the measured values showing a good linear adjustment with very small intercepts and slopes practically equal to 1. The output implies that the designed network was able to generalize the relationship between the input and output parameters without overtraining. The trained ANN models were able to predict the grain losses with RMSE of 0.1582 for cutter bar loss, 0.1299for threshing loss , and 0.1321for separation loss. The maximum variation was observed in the separation loss, which might be due to more straw moisture during harvesting that affects the unthreshing of the grain as well as more breakage of the grain. But, the ANN models for the cutter bar and threshing losses exhibited very small differences between the predicted and the observed values. It was evident that a simple ANN model could be more powerful for measuring the grain losses using paddy combine harvester in the field condition. It was also noticed that the ANN parameters, including the number of neurons per layer, number of epochs and learning rate effects significantly to ANN performance.

Fig. 3a. Cutter bar losses

Artificial Neural Network for Assessment of Grain Losses

Fig. 3b. Threshing losses

Fig. 3c. Separation losses

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4 Conclusion The ANN model was developed in this research by considering the four primary factors viz., moisture content of straw, moisture content of grain, width of cut and time taken shows the significantly (P = 0.01) influenced with cutter bar, threshing and separation losses. The standard deviation for threshing loss were higher (0.51) shows that maximum losses might be due to higher moisture content of grain during harvesting, a non-significant trend was observed. However, the data showed that the moisture content around 15.05 to 19.68 %. The data showed that there was strong relationship between the input variables regardless to the parameters. The artificial neural network powerful tool was useful to estimate and predict the post harvest grain losses using paddy combine harvester in the field condition. The ANN model confirmed the relationship between the four input parameters and three outputs. In comparison with developed model for observed and estimated values would helps to predict and judge the post harvest losses in the field condition using combine harvester. These results confirmed that a properly training ANN can be used to produce simultaneously more than one output as compare to mathematical models where two regression models were needed for individual output. Acknowledgements. The authors would like to express their thanks to the University of Agricultural Sciences, Raichur, Karnataka, India for funding. The authors would also like to thank the University of Saskatchewan, Canada for their technical support, which made this possible.

References 1. Pillai, K.G.: Current scenario in rice and prospects for sustainable rice production. Fertilizer News 41(2), 15–23 (1996) 2. Yadav, R.N.S., Yadav, B.G.: Design and development of CIAE bullock drawn reaper. AMA 23(2), 43–51 (1992) 3. He, Y., Yidan, B., Xiansheng, L., Algader, A.H.: Grain post production practices and loss estimates in south China. AMA 28(2), 37–40 (1997) 4. Fouad, H.A.: Agricultural mechanization in Egypt and socio economical conditions in mutual effect: Mercoop Faculty of Agriculture, Amman, Jordan (1984) 5. Goel, A.K., Darh, R.C., Swain, S., Behera, B.K.: Feasibility study of different mode of paddy harvesting in Orissa. Paper Presented in 36th ISAE Conference held at IIT Kharagapur on January 28-30 (2002) 6. Hall, J.W.: Emulating Human Process Control Functions with Neural Networks. Ph. D. thesis. Dept. of Mech. Eng., Uni. of Illinois, Urbana, IL (1992) 7. Kushwaha, R.L., Zhang, Z.X.: Evaluation of factors and current approaches related to computerized design of tillage tools: a review. J. Terramech. 35(2), 69–86 (1998) 8. Ma, F.L., Li, S.P., He, Y.L., Liang, S., Hu, S.S.: Knowledge acquisition based on neural networks for performance evaluation of sugarcane harvester. In: Wang, J., Yi, Z., Zurada, J.M., Lu, B.L., Yin, H. (eds.) Advances in Neural Networks, pp. 1270–1276. Springer, Heidelberg (2006)

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9. Hosseini, S.M.T., Sioseh mardeh, A., Parviz, F., Sioseh Mardeh, M.: Application of ANN and multiple regressions for estimating assessing the performance of dry farming wheat yield in Ghorveh region, Kurdistan province. Agricultural Research 7(1), 41–54 (2007) 10. Kumar, G.V.P., Srivastava, B., Nagesh, D.S.: Modeling and optimization of parameters of flow rate of paddy rice grains through the horizontal rotating cylindrical drum of drum seeder. Comput. Electron. Agric. 65(1), 26–35 (2009) 11. Anantachar, M., Kumar, P.G.V., Guruswamy, T.: Neural network prediction of performance parameters of an inclined plate seed metering device and its reverse mapping for the determination of optimum design and operational parameters. Computers and Electronics in Agriculture 72, 87–98 (2010) 12. FMO: Combine harvesting, Fundaments of machine operation, 3rd edn. Deere and Company service training, Moline, Illinois, U.S.A (1987) 13. Neural Network Toolbox TM : v. 7, R2010b. The MathWorks, Inc., Natick, MA (2010) 14. Zurada, J.M.: Introduction to Artificial Neural Systems. West, St. Paul (1992) 15. Shrestha, B.L.: A neural network based machine vision system for identification of malting barley varieties. M. Sc. Thesis, University of Saskatchewan, Canada (1996) 16. Uno, Y., Prasher, S.O., Lacroix, R., Goel, P.K., Karimi, Y., Viau, A., Patel, R.M.: Artificial neural networks to predict corn yield from compact airborne spectrographic imager data. Comput. Electron. Agric. 47, 149–161 (2005)

Implementation of Elman Backprop for Dynamic Power Management P. Rajasekaran1, R. Prabakaran2, and R. Thanigaiselvan2 1

M.E(Pervasive Computing Technologies)/CCT, Anna University of Technology, Tiruchirappalli, Tamil Nadu, India [email protected] 2 Assistant Professor, Department of Electrical and Electronics Engineering, Anna University of Technology, Tiruchirappalli, Tamil Nadu, India [email protected], [email protected]

Abstract. Dynamic power management is a technique used to save power when the system is idle. Earlier it was assumed that the prediction can be done only in long range dependent systems. But a single user will not work similarly the next time, so a single assumption will not hold good. To overcome the above assumptions, we propose an Elman Model which uses Moving Average, Elman Backprop network and random walk model to predict the idle period. Here we use Artificial Neural Network (ANN) in which we train the neurons in a particular way the user desires, replacing neurons by time series we can calculate how much power is saved. This model utilizes both long range dependency and central tendency to predict the past idle periods, by which we predict the future idle period. By simulation we can show that this method achieves higher power saving compared to other methods. Keywords: Dynamic Power Management, Elman Backprop Neural Network, Hurst Exponent.

1 Introduction Power utilization has become a major issue in portable designs, since its battery storage is less compared to its usage. One of the popular techniques to solve this problem is to use Dynamic Power Management (DPM) at the system level. It is observed that most of the time the system is idle and it is busy only when the user uses it, DPM is based on this factor. It will create huge power loss if these systems are active all the time. By putting these portable systems in power saving mode during its idle period, DPM is achieved [1]. Actually the idle period should be greater than breakeven time (Tb) Breakeven Time = Transition Time + (Wake up energy + Shutdown energy) Power consumed P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 232–240, 2011. © Springer-Verlag Berlin Heidelberg 2011

(1)

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Power down and power up processes introduce latency, initially idle period is shorter than breakeven time, and therefore turning off the device is not a wise choice. It is obviously a main challenge in DPM is to choose the best prediction method. Existing methods for implementation of DPM are, a) Hybrid model: It is always not wise to predict future idle periods with single assumption which assumes memory component to be constant. Here we have performed prediction using central tendency test and long range dependency. This method performs better compared to many other techniques. b) Timeout: It is simple compared to other techniques [2]. This is also called Competitive Analysis (CA). An adaptive timeout period is increased if predictions are too eager, and decreased if they are too conservative [3]. This is not suitable because it needs to wait for timeout period to elapse before entering low power state. c) Stochastic Method: Here we assume the user request and model the device in stochastic processes. Discrete Time Markov Decision Process assumes geometric distribution [4]. By using sliding window technique we deal with non Stationery requests [5]. Continuous Time Markov Process (CM) used here is an asynchronous process [6]. Both DM and CM use memory less distribution. Semi Markov model (SM) is assumed to have uniform distribution. [7]For Pareto distributions we use Time Indexed Semi Markov (TISM). Idle periods that are short, separated from each other by long idle period are called sessions [8]. Session length is predicted using an algorithm. In Artificial Neural Network (ANN) next idle period is predicted using past idle periods. The proposed method is tested with uniform distribution. The algorithm used here is Genetic Algorithm (GA). This method works well for long range dependent (LRD) series. AR prediction method fits well for short range dependent (SRD). Another technique used to predict future idle periods using past history is Online Probabilistic Based Algorithm (OPBA). Here, when a new idle period is entered oldest idle period will be deleted, this makes the algorithm adaptive. EA is designed for X-server and Telnet because it is not suitable for hard disk idle period prediction [9]. Workload which is LRD alone can be predicted using ANN and GA, for SRD we use Autoregressive (AR) and Autoregressive Moving Average (ARMA). Since Markov model is a memory less distribution, future periods are not dependent on past idle periods.First we have to decide whether the user idle period is LRD, random walk, SRD or combination of the three. To check LRD we perform Rescaled Range Analysis(R/S). There are many methods available to perform Fractal Analysis and we have chosen R/S analysis [10]. LRD is unique for every user. Similarly Hurst Exponent varies in different places in time series. We can infer from this that according to usage pattern the workload characteristic varies. Therefore, it is not enough to represent workload characteristic with a single model alone. The Elman Model proposed here achieves better prediction in different scenarios; this will be explained in Section 2. R/S analysis and ANN will be discussed in section 3 and 4 respectively. In Section 5 result is presented.

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2 Elman Model for DPM In this paper we take two things into consideration, they are central tendency and LRD, by which we choose prediction method. EBNN is used for idle periods that are LRD. Similarly random walk model is used for mean reverting idle periods. Central tendency (CT) is the ratio between mean and standard deviation. CT = Standard Deviation Mean

(2)

If a time series has large CT, then the series is said to be deviating from mean. In contrast if it has a small CT, then the series is said to be near mean value. LRD is found using R/S analysis. The proposed model is described below: 1. CT is performed in idle period history. If CT is less than 1, then it is said to be deviating from mean, in this situation we use Moving Average (MA) to predict idle period. If CT is greater than 1, the series will be found near mean and a simple method will not hold good, therefore go to step 2. 2. R/S analysis is performed here. If H > HT, the series will show strong LRD, therefore EBNN will be used. EBNN will update weights while training. If H < HT, the series is mean reverting and random walk model is used as predictor. 3. Here queue is followed, oldest idle periods will be replaced by new idle periods, and hence the history length will be the same. Now we choose the technique for prediction.

Fig. 1. Work flow for Proposed Elman Model

Timeout method is also used here as it has high power saving ability. If the predicted value is less than threshold value, then timeout is used.

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3 Rescaled Range Analysis When a series is LRD, it is wise to use R/S analysis to find Hurst Exponent [11]. R/S analysis is found by a hydrologist H.E.Hurst. For a given time series x1, x2,xn, R/S method is as follows: (for all equations t = 1,2,…n) 1.

Calculate mean (M) M=

2.

∑

(3)

Calculate Mean adjusted series (Y) Yt = Xt – M

3.

(4)

Calculate cumulative deviation series (Z) Zt = ∑

4.

(5)

Range series R Rt = max (Z1, Z2..., Zt) – min (Z1, Z2...,Zt)

5.

Standard deviation S St =

6.

(6)

∑

2

(7)

where u= mean value from X1 to Xt Rescaled Range Analysis (R/S) (R/S) t = Rt / St

(8)

A value of t is chosen such that it is divisible by n. Hurst found that R/S is similar to power law, (R/S)t = c* tH

(9)

where c is a constant and H is called Hurst Exponent. We estimate R/S and t in log-log scale to find H. The slope in the plot is said to be H. Table 1. Hurst Exponent Interpretation

H value Lies between 0 and 0.5 Equal to 0.5 Lies between 0.5 and 1

Characteristics Anti persistent, mean reverting Random series Persistent, trend reinforcing

Artificial Neural Network can be used to predict the idle period, since it is used in financial time series which have H > 0.5.

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4 Elman Backprop Neural Networks A Neural Network is an interconnection of processing nodes. In Neural Network method, the first step is data preparation and pre-processing [9]. Neural Networks are also named Universal Function Approximators. Most of the networks use single hidden layer. Elman Network is one of the recurrent networks. These networks are two layer back propagation networks, a feedback connection is taken from hidden layer output to input. This feedback connection enables them to learn, to recognize, to generate temporal and spatial patterns. Elman Network is an extension of two layer sigmoid architecture. Elman Network has tansig neurons in hidden layer and purelin neurons in output layer. This network differs from other conventional two layer networks in that the first layer has the recurrent connection. If two Elman Networks are given same inputs, weights and biases, their outputs differ because of the feedback states. The back propagation training function used here is trainbfg. Compared to other methods Elman Network needs more hidden neurons to learn a problem. This network is useful in signal processing and prediction. Output Y (t)

Hidden Z (t) Copy Z

Input X (t)

Hidden Z (t-1)

Fig. 2. Basic Structure of Elman Network

Least mean Square (LMS) algorithm is an example of supervised training. To reduce the Mean Square Error (MSE), LMS algorithm adjusts the weights and biases of the linear network. For every input, there exists an output which is compared to the target output. Error is the difference between the target output and the network output. Adaptive networks use LMS or Widrow Hoff learning algorithm. These equations form the basis for LMS algorithm. W (k+1) = W (k) + 2αe (k) pT (k) b (k+1) = b (k) + 2αe (k)

(10) (11)

where e is the error, bias b are vectors and α is the learning rate. If α is high, learning occurs quickly.

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5 Experimental Setup Simulation is performed for the proposed Elman Method based on the usage of PC/laptop by an under graduate student. Breakeven Time (Tb) Calculation: Breakeven time is defined as the transition time plus the ratio between energy latency and power saved. Steps to calculate Tb is as follows: 1.

Calculate Energy latency, which is the sum of wake up energy and shut down energy: El = Ew + Es

2.

(12)

Calculate the power saved, which is the difference when the system is in low power state and the active state: Ps = Pa - Plp

3.

(13)

where Ps is the power saved, Pa is active power and Plp is power in low power state Find transition time Tt by using (1).

5.1 Experimentation Laptop LCD is used as power manageable component. Tb is found as shown below. Energy consumed during turn on and turn off is 15.2J. Therefore, El = 15.2 + 15.2 = 30.4J When LCD is off, Plp = 17.1W, when it is on Pa = 21.1W. Therefore, Ps = 21.1 – 17.1 =4W Then, t = El

= 30.4

= 7.6s

Ps 4 Tb= t + Tt = 7.6 + 8 = 15.6s

≈ 16s

If the idle period is greater than 16s, it is better to turn off. Table 2. Power Consumption for LAPTOP

STATE

POWER CONSUMED (%)

SLEEP STAND - BY

0 0

IDLE ACTIVE

4 32

In the above table, power consumption is checked for every one hour. When a request arrives, when the system is in sleep state, the disk will go to active state directly.

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Fig. 3. View of the Elman Backprop network

Fig. 4. Training parameters

Transition from higher power state to lower power state consumes negligible power. Thus the proposed method is able to predict idle period from past idle period history. In Matlab simulation we have to give the input and target data to the network. Select the neural network through which we have to train the neurons. Training function selected is traingdm and learning function is said to be learngdm. The performance function used is Mean Square Error (MSE) function. Number of layers used here is 2. Layer 1 has 10 neurons and transfer function is tansig. In layer 2, user can specify number of neurons and transfer function is same. This is how the proposed network looks like. To train the network, the maximum epoch 1000 is specified. Training is done with Gradient Descent back propagation with adaptive learning rate. The delay states are taken from the same network itself. By simulation we obtain the required output, error and delay states.

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Fig. 5. Performance plot for the network

Fig. 6. Training State

6 Results and Discussions The experiment is done with the proposed method and comparison is done with other DPM techniques. Performances are compared on the basis of Competitive Ratio (CR). It is a method to analyse online algorithms, where the performance of an online algorithm is compared to the performance of an optimal algorithm. Time out: It is a static technique. Hybrid Method: This method uses Moving Average, Time Delay Neural Network and Last Value as predictor. We assume maximum training epoch as 1000 for every training round during which laptop is running in full speed. Comparison chart for three DPM techniques are given below based on Competitive Ratio. Table 3. Comparison Chart Techniques Time out Hybrid Method Elman Model

CR 1.51 1.18 1.04

Time out produces average result in most of the cases and performs worst when the series have small idle periods by which it fails to save power. Hybrid method

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performs well all cases but yields high Competitive Ratio. Elman Model is best among all DPM techniques, since it achieves most power saving.

7 Conclusion In portable devices workload characteristics vary according to usage pattern which causes the memory component to change. Therefore, it is not wise to predict the future idle period based on single technique. By estimating Central Tendency and Long Range Dependency we have proposed a more effective Elman Model to predict the future idle period. Simulation results show that our method can perform better than many other available DPM techniques.

References 1. Lee, W.-K., Lee, S.-W., Siew, W.-O.: Hybrid model for Dynamic Power Management. IEEE Transaction on Consumer Electronics 55(2), 650–655 (2009) 2. Greenwalt, P.: Modelling Power Management for Hard Disks. In: Proceedings of International Workshop Modelling, Analysis and Simulation for Computer and Telecommunication Systems, pp. 62–65 (1994) 3. Golding, R., Bosh, P., Wilkes, J.: Idleness Is Not Sloth. In: Proceeding of Winter USENIX Technical Conference, pp. 201–212 (1995) 4. Benini, L., Bogliolo, A., Paleologo, G.A., Micheli, G.D.: Policy Optimization For Dynamic Power Management. IEEE Transaction on Computer-aided Design of Integrated Circuits and System 18(6), 813–833 (1999) 5. Chung, E.Y., Benini, L., Bogliolo, A., Lu, Y.H., de Michelli, G.: Dynamic Power Management for Non-stationary Service Requests. IEEE Transaction on Computers 51(11), 1345–1361 (2002) 6. Qiu, Q., Pedram, M.: Dynamic Power management Based on Continuous Time Markov Decisoin Processes. In: Design Automation Conference, pp. 555–561 (1999) 7. Simunic, T., Benini, L., Glynn, P., de Micheli, G.: Dynamic Power Management of Laptop Hard Disk. In: IEEE Proceedings of Design Automation and Test Conference and Exhibition in Europe, p. 736 (2001) 8. Lu, Y.H., De Micheli, G.: Adaptive Hard Disk Power Management on Personal Computers. In: Proceeding of Great Lakes Symposium VLSI, pp. 50–53 (1999) 9. Qian, B., Rasheed, K.: Hurst Exponent and Financial Market Predictability. In: Proceedings of FEA – Financial Engineering and Applications, pp. 437–443 (2004) 10. Lee Giles, C., Lawrence, S., Tsoi, A.C.: Noisy Time Series Prediction using a Recurrent Neural Network and Grammatical Inference. Machine Learning 44, 161–183 (2001) 11. Karagiannis, T., Faloutsos, M., Riedi, R.H.: Long range dependence: Now you see it, now you don’t. In: IEEE Global Telecommunications Conference, vol. 3, pp. 2165–2169 (2002)

Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer W. Abdul Hameed1 and M. Bagavandas2 1

Department of Mathematics, C. Abdul Hakeem College, Melvisharam, India [email protected] 2 School of Public Health, SRM University, Chennai-603203, India [email protected]

Abstract. The automatic diagnosis of breast cancer is an important, real-world medical problem. A major class of problems in Medical Science involves the diagnosis of disease, based upon various tests performed upon the patient. When several tests are involved, the ultimate diagnosis may be difficult to obtain, even for a medical expert. This has given rise, over the past few decades, to computerized diagnostic tools, intended to aid the Physician in making sense out of the confusion of data. This Paper carried out to generate and evaluate both fuzzy and neural network models to predict malignancy of breast tumor, using Wisconsin Diagnosis Breast Cancer Database (WDBC). Our objectives in this Paper are: (i) to compare the diagnostic performance of fuzzy and neural network models in distinction between malignance and benign patterns, (ii) to reduce the number of benign cases sent for biopsy using the best model as a supportive tool, and (iii) to validate the capability of each model to recognize new cases. Keywords: Breast cancer, neural network, fuzzy logic, learning vector quantization, fuzzy c-means algorithm.

1 Introduction Cancer, in all its dreaded forms, causes about 12 per cent of deaths throughout the world. In the developed countries, cancer is the second major cause of death, accounting for 21 per cent of all mortality. In the developing countries, cancer ranks third major cause of death accounting for 9.5 per cent of all deaths (ICMR, 2002). Cancer has become one of the ten main causes of death in India. As per the statistics, there are nearly 1.5 to 2 million cancer cases in the country at any given point of time. Over 7 lakh new cases of cancer and 3 lakh deaths occur annually due to cancer. Nearly 15 lakh patients require facilities for diagnosis, treatment and follow-up procedure (ICMR, 2001). Despite a great deal of public awareness and scientific research, breast cancer continues to be the most common cancer and the second largest cause of cancer deaths among women (Marshall E, 1993). In the last decade, several approaches to classification had been utilized in health care applications. A woman’s chances for long term survival are improved by early detection of the cancer, and early detection P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 241–248, 2011. © Springer-Verlag Berlin Heidelberg 2011

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is enhanced by accurate diagnosis techniques. Most breast cancers are detected by the patient or by screening as a lump in the breast. The majority of breast lumps are benign. And therefore, it is binding to diagnose breast cancer, that is, to distinguish benign lumps from malignant ones. In order to diagnose whether the lump is benign or malignant, the Physician may use mammography, fine needle aspirate (FNA) with visual interpretation or surgical biopsy. The reported ability for accurate diagnosis of cancer when the disease is prevalent is between 68% - 79%, in case of mammography (Fletcher SW et al., 1993); 65% - 98% in case FNA technique is adopted (Giard RWM et al., 1992), and close to 100% if a surgical biopsy is undertaken. And from this it is clear that mammography lacks sensitivity; FNA sensitivity varies widely and surgical biopsy although accurate is invasive, time consuming, and expensive. The goal of the diagnostic aspect of this Paper is to develop a relatively objective system to diagnose FNA with an accuracy that is best achieved visually. In this Paper, it is compared how fuzzy and neural network models can diagnose whether the lump in the breast is cancerous or not, in the light of the patient’s medical data.

2 Material This Paper makes use of the Wisconsin Diagnosis Breast Cancer Database (WDBC) made publicly available at http://ftp.ics.uci.edu/pub/machine-learning-database/ breastcancer-wisconsin/. This data set is the result of efforts made at the University of Wisconsin Hospital for the diagnosis of breast tumor, solely based on the Fine Needle Aspirate (FNA) test. This test involves fluid extraction from a breast mass using a small gauge needle and then a visual inspection of the fluid under a microscope. The WDBC dataset consists of 699 samples. Each sample consists of visually assessed nuclear features of FNA taken from patient’s breast. Each sample has eleven attributes and each attribute has been assigned a 9-dimensional vector and is in the interval, 1 to 10 with value ‘1’ corresponding to a normal state and ‘10’ to the most abnormal state. Attribute ‘1’ is sample number and attribute ‘11’ designates whether the sample is benign or malignant. The attributes 2 to 10 are: clump thickness, uniformity of cell size, uniformity of cell shape, marginal adhesion, single epithelial cell size, bare nuclei, blend chromatin, normal nucleoli and mitosis. There had 16 samples that contained a single missing (i.e., unavailable) attribute value and had been removed from the database, setting apart the remaining 683 samples. Each of these 683 samples has one of two possible classes, namely benign or malignant. Out of 683 samples, 444 have been benign and the remaining 239 have been malignant, as given by WDBC dataset.

3 Neural Network Model Research on neural network dates back to the 1940s when McCulloch and Pitts found that the neuron can be modelled as a simple threshold device to perform logic function (McCulloch WS and Pitts W, 1943). The modern era of neural network research

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is commonly deemed to have commended with the publication of the Hopfield network in 1982. This Paper has been worked adopting vector quantization as a technique for breast cancer diagnosis. Learning vector quantization (LVQ) was developed by Kohonen (1986) and in the year 1990 (Kohonen T), who summarizes three versions of the algorithm. This is a supervised learning technique that can classify the input vectors based on vector quantization. The version of LVQ incorporated in this Paper is LVQ1 - Kohonen’s first version of Learning Vector Quantization (Kohonen T, 1986). Given an input vector xi to the network, the “input neuron” (i.e., the class or category) in LVQ1 is deemed to be a “winner” according to min d(xi, wj)=min || xi-wj||2

(1)

The set of input vectors have been denoted as {xi}, for i =1, 2,…,N, and the network synaptic input vectors has been denoted as {wj}, for j = 1, 2, …,m. Cwj has been the class (or category) that has been associated with the (weight) Voronoi vector wj, and Cxi as the class label of the input vector xi to the network. The weight vector wj has been adjusted in the following manner: If the class associated with the weight vector and the class label of the input are the same, that is, Cwj = Cxi, then wj(k + 1) = wj( k ) + μ(k) [xi - wj( k )]

(2)

where 0< μ(k)<1 (the learning rate parameter). But if Cwj ≠ Cxi, then wj(k + 1) = wj( k ) - μ(k) [xi - wj( k )]

(3)

and the other weight vectors are not adopted. Therefore, the update rule for modifying a weight vector in (2) is absolutely standard, if the class is correct. In other words, according to the learning rule in (2), the weight vector wj is moved in the direction of the input xi, if the class labels of the input vector and of the weight vector agree. However, if the class is not correct, the weight vector is moved in the opposite direction away from the input vector, according to (3). The learning rate parameter μ(k) gets monotonically decreased in accordance with the discrete time index k; e.g., linearly decreased in time, starting at 0.01 or 0.02 (Kohonen T, 1990); however , many times 0.1 is used as the initial value. The convergence properties of LVQ1 have been studied by Baras and La Vigna (1990) and their approach is based on stochastic approximation theory .The weights can be initialized by using several methods. In this Paper, the first m (total number of classes) vectors from the set of training vectors have been used to initialize the weight vectors, that is, wj(0) for j = 1,2,…,m. and the stopping condition is the total number (predefined) of iterations.

4 Fuzzy Model An iota of uncertainty is present in almost every sphere of our daily routine. Traditional mathematical tools are not enough to find solution to all the practical problems

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in fields such as Medical Science, Social Science, Engineering, Economics, etc., which involve uncertainty of various types. Zadeh (Zadeh L A, 1965) in 1965 was the first to come up with his remarkable theory of fuzzy set for dealing these types of uncertainties where conventional tools fail. 4.1 Fuzzy C-Means Algorithm Fuzzy c-means clustering is an easy and a well improved tool which has its application in several medical fields. However, in c-means algorithms, like in all other optimization procedures, which look for the global minimum of a function, there is the risk of coming down to local minima. Therefore, the result of such a classification has to be regarded as an optimum solution with a determined degree of accuracy (Bezdek J, 1974).Classes in which each member has full membership are called discontinuous or discrete classes. On the other-hand, classes in which each member belongs to some extent to every cluster or partition are called continuous classes (McBratney, AB., and JJ. de Gruijter, 1992). Continuous classes are a generalization of discontinuous classes when the indicator function of conventional set theory, with values 0 or 1, is replaced by the membership function of fuzzy set theory, with values in the range of 0 to 1. Let us take the partition of a set of n-individuals in c-discontinuous classes. According to such partition, each individual is a member of exactly one class. This can be numerically represented by a (n x c) membership matrix U = (µ ij) where µ ij = 1 if the individual belongs to class j and µ ij = 0 otherwise. In order to ensure that the classes are mutually exclusive, jointly exhaustive and non-empty, the following conditions are to be applied to U: c

∑ µ ij = 1,

i =1, 2, …, n

(4)

j =1, 2, …, c

(5)

j=1

n

∑ µ ij = 1,

i=1

µ ij ε{0, 1}

i =1, 2, …, n

j =1, 2, …, c

(6)

Condition (6) corresponds to all-or-nothing status of the membership in discrete classes. According to the fuzzy set theory, this condition is relaxed in such a way that partial memberships are allowed i.e., to take any value between and including 0 and 1. Thus, condition (6) for continuous classes becomes: µ ij ε[0, 1]

i =1, 2, …, n

j =1, 2, …, c

(7)

Any (n x c) matrix U satisfying (4), (5), and (7) represents a so-called fuzzy partition of the n-individuals into c-classes. Partitions satisfying (4), (5), and (6) are referred to as hard partitions. As condition (7) implies (6), hard partitions are special cases

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of fuzzy partitions. A special case of Fuzzy C-Means Algorithm (FCM) was first reported by Dunn in 1973. His algorithm was later generalized by Bezdek (1980), Bezdek et al., (1984) and Kent et al., (1988). These methods use an (n x p) data matrix X = (xi) as input, where p denotes the number of variables and xi denotes the value of individual i. The most popular fuzzy clustering method to date is the fuzzy c-means which is a generalization of the hard c-means clustering. The hard c-means method minimizes the function J(U, V), which represents the within-class sum-ofsquare errors between classes under conditions (4), (5), and (6): n

c

i=1

j=1

J(U, V) = ∑ ∑ µ ij d2(xi, vj)

(8)

where V=(vj) is an (c x p) matrix of centre of classes, vj denotes the value of the centre of the jth class. xi = (xi1, xi2, …, xip)T is the vector which represents the individual i, vj = (vj1, vj2, …., vjp) T is the vector which represents the centre of class j, and d2(xi, vj) is the square of the distance between xi, and vj according to a given definition of distance, further denoted by dij2 to simplify. J (U, V) is the sum of the square errors of the distance of each individual from the centre of the given classes (Kolen J. Hutcheson T, 2002). A fuzzy generalization of J (U, V) is obtained by a modification of the membership with an exponent. This weighting exponent controls the extent of membership sharing, or the “degree of fuzziness”, among the resulting clusters. Hence, a new JF (U, V) function is defined as: n

JF(U, V) = ∑

i=1

c

∑ µ ijm d2ij

(9)

j=1

This function is minimized under conditions (4), (5), and (6). The value of m is chosen from (0, ∞). If m =1, the solution of (9) is a hard partition, i.e., the result is not fuzzy. As m → ∞, the solution approaches its maximum degree of fuzziness, with µ ij = 1/c for every pair of i and j. There is no theoretical basis for an optical selection of m. It is often chosen on empirical grounds to be equal to 2. If m>1, equation (9) could be minimized by Picard iteration (Bezdek, 1984): c

µ ij = 1 / ∑ { dij / dik ) 2/(m-1) } i=1,2, …, n

j=1,2, ….., c

k=1

n

n

vj = ∑ (µ ij )m xi / ∑ (µ ij )m i=1

j=1, 2, ….., c

i=1

5 Methods This Paper has compared two models namely, Neural Network model and Fuzzy model for breast cancer diagnosis. To determine the performance of these models in

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practical usage, the database has been divided randomly into two separate sets, one for training and another for validation: (a) the training samples comprising 500 patients records (303 benign, 197 malignant) and (b) the validation samples comprising 183 patients records (141 benign, 42 malignant). Using the patients’ records in training sample the models have been trained by adjusting the weight values for interconnection limits for the neural network model and for estimating the membership function needed to establish the classification rules for fuzzy model. Then, the patients’ record in validation samples (n = 183) have been utilized to evaluate the generalizing ability of the above said models, separately. The best of these models has been compared in terms of accuracy, sensitivity, specificity, false positive and false negative.

6 Experiment and Results Out of 683 samples of Wisconsin Diagnosis Breast Cancer data set, 500 samples have been used to train the neural network using the Kohonen’s first version of Learning Vector Quantization (LVQ1) technique and the remaining 183 samples have been used to test the sample data. The learning rate parameter has been initialized to μ(k=1) = μ(1) = 0.1 and decreasing it by k at every training epoch. For example μ(2) = μ(1)/2, μ(3) = μ(2)/ 3, etc. To initialize the codebook vectors, this study uses one benign and one malignant as the first two samples. The associated classes for them are ‘1’ and ‘2’ respectively. The Matlab software has been used with the number of training epochs set to 35,000 to find the codebook vectors. If we calculate the minimum distance between each of the remaining 183 samples and the computed weight vectors of the codebook, we will know the class to which each of the remaining samples belongs. Table-1 presents the true positive (TP), false positive (FP), true negative (TN) and false negative (FN) results. Thus, having a priori known set of 444 benign and 239 malignant instances, the neural network has successfully identified 435 (96.67%) instances as negative and 224 (96.14%) instances as positive. According to these observations, Table-2 presents the sensitivity, specificity and efficiency of the neural network model using LVQ1 technique, as well as the predicted values of a positive/negative test results. As a second model we have applied fuzzy classification using fuzzy c-means algorithm to the same 683 samples of WDBC data set. Table-1 presents results that have been obtained using this model. Out of 444 benign and 239 malignant instances, the fuzzy model has successfully identified 436 (93.16%) instances as negative and 217 (96.44%) instances as positive. The same table also presents the true positive (TP), false positive (FP), true negative (TN) and false negative (FN) results. According to these observations, Table-2 shows the sensitivity, specificity and efficiency of the fuzzy model using fuzzy c-means algorithm. The table presents the predicted values of a positive/negative test results.

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Table 1. Comparative performance of the models Predicated instances Data

Training Data (500) Testing Data (183) Total Data (683)

Actual Instances

Group

Benign

303

Malignant

197

Benign

141

Malignant

42

Benign

444

Malignant

239

Neural Network

Fuzzy Logic

Positive

Negative

Positive

Negative

09 (FP) 182 (TP) 0 (FP) 42 (TP) 09 (FP) 224 (TP)

294 (TN) 15 (FN) 141 (TN) 0 (FN) 435 (TN) 15 (FN)

08 (FP) 175 (TP) 0 (FP) 42 (TP) 8 (FP) 217 (TP)

295 (TN) 22 (FN) 141 (TN) 0 (FN) 436 (TN) 22 (FN)

Table 2. Comparative results of the models Data

Measurements Sensitivity Specificity Efficiency Predictive Value (Positive) Predictive Value (Negative)

Neural Network 92.39 97.03 95.20 95.29 95.15

Fuzzy Logic 88.83 97.36 94.00 95.60 93.05

Testing Data (183)

Sensitivity Specificity Efficiency Predictive Value (Positive) Predictive Value (Negative)

100 100 100 100 100

100 100 100 100 100

Total Data (683)

Sensitivity Specificity Efficiency Predictive Value (Positive) Predictive Value (Negative)

93.72 97.97 96.49 96.14 96.67

90.80 98.20 95.60 96.44 93.16

Training Data (500)

7 Conclusion This Paper has compared two models namely neural Network model and fuzzy model for the diagnosis of breast cancer. The main aim of this Paper is to investigate which model obtains more reasonable specificity while keeping high sensitivity. The benefit is that the number of breast cancer patients for biopsy can be restricted. The seriousness of the ailment can easily be assessed. The output of the artificial neural network has yielded a perfect sensitivity of 93.72%, specificity of 97.97% and high efficiency of 96.49% for the total data set. The output of the fuzzy model has yielded a sensitivity of 90.8%, maximum specificity of 98.2% and efficiency of 95.6% for the total

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data set, demonstrating that fuzzy model also gives similar result as neural network model differentiate a malignant from a benign tumor. The results of this comparative study suggest that the diagnostic performance of neural network model is relatively better than the fuzzy model. This Paper developed a diagnostic system that performs at or above an accuracy level in any procedure short of surgery. The results have also suggested that neural networks and fuzzy models are a potentially useful multivariate method for optimizing the diagnostic validity of laboratory data. The Physicians can combine this unique opportunity extended by fuzzy and neural network models with their expertise to detect the early stages of the disease.

References 1. Barras, J.S., Vigna, L.: Convergence of Kohonen’s Learning Vector Quantization. In: International Joint Conference on Neural Networks, San Diego, CA, vol. 3, pp. 17–20 (1990) 2. Bezdek, J.: Cluster validity with fuzzy sets. J. Cybern. (3), 58–71 (1974) 3. Bezdek, J.C.: A Convergence Theorem for the Fuzzy ISODATA Clustering Algorithms. IEEE Trans. Pattern Anal. Machine Intell. PAM 1-2(1), 1–8 (1980) 4. Bezdek, J.C., Ehrlich, R., Full, W.: FCM: the fuzzy c-means clustering algorithm. Computer & Geosciences 10, 191–203 (1984) 5. Fletcher, S.W., Black, W., Harrier, R., Rimer, B.K., Shapiro, S.: Report of the International workshop on screening for breast cancer. Journal of the National Cancer Institute 85, 1644–1656 (1993) 6. Giard, R.W.M., Hermann, J.: The value of aspiration cytologic examination of the breast: A statistical review of the medical literature. Cancer 69, 2104–2110 (1992) 7. ICMR, National Cancer Registry Programme, Consolidated report of the population based cancer registries,1997 Indian Council of Medical Research, New Delhi (2001) 8. ICMR, National Cancer Registry Programme, 1981-2001, An Overview. Indian Council of Medical Research, New Delhi (2002) 9. Kent, J.T., Mardia, K.V.: Spatial Classification Using Fuzzy Memberships Models. IEEE Trans. on PAMI 10(5), 659–671 (1988) 10. Kohonen, T.: Learning Vector Quantization for Pattern Recognition. Technical Report TKK-F-A601, Helsinki University of Technology, Finland (1986) 11. Kohonen, T.: Improved Version of Learning Vector Quantization. In: Proceedings of the International Joint Conference on Neural Networks, San Diego, CA, vol. 1, pp. 545–550 (1990) 12. Kolen, J.F., Hutcheson, T.: Reducing the time complexity of the fuzzy c-means algorithm. IEEE Trans. Fuzzy Syst. 10(2), 263–267 (2002) 13. Marshall, E.: Search for a Killer: Focus shifts from fret to hormones in special report on breast cancer. Science 259, 618–621 (1993) 14. McBratney, A.B., de Gruijter, J.J.: A Continuoum Approach to Soil Classification by Modified Fuzzy k-Means with Extra-grades. Journal of Soil Sciences 43, 159–175 (1992) 15. McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. of mathematical Biophysics 5, 115–133 (1943) 16. Zadeh, L.A.: Fuzzy Sets, Information and Control, vol. 8, pp. 338–353 (1965)

A Secure Key Distribution Protocol for Multicast Communication P. Vijayakumar1, S. Bose1, A. Kannan2, and S. Siva Subramanian3 1

Department of Computer Science & Engineering, Anna University, Chennai-25 [email protected], [email protected] 2 Department of Information Science & Technology, Anna University, Chennai-25 [email protected] 3 Department of Mathematics, University College of Engineering Tindivanam, India [email protected]

Abstract. Providing efficient security method to support the distribution of multimedia multicast is a challenging issue, since the group membership in such applications requires dynamic key generation and updation which takes more computation time. Moreover, the key must be sent securely to the group members. In this paper, we propose a new Key Distribution Protocol that provides more security and also reduces computation complexity. To achieve higher level of security, we use Euler’s Totient Function and in the key distribution protocol. Therefore, it increases the key space while breaking the re-keying information. Two major operations in this scheme are joining and leaving operations for managing group memberships. An N-ary tree is used to reduce number of multiplications needed to perform the member leave operation. Using this tree, we reduce the computation time when compared with the existing key management schemes. Keywords: Multicast Communication, Key Distribution, GCD, One Way hash Function, Euler’s Totient Function, Computation Complexity.

1 Introduction Multimedia services, such as pay-per-view, videoconferences, some sporting event, audio and video broadcasting are based upon multicast communication where multimedia messages are sent to a group of members. In such a scenario, group can be either opened or closed with regard to senders. In a closed group, only registered members can send messages to this closed group. In contrast, data from any sender is forwarded to the group members in open groups. Group can be classified into static and dynamic groups. In static groups, membership of the group is predetermined and does not change during the communication. In dynamic groups, membership can change during multicast communication. Therefore, in dynamic group communication, members may join or depart from the service at any time. When a new member joins into the service, it is the responsibility of the Group Centre (GC) to disallow new members from having access to previous data. This provides backward secrecy in P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 249–257, 2011. © Springer-Verlag Berlin Heidelberg 2011

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a secure multimedia communication. Similarly, when an existing group member leaves from any group, he/she do not have access to future data. This achieves forward secrecy. GC also takes care of the job of distributing the Secret key and Group key to group members. In this paper, we propose a key Distribution scheme that reduces the computational complexity at the same time; it increases security by proving a large key space. The remainder of this paper is organised as follows: Section 2 provides the features of some of the related works. Section 3 discusses the proposed key distribution protocol and a detailed explanation of the proposed work. Section 4 shows the simulation results. Section 5 gives the concluding remarks and suggests few possible future enhancements.

2 Literature Survey There are many works on key management and key distribution that are present in the literature [1],[2],[12]. In most of the Key Management Schemes, different types of group users obtain a new distributed multicast key for every session update. Among the various works on key distribution, Maximum Distance Separable (MDS) [6] method focuses on error control coding techniques for distributing re-keying information. In MDS, the key is obtained based on the use of Erasure decoding functions [7] to compute session keys by the group members. Here, Group center generates n message symbols by sending the code words into an Erasure decoding function. Out of the n message symbols, the first message symbol is considered as a session key and the group members are not provided this particular key alone by the GC. Group members are given the (n-1) message symbols and they compute a code word for each of them. Each of the group members uses this code word and the remaining (n-1) message symbols to compute the session key. The main limitation of this scheme is that it increases both computation and storage complexity. The computational complexity is obtained by formulating lr+(n-1)m where lr is the size of r bit random number used in the scheme and m is the number of message symbols to be sent from the group center to group members. If lr=m=l, computation complexity is nl. The storage complexity is given by [log2L]+t bits for each member. L is number of levels of the Key tree. Hence Group Center has to store n ([log2L]+t ) bits. The Data Embedding Scheme proposed in [3, 4] is used to transmit rekeying message by embedding the rekeying information in multimedia data. In that scheme, the computation complexity is O(log n). The storage complexity also increases to the value of O(n) for the server machine and O(log n) for group members. This technique is used to update and maintain keys in secure multimedia multicast via media dependent channel. One of the limitations of this scheme is that a new key called embedding key has to be provided to the group members in addition to the original keys, which causes a lot of overhead. Key management using key graphs [11] has been proposed by Wong Gouda which consists of creation of secure group and basic key management graphs scheme using Star based, Tree based method. The limitation of this approach is that scalability is not achieved. A new group keying method that uses one-way functions [8] to compute a tree of keys, called the One-way Function Tree (OFT) algorithm has been proposed by David and Alan. In this method, the keys are

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computed up the tree, from the leaves to the root. This approach reduces re-keying broadcasts to only about log n keys. The major limitation of this approach is that it consumes more space. However, time complexity is more important than space complexity. In our work, we focused on reduction of time complexity. Wade Trappe and Jie Song proposed a Parametric One Way Function (POWF) [5] based binary tree key Management. Each node in the tree is assigned a Key Encrypting Key (KEK) and each user is assigned to a leaf is given the IKs of the nodes from the leaf to the root node in addition to the session key. These keys must be updated and distributed using top down or bottom up approach. The storage complexity is given by logan+2 keys for a group centre. The amount of storage needed by the individual user is given as S=aL+1-1/a-1Keys. Computation time is represented in terms of amount of multiplication required. The amount of multiplication needed to update the KEKs using bottom up approach is 1. Multiplication needed to update the KEKs using top down approach is Ctu= (a-1)logan(logan+1)/2.This complexity can be reduced substantially if the numbers of multiplications are reduced. Therefore, in this paper we propose a new N-ary tree based key Management Scheme using Euler’s Totient Function φ n and gcd φ n which provides more security by increasing the key space and reduces computation time by reducing the number of multiplications required in the existing approaches.

3 Key Distribution Protocol Based on Greatest Common Divisor and Euler’s Totient Function (Proposed Method) 3.1 GC Initialization Initially, the GC selects a large prime number P. This value, P helps in defining a multiplicative a group Zp* and a secure one-way hash function H(.). The defined function, H(.) is a hash function defined from where X and Y are nonidentity elements of Zp*. Since the function H(.) is a one way hash function, x is computationally difficult to determine from the given function Z = y x (mod p) and y. 3.2 Member Initial Join Whenever a new user i is authorized to join the multicast group for the first time, the GC sends it (using a secure unicast) a secret key Ki which is known only to the user Ui and GC. Ki is a random element in Zp*. Using this Ki the Sub Group Keys (SGK) or auxiliary Keys and a Group key Kg are given for that user ui which will be kept in the user ui database. 3.3 Rekeying Whenever some new members join or some old members leave the multicast group, the GC needs to distribute a new Group key to all the current members in a secure way with minimum computation time. When a new member joins into the service it is easy to communicate the new group key with the help of old group key. Since old

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group key is not known to the new user, the newly joining user can not view the past communication. This provides backward secrecy. Member Leave operation is completely different from member join operation. In member leave operation, when a member leaves from the group, the GC must avoid the use of old Group key/SGK to encrypt new Group key/SGK. Since old members, knows old GK/SGK, it is necessary to use each user’s secret key to perform re-keying information when a member departs from the services. In the existing key management approaches, this process increases GC’s computation time since aims only the security level. However, the security levels achieved in the existing works are not sufficient with the current computation facilities. Therefore this work focuses on increasing the security level as well as attempts to reduce the computation time. The GC executes the rekeying process in the following steps: 1.

GC defines a one way hash function h(ki,y) where ki is the users secret information, y is the users public information and computes its value as shown in equation (1). ,

2.

(1)

GC computes Euler’s Totient Function φ n [9] for the user ui using the function , as shown in equation (2). Next it can compute , for the user uj. Similarly it can compute Totient value for ‘n’ numbers of user if the message has to be sent to ‘n’ numbers of user. ,

3.

,

(2)

It defines a new function g(ki,y) which is obtained by appending , with the GCD (Greatest Common Divisor) [10] of , , , as shown in equation (3). If the total number of values produced by the func, tion is one, then by default its GCD value is considered as 1. If , , , , then , and , are said to be rela1= gcd tively prime. In such cases, the security level is equivalent to the security level provided in [5]. ,

4.

gcd

,

,

,

(3)

, , , with The purpose of concatenating the value of gcd , is to increase the security and each user can recover the original keying information. for the new GK to be GC computes the new keying information sent to the group members as shown below. ∏

5.

,

GC sends the newly computed GCD value and members.

,

(4) to the existing group

Upon receiving the GCD value and the encoded information from the GC, an authorized user u of the current group executes the following steps to obtain the new group key.

A Secure Key Distribution Protocol for Multicast Communication

1. 2. 3. 4.

253

where Ki is user’s secret key and Calculate the value , y is the old keying information which is known to all the existing users. , , Compute , . Append the received GCD value from GC in front of A legitimate user ui may decode the rekeying information to get the new group key by calculating the following value. ,

(5)

The proposed method uses the Euler’s Totient Function together with GCD function for improving security and computation time. By defining this function, it is possible to overcome the pitfall discussed in the existing key distribution protocol [5] [7] [11]. The key space in our proposed work is increased by ten times in comparison with the original key space indicated in the earlier approach. For instance, if the key values range from 1 to 10 (maximum size=10) in the existing work, the key values range from 1 to 100 (maximum size=100) in the proposed approach. This increases the key space from 101 to 102 for 8-bit keying information. Thus it provides more difficulty for an adversary to break the keying information. 3.4 Tree Based Approach Scalability can be achieved by employing the proposed approach in a key tree based key management scheme to update the GK and SGK. Fig.1. shows a key tree in which, the root is the group key, leaf nodes are individual keys, and the other nodes are auxiliary keys (SGK).

Fig. 1. Key Tree based Key management Scheme

In a key tree, the k-nodes and u-nodes are organized as a tree. Key star is a special key tree where tree degree equals group size [11]. In this paper we have discussed about N-ary based key tree (N=3) wherein the rekeying operation used for member leave case is alone considered. For Example, if a member M9 from the above figure leaves from the group, the keys on the path from his leaf node to the tree’s root should

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be changed. Hence, only the keys k 7, 9 , k 1, 9 will become invalid. Therefore, these keys must be updated. In order to update the keys, two approaches namely top-down and bottom-up are used in the members departure (Leave) operation. In the top-down approach, keys are updated from root node to leaf node. On the contrary, in the bottom-up approach, the keys are updated from leaf node to root node. When member M9 leaves from the group, GC will start to update the keys k 7, 9 , k 1, 9 using bottom-up approach. In top-down approach, the keys are updated in the order k 1, 9 , k 7, 9 . The number of multiplications required to perform the rekeying operation is high in top down approach than bottom up approach. This is evident from the complexities shown in equation (10) and (11). So it is a good choice to use bottom up approach in key tree based key management scheme. In binary tree based approach, three updating are required for a group with a size of 8 members. Only two keys are updated in N-ary tree for the group size 9. In this way, this N-ary tree approach reduces the computation time when the group has large number of group members. The working principle of the top-down approach process can be described as follows: When a member M9 leaves from the service, GC computes the Totient value for the remaining members of the group. For the computed Totient value it computes GCD value and it is sent to all the existing group members accordingly. For simplicity, GC chooses K1,9 (old Group Key) as y. Next, GC forms the message given in equation (6) and sends the message to all the existing members in order to update the new Group Key. ,

g k

,

,

, y

g k

,

, y

g k , y

g k , y

(6)

After the successful updation of Group key, GC will update K7, 8 using the formula, ,

,

g k , y

g k , y

(7)

In the bottom up approach, the updation of keys follows an organized procedure and the keys are updated using the formula, ,

∏

,

,

(8)

The next key to be updated is K1, 8. This is performed by using the formula, ,

,

g k

,

, y

g k

,

, y

g k

,

, y

(9)

After updating all the above keys successfully, data can be encrypted using the new Group Key K7, 8. Now the remaining members of the group can decrypt the data using the new Group key K7,8. In the N-ary tree based method, the number of multiplications needed to compute the re-keying information using the bottom up approach is given by the relation, 2

2

Where, B=the number of multiplications needed in bottom up approach, a= the number of children for each node of the tree, l = Level of the tree (0 …K), n= number of registered users.

(10)

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Similarly, the number of multiplications m needed for top down approach in the N--ary based method to compute th he re-keying information can be given by the relation, 1

((11)

4 Simulation Resultss The proposed method has been simulated in NS-2 for more than 500 users and we have analyzed the computation time with existing approaches to perform the rekeyying operation. The graphical reesult shown in Fig.2.is used to compare the key compuutation time of proposed meth hod with the existing methods. It compares the results obtained from GCD based keey distribution scheme with the binary tree-based, erassure encoding and some of the cryptographic methods (RC-4 and AES).

Fig. 2. GC Key Co omputation Time of various Key Distribution schemes

From this, it is observed that when the group size is 600, the key computation tiime is found to be 12µs in ourr proposed approach, which is better in comparison w with existing schemes. Moreoveer if the number of members who are joining and leavving increases the computation time t proportionately increases. However it is less in coomparison with the existing ap pproaches. Computation Time(µs)

15

Binary Tree

10

Erasure method gcd method

5

AES 0

RC-4

Group Size 100

2000

300

400

500

600

Fig. 3. User Key computation for getting the new key value

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The results shown in Fig.3.is used to compare the user’s key computation time of our proposed method with the existing methods. It compares the results obtained from GCD based key distribution scheme with existing approaches and it is observed that when the group size is 600, the key recovery time of a user found to be 2µs in our proposed approach, which is better in comparison with existing schemes.

5 Conclusion In this paper, a new N-ary tree based key distribution protocol for n bit numbers as the key value has been proposed for providing effective security in multicast communications. The major advantages of the work are provision of large key space to improve the security. In order to do that we introduced Euler’s Totient function and GCD function to achieve higher level of security. But it will increase the computation time. To overcome this, we introduced the N-ary tree based approach that reduces the computation time (computation complexity), because it takes less number of multiplications when a key is updated. Further extensions to this work are to devise techniques to reduce the storage complexity which is the amount of storage required to store the key related information, both in GC and group members’ area. Rekeying cost also plays a vital role in performing the rekeying for batch join and batch leave operations. Our next work is to device some algorithms to reduce rekeying cost for batch join and batch leave operations.

References 1. Li, M., Poovendran, R., McGrew, D.A.: Minimizing Center Key Storage in Hybrid OneWay Function based Group Key Management with Communication Constraints: Elsevier. Information Processing Letters, 191–198 (2004) 2. Lee, P.P.C., Lui, J.C.S., Yau, D.K.Y.: Distributed Collaborative Key Agreement Protocols for Dynamic Peer Groups. In: Proceedings of the IEEE International Conference on Network Protocols, p. 322 (2002) 3. Trappe, W., Song, J., Poovendran, R., Ray Liu, K.J.: Key Distribution for Secure Multimedia Multicasts via Data Embedding. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, pp. 1449–1452 (2001) 4. Poovendran, R., Baras, J.S.: An Iinformation-Theoretic Approach for Design and Analysis of Rooted-Tree-Based Multicast Key Management Schemes. IEEE Transactions on Information Theory 47, 2824–2834 (2001) 5. Trappe, W., Song, J., Poovendran, R., Ray Liu, K.J.: Key Management and Distribution for Secure Multimedia Multicast. IEEE Transactions on Multimedia 5(4), 544–557 (2003) 6. Blaum, M., Bruck, J., Vardy, A.: MDS Array Codes with Independent Parity Symbols. IEEE Transactions on Information Theory 42(2), 529–542 (1996) 7. Xu, L., Huang, C.: Computation-Efficient Multicast Key Distribution. IEEE Transactions on Parallel and Distributed Systems 19(5), 577–587 (2008) 8. David, A., McGrew, Sherman, A.T.: Key Establishment in Large Dynamic Groups using One-Way Function Trees. Cryptographic Technologies Group, TIS Labs at Network Associates (1998)

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9. Apostol, T.M.: Introduction to Analytic Number Theory,Springer International Students Edition, vol. 1, pp. 25–28 (1998) 10. Trappe, W., Washington, L.C.: Introduction to Cryptography with Coding Theory, 2nd edn., pp. 66–68. Pearson Education, London (2007) 11. Wong, C., Gouda, M., Lam, S.: Secure Group Communications using Key Graphs. IEEE/ACM Transactions on Networking 8, 16–30 (2000) 12. Zhou, Y., Zhu, X., Fang, Y.: MABS: Multicast Authentication Based on Batch Signature. IEEE Transactions on Mobile Computing 9, 982–993 (2010)

Security-Enhanced Visual Cryptography Schemes Based on Recursion Thomas Monoth and P. Babu Anto Department of Information Technology Kannur University Kannur-670567, Kerala, India [email protected]

Abstract. In this paper, we propose a security-enhanced method for the visual cryptography schemes using recursion. Visual cryptography is interesting because decryption can be done with no prior knowledge of cryptography and can be performed without any cryptographic computations. The proposed method, the secret image is encoded into shares and subshares in a recursive way. By using recursion for the visual cryptography schemes, the security and reliability of the secret image can be improved than in the case of existing visual cryptography schemes. Keywords: visual cryptography, recursive visual cryptography, visual secret sharing, visual cryptography schemes.

1 Introduction Cryptography is the study of mathematical techniques related to the aspects of information security [1]. The cryptography techniques provide information security, but no reliability. That is, having only one copy of this information means that, if this copy is destroyed there is no way to retrieve it. Replicating the important information will give greater chance to intruders to access it. Thus, there is a great need to handle information in a secure and reliable way. In such situations, secret sharing is of great relevance. An extension of secret sharing scheme is visual cryptography [2]. Visual cryptography (VC) is a kind of secret image sharing scheme that uses the human visual system to perform the decryption computations.Recursive hiding of secrets in visual cryptography and recursive threshold visual cryptography proposed by A. Parakh and S. Kak [3,4]. When Recursive threshold visual cryptography is used in network application, network load is reduced.

2 Visual Cryptography Visual cryptography, proposed by Naor and Shamir [2], is one of the cryptographic methods to share secret images (SI). In visual cryptography scheme (VCS)[5], the SI is broken up into n pieces, which individually yield no information about the SI. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 258–265, 2011. © Springer-Verlag Berlin Heidelberg 2011

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These pieces of the SI, called shares or shadows, may then be distributed among a group of n participants/dealers. By combining any k (k ≤ n) of these shares, the SI can be recovered, but combining less than k of them will not reveal any information about the SI. The scheme is perfectly secure and very easy to implement. To decode the encrypted information, i.e., to get the original information back, the shares are stacked and the SI pops out. VCS is interesting because decryption can be done with no prior knowledge of cryptography and can be performed without any cryptographic computations. 2.1 The Existing Model The existing model for black-and-white visual cryptography schemes are defined by Naor and Shamir [2],[6],[7]. In this model, both the original secret image and the share images contain only black and white pixels. The formal definition for blackand-white visual cryptography schemes is: Definition 1: A solution to the k-out-of-n visua1 cryptography scheme consists of two collections of n x m Boolean matrices C0 and C1. To share a white pixel, the dealer randomly chooses one of the matrices in C0 and to share a black pixel, the dealer randomly chooses one of the matrices in C1. The chosen matrix defines the color of the m subpixels in each one of the n transparencies. The solution is considered valid if the following three conditions are met [8][9]: 1). For any S ∈ C0, the OR m-vector V of any k of the n rows in S satisfies H (V) ≤ d – α .m. 2). For any S ∈ C1, the OR m-vector V of any k of the n rows in S satisfies H (V) ≥ d. 3). For any set {r1, r2, . . . , rt} ⊂ {1,2,. . .,n} with t
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Share1

Share2

Share1+ Share2

dealer randomly chooses one row from the first two rows of Table 1. Similarly, if the pixel in SI is white, the dealer randomly chooses one row from the last two rows of Table 1. To analyse the security of the 2-out-of-2 VCS, the dealer randomly chooses one of the two pixel patterns (black or white) from the Table1 for the shares 1 and 2. The pixel selection is random, so that the shares 1 and 2 consist of equal number of black and white pixels. Therefore, by inspecting a single share, one cannot identify the secret pixel as black or white. Therefore, this method provides perfect security. The two participants can recover the secret pixel by superimposing the two shared subpixels. If the superimposition results in two black subpixels, the original pixel was black; if the superimposition creates one black and one white subpixel, it indicates that the original pixel was white. We mathematically represent the white pixel by 0 and the black pixel by 1. For the 2-out-of-2 VCS, the basis matrices, S0 and S1 are designed as follows:

⎡1 ⎣1 ⎡1 S1= ⎢ ⎣0 S0= ⎢

0⎤ 0⎥⎦ 0⎤ 1⎥⎦

There are two collections of matrices, C0 for encoding white pixels and C1 for encoding black pixels are obtained by permuting the columns of basis matrices S0 and S1 respectively. Let C0 and C1 be the following two collections of matrices: C0 = {

⎡1 0⎤ ⎡0 1⎤ ⎡1 0⎤ ⎡0 1⎤ ⎢1 0⎥ , ⎢0 1⎥ } and C1 = { ⎢0 1⎥ , ⎢1 0⎥ } ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

To share a white pixel, the dealer randomly selects one of the matrices in C0, and to share a black pixel, the dealer randomly selects one of the matrices in C1. The first row of the chosen matrix is used for share 1(S1) and the second row is used for share 2 (S2).

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Example Figure 1 depicts example for 2-out-of-2 VCS with two subpixel layouts.

(a)

(b)

(c)

(d)

Fig. 1. (a) SI,(b) S1, (c) S2, and (d) superimposed S1 and S2

The problem that arises with this scheme is that for every pixel encoded from the original image into two subpixels and placed on each share in a horizontal or vertical fashion (here horizontal), the shares have a size of s x 2s if the secret image is of size s x s. Hence there is distortion. It is also possible to do visual cryptography with consistent share size using random basis column pixel expansion technique [10]. In this technique, the shares have a size s x s if the secret image is of the size s x s. That is, the reconstructed image is identical to the original image. 2.3 The Consistent Image Size Visual Cryptography Scheme In order to demonstrate the consistent image size visual cryptography scheme, we use 2-out-of-2 VCS with the random basis column pixel expansion technique [10]. This method, one of the columns is randomly chosen from S0 for encoding white pixels and one of the columns from S1 is taken for encoding black pixels. That is, the chosen column vector V = [vi]: V=

⎡ v1 ⎤ ⎢v 2 ⎥ ⎣ ⎦

The ith element determines the color of the pixel in the ith share image; that is, the element v1 is interpreted as a pixel in the first share image, and element v2 is interpreted as a pixel in the second share image and so on. For example, in 2-out-of-2 VCS, to share a white pixel, one of the column matrix in S0 is chosen at random. Suppose the chosen column vector from S0 as

⎡1⎤ ⎢1⎥ ⎣⎦ In the column vector, the 1st element determines the color of the pixel in the 1st shared image; that is, the element 1 represents the black pixel in the first share image and the

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2nd element 1 represents the black pixel in the second share image. Similarly, to share a black pixel, one of the column matrix in S1 is chosen at random. 2.4 Security Analysis of the Existing VCS Security is one of the most important parameters in visual cryptography schemes as a performance measure. By analyzing the security of the VC, consider 2-out-of-2 VCS with the random basis column pixel expansion technique. We have noted that the consistent image size visual cryptography scheme does not satisfy the security conditions of the traditional VCS. But this does not affect the final security of the proposed VCS; because it employs multilevel encryption method.

3 Proposed Method In order to demonstrate the proposed scheme, some experiments were conducted using 2-out-of-2 VCS. In the proposed scheme, the secret image (SI) is encoded into two shares. The each of the two shares are further encoded into two subshares in recursive manner and so on. That is, SI = { S11, S12 } S11 = { S21, S22 } S12 = { S23, S24 } . Skm = { Sk1, Sk2, Sk3 ………..,Skm} where SI be the secret image, S11 and S12 are the two subshares generated from SI. The subshare S11 and S12 again enoded into two subshares and so on. The Skm represents the subshare, where k is the level and m is the subshare number. This recursive process is repeated until the required levels of subshares are produced. The proposed model is also represented by using binary tree structures. That is: SI

Level 1 S11

S21

S12

S22

S23

S24

Level 2 Level 3

S31 S32 S33 S34 S35 S36 S37

S38

Fig. 2. Tree representation of proposed recursive method

The number of level of encryption used depends on the security and reliability requirements. Increasing the level of encryption will increase the level of security and reliability of the secret image. On the decryption process, there are different ways to reconstruct the SI. That is:

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SI = S11 + S12 SI = S21+S22+S23+S24 .

SI = Sn1+Sn2+Sn3 + ………..+Snm We mathematically represent the proposed method as: 2k

SI =

∑S i =1

ki

k≥1

where k is the level of encryption required. That is, If k = 1, then 2

SI =

∑S i =1

ki

= S11 + S12 If k = 2, then 4

SI =

∑S i =1

ki

= S21 + S22+ S23 + S24 ..

If k = n, then 2n

SI =

∑S i =1

ki

= Sn1 + Sn2+ Sn3 ……….. + Sn2n 3.1 Experimental Results The proposed scheme is feasible; experiments were conducted using 2-out-of-2 VCS with three level of encryptions (i.e. k= 3). In the first level of encryption, the secret image (SI) is encoded into two shares, S11 and S12 respectively as shown in figure 3.

(a)

(b)

(c)

(d)

Fig. 3. A 2-out-of-2 VCS: (a) SI, (b) S11, (c) S22, and (d) the superimposed S11 and S12

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In the second level of encryption, the shares, S11 and S12 are further encoded into two shares each. That is, S21, S22, S23 and S24 respectively. The experimental results are shown in figure 4.

(a)

(b)

(c)

( d)

(e)

Fig. 4. A 2-out-of-2 VCS: (a) S21, (b) S22, (c) S23, (d) S24, and (e) the superimposed four shares

In the third level of encryption, the shares, S21, S22, S23 and S24 are further encoded into two shares each. That is, S31, S32, S33,S34,S35,S36 ,S37 and S38 respectively. The experimental results are shown in figure 5.

(a)

(f)

(b)

(g)

(c)

(h)

(d)

(e)

(i)

(j)

Fig. 5. A 2-out-of-2 VCS : (a) S31, (b) S32,(c) S33, (d)S34 ,(e) S35, (f) S36,(g) S37, (h) S38, (i) the superimposed eight shares, and (j) the superimposed eight shares using XOR operations

3.2 Analysis of Experimental Results Analyze the security of the proposed scheme by comparing with traditional VCS. When we analyze the figure 2, we can see that the different levels (here three level) of encryption is used. When the number of encryption has been increased, the level of security of encrypted secret image also increased when compared with traditional VCS (in existing VCS only one level of encryption is used). By using different level of encryption, the cryptanalysis make very complex or even impossible. Also, comparing the reconstructed images in Figure 4(e) and Figure 5(i) with figure 3(d), we can see that proposed method achieves almost same contrast. In order to overcome the loss in contrast of reconstructed secret image, the proposed method for the reconstruction of secret image is based on XOR operation. This will enable one to obtain back near perfect secret image. The figure 5(j) shows decrypted secret image using XOR operation.

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4 Conclusions This paper proposes a new method for security-enhanced secret image sharing mechanism. This work shows, using a simple example, the application of 2-out-of-2 VCS. Different variations of this method are also possible. The proposed method has no pixel expansion and almost same contrast compared with traditional VCS. Using our new encryption and decryption method, any existing VCS can be easily modified. The work also leaves open the scope for future expansions which addresses emerging dependence on computers at all levels of our life. Our future work will focus on improving the contrast, the reliability etc. of the VCS proposed here.

References 1. Pieprzyk, J., Hardjono, T., Sberry, J.: Fundamentals of Computer Security. Springer, Heidelberg (2003) 2. Naor, M., Shamir, A.: Visual Cryptography. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 1–12. Springer, Heidelberg (1995) 3. Gnanaguruparan, M., Kak, S.: Recursive hiding of secrets in visual cryptography. Cryptologia 26, 68–76 (2002) 4. Parakh, A., Kak, S.: A recursive threshold visual cryptography scheme. Cryptology ePrint Archive, 535 (2008) 5. Furht, B., Muharemagic, E., Socek, D.: Multimedia Encryption and Watermarking. Springer, Heidelberg (2005) 6. Droste, S.: New results on visual cryptography. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 401–415. Springer, Heidelberg (1996) 7. Naor, M., Shamir, A.: Visual Cryptography II: Improving the Contrast Via the Cover Base. In: Security in Communication Networks, pp. 197–202 (1996) 8. Ateniese, G., Blundo, C., Santis, A.D., Stinson, D.: Constructions and bounds for visual cryptography. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 416–428. Springer, Heidelberg (1996) 9. Wang, D., Yia, F., Li, X.: On general construction for extended visual cryptography schemes. J. Pattern Recognition 42, 3071–3082 (2009) 10. Monoth, T., Babu Anto, P.: Recursive Visual Cryptography Using Random Basis Column Pixel Expansion. In: 10th IEEE International Conference on Information Technology, pp. 41–43 (2007)

Context-Aware Based Intelligent Surveillance System for Adaptive Alarm Services Ji-hoon Lim and Seoksoo Kim* Dept. of Multimedia, Hannam Univ., 133 Ojeong-dong, Daedeok-gu, Daejeon, Korea [email protected], [email protected] Abstract. In this paper, we suggest an intelligent surveillance system that detects an intruder, using images received in real-time, and offer accurate and adaptive alarm services based on context-aware technology. A surveillance system, based on only image processing, may detect a target other than an intruder as a moving object, providing users unnecessary information. In addition, face recognition may not be accurate or possible. Therefore, we designed a system that detects an intruder through context information such as positions of a camera, directions of movement, time, space and so on, as well as image processing so as to offer adaptive alarm services. Keywords: Adaptive Alarm Services, Context-Aware, Intelligent Surveillance System.

1 Introduction As violent crimes are on the rise, the demand for security systems such as CCTV and cameras is increasing as well. Most surveillance systems, however, requires an operator who keeps watching the screen and he needs to view recorded images if necessary. In order to solve such inconvenience, a lot of research efforts are being made on intelligent surveillance systems that can issue an alarm for abnormal behaviors by automatically detecting a moving object, its movement, and its route based on image processing[1,2]. Yet, existing intelligent surveillance systems have their limitations in that they detect not an intruder but other targets as a moving object or accurate face recognition is not possible, only depending on image processing. In this paper, therefore, we designed an intelligent surveillance system based on context-aware technologies as well as image processing, which can offer users adaptive alarm services and make prompt decisions. The system targets objects inside rather than outside.

2 Related Works 2.1 Detection of a Moving Object Detection of a moving object using image processing is essential to surveillance systems, and researchers are studying methods of detecting objects by dividing *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 266–270, 2011. © Springer-Verlag Berlin Heidelberg 2011

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continuous image sequences into movement areas and background areas. Largely, there are two detection methods: one using background images [3,4] and the other not using background images [5,6]. In this paper, we used the former, the different image method [7]. As described in Figure 1, the method detects a movement area by comparing the current frame and the previous frame.

Fig. 1. Difference Image

2.2 Face Recognition The face recognition in this research is used in order to accurately detect a moving object and to find out an intruder. Once a detected object is considered a person, the system defines ROI (Region of Interest) and extracts features. The extracted features are compared with the data of facial images of internal staffs saved in the intelligent surveillance system. Face recognition technologies are being applied to a wide range of areas including school attendance, bank/shop security systems, face restoration, and so on [8-10]. 2.3 Context-Aware Context-aware services are offered based on context information such as positions of a camera, movements of an object, conditions, time, space, and so on. The system using context-aware technology can make possible more intelligent surveillance services compared to a system based on only image processing and provide alarm services according to situations. Context information sensed by heterogeneous sensors require context data model for storage and management. In this paper, therefore, we used ontology-based context information modeling [11] that can collect high-level context data through inferences.

3 Intelligent Surveillance System Suggested 3.1 System Structure The intelligent surveillance systems for adaptive alarm services are comprised of a system processing server, a remote device, a camera, a heterogeneous sensor, and so on, as described in Figure 2. The system processing server analyzes intruders and offers adaptive alarm services, using context data sensed by heterogeneous sensors and image data received from cameras.

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Fig. 2. Intelligent Surveillance System Structure

Supposing that a user has a remote device allowing him to use wireless network, he can receive an alarm message from the server in case the system detects an intruder. Then, he can monitor a facility with a remote device in real-time and report to the police if necessary. 3.2 Context Information Agent for Adaptive Alarm Services Figure 3 describes the structure of context-information agent that offers context information inference services. The agent is largely divided into a context framework module and an inference engine module. The process of each module is as follows.

Fig. 3. Context Inference Process

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‘Context acquisition’ of the context framework module collects context data from heterogeneous sensors and image data analyzed by image processing. ‘Preprocess’ processes high-level context information and transmits it to an inference engine module. Then, the inference engine module saves the high-level context data in ‘Work Memory’ and compares them with predefined rules through ‘Pattern matching’. After the analysis, the results of the inference are sent back to the context framework module. The result is defined as a new pattern to be used for the next inference process.

4 Conclusion As violent crimes are on the sharp rise, the demand for security systems is increasing also. Most surveillance systems, however, requires an operator who keeps watching the screen in real-time, which has called for intelligent surveillance systems. In this paper, we designed an intelligent surveillance system that targets objects inside rather than outside, allowing users to receive accurate alarm services through context-aware technologies. The system is based on context-aware technologies as well as image processing, which makes possible accurate and smart services. However, the system requires processing of a huge amount of data, which may slow down the speed of data processing. Hence, follow-up studies need to solve the problem.

Acknowledgement This paper has been supported by the 2010 Preliminary Technical Founders Project fund in the Small and Medium Business Administration.

References 1. Moncrieff, S., Venkatesh, S., West, G.A.W.: Dynamic Privacy in Public Surveillance, vol. 42, pp. 22–28. IEEE Computer Society, Los Alamitos (2009) 2. Gorelick, L., Blank, M., Shechtman, E., Irani, M., Basri, R.: Actions as Space-Time Shapes. IEEE Trans. on Pattern Analysis and Machine Intelligence 29(12), 2247–2253 (2007) 3. Hariataoglu, I., Harwood, D., Davis, L.S.: W4: A real time surveillance of people and their activities. Proc. IEEE Trans. on Pattern Analysis and Machine Intelligence 22(8), 809–830 (2000) 4. Stauffer, C., Grimson, W.E.L.: Adaptive background mixture models for real-time tracking. In: Proc. of the IEEE International Conference on Computer Vision and Pattern Recognition, pp. 246–252 (1999) 5. Stein, A.N., Herbert, M.: Local Detection of Occlusion Boundaries in Video. Image and Vision Computing 27(5), 514–522 (2009) 6. Kim, S.H.: A Novel Approach to Moving Edge Detection Using Cross Entropy. In: GVIP 2005 Conference, pp. 418–421 (2005) 7. Piccardi, M.: Background subtraction techniques: a review. In: IEEE Int. Conf. on Systems, Man and Cybernetics, pp. 3099–3104 (2004)

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8. Lee, W.B.: A Attendance-Absence Checking System using the Self-organizing Face Recognition. The Journal of the Korea Contents Association 10(3), 72–79 (2010) 9. Hwang, B.W., Lee, S.W.: Reconstruction of Partially Damaged Face Images Based on a Morphable Face Model. IEEE Trans. on Pattern Analysis and Machine Intelligence 25(3), 365–372 (2003) 10. Yang, J., Zhang, D., Frangi, A.F., Yang, J.: Two-Dimensional PCA: A New Approach to Appearance-Based Face Representation and Recognition. IEEE Trans on Pattern Analysis and Machine Intelligence 26(1), 131–137 (2004) 11. Noy, N.F., Hafner, C.D.: The State of the Art in Ontology Design: A Survey and Comparative Review. AI Magazine 18(3), 53–74 (1997)

Modified Run-Length Encoding Method and Distance Algorithm to Classify Run-Length Encoded Binary Data T. Kathirvalavakumar and R. Palaniappan Department of Computer Science, V.H.N.Senthikumara Nadar College, Virudhunagar, 626001 Tamilnadu, India [email protected], [email protected]

Abstract. In this paper, we have proposed a modiﬁed run-length encoding (RLE) method for binary patterns. This method of encoding considers only the run-length of ’1’ bit sequences. The proposed distance algorithm considers binary patterns in the proposed encoded form and computes the distance between patterns. Handwritten digit data in the binary sequence is encoded using our proposed encoding method and the proposed distance algorithm is used to classify them in the encoded form itself. It has been observed that, there is a signiﬁcant reduction in the computation time, and the amount of memory (Run dimension) required for the proposed work. Keywords: Run-length; Encoding; Distance measure; Classiﬁcation; KNearest Neighbor.

1

Introduction

”The most valuable talent is that of never using two words when one will do” is a famous quote by Thomas Jeﬀerson, expressing the need for representing anything in a short and understandable form. Data encoding is such an important ﬁeld of study. In the ﬁeld of pattern recognition and data classiﬁcation, there is a need for advanced encoding and decoding techniques to compress and decompress massive data sets, which deals with multi-gigabyte or tera bytes of data. To process and manipulate these large volume of data sets, several operations are required for transferring them from secondary storage to primary storage and vice-versa. Even though the latest primary memories are large in storage capacity, we cannot keep the entire data set in it throughout the process of execution. Therefore, it is very much essential to use an encoding technique to encode the large data set, and process them in the encoded form itself with no loss of the original data. According to Storer [15], there are several encoding and compression techniques available, but unfortunately, no algorithm is eﬀective in compressing or encoding all types of ﬁles. According to Hsu and Zwaarico[6], dynamic Huﬀman coding works best on data ﬁles with a high variance in the frequency of individual characters (including some graphics and audio data), achieves mediocre P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 271–280, 2011. c Springer-Verlag Berlin Heidelberg 2011

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performance on natural language text ﬁles and performs poorly on high redundancy binary data. On the other hand, Run-Length Encoding works well on high - redundancy binary data, but performs poorly on text ﬁles. Sedgewick[14] implemented a straightforward RLE algorithm which compresses data by replacing contiguous occurrences of single-unit symbol(either bit or byte) by an eﬃciently coded count of these runs, usually a single occurrence of the symbol and the number of occurrences. RLE is very useful for solid black picture bits[2],[5]. RLE can be applied to any information source, provided that it contains large number of sequences of consecutive identical symbols in order to achieve good data compression. This technique is very useful and very easy to implement when long sequences of black and white pixels can be found[7],[10]. Handling binary images in runs, rather than in pixels, can bring a considerable reduction in computation because the number of runs is usually much smaller than the number of pixels [9]. Piper and Tang [12] introduced the concept of using RLE for binary morphological operations. According to Ryan [13], the low bit-depth images can be more compactly expressed using run-length representation i.e., representation where one code can represent a sequence of pixels. Search and pattern matching operations can for instance be performed more eﬀectively when data has a run-length representation. Run-length based processing can play an important role in Internet Map applications. Ryan[13] used only 5 values to represent 100 pixels, which was the key to speed-up compression and processing. Kim et al.[8] proposed fast algorithms for 3D morphological operations using RLE based morphological ﬁltering of 3D dilation or erosion, which reduced the amount of computational time over pixel-wise computation. These algorithms can process any dimensional data eﬃciently. Akimov et al.[1] extracted chain codes from a vector or raster map image of dimension 5000 * 5000 pixels by encoding it into four connected diﬀerential chain codes (CC4) or eight connected diﬀerential chain codes (CC8), and showed the ability of using the chain coding techniques for compressing map images. Zahir and Naqvi[16] proposed a near minimum sparse pattern coding scheme for binary image compression. They developed matrix ﬂipping method of block coding and showed that the method performed well when most of the non-zero elements in a matrix are on the left side of the image. Compressing the data for clustering by means of abstraction was earlier carried out by Zhang et al.[17]. Intelligent compression techniques were developed by Fung[4], which were used for clustering. Makinen et al.[11] proposed an algorithm to compute the approximate and exact edit distances of run-length encoded string. Babu et al.[3] proposed a technique of classifying the run length encoded binary data in the Compressed Data Representation (CDR) form itself. For example, in CDR the encoding of a binary pattern is performed as follows: 1) ”0111011101011110” (binary pattern) CDR – ”0131311141” —– length of run string = 10

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2) ”1100111011010011” (binary pattern) CDR – ”223121122” —– length of run string = 9 In the proposed work, we have modiﬁed the run-length encoding of the CDR by considering only the run-length of 1s in a binary pattern, and propose a new distance algorithm to compute the pattern proximity between any two binary patterns which are in the proposed encoded form. Based on these modiﬁed encoding and distance algorithm we have classiﬁed the handwritten digit data. Compared to the CDR, the proposed work has reduced the run dimension and computing time. The remainder of this paper is organized as follows: in Section 2, we have proposed our modiﬁed encoding method is described. In section 3, the distance measuring algorithm for the data in the proposed encoded form is described. In Section 4, the results of the classiﬁcation of Handwritten digit data set is described and ﬁnally we conclude in Section 5.

2

Proposed Run-Length Encoding Method

A pattern in the binary form can be encoded using RLE technique and then, it can be decoded without any loss of information. In the proposed encoding method, the run-lengths of 1s in the binary pattern are only taken for encoding and decoding with no loss of information. For the proposed encoding method, we use two important information of the binary string: the starting position of the run-length of continuous 1s, and the number of continuous 1s from that position. These two information are transformed into a single value in the encoded form. The formula for performing this encoding is: (starting position of the continuous 1s) × 100 + (number of continuous 1s). For example, in a pattern, if a bit 1 occurs continuously 12 times from a location 7, then its encoded form is 7 × 100 + 12 = 712. Suppose, if a bit 1 occur only one time at a location 5, then, this is encoded as 5 × 100 + 1 = 501. Let us consider a binary pattern, 0011110100011010 for encoding. In this binary pattern, 1s are occurring continuously in four diﬀerent places, that is at the 3rd position, 8th position, 12th position and at the 15th position. Number of individual continuous 1s in a pattern is deﬁned as Run dimension. In this example, the Run dimension is 4. The encoding is as follows: (i) Four 1s occur continuously from the position 3, its encoding is 3 × 100 + 4 = 304, (ii) Single 1 occur at the position 8, its encoding is 8 × 100 + 1 = 801, (iii) Two 1s occur continuously from the position 12, its encoding is 12 × 100 + 2 = 1202, (iv) Single 1 occur at the position 15, its encoding is 15 × 100 + 1 = 1501. For our algorithmic convenience, it has been proposed to have the last bit of a pattern to be a 1. If the pattern ends with a 0 bit, then an additional bit ’1’ must be appended with the last position of the original pattern. In the above illustration, as the bit pattern ends with a 0 bit, it must be appended with an additional bit 1 before the encoding is carried out. The last appended bit occur at a position 17 and its encoded value is 17 × 100 + 1 = 1701. Now, the Run dimension of the

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pattern is 5. The above encoded values of the pattern can be stored in an array namely ’a’ for further processing as a[1] = 304, a[2] = 801, a[3] = 1202, a[4] = 1501 and a[5] = 1701. The decoding process uses an integer division by 100 and a modulo operation by 100 to extract the encoded values. The quotient of the division operation gives the starting position of continuous 1s. The remainder of the modulo operation gives the number continuous 1s from that starting position. In the above example, let us take the ﬁrst value in the array, that is a[1] = 304. By performing integer division on a[1] we get 3 ( that is, a[1] / 100 = 304 / 100 = 3 (quotient)), the starting position, and by performing modulo division, we get 4 (that is, a[1] % 100 = 304 % 100 = 4 (remainder)), the number of continuous 1s from the starting location 3. 2.1

Modified RLE Algorithm

step 1. ﬂag =0; count = 0; j =0; n= number of bits in the pattern step 2. Read the bits of the pattern one by one step 3. If the bit = 1 and ﬂag = 0 then ﬂag =1; count ++; pos = position of that bit in the pattern; step 4. If the bit = 1 and ﬂag = 1 then count ++; step 5. If the bit = 0 and ﬂag =1 then ﬂag =0; count =0; b[j] = pos * 100 + c; j++; step 6. Repeat the steps 2 to 4 for all the bits in the pattern. step 7. If ﬂag = 0 then b[j] = (n+1) * 100 + 1; else b[j] = pos * 100 +c; step 8. End.

3

Description of the Distance Measure

The proposed distance measuring algorithm calculates the diﬀerence or the dissimilarity between any two patterns. Let p1 and p2 are encoded data of two patterns, stored in arrays ’a’ and ’b’ respectively. Each array is pointed with an individual pointer namely ap and bp respectively. The pointers of the arrays point to the starting position of continuous 1s in the respective patterns. The length of every run-length in ’a’ and ’b’ are represented by ’apl’ and ’bpl’ respectively. Based on the position of ’ap’ and ’bp’ the distance measuring procedure

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is carried out corresponding to any one of the following condition : ap = bp, ap > bp and ap < bp. When ap = bp, distance to be measured based on the following conditions: apl = bpl, apl > bpl and apl < bpl. The following illustrations explains these categories and the corresponding distance calculations. case ’A’: (when apl = bpl) p1: 001110, a[1] = 303. Here ap = 3 and apl = 3. p2: 001110, b[1] = 303. Here bp = 3 and bpl = 3. In this case, there is no dissimilarity between the run-lengths of the above two patterns. Now, the two pointers ’ap’ and ’bp’ are to be moved to the next runlength of 1s in the corresponding patterns. case ’B’: (when apl > bpl) p1: 001110, a[1] = 303. Here ap = 3 and apl = 3 p2: 00110, b[1] = 302. Here bp = 3 and bpl = 2 In this case, the pointer ’ap’ has to be moved to the position ap + bpl and the length of ’apl’ has to be decreased correspondingly, that is apl = apl − bpl. The pointer ’bp’ has to be moved to the next run-length of 1s. case ’C’: (when apl < bpl) p1: 001110, a[1] = 203. Here ap = 3 and apl = 3 p2: 0011110, b[1] = 304. Here bp = 3 and bpl = 4 In this case, ’ap’ has to be moved to the next run-length of 1s and ’bp’ has to be moved to the position bp + apl and the value of ’bpl’ has to be decreased correspondingly,that is bpl = bpl − apl . When ap > bp, distance to be measured based on the following conditions: (A) the position of ’ap’ = the last position of continuous 1s of ’b’ (B) the position of ’ap’ > the last position of continuous 1s of ’b’ (C) the position of ’ap’ < the last position of continuous 1s of ’b’ case ’A’: p1: 00001, a[1] = 501. Here ap = 5 p2: 011110, b[1] = 204. Here bp =2 and bpl = 4 In the above, there is dissimilarity between the run-length of the patterns from the starting position of ’bp’ to a position before ’ap’. This dissimilarity is calculated by ”distance = distance + (ap − bp)”. The value of ’bpl’ is decreased by ap − bp and the position of ’bp’ is moved to the position of ’ap’.

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case ’B’: p1: 0001, a[1] = 401. Here ap = 4 p2: 0110, b[1] = 202. Here bp = 2 and bpl = 2 In the above case, the dissimilarity is the length of continuous 1s from ’bp’, that is ’bpl’. The pointer ’bp’ has to be moved to the starting position of next continuous 1s. case ’C’: p1: 001, a[1] = 301. Here ap = 3 p2: 011110, b[1] = 204. Here bp = 2 and bpl = 4 In the above case, there is dissimilarity between the run-length of the patterns from the starting position of ’bp’ to a position before ’ap’. The length of ’bpl’ has to be decreased by ap − bp and the pointer ’bp’ has to be moved to the position of ’ap’. When ap < bp, distance to be measured based on the following conditions: (A) the position of ’bp’ = the last position of continuous 1s of ’a’ (B) the position of ’bp’ > the last position of continuous 1s of ’a’ (C) the position of ’bp’ < the last position of continuous 1s of ’a’ case ’A’: p1: 011110, a[1] = 204. Here ap = 2 and apl = 4 p2: 00001, b[1] = 501. Here bp =5 In the above, there is dissimilarity between the run-length of the patterns from the starting position of ’ap’ to a position before ’bp’. This dissimilarity is calculated by ”distance = distance + (bp − ap)”. The value of ’apl’ is decreased by bp − ap and the position of ’ap’ is moved to the position of ’bp’. case ’B’: p1: 01110, a[1] = 203. Here ap = 2 and apl = 3 p2: 00001, b[1] = 501. Here bp = 5 In the above case, the dissimilarity is the length of continuous 1s from ’ap’, that is ’apl’. The pointer ’ap’ has to be moved to the starting position of next continuous 1s. case ’C’: p1: 011110, a[1] = 204. Here ap = 2 and apl = 4 p2: 001, b[1] = 301. Here bp = 3

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In the above case, there is dissimilarity between the run-length of the patterns from the starting position of ’ap’ to a position before ’bp’. The length of ’apl’ has to be decreased by bp − ap and the pointer ’ap’ has to be moved to the position of ’bp’.

4

Results and Discussion

The proposed modiﬁed RLE and the corresponding distance measure algorithms are demonstrated by classifying the Handwritten Digit Data set comprising of 10 digits ( 0 - 9 ). We have taken 10000 patterns each with 192 attributes. Among these, 6670 patterns are used for training and 3330 patterns are used for testing, with equal proportion of patterns in each digit. The above data set is classiﬁed using the following steps: Step 1: All training patterns from the data ﬁle are read one by one and the patterns are encoded using the proposed algorithm and stored in an array. Step 2: Read a test pattern from the ﬁle and encode it using the proposed algorithm and store it in an array. Step 3: Classify the encoded test pattern using the K-Nearest Neighbor (KNN) algorithm. Proposed distance measure algorithm is used to ﬁnd the pattern proximity of the data in the encoded form. Step 4: Step 2 and Step 3 are repeated for all the test patterns. The above procedure is repeated for diﬀerent K values (1 - 20) of KNN for the Handwritten Digit Data. The classiﬁcation accuracy(CA) obtained for runlength encoded binary data of Babu et al.[3] using KNN algorithm is matched with the proposed procedure. It has been found that the CA of proposed procedure with modiﬁed RLE is same as Babu et al.[3]. Fig. 1 shows the CA obtained for diﬀerent K values of the proposed procedure. The best CA obtained is 92.52% for k=7. The processing time of the proposed procedure is comparatively lesser than Babu et al.[3]. when executed in the Intel Core2Duo 2.66GHz system. The time requirement for KNN classiﬁcation using the proposed modiﬁed RLE and the Run-length encoded CDR of Babu et al.[3]. are shown in Fig. 2. The average memory requirement of the proposed modiﬁed RLE and the CDR of Babu et al.[3] for each digit of the Handwritten Digit Data are shown in Fig. 3. It is found that, a signiﬁcant amount of data size has been reduced by virtue of the proposed encoding technique. Handwritten Digit Data set needs totally 1280640 number of memory locations (6670 x 192), but the proposed encoded representation occupies only 139054 number of memory locations(Run dimension). It has also been observed that the memory requirement(Run dimension) of the proposed

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93

92.5 CA (%) 92

Classification Accuracy (% )

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91

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88 0

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Fig. 3. Memory Requirement - Proposed vs. CDR Data

encoded data is approximately 50% of the memory requirement(Run dimension) of the run-length encoded CDR of Babu et al.,[3]. In Table 1 we have shown the Run dimension (memory requirement) for the proposed RLE and CDR of Babu et al., [3] with six diﬀerent possible binary patterns.

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Table 1. Comparison of Proposed RLE and the CDR in terms of Run Dimension Sl. No. Pattern CDR Run Proposed RLE Run Run (16 Bits) String Run String Dimension Dimension CDR Proposed 1 111000111 {3,3,5,2,3} {103,705,1403} 5 3 1100111 2 010101100 {0,1,1,1,1, {201,401,602, 12 6 1101000 2,2,2,1,1,3} 1002,1301,1701} 3 101010101 {1,1,1,1,1, {101,301,501, 16 9 0101010 1,1,1,1,1, 701,901,1101, 1,1,1,1,1,1} 1301,1501,1701} 4 010101010 {0,1,1,1,1, {201,401,601, 17 8 1010101 1,1,1,1,1,1 801,1001,1201, 1,1,1,1,1,1} 1401,1601} 5 000000000 {0,16} {1701} 2 1 0000000 6 111111111 {16} {116} 1 1 1111111

5

Conclusion

An eﬃcient modiﬁed run-length encoding technique for classifying binary data is proposed. In this method, only the run-lengths of 1’s of a pattern is considered. Pattern proximity is computed over the run-length of encoded data using the proposed distance algorithm for classifying binary data. Proposed encoding method and corresponding distance algorithm are used to classify handwritten digit data set. KNN algorithm is used for classiﬁcation. The tabulated values and ﬁgures show the eﬃciency of the proposed method in terms of memory and time.

References 1. Akimov, A., Kolesnikov, A., Franti, P.: Lossless compression of map contours by context tree modelling of chain codes. Pattern Recognition 40, 944–952 (2007) 2. Al-laham, M., Emary, I.: Comparative study between various algorithms of data compression techniques. IJCSNS, International Journal of Computer Science and Network Security 7(4), 284–290 (2007) 3. Babu, T.R., Murty, N., Agarwal, V.K.: Classiﬁcation of run-length encoded binary data. Pattern Recognition 40, 321–323 (2007) 4. Fung, B.C.M.: Hierarchical document clustering using frequent item-sets, Thesis submitted in Simon Fraser University (1999), http://citeseer.ist.psu.edu/581906.html 5. Gilbert, H.: Data and image compression tool and techniques, pp. 68–343. Wiley, Chichester (1996) 6. Hsu, H.W., Zwaarico, E.: Automatic synthesis of compression techniques for heterogeneous ﬁles. Software-Practice and Experience 25(10), 1097–1116 (1995)

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7. Jones, C.B.: An eﬃcient coding system for long source sequence. IEEE Trans. Information Theory 27, 280–291 (1981) 8. Kim, W.-J., Kim, S.-D., Radha, H.: 3D binary morphological operations using run-length representation. Signal Processing: Image Communication 23, 442–450 (2008) 9. Kim, W., Kim, S., Kim, K.: Fast algorithm for binary dilation and erosion using run-length encoding. ETRI Journal 27, 814–817 (2005) 10. Langdon, G.G., Rissannen, J.J.: A simple general binary source code. IEEE Trans. Information Theory 29, 858–867 (1982) 11. Makinen, V., Navano, G., Ukkonen, E.: Approximate matching of run-length compressed strings. Algorithmica 35(4), 347–369 (2003) 12. Liang, J., Piper, J., Tang, J.Y.: Erosion and Dilation of binary images by arbitrary structuring elements using interval coding. Pattern Recognition Letters 9, 201–209 (1989) 13. Ryan, O.: Runlength-based Processing Methods for Low Bit-depth Images. IEEE Transactions on Image Processing 18, 2048–2058 (2009) 14. Sedgewick, R.: Algorithms, 2nd edn. Addision-Wesly, Reading (1988) 15. Storer, J.A.: Data compression: Methods and Theory. Computer Science Press, Rockville (1988) 16. Zahir, S., Naqvi, M.: A Near Minimum Sparse Pattern Coding Based Scheme for Binary Image Compression. In: IEEE-ICIP 2005, Genoa, Italy (September 2005) 17. Zhang, T., Ramakrishnan, R., Linvy, M.: BIRCH: An eﬃcient data clustering method for very large databases. In: Proceedings of ACM SIGMOD, International Conference on Management of Data. ACM Press, New york (1996)

A Fast Fingerprint Image Alignment Algorithms Using K-Means and Fuzzy C-Means Clustering Based Image Rotation Technique P. Jaganathan and M. Rajinikannan Department of Computer Applications, P.S.N.A. College of Engineering and Technology, Dindigul, Tamilnadu-624622, India [email protected], [email protected]

Abstract. The Fingerprint recognition system involves several steps. In such recognition systems, the orientation of the fingerprint image has influence on fingerprint image enhancement phase, minutia detection phase and minutia matching phase of the system. The fingerprint image rotation, translation, and registration are the commonly used techniques, to minimize the error in all these stages of fingerprint recognition. Generally, image-processing techniques such as rotation, translation and registration will consume more time and hence impact the overall performance of the system. In this work, we proposed fuzzy c-means and k-mean clustering based fingerprint image rotation algorithm to improve the performance of the fingerprint recognition system. This rotation algorithm can be applied as a pre-processing step before minutia detection and minutia matching phase of the system. Hence, the result will be better detection of minutia as well as better matching with improved performance in terms of time. Keywords: k-means clustering, fuzzy c-means clustering, Fingerprint Image Enhancement, Fingerprint Image Registration, Rotation, Alignment, Minutia Detection.

1 Introduction Fingerprint recognition is part of the larger field of Biometrics. Other techniques of biometrics include face recognition, voice recognition, hand geometry, retinal scan, ear surface and so on. 1.1 The Fingerprint Image and Minutia A fingerprint is the feature pattern of one finger. Fingerprints are believed to be unique across individuals, and across fingers of the same individual [10]. Fingerprintbased personal identification has been used for a very long time [7]. It is believed that each person has his own fingerprints with the permanent uniqueness. Hence, fingerprints have being used for identification and forensic investigation for a long P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 281–288, 2011. © Springer-Verlag Berlin Heidelberg 2011

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time. Minutia points occur either at a ridge ending or a ridge bifurcation. A ridge ending is defined as the point where the ridge ends abruptly and the ridge bifurcation is the point where the ridge splits into two or more branches. A fingerprint expert is often able to correctly identify the minutia by using various visual clues such as local ridge orientation, ridge continuity, ridge tendency, etc., as long as the ridge and furrow structures are not corrupted completely [1].

Fig. 1. The Ridge Endings and Bifurcation

Fig. 2. The Characteristic of a Minutia

The above diagram shows the clear view of a minutia and its characteristic attributes. 1.2 The Importance of Fingerprint Image Alignment Algorithms A typical fingerprint recognition system, have several types of algorithms in each stage. We can broadly reduce these stages of algorithms as Fingerprint Image Enhancement (Ridge and Valley Enhancement) Algorithms, Minutia Detection Algorithms and Minutia Matching Algorithms. Fingerprint Image alignment algorithm is the most important step, since the results of Minutia extraction Algorithms and the Minutia matching Algorithms are very much depend on the clarity and orientation of the fingerprint image. Point pattern matching is generally intractable due to the lack of knowledge about the correspondence between two point’s sets. To address this problem, Jain et al. proposed alignment-based minutia matching [4], [5]. Two sets of minutia are first aligned using corresponding ridges to find a reference minutia pair, one from the input fingerprint and another from the template fingerprint, and then all the other minutia of both images are converted with respect to the reference minutia. Jiang et al. also proposed an improved method [6], using minutia together with its neighbors to find the best alignment. In general, minutia matcher chooses any two minutias’s as a reference minutia pair and then matches their associated ridges first. If the ridges match well [11], two fingerprint images are aligned and matching is conducted for all remaining minutia. In any of this matching policy, the fingerprint image alignment will play significant role and will have influence on accuracy. In the following sections, the previous work on fingerprint alignment and the proposed algorithms are discussed.

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2 Previous Works 2.1 Registration Using Mutual Information The Registration based method called ‘Automatic Image Registration’ by Kateryna Artyushkova, University of New Mexico, has been used for comparing the performance of the proposed fuzzy c-means and k-means based image alignment algorithm. Registration or alignment is a process through which the correct transformation is determined. It uses mutual information as the similarity measure and aligns images by maximizing mutual information between them under different transformations [8]. Mutual information describes the uncertainty in estimating the orientation field at a certain location given another orientation field. The more similar or correlated the orientation fields are, the more mutual information they have. MI measures the statistical dependency between two random variables. The physical meaning of MI is the reduction in entropy of Y given X. Viola et al. proposed [12] that registration could be achieved by maximization of mutual information. In this paper, we apply MI to fingerprint registration. Here we use MI between template and input’s direction features to align the fingerprints. In this registration algorithm, we assume that distortion is not very large. 2.2 The Proposed Fingerprint Image Alignment Algorithms 2.3 Pre-processing the Input Fingerprint Image Initially, the input fingerprint image is pre–processed by changing the intensity value to 0 for the pixel, whose intensity value is less than 32. Now, only the ridges and valleys of the fingerprint image are considered. Except ridges and valleys the other parts of the fingerprint image pixels intensity values are changed to 0. Finally the binary image will be constructed. The co-ordinates of (x,y) the one valued pixels are considered as two attributes and passed as Input to the K-Means algorithm. 2.4 K-Means Algorithm K-Means algorithm is very popular for data clustering. Generally, K-Means algorithm is used in several iterations to cluster the data since the result is very much depend on the initial guess of the cluster centres. The Algorithm goes like this 1. Start iteration 2. Select k Center in the problem space (it can be random). 3. Partition the data into k clusters by grouping points that are closest to that k centers. 4. Use the mean of these k clusters to find new centers. 5. Repeat steps 3 and 4 until centers do not change, for N iterations. 6. Among the N results, find the result with minimum distance. 7. Display the results corresponding to that minimum distance. 8. Stop iterations.

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2.5 Fuzzy C-Means Algorithm Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. This method was developed by Dunn [3] in 1973 and improved by J.C. Bezdek [2] in 1981 is frequently used in pattern recognition. Fuzzy c-means (FCM) is a data clustering technique wherein each data point belongs to a cluster to some degree that is specified by a membership grade. The Fuzzy c-means algorithm starts with an initial guess for the cluster centers. The initial guess for these cluster centers is most likely incorrect. By iteratively updating the cluster centers and the membership grades for each data point, Fuzzy c-means algorithm iteratively moves the cluster centers to the right location within a data set. This iteration is based on minimizing an objective function that represents the distance from any given data point to a cluster center weighted by that data point's membership grade. The Algorithm is composed of the following steps. 1. 2.

Initialize U=[uij] matrix, U(0) At k-step: calculate the centers vectors C(k)=[ c j ] with U(k)

3. 4.

Update U(k) , U(k+1) If || U(k+1) - U(k)||< ε then STOP; otherwise return to step 2.

Finally, the above algorithm gives the two cluster points C1 and C2. After getting the cluster points C1 and C2 the following steps will be followed to rotate the image to 90 degrees. 2.6 K-Means and Fuzzy C-Means Algorithm for Fingerprint Image Rotation Let us consider the pixels of the two-dimensional finger print image as the plotted three data points of x, y, Gray level. 1. The fingerprint image pixels are assumed as data points in 2D space.

2. The data points were clustered in to two groups using k-means and fuzzy c means clustering algorithm. The green line showing the cluster boundary. 3. The points C1 and C2 are the centers of the two clusters.

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4. A line connecting C1 and C2 will be almost equal to the inclination of the fingerprint image. 5. The inclination angle θ can be measured from the base of the image. θ = atan ( (x1-x2) / (y1-y2 ) ) (in radians) θ = θ *(180/pi) (in degree) if θ < 0 θ = 90+ θ else θ = - ( 90- θ) end 6. Now rotating the image by angle θ. 7. The direction of rotation can be decided with respect to the location of the point C1 in the top two quadrants of the four Quadrants. 8. This will finally give the well aligned image.

3 Implementation Results and Analysis The following diagram shows the implemented model of the fingerprint alignment system

Read Input Fingerprint Image

Enhance the Fingerprint Image

Estimate the Inclination angle using k-means and fuzzy c-Means

Rotate the image by angle θ

Fig. 3. The fingerprint alignment system

3.1 Fingerprint Database Used for Evaluation A fingerprint database from the FVC2000 [9] (Fingerprint Verification Competition 2000) is used to test the experiment performance. FVC2000 was the First International Competition for Fingerprint Verification Algorithms. This initiative is organized by D. Maio, D. Maltoni, R. Cappelli from Biometric Systems Lab (University of Bologna), J. L. Wayman from the U.S. National Biometric Test Center (San Jose State University) and A. K. Jain from the Pattern Recognition and Image Processing Laboratory of Michigan State University. Sets of few selected images from the above database were used to evaluate the performance of the algorithms. 3.2 Sample Set of Results The following outputs show some of the inputs as well as the corresponding outputs. In the second column images, the angle of rotation estimated by the k-means based algorithm also given below the image.

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k-means : θ = 26.5651 FCMeans : θ = 26.9395

k-means : θ = -14.0362 FCMeans : θ = -13.4486

k-means : θ = -33.2936 FCMeans : θ = -33.6901

k-means : θ = -6.9343 FCMeans : θ = -7.1250

k-means : θ = 21.9672 FCMeans : θ = 23.0026 Fig. 4. The Results of Rotation

The following tables show the performance of the algorithms in terms of time. The registration based image alignment algorithm will use two images. One will be considered as reference image to which that other image has to be aligned. The registration-based method has been used for comparing the performance of the proposed fuzzy c-means and k-means based image alignment algorithm. The results of all the three algorithms are shown in the following table.

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Table 1. The average time taken for Alignment

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Table 1 shows the average time taken (in seconds) by the registration based method, proposed K-Means Algorithm and Fuzzy C-Means Algorithm. The result shows that the average time taken by the proposed K-Means Algorithm of fingerprint alignment is approximately 0.946 seconds, whereas the Fuzzy C-Means Algorithm takes 2.558 seconds and the registration based method takes 17.6 seconds. The following chart compares the average time taken for alignment of the fingerprint image.

Time(in seconds)

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Fig. 5. The average time taken for Alignment

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4 Conclusion and Future Work We have successfully implemented and evaluated the proposed k-means and fuzzy cmeans based fingerprint image alignment algorithm under Matlab 6.5 on Intel Core 2 Duo processor and RAM size is 2 GB DDR2. Its performance is compared with normal registration based alignment algorithm. The arrived results were significant and more comparable. It is clear that the performance of k-means algorithm is better than that of Registration based and fuzzy c-means algorithms. If we use the aligned fingerprint image for the detection of minutia and minutia matching, then we may expect better accuracy in recognition. Future works may evaluate the difference in recognition with and without the proposed fingerprint image alignment phase.

References 1. Hong, L., Wan, Y., Jain, A.K.: Fingerprint image enhancement: Algorithm and performance evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 777– 789 (1998) 2. Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981) 3. Dunn, J.C.: A Fuzzy Relative of the ISODATA Process and its use in Detecting Compact Well – Separated Clusters. Journal of Cybernetics 3, 32–57 (1973) 4. Jain, A.K., Hong, L., Pankanti, S., Bolle, R.: An Identity-authentication system using fingerprints. Proc. IEEE 85(9), 1365–1388 (1997) 5. Jain, A.K., Hong, L., Bolle, R.: On-Line fingerprint verification. IEEE Trans. Pattern Anal. Mach. Intell. 19(4), 302–314 (1997) 6. Jiang, X., Yau, W.: Fingerprint minutia matching based on the local and global structures. In: Proc. 15 th Int. Conf. Pattern Recognition, Barcelona, Spain, vol. 2, pp. 1042–1045 (September 2000) 7. Lee, H.C., Gaensslen, R.E. (eds.): Advances in Fingerprint Technology. Elsevier, New York (1991) 8. Liu, L., Jiang, T., Yang, J., Zhu, C.: Fingerprint Registration by Maximization of Mutual Information. IEEE Transactions on Image Processing 15, 1100–1110 (2006) 9. Maio, D., Maltoni, D., Cappelli, R., Wayman, J.L., Jain, A.K.: FVC2000: Fingerprint Verification Competition. In: 15th ICPR International Conference on Pattern Recognition, Spain, September 3-7 (2000), http://bias.csr.unibo.it/fvc2000/ 10. Pankanti, S., Prahakar, S., Jain, A.K.: On the individuality of fingerprints. IEEE Trans. Pattern and Mach. Intell. 24, 1010–1025 (2002) 11. Singh, R., Shah, U., Gupta, V.: Fingerprint Recognition, Student project, Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, India (November 2009) 12. Viola, P.: Alignment by Maximization of Mutual Information. Ph. D thesis, M.I.T. Artificial Intelligence Laboratory (1995)

A Neuro Approach to Solve Lorenz System J. Abdul Samath1, P. Ambika Gayathri1, and A. Ayisha Begum2 1

Department of Computer Applications, Sri Ramakrishna Institute of Technology, Coimbatore-641010, Tamilnadu, India [email protected], [email protected] 2 Department of Information Technology, VLB Janakiammal College of Engineering and Technology, Coimbatore-641008, Tamilnadu, India [email protected]

Abstract. In this paper, Neural Network algorithm is used to solve Lorenz System. The solution obtained using neural network is compared with Runge-Kutta Butcher (RK Butcher) method and it is found that neural network algorithm is efficient than RK method. Keywords: Lorenz System, Runge-Kutta Butcher method, Neural Network.

1 Introduction The earth’s atmosphere is approximately a system of Rayleigh-Benard heat convection that is a layer of radially flowing fluid bounded by two boundaries of different temperature: the warm crust and the chilling upper stratosphere. In 1963, Edward Lorenz attempted to model this convection occurring in the earth’s atmosphere as a simple set of three first-order non-linear ordinary differential equations, called the Lorenz system. Like the weather, this system is highly sensitive to its initial conditions. Due to the non-linearity of the equations, an exact form of the solution cannot be calculated. Instead, numerical methods, such as Euler’s and Runge-Kutta methods, are employed to calculate approximations to the system solution through iteration. Morris Bader [2, 3] introduced the RK-Butcher algorithms for finding the truncation error estimates and intrinsic accuracies and the early detection of stiffness in coupled differential equations that arises in theoretical chemistry problems. Most recently, Murugesan et al [1] and Park et al [4] applied the RK-Butcher algorithm for an industrial robot arm control problem and optimal control of linear singular systems. Many methods have been developed so far for solving differential equations. Some of them produce a solution in the form of an array that contains the value of the solution at a selected group of points. Others use basis-functions to represent the solution in analytic form and transform the original problem usually to system of algebraic equations. The solution of a linear system of equations is mapped onto the architecture of a Hopfield neural network. The minimization of the network’s energy function provides the solution to the system of equations [5, 6, and 7]. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 289–296, 2011. © Springer-Verlag Berlin Heidelberg 2011

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The dynamic of a physical system is usually determined by solving a differential equation. It normally proceeds by reducing the differential equation to a familiar form with known solutions using suitable coordinate transformations [8, 9]. In nontrivial cases however, finding the appropriate transformations is difficult and numerical methods are used. A number of algorithms (e.g., Runge-Kutta, finite difference, etc.) are available for calculating the solution accurately and efficiently [10]. Neural network or simply neural nets are computing systems, which can be trained to learn a complex relationship between two or many variables or data sets. Having the structures similar to their biological counterparts, neural networks are representational and computational models processing information in a parallel distributed fashion composed of interconnecting simple processing nodes. In this paper, Neural Network algorithm is used to solve Lorenz System. The solution obtained using neural network is compared with Runge-Kutta Buthcer(RK Butcher) method and it is found that neural network algorithm is efficient than RK Butcher method.

2 Statement of the Problem The Lorenz model is a dissipative system, with some property of the flow, such as total energy is conserved. It consists of three first-order nonlinear ordinary differential equations with no exact solution.

.

.

.

. .

(1)

.

where x,y,z are respectively convective velocity, the temperature differences between descending and ascending flow and the mean convective heat flow. The parameter Pr, R and b are kept constant within integration, but they can be changed to form a family of solutions of the dynamical system defined by the differential equations. The particular values chosen by Lorenz were Pr=10, R=28, and b=8/3, which result in nonperiodic or chaotic solutions.

3 Runge - Kutta Butcher Solution RK Butcher algorithm is normally considered as sixth order, since it requires six functions evaluation. Even though it seems to be sixth order method, it is only a fifth order method. The accuracy of this algorithm is considered better than other algorithms. The system of non-linear differential equation (1) is solved as follows: 1

90 1

7 1

90

71

32 3

12 4

35 5

32 3

12 4

35 5

7 6 76

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291

Where , ,

,

, ,

, ,

,

,

, ,

8 2

, ,

,

2 3

, ,

1

,

9 16 1 9 16 1 2

3

,

2 2

3, 1

3

12 8

3,

2 2

3 , 4 ,

, 2

1

,

2 ,

,

, 4

,

,

3 4

,

4 2

,

3

,

1

9

4

16

12

3

12

4

8

5

7 3

1

2

2

12

3

12

4

,

8

, 5

,

7 3

1

2

2

12

3

12

4

8

5

7

4 Neural Network Solution In this approach, new feedforward neural network is used to change the trail solution of Eq (1) to the neural network solution of (1). The trial solution is expressed as the difference of two terms as below. Vi ,Ij a

,

-

(2)

The first term satisfies the TCs and contains no adjustable parameters. The second correspond to the term employs a feedforward neural network and parameters weights of the neural architecture. Consider a multi layer perception with n input units, one hidden layer with n sigmoidal units and a linear output unit. The extension to the case of more than one hidden layer can be obtained accordingly. For a given input vector, the output of the network is Nij = ∑ where zi =∑ hidden unit i,

i)

(zi)

, denotes the weight from the input unit j to the ij)tj + denotes the weight from the hidden i to the output, denotes the

bias of the hidden unit i and

is the sigmoidal transfer function.

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The error quantity to be minimized is given by E =∑ ,

i,

j)a

-

ij

(t, (

i,

j)a))

2

.

(3)

The neural network is trained till the error function (3) becomes zero. Whenever E becomes zero, the trail solution (2) becomes the neural network solution of Eq (1). 4.1 Structure of the FFNN The architecture consists of n input units, one hidden layer with n sigmoidal units and a linear output. Each neuron produces its output by computing the inner product of its input and its appropriate weight vector.

Fig. 1. Neural Network Architecture

During the training, the weights and biases of the network are iteratively adjusted by Nguyen and Widrow rule. The neural network architecture is given in Fig. 1 for computing Nij. The neural network algorithm was implemented in MATLAB on a PC, CPU 1.7 GHz for the neuro computing approach. Neural Network Algorithm Step 1: Feed the input vector tj Step 2: Initialize randomized weight matrix wij and bias ui. Step 3: Compute =∑ Step 4: Pass zi into n sigmoidal functions. Step 5: Initialize the weight vector vi from the hidden unit to output unit. Step 6: Calculate Nij=∑ Step 7: Compute purelin function (Nij) Step 8: Repeat the neural network training until the following error function E=∑ ,

i,

j) a -

ij

(t,(

i,

j) a))

2

= 0.

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5 Results and Discussion MATLAB Neural Network Toolbox has been used to solve the Lorenz system where Pr and b is set to be 10 and 8/3 respectively for which the initial condition is given by x(0)=-15.8, y(0)=-17.48 and z(0)=35.64. The methods were examined in time range [0, 1] with two time steps ∆t=0.01 and ∆t=0.001. 5.1 Non-chaotic Solution For R=23.5, we determine the accuracy of NN, and RK Butcher with time steps ∆t=0.01 and ∆t=0.001. The results are presented in Table1-3 and the corresponding graph in Fig. 2-4. Table 1. x-direction difference between all methods for R=23.5 x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

NN0.01 − NN0.001 0.00E+00 5.64E-02 2.23E-02 5.05E-03 7.37E-03 6.76E-04 1.27E-02 1.13E-02 1.94E-02 2.85E-02

Absolute error for x

RKButcher0.01 − RKButcher0.001 0.00E+00 1.92E-01 7.95E-02 1.24E-02 2.03E-02 2.33E-03 4.39E-02 3.44E-02 7.00E-02 9.68E-02

1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01

Absolute error

t

NN

1.00E-02

RK Butcher

1.00E-03

1.00E-04 Time, t

Fig. 2. Absolute error of all methods for x

Table 2. y-direction difference between all methods for R=23.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 4.52E-02 1.60E-02 1.04E-02 2.44E-03 8.88E-03 2.02E-02 1.05E-02 4.02E-02 2.15E-02 3.91E-03

y RKButcher0.01 − RKButcher0.001 0.00E+00 1.52E-01 5.35E-02 3.14E-02 3.69E-03 3.41E-02 6.64E-02 4.31E-02 1.38E-01 6.83-02 8.39E-03

Absolute error for y 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

t

NN RK Butcher 1.00E-02

1.00E-03 Time,t

Fig. 3. Absolute error of all methods for y

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Table 3. z-direction difference between all methods for R=23.5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3.25E-02 2.17E-02 2.19E-02 1.88E-02 6.75E-03 3.81E-02 1.79E-03 3.34E-02 4.13E-02

1.15E-01 7.52E-02 7.30E-02 5.63E-02 3.18E-02 1.28E-01 1.83E-03 1.18E-01 1.40E-01

t

Absolute error for z 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

0 0.1

NN0.01 − NN0.001 0.00E+00 5.78E-02

z RKButcher0.01 − RKButcher0.001 0.00E+00 2.07E-01

NN RK Butcher

1.00E-02

1.00E-03 Time, t

Fig. 4. Absolute error of all methods for z

5.2 Chaotic Solutions

For a chaotic system, R=28, it can be seen that once again NN shows a better accuracy as compared to the other two methods for simple time steps ∆t=0.01 and ∆t=0.001. This can be observed from Table 4-6 and Fig 5-7 below: Table 4. x-direction difference between all methods for R=28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 4.53E-02 2.76E-02 1.61E-03 2.97E-03 2.19E-03 4.75E-03 2.86E-02 4.47E-02 7.33E-02 3.85E-02

Absolute error for x 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01

Absolute error

t

x RKButcher0.01 − RKButcher0.001 0.00E+00 1.53E-01 9.73E-02 8.93E-03 5.94E-03 3.37E-04 2.57E-02 9.64E-02 1.62E-01 2.50E-01 1.29E-01

NN

1.00E-02

RK Butcher

1.00E-03

1.00E-04 Time, t

Fig. 5. Absolute error of all methods for x

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Table 5. y-direction difference between all methods for R=28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 5.48E-02 6.86E-03 4.84E-03 7.05E-04 2.26E-03 1.62E-02 9.18E-03 1.28E-01 3.86E-02 2.13E-02

y RKButcher0.01 − RKButcher0.001 0.00E+00 1.84E-01 2.33E-02 1.44E-02 2.69E-03 1.81E-02 6.72E-02 2.07E-02 4.33E-01 1.24E-01 6.60E-02

Absolute error for y 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

t

NN

1.00E-02

RK Butcher

1.00E-03

1.00E-04 Time,t

Fig. 6. Absolute error of all methods for y

Table 6. z-direction difference between all methods for R=28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 4.62E-02 4.71E-02 3.09E-02 2.42E-02 2.08E-02 1.33E-02 6.82E-02 2.24E-03 8.21E-02 6.61E-02

Absolute error for z 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

t

z RKButcher0.01 − RKButcher0.001 0.00E+00 1.64E-01 1.65E-01 1.08E-01 8.23E-02 6.58E-02 2.77E-02 2.48E-01 1.09E-02 2.87E-01 2.28E-01

NN RK Butcher

1.00E-02

1.00E-03 Time, t

Fig. 7. Absolute error of all methods for z

References 1. Murugesan, K., Sekar, S., Murugesh, V., Park, J.Y.: Numerical solution of an industrial robot arm control problem using RK-Butcher algorithm. International Journal of Computer Applications in Technology 19, 132–138 (2004) 2. Bader, M.: A comparative study of new truncation error estimates and intrinsic accuracies of some higher order Runge-Kutta algorithms. Comput. Chem. 11, 121–124 (1987) 3. Bader, M.: A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge-Kutta algorithms. Theor. Chem. Acc. 99, 215–219 (1998) 4. Park, J.Y., Evans, D.J., Murugesan, K., Sekar, S., Murugesh, V.: Optimal control of singular systems using the RK -Butcher algorithm. Intern. J. Comp. Math. 81(2), 239–249 (2004)

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5. Lee, H., Kang, I.: Neural algorithms for solving differential equations. J. Comput. Phys. 91, 110–117 (1990) 6. Yentis, R., Zaghoul, M.E.: VLSI implementation of locally connected Neural Network for Solving Partial Differential Equations. IEEE Trans. Circuits Syst. I 43(8), 687–690 (1996) 7. Wang, L., Mendel, J.M.: Structured Trainable Networks for Matrix Algebra. In: IEEE Int. Joint Conf. Neural Networks, vol. 2, pp. 125–128 (1990) 8. Joes, G., Freeman, I.: Theoretical Physics. Dover Publication, New York (1986) 9. Arfken, G., Weber, H.: Mathematical Methods for Physicists, 4th edn. Academic Press, New York (1995) 10. Press, W., Flannery, S., Teakolesky, S., Vetterling, W.: Numerical Recipies: The Art of Scientific Computing. Cambridge University Press, New York (1986)

Research on LBS-Based Context-Aware Platform for Weather Information Solution Jae-gu Song and Seoksoo Kim* Department of Multimedia, Hannam University, 133 Ojeong-dong, Daedeok-gu, Daejeon-city, Korea [email protected], [email protected]

Abstract. Mobile solutions using LBS (Location based Service) technology have significantly affected development of various application programs. Particularly, application technology using GPS-based location information and load map is essential to smartphone application. In this study, we designed a context-aware processing module using location and environment information in order to apply weather information to LBS-based services. This research is expected to provide various application services according to personal weather environments. Keywords: LBS, Context-aware, weather information, smartphone appliciation.

1 Introduction LBS-based service is a technology of providing various application services to users utilizing location information obtained from GPS or mobile communication network. The service is of growing interest as major application in the mobile communication industry, promoted by rapid technology development and fast distribution of PDA, smartphones, notebooks, and so on [1, 2]. LBS is a service of providing varied information related with user location through wireless communication, mainly used for effectively managing people or vehicles on the move. More recently, the service is used for locating those who need care and protection such as children and the elderly or tracking sex offenders. LBS involves previous technologies such as GIS (geographical information system), GPS (global positioning system), and telematics, applied to a vide range of areas. Basic technologies for providing LBS services include positioning technology, LBS platform technology for location-based services, and various LBS application technologies [3, 4]. In this study, we suggest LBS application technology, which is a solution technology for location-based services. The service is divided into a general consumer service and a corporate service while we carried out research on services for general consumers utilizing common information services. *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 297–302, 2011. © Springer-Verlag Berlin Heidelberg 2011

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2 Background Research 2.1 LBS Application Technology LBS offers an additional application service by combining location information of a user on the move with other information. Effective application service requires a mobile device, a wireless network, positioning technology, a solution for data processing, and content/application for providing additional services. More recently, smartphones support Wi-Fi based wireless services, making possible more complicated services to be offered [5]. Generally, the architecture of LBS platform is as follows, depicted in Figure 1. The system includes Positioning, Location Managing, Location based Function, Profile Management, Authentication and security, Location based billing, and Information roaming between carriers.

Fig. 1. Referential architecture of LBS Platform

2.2 Context-Aware Technology Context-awareness is software which is adaptive to places and surrounding people/objects as well as can accept changes of objects over time [6]. Such contextawareness technology is considered essential to processing data in various application service areas. Major research efforts are being made in areas of advertisement, shopping, and user preferences [7]. Context awareness technology should take into consideration the following elements.

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Table 1. Categories of context Primary context Location(where) Identity(who)

Secondary context Person’s identity (age, phone number, address) Entity’s location (e.g. what object are near, what activity is occurring near the entity)

Time(when) Activity(what) Table 2. Another categories of Context External (physical) context Measured by h/w sensors (e.g. location, light, sound, movement, touch, temperature)

Internal (logical) context Specified by users or captured monitoring the user’s interaction (e.g. user’s goal, tasks, user’s emotional state)

In this study, we designed a context information processing module using primary context, weather information, and location information (based on the Web and GPS).

3 LBS-Based Context-Aware Platform for Weather Information Solution In this research, we designed a system of providing weather information to users by obtaining mobile location information. A user can employ a social network to send his context information and receive more accurate real-time weather information. Figure 2 shows the structure of LBS-based context-aware platform for weather information solution. The information of the system is largely composed of Weather information, Location information, Smartphone apps, and User information. The information is divided into user-based application service and system-based application through context information matching. User-based application is a method of responding to information frequently changing through user feedback in case patterns are not defined. For example, an alarm application program, combining information of destination and weather, provides users the estimated time required and receives feedback from users for application when a user needs to decide time to leave for an appointment. The program combines location and map information using GPS with sensor data in order to accurately find out locations and calculate the time required through weather and vehicle information. Then, the system asks user to confirm the time to depart.

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Fig. 2. LBS-based Context-Aware Platform for Weather Information Solution

In case service use patterns are clear, system-based application produces an outcome based on the patterns to offer services to users. When a user’s current location is clear, the program provides the time required taking into account weather and vehicle information.

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Fig. 3. Structure of LBS-based Context-Aware Platform for Weather Information Solution

4 Conclusion This research on the LBS-based context-aware platform for weather information solution classifies elements required by context-awareness into a group of information and explains application methods. In particular, it designs a system to apply location and weather information to application programs. In view of the nature of contextawareness, it is necessary to effectively find out location of users on the move and align the information with related context information to provide useful information. To that end, we divided the services into user-based and system-based services in order to receive feedback on changing context from users. It is expected that this study can provide the basis for applying weather and LBS information to related application programs.

Acknowledgement This paper has been supported by the 2010 Preliminary Technical Founders Project fund in the Small and Medium Business Administration.

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References 1. Barnes, S.J.: Developments in the m-commerce value chain: Adding value with location based services. Geography 88(4), 277–288 (2003) 2. Beinat, E.: Location-based Services - Market and Business Drivers. GeoInformatics 4(3), 6–9 (2001) 3. Berger, S., Lehmann, H., Lehner, F.: Location-Based Services in the Tourist Industry. Information Technology & Tourism 5(4), 243–256 (2003) 4. Barkuus, L., Dey, A.: Location-Based Services for Mobile Telephony: a Study of Users’ Privacy Concerns. In: Proceedings of the INTERACT 2003, 9TH IFIP TC13 International Conference on Human-Computer Interaction, IRB-TR-03-024 (July 2003) 5. Krumm, J., Horvitz, E.: LOCADIO: Inferring Motion and Location from Wi-Fi Signal Strength. In: First Annual International Conference on Mobile and Ubiquitous Systems: Networking and Services, Boston, MA, USA, pp. 4–14 (2004) 6. Schilit, W.N.: A System Architecture for Context-Aware Mobile Computing. PhD Thesis, Columbia University (1995) 7. Song, J.-g., Kim, S.: A study on applying context-aware technology on hypothetical shopping advertisement. Information Systems Frontiers 11(5), 561–567 (2009)

A New Feature Reduction Method for Mammogram Mass Classification B. Surendiran and A. Vadivel Department of Computer Applications, National Institute of Technology, Tiruchirappalli, India [email protected], [email protected]

Abstract. In this paper, we present a new dimension reduction method using Wilk’s Lambda Average Threshold (WLAT) for classifying the masses present in mammogram. According to Breast Imaging Reporting and Data System (BIRADS) benign and malignant can be differentiated using its shape, size and density, which is how radiologist visualize the mammograms. Measuring regular and irregular shapes mathematically is found to be a difficult task, since there is no single measure to differentiate various shapes. Various shape and margin features of masses are extracted, which are effective in differentiating regular from irregular polygon shapes. It has been found that not that all the features are equally important in classifying the masses. In our experiment, we have used Digital Database for Screening Mammography database (DDSM) database and the classification accuracy obtained for WLAT selected features is better than Principal Component Analysis (PCA) and Image Factoring dimension reduction methods. The main advantage of proposed WLAT method are i) the features selected can be reused when the database size increases or decreases, without the need of extracting components each and every time; ii) WLAT method considers grouping class variable for finding the important features for dimension reduction. Keywords: Digital Mammogram, Shape and margin features, Benign and Malignant, WLAT, Feature selection, Dimension Reduction, PCA.

1 Introduction The Breast cancer is the leading cause of death in female population. Every 3 minutes, a woman is diagnosed with breast cancer, and in every 13 minutes a woman dies from the disease [1]. The exact cause of breast cancer is unknown and best known prevention is precautious diagnosis. Mammography is one of the best known techniques for early detection of breast cancer. Breast cancer death rates have been dropping steadily since 1995 due to earlier detection and increased use of mammography [1]. Computer Aided Detection (CAD) systems have been developed to aid radiologists in diagnosing cancer from digital mammograms. Several studies have proved that CAD improves breast cancer diagnostic accuracy rate by 14.2% [2]. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 303–311, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Malignant and benign masses are abnormal growth of tumor cells present in mammogram. While malignant are considered as cancerous tumors and benign are noncancerous. According to BI-RADS system, the masses can be characterized by shape, size, margins (borders) and density [3]. Benign masses are round and oval in shape and have smooth, circumscribed margins. The malignant masses are irregular shape and illdefined, microlobulated or spiculated margins. It has been observed that shape and margin characteristics can be effectively used for classifying the masses either as benign or malignant. Based on shape and margin features, some of the known approaches, which classify the abnormalities based on BI-RADS system, have been giving accurate results [4, 5]. Thus, in this paper, we have used various shape and margin features of the masses. These simple and yet effective geometric shape and margin features visualize the masses as radiologists visualize the mammograms. Researchers have proposed and used various features for classifying the masses in mammograms. The statistical measures such as variance, uniformity, smoothness, third moment etc [6, 7, 8, and 9] have been used for classifying masses. Various statistical measures such as mean, variance, skewness and kurtosis are used in [6]. In [6], they had used only few masses for classification and able to obtain only 74% for normal vs malignant and similarly for benign only 15 are correctly classified and accuracy is 68.18%. The test was not carried out for malignant vs. benign due to the complex characteristics of benign and malignant masses. In [7], Sheshadri et al, have used various statistical measures to classify the breast tissue in to four basic categories like fatty, uncompressed fatty, dense and high density and effect of these measures for classifying masses is not known. Similarly in [8, 9] statistical measures are used for classifying normal vs. micro-calcifications, but the effect of these measures for mammogram masses is not known. In [9], a total of 164 features are extracted to classify normal vs. micro-calcification, which seems to be very large set of features. These statistical measures are based on gray value or histogram of mammogram masses. However, the gray values of mammogram tend to change, due to overenhancement or due to the presence of noise. Most of the existing works have been concentrated on classifying the region using shape features [10, 11]. In [10], Flores et al have used various geometric shape features for classifying it normal vs. calcifications and have obtained better results. In [11], Sun et al have used multi-resolution features for classifying the region either as normal or abnormal. In [12], a complex Bayesian Neural Networks classifier with co-occurrence matrix for 5 statistical measures has been used to classify the masses with small dataset containing only 17 sample mammograms and have achieved maximum of 81% accuracy. One may assume that the use of more features will automatically improve the classification power of the classifier. However the number of samples needed to train a classifier with a certain level of accuracy increases exponentially with the number of features. Thus it is vital to reduce the feature dimension. For dimension reduction, some of the commonly used techniques in the literature are PCA and Image factoring [13]. The main disadvantage of the PCA and Image factoring method is that it extracts components without considering the grouping class variable. And these methods do

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not identify the specific features as best performing features [14]. The various researchers have used PCA for dimension reduction and it can be found in [13, 15, 16, and 17]. The combination PCA and SVM classifier is used in medical imagery, where PCA components are extracted and are used to train SVM classifier [15]. Main disadvantage of PCA method is i) the components have to be extracted each and every time, whenever the database size increases or decreases, ii) does not consider the grouping class variable for constructing components. To overcome limitation of PCA, we had used supervised dimension reduction technique called discriminant analysis. In [16], Samulski had used Linear Discriminant Analysis (LDA) along with PCA, and found that LDA works better than PCA for classifying mammograms. But a very large set of 81 features are extracted for each mammogram. In [17], Joo had used combination for PCA and Fisher Discriminant Analysis (FDA) for face recognition and obtained good results. It can be observed that all the features are not important. The number of features or dimension can be reduced, so that lowers the classifier complexity and computation time. In this paper, we present feature selection and feature reduction methods using ANOVA discriminant analysis Wilk’s lambda statistic measure, which is extension of LDA [13, 16, and 17]. Proposed WLAT method is found to be giving encouraging results compared to PCA and Image factoring. This paper is organized as follows. In Section 2, we present techniques for feature extraction using shape features. We discuss univariate Discriminant Analysis classification method in section 3. In section 4, we present the experimental results with feature selection method using proposed WLAT method. We conclude the paper in last section.

2 Shape and Margin Features of Mass A. Shape Characteristics of mass According to BIRADS system, the shapes of the masses are characterized as round, oval, lobular, and irregular. Similarly, margin of the masses are characterized as circumscribed, obscured, micro-lobulated, and spiculated. While, benign masses have round, oval and lobular shapes with circumscribed margin, the malignant masses have lobular and irregular shape with ill-defined, microlobulated or spiculated margins. These shape features can be used for classifying the masses present in mammogram. B. Shape Features We have used mammograms with ground truth from DDSM Database [19] for classification. Total of 300 mammograms masses are considered for experiment. Out of 300, 150 are benign masses and remaining 150 are malignant masses. The masses used in this experiment contain various characteristics such as oval, round, lobular, irregular, architectural distortion, asymmetric density etc. Most of the earlier works on mammograms have not considered mass shapes such as irregular, lobular, architectural distortion, as it is difficult to classify these mass types compared to round, oval masses as benign or malignant. In Table 1, we present various shape and margin features used for classification.

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B. Surendiran and A. Vadivel Table 1. Shape and margin Features

S.no 1.

Measure Area(A)

2.

Perimeter(P)

3.

Max Radius (Rmax)

4.

Min Radius (Rmin)

5.

Euler Number (EULN) Eccentricity

6.

(Ect)

7.

Equivdiameter (Eqd)

8.

11.

13.

14.

15.

16.

Total Pixels in Edge border of mass Maximum distance between center to edge of the mass Minimum distance between center to edge of the mass

To differentiate regular polygons from irregular polygons. For irregular polygon shapes, the max radius will be high compared regular polygons with same area of irregular polygon.

Number of connected components – Number of holes

The Euler number is the difference between the number of disjoint regions and the number of holes in an image. Around 0 for round/oval masses. Varying values for irregular shapes

⎛ MinRadius ⎞ 1− ⎜ ⎟ ⎝ MaxRadius ⎠ 4*

2

For identifying elliptical characteristics of a masses. For differentiating round/oval masses from

Area

π

⎛ ⎞ Area ⎜⎜ ⎟ 2 ⎟ ( 2 *MaxRadius ) ⎝ ⎠

Entropy (Entpy)

∑ ( p * log ( p )) 2

Measures the amount of disorder. Measures the amount of spiculations present in mass. Useful for identifying malignant masses.

Circularity1 (C1)

Area π*Max Radius 2

Circularity2 (C2)

(CN)

MinRadius MaxRadius ⎛ 2 * Area * π ⎜ ⎜ Perimeter ⎝

⎞ ⎟ ⎟ ⎠

Useful for differentiating circular/oval benign masses from irregular malignant masses.

Compactness is independent of linear transformations. Measures the degree of roughness of a region

Dispersion(Dp)

⎛ MaxRadius ⎞ ⎜ ⎟ Area ⎝ ⎠

For identifying malignant irregular shape characteristics

Thinness Ratio (TR)

⎛ 4 * π * Area ⎞ ⎜⎜ ⎟ 2 ⎟ ⎝ perimeter ⎠

This measure differentiates line from other shapes

Std Deviation Of mass (SD)

1 n

Standard Deviation Of Edge (Esd) Shape Index(SI)

17.

Malignant masses are bigger in size compared to benign masses.

Elongatedness (En)

Compactness 12.

Total Pixels in mass

irregular/lobular mass Max value 1 for square. Useful for differentiating regular oval benign masses from irregular malignant masses.

9. 10.

Impact

Expression

∑ ( xi − x ) 2

A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation

1 m ∑ ( yi − y ) 2 m i =1

A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation

⎛ Perimeter ⎞ ⎜ ⎟ ⎝ 2 * MaxRadius ⎠

This property provides information about margin characteristics. For differentiating skinny polygons from the regular polygons.

n

i =1

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For classification of masses, we have used 17 shape and margin features of masses present in mammograms [18]. These shape and margin features are newly derived measures based on Area, Perimeter, max radius and min radius, which are effective in differentiating regular polygon from irregular polygon shapes. Out of 17 shape features, basic shape features are Area, Perimeter, Max Radius, Min Radius and remaining other features are advanced shape features. For each mass, the shape based features are extracted and constructed as feature vector. The shape based features are used as a Shape Feature Vector (SFV) = {mammogram, pt, mass_type}, where t=1…17 and pt is the shape property. Figure 1 shows the extracted masses from sample mammograms from DDSM Database.

(a) Benign Masses

(b) Malignant Masses

Fig. 1. Sample benign and Malignant Masses

C. ANOVA Discriminant Analysis ANOVA DA uses single dependent continuous variable and use more than one independent categorical variable. ANOVA DA is an excellent statistical method for classification as it gives the relation of features between groups and within-groups. ANOVA is a special case of the General Linear Model and it can be written as y = Xb + e

(1)

In above equation, y is a Dependent Variable (DV), X is a matrix of predictors or Independent Variables (IVs), b is a vector of regression coefficients (weightings) and e is a vector of error terms. The ANOVA is a procedure that determines whether differences exist between two or more population means by analyzing the within-group and between group variances. For classification, we have used SPSS tool [20, 21], where a dependent variable signify 0 for benign and 1 for malignant. The 17 shape features are used as independent categorical variables. The ANOVA DA classifier predicts the discriminating power of each feature for classifying the masses either as benign or malignant. In our paper, Wilk’s lambda measure is used for examining the classification capability various shape features. Wilks lambda is used to test the null hypothesis that the populations have identical means on D (discriminating function). Wilks’ lambda is defined as the ratio of within-group sum of squares to the total sum of squares. Wilk’s lambda, Λ =

SS within _ groups SS total

(2)

For each shape and margin features, Wilk’s lambda measure is calculated using Eq. (2) are shown in Table 2. Lower the Wilk’s lambda, higher the classification capability of the property. Thus, the Wilk’s lambda statistic measure produced by ANOVA DA is used for determining the classification capability of each feature and its significance in classification accuracy.

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B. Surendiran and A. Vadivel Table 2. Test of Equality of Group Means

Features

Λ

Area .698

Rmax .663

Rmin .857

ECT .971

Eqd .558

En .968

Entpy .667

SD .532

Esd .616

D. WLAT Feature Reduction The objective of this proposed method is to reduce the number of features used for classification of masses. In this method, the Wilk’s Lambda statistics for all shape and margin features are calculated and average is found. The average Wilk’s lambda statistic measure calculated as shown below, Λavg =

Λ f 1 + Λ f 2 + .... + Λ fn

,

(3)

n

where Λfi is Wilk’s lambda statistic for each property. Using the calculated average value, the number of features required for classification is reduced. The steps followed for feature reduction is shown below WLAT Steps:

1. Find Λavg using Λ value of all features. 2. Reduce the feature set by considering only those features whose Λfi < Λavg. Here Λavg is considered as a threshold. Store the reduced feature set as R={set of reduced features} 3. If the reduced feature set is giving good classification result then find Λavg for the reduced feature set. Repeat the step 2 to 3 recursively. 4. Else consider the previously stored reduced feature set as optimal feature set. In WLAT Level 1: Λavg = 0.811 Number of features (Reduced set) = 8 (out of 17) List of reduced features set = Area, Perimeter, Rmax, Eqd, Entpy, Dp, SD

and Esd. In WLAT Level 2: Λavg = 0.657 Number of features (Reduced set) = 3 (out of 8) List of reduced features set = Eqd, SD and Esd.

The number of features used by WLAT method is only three. This will increase the classification speed and relatively reduce the computational speed and complexity.

3 Experimental Results A. Comparison Between PCA, Image Factor Analysis, and proposed WLAT Method In this section, we present the experimental results carried out by using benchmark DDSM mammogram database. The number of samples needed to train a classifier with a certain level of accuracy increases exponentially with the number of features. Thus it

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is vital to reduce the feature dimension. The commonly used dimension reduction techniques are PCA and image factoring. These methods are unsupervised and their main drawbacks are they don’t consider the grouping class variable and components has to be extracted if database size gets changed [22]. The assumption made in PCA is that most of the information is carried in the variance of the features: the higher the variance in one dimension (feature), the more information is carried by that feature. The general idea is therefore to preserve the most variance in the data using the least number of dimensions. Total of 4 components are extracted for PCA and 4 components are extracted for Image Factoring methods. Only 4 components are considered whose Eigen value is greater than 1. The classification accuracy using ANOVA DA for 4 extracted PCA and Image factor components is shown in Table 3. The performance comparison between PCA, Image factor analysis and proposed WLAT method is shown in Table 3 and in Figure 2. In this table, the number of features used for the classification is given in 2nd column along with the total number features. A Leave One Out Cross Validation (LOOCV) is done using ANOVA DA classifier. Table 3. Comparison of PCA vs Image Factor vs proposed WLAT methods

Method

Reduced and Original Features

Classification Accuracy

PCA

4 (17)

82.0%

Image Factoring

4 (17)

81.7 %

Proposed WLAT – Level 1

8 (17)

86.3 %

Proposed WLAT – Level 2

3 (17)

84.3%

From Figure 2, it can be observed that proposed WLAT methods achieve better accuracy than PCA and Image factor Analysis. . The main advantage of proposed WLAT method is that the features selected can be reused when the database size increases or decreases, without the need of extracting components each and every time.

Accuracy

87.00% 84.00% 81.00% 78.00% PCA

Image Factoring

WLAT – Level 1

WLAT – Level 2

Fig. 2. Performance comparison of various dimension reduction methods

4 Conclusion We have used WLAT feature dimension reduction approach for effectively reducing the number of shape and margin features used for mass classification. The number of

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features required for mass classification is reduced in steps. The proposed WALT method has reduced 17 features to 8 (level 1) and 3 (Level 2) features set and the classification accuracy obtained using DDSM database is encouraging. The classification accuracy is compared with recently proposed dimension reduction techniques such as PCA and Image factor analysis. The proposed method produces better classification accuracy compared to other similar methods. The main advantage of proposed WLAT method is that the selected features can be reused when the database size increases or decreases, without the need of extracting components each and every time.

Acknowledgment The work done by Dr. A.Vadivel is supported by research grant from the Department of Science and Technology, India, under Grant SR/FTP/ETA-46/07 dated 25th October, 2007.

References [1] A.C.S. (AMS).: Learn about breast cancer, http://www.cancer.org (accessed on May 2010) [2] American Journal of Roentgenology (AJR), Computer-aided Detection Improves Early Breast Cancer Identification, http://www.ajronline.org (accessed on February 5, 2010) [3] American College of Radiology (ACR). Breast Imaging Reporting and Data System (BI-RADS). 3rd ed. American College of Radiology, Reston, Va (1998) [4] Markey, M.K., Lo, J.Y., Tourassi, G.D., Floyd, C.E.: Cluster analysis of BI-RADS descriptions of biopsy-proven breast lesions. In: Medical Imaging: Image Processing, Proceedings of SPIE, vol. 4684, pp. 363–370 (2002) [5] Sampat, M.P., Alan, C., Bovik, B., Markey, M.K.: Classification of mammographic lesions into BI-RADS shape categories using the Beamlet Transform. In: Medical Imaging: Image Processing, Proc. of the SPIE, vol. 5747, pp. 16–25 (2005) [6] Vibha, L., Harshavardhan, G.M., Pranaw, K., Deepa Shenoy, P., Venugopal, K.R., Patnaik, L.M.: Classification of Mammograms Using Decision Trees. In: 10th International Database Engineering and Applications Symposium (IDEAS 2006), pp. 263–266. IEEE, Los Alamitos (2006) [7] Sheshadri, H.S., Kandaswamy, A.: Breast Tissue Classification Using Statistical Feature Extraction Of Mammograms. Medical Imaging and Information Sciences 23(3), 105–107 (2006) [8] Zhang, X.: Boosting Twin Support Vector Machine Approach for MCs Detection. In: Asia-Pacific Conference on Information Processing, pp. 149–152 (2009) [9] Zhang, X., Gao, X., Wang, Y.: MCs Detection with Combined Image Properties and Twin Support Vector Machines. JCP 4(3), 215–221 (2009) [10] Flores, B.A., Gonzalez, J.A.: Gonzalez,:Data Mining with Decision Trees and Neural Networks for Calcification Detection in Mammograms. In: Third Mexican International Conference on Artificial Intelligence, Proceedings. LNCS, pp. 232–241. Springer, Heidelberg (2004) [11] Sun, Y., Babbs, C., Delp, E.: Normal Mammogram Classification Based on Regional Analysis. In: Proceedings of the IEEE Midwest Symposium on Circuits and Systems, vol. 45, pp. 375–378 (2002)

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[12] Martins, L.d.O., dos Santos, A.M., Silva1, A.C., Paiva1, A.C.: Classification of Normal, Benign and Malignant Tissues Using Co-occurrence Matrix and Bayesian Neural Network in Mammographic Images. In: Proceedings of the Ninth Brazilian Symposium on Neural Networks (SBRN 2006), pp. 24–29. IEEE computer society, Los Alamitos (2006) [13] Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. John Wiley, Chichester (2001) [14] Mu, T., Nandi, A.K., Rangayyan, R.M.: Classification of breast masses using selected shape, edge-sharpness, and texture properties with linear and kernel-based classifiers. Journal of Digit Imaging, 153–169m (2008) [15] Li, S., Fevens, T., Krzyzak, A.: Automatic clinical image segmentation using pathological modeling, pca and svm. Engineering Applications of Artificial Intelligence 19(4), 403–410 (2006) [16] Samulski, M.R.M.: Classification of Breast Lesions in Digital Mammograms. Master Thesis (2006) [17] Joo, S.: Face recognition using PCA and FDA with intensity normalization, Technical report, Department of Computer Science. University of Maryland (December 2003) [18] Surendiran, B., Sundaraiah, Y., Vadivel, A.: Classifying Digital Mammogram Masses using Univariate ANOVA Discriminant Analysis. In: International Conference on Advances in Recent Technologies in Communication and Computing - ARTCom 2009, IEEE explore (October 2009) (Best Paper Award) [19] Rose, C., Turi, D., Williams, A., Wolstencroft, K., Taylor, C.J.: Web Services for the DDSM and Digital Mammography Research, pp. 376–383 (2003) [20] Field, A.: Discovering Statistics usng SPSS, 3rd edn. SAGE Publications Ltd., Thousand Oaks (2009) [21] Spss : SPSS Data Mining, Statistical Analysis Software, Predictive Analysis, Predictive Analytics, Decision Support Systems, http://www.spss.co.in/ [22] Samulski, M., Karssemeijer, N., Lucas, M.D., Groot, P.: Classification of mammographic masses using support vector machines and Bayesian networks. In: Proceedings of SPIE. Medical Imaging 2007: Computer-Aided Diagnosis, vol. 6514 (2007)

Chaos Based Image Encryption Scheme P. Shyamala Department of CSE, Anna University of Technolgy, Tiruchirappalli, India [email protected]

Abstract. Image encryption is different from text encryption due to some inherent features of image such as bulk data capacity and high correlation among pixels, which are generally difficult to handle by conventional methods. The desirable cryptographic properties of chaotic maps such as initial conditions and random-like behavior can be used to develop new encryption algorithms. The chaos – based cryptographic algorithms have suggested new ways to develop efficient image encryption schemes. The random-like nature of chaos is effectively spread into encrypted images. The proposed method transforms the statistical characteristic of original image information. So, it increases the difficulty of an unauthorized individual to break the encryption. The proposed symmetric image encryption algorithm provides an effective way for real-time applications and transmission. Keywords: Chaotic maps, Image encryption, Information security.

1 Introduction With the rapid development of the Internet and multimedia technology, multimedia communication has become an important means of information communication. An image contains a large amount of information and is widely used and transferred over the network. Therefore, the security of private images stored in a specific way or transmitted over the internet has become a major issue of concern. Image encryption technology is an effective measure to ensure the information security of private images, because it keeps the private digital images encrypted before storing for transmission. Thus, Eve cannot identify the original information from the encrypted image information, and only the legitimate receivers or users can decrypt it correctly and effectively with private cipher keys. In the past, many classical encryption algorithms were proposed such as Arnold transform, Josephus traversing, chaos map [1], [2], [3], [4], [5], [6], [7]. All these algorithms could encrypt images better than traditional cipher theory, but, many of them were to consider the position scrambling encryption, but the pixel values did not change. So they are not ideal algorithms considering statistical characteristic, which is the same in original image and the cipher image Gray encryption [2], [6], [7] was also adopted to enhance the security further. Chaos-based cryptographic algorithm [8] is an efficient encryption first proposed in 1989. It has many unique characteristics different from other algorithms such as the sensitive dependence on initial conditions, non-periodicity, non-convergence and P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 312–317, 2011. © Springer-Verlag Berlin Heidelberg 2011

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control parameters [9], [10].The one-dimensional chaos system has the advantage of simplicity and high security [7]. And many studies [2], [3], [4], [5], [6], [7] were proposed to adopt and improve it. Some of them use high-dimensional dynamical systems, other studies use couple maps [11]; however, all research is mainly to increase parameters to enhance the security. As we know, a good encryption method should be sensitive to keys and space should be large enough [12], moreover, it must have some ability to resist outer attack from statistics analysis, i.e., gray distribution. However, too many parameters will influence the implementation and increase computation; for example, there are eight parameters in [7]. The efficiency will also be low using more than one encryption algorithm to an image. An image scrambling encryption algorithm of pixel bit based on chaos map is reported. On one hand, the algorithm uses the unique significant features such as sensitivity to initial condition, control parameters to permute efficiently the pixel positions of original image. On the other hand, the gray scrambling encryption is carried out at the same time according to pixel bit. The gray distribution in our algorithm is changed in different iteration. The rest of this paper is organized as follows. Section 2 introduces chaos and its unique characteristics. In Section 3, the image encryption based on chaos is proposed including new algorithm steps. The experimental results are tested in Section 4. And in Section 5, the performance of new algorithm is analyzed. Finally conclusions will be discussed in Section 6.

2 Chaos and Its Unique Characteristics Chaos is a field of study in mathematics, physics and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. Small differences in initial conditions yield widely diverging outcomes for chaotic systems. This happens even though these systems are deterministic, meaning that their initial conditions with no random elements involved. Chaotic systems have a number of interesting properties such as sensitivity to initial conditions (SIC) and system parameter, ergodicity and mixing (stretching and folding) properties, etc. These properties make the chaotic systems a worthy choice for constructing the cryptosystems as sensitivity to the initial condition/system parameter and mixing properties are analogous to the confusion and diffusion properties of a good cryptosystem. In an ideal cryptosystem, confusion reduces the correlation between the plaintext and cipher text while diffusion transposes the data located at some co-ordinates of the input block to other co-ordinates of the output block. Diffusion changes the position of data while the data itself is modified during the confusion process.

3 Proposed Image Encryption Algorithm Based on Chaotic Map 1. Generation of the cryptosystem keys The proposed image encryption process utilizes a 128-bit long external secret key (P). This key is generated using a reference image of size X x Y. The reference image is divided into blocks of size M x N (M < X and N < Y), (let number of blocks is A).

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From each pixel of each block the R, G, B values are extracted and their mean values are calculated. The mean R, G, B values of each blocks are EX-OR ed. Finally A values are extracted, one value from each block. This is a 128-bit secret key (P). Shared key is generated using external secret key and the private and public keys of sender and receiver. Sender’s public key , where is the sender’s private key is sent to receiver. Receiver can then generate the common shared key , , where is the receiver’s private key, by multiplying its private key with the sender’s public key. In the same way receiver’s public key , where is the receiver’s private key is sent to sender. Sender can then calculate the . Shared key if of 128-bit long. Only on common shared key , the generation of correct shared key the image can be decrypted correctly. 2. Pixel bit shuffling In existing systems only the pixels are scrambled. Where as in proposed system, pixels are decomposed into 8-bit sequences. Each bit is scrambled using Arnold cat map. 1

′ ′

1

Where p and q are positive integers, det (A) = 1. The map is area-preserving since the determinant of its linear transformation matrix equals 1. The (x’, y’) is the new position of the original pixel bit position (x,y) when Arnold cat map is performed once. 3. Proposed encryption algorithm and encryption steps In the new algorithm, the chaos-based image encryption is combined with pixel bit. Before applying the new algorithm, we decompose the pixel values into bits. Let , denote the pixel in the ith row and jth column of original image, and let , denote the t(t=0,1,…7) digit after transforming , . Then ,

,

1 0

(1)

So, the gray image is transformed into bit matrix composed of zero and one by formu, (t=0,1,2…7) cab be transformed into gray bits , using la (1). Contrarily the formula: ,

∑

2

,

(2)

The encryption process for a gray image using the new algorithm is performed according to the following steps: Step 1. Reading a gray image and expressing it with a decimal matrix A of size M x N. Step 2. Transforming all pixels in A into bit pixels by formula (1) and noting it as matrix B of size M x 8N. Step 3. Applying chaos encryption to rows and columns of matrix B, then matrix C is got. Step 4. Transforming contrarily the encrypted matrix C by formula (2) and getting decimal matrix D of size M x N. Step 5. The encryption process is complete and a gray image is generated.

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4 Experimental Results In this section, the experiments for validating the security and practicability of proposed algorithm.

(a)

(b)

(c)

(d)

Fig. 1. Image encryption and decryption test using new algorithm: (a) original image, (b) encrypted image, (c) decrypted image with correct key, and (d) decrypted image with wrong keys

5 Performance Analysis (1) Gray difference degree The gray difference of a pixel in image is computed as follows: ′

,

,

, ′

4

Where G(x,y) denotes the gray value in position (x,y). The average neighborhood gray difference of the whole image can be computed by formula (3). ∑

,

∑

,

(3)

And the gray value degree is defined by ′ ′

, ,

, ,

Where AN and AN’ denote the average neighborhood gray difference of image before and after being encrypted separately.

316

P. Shyamala Table 1. Gray value degree of the lenna image

IMAGE LENNA.BMP

GVD 0.9813440719066054

(2) Histogram analysis A histogram is a graphical representation, showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable. Histogram, also called a gray distribution picture, is a chart of the pixel distribution order for an image. It reflects the distribution of pixels in an image after being disposed.

(a)

(b)

Fig. 2. a. Is the histogram of lenna image, b. Denote the histogram of two encrypted images by the proposed method

Histogram of the encrypted image ensures that, confusion and diffusion processes are performed better in proposed algorithm, which gives better distribution of image pixels and hard to find the original image without knowing the exact shared key. (3) Correlation analysis of two adjacent pixels The following formula (4) is commonly applied to test the correlation between two adjacent pixels in original image and encrypted image. Two adjacent image pixels including horizontal, vertical and diagonal directions from lenna image. The result can be seen in Table 2, from which we can find that the correlation is almost zero. ,

∑

,

Here ∑

(4) ,

∑

, and

.

Table 2. Correlation coefficients of two adjacent pixels in lenna image and its encrypted image

Model Horizontally Vertically Diagonally

Original image 0.9690 0.9637 0.9492

By proposed method 0.2881 -0.03247 0.06600

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6 Conclusion An image encryption scheme based on chaos map together with pixel bit scrambling is presented in this paper. The algorithm proposed in this paper can also be extended to high-dimensional chaos map and can be applied to 3D images. It is an efficient encryption algorithm that can be used directly to transmit securely all kinds of image information over the Internet. Since, bit shuffling is employed (which changes the statistical properties of the image) security is ensured. Since, shuffling provides encryption along with bit shuffling, time complexity for encryption is reduced.

References 1. Guan, Z.H., Huang, F.J., Guan, W.J.: Chaos-based image encryption. Phys. Lett. A 346, 153–157 (2005) 2. Zhang, L., Liao, X., Wang, X.: An image encryption approach based on chaotic maps. Chaos Soliton Fract. 24, 759–765 (2005) 3. Fang, Z.J., Lu, X., Wei, W.M., Wang, S.Z.: Image scrambling based on bit shuffling pixels. J. Optoelectron, laser 18(12), 1486–1488 (2007) 4. Fang, Z.Y., Tong, W.Q.: Image scrambling algorithm based on chaos mapping. Modern Comput. 10, 51–53 (2007) 5. Li, Y., Fan, Y.Y., Hao, C.Y.: Information hiding technology based on image second scrambling. J. Image Graph. 11(8), 1088–1091 (2006) 6. Pareek, N.K., Patidar, V., Sud, K.K.: Image encryption using chaotic logistic map. Image and Vision Computing 24, 926–934 (2006) 7. Gao, T.G., Chen, Z.Q.: Image encryption based on a new total shuffling algorithm. Chaos, Solitons Fract. 38, 213–220 (2008) 8. Matthews, R.: On the derivation of a chaotic encryption algorithm. Cryptologia XII(1), 29–42 (1989) 9. Baptita, M.S.: Cryptography with chaos. Phys. Lett. A 240, 50–54 (1999) 10. Chung, K.L., Chang, L.C.: Large encryption binary images with higher security. Pattern Recognition Lett. 19, 461–468 (1998) 11. Behnia, S., Akhshani, A., Mahmodi, H., Akhavan, A.: A novel algorithm for image encryption based on mixture of chaotic maps. Chaos, Solitons Fract. 35, 408–419 (2008) 12. Schneier, B.: Applied cryptography: protocols, algorithms and source code in C. John Wiley and Sons, New York (1996)

Weighted Matrix for Associating High-Level Features with Images in Web Documents for Image Retrieval P. Sumathy and P. Shanmugavadivu Department of Computer Science and Applications Gandhigram Rural Institute, Dindigul, Tamilnadu, India [email protected], [email protected]

Abstract. One of the main obstacles in capturing semantic of images present in the Web document is that it is difficult to describe the semantics. The high-level textual information of images can be extracted and associated with the textual keywords for narrowing the search space and improving the precision of retrieval. In this paper, we propose a weighted matrix approach for associating keywords and high-level semantics of images to use their complementing strengths. The high-level semantics of the image is described in the HTML documents in the form of image name, optional description and caption. A weighted matrix is constructed using both textual keywords and image information for measuring the association between the keywords and the HTML page. The frequency of occurrence of words can be calculated and used for measuring the relevance of keywords to images. The retrieval performance is compared with various recently proposed techniques and is found to be encouraging. Keywords: Web Semantics, Web Image Retrieval, High-level Features, Textual Keywords.

1 Introduction In domain-specific image retrieval applications, the low-level features such as colour, texture and shape can be used for image retrieval. Colour histograms such as Human Colour Perception Histogram (HCPH) [11] and Gevers and Stokman [3] as well as colour-texture features like Integrated Colour and Intensity Co-occurrence Matrix (ICICM) [10] and Palm[7] show high precision of retrieval in such applications. However, in generic applications, such as retrieval of relevant images from the World Wide Web (WWW), low-level features may not be adequate for representing and capturing the semantic content of images effectively. In addition, representing and storing huge voluminous low-level features of images may be a costly affair in terms of processing time and storage space. This situation can be effectively handled by using keywords along with the most relevant textual information of images to narrow down the search space. For annotating images, which are embedded into the HTML pages; it is felt that the textual content of the document can be used without any need for manual intervention. Often, it is found that the text in HTML documents describes images, which is situational and is more relevant. Thus, keywords extracted by taking into account their frequency of occurrence, name of the image, caption and optional P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 318–325, 2011. © Springer-Verlag Berlin Heidelberg 2011

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information of images from HTML documents can be used for labelling images for improving precision of retrieval. An approach for unifying keywords and visual features of images [6] has been proposed and in this method, authors have assumed that some of the images in the database have been already annotated in terms of short phrases or keywords. These annotations are assigned either using surrounding texts of the images in HTML pages. During retrieval, user’s feedback is obtained for semantically grouping keywords with images. An attempt has been made to improve the semantic correlation between keywords in the document title in HTML documents and image features for improving retrieval in news documents [15]. This method has used a collection of 20 documents from a news site. From the results, it is found that the proposed technique performs well for a small number of web pages and images. In general, image search results returned by an image search engine contain multiple topics. Organizing the results into different semantic clusters helps user. A method has been proposed for analyzing the retrieval results from a web search engine [2]. This method has used Vision-based Page Segmentation (VIPS) [1] to extract semantic structure of a web page based on visual presentation. The semantic structure is represented as a tree with nodes, where every node represents the degree of coherence to estimate the visual perception. Tollari et al [9] have proposed a method for mapping textual keywords with visual features of images. In order to map textual and visual information, semantic classes containing a few image samples are obtained. Each class is initialized with exactly one image and then two classes are merged based on a threshold value on distance between them. The merging process is stopped when distances between all classes are higher than a second threshold. Finally, the semantic classes are divided into two partitions as reference set and test set. The reference partition provides example documents of each class, which is used to estimate the class of any image of test partition, either using textual, visual or combination of textual and visual information. The performance of this proposed method is high with less number of images with corresponding textual annotations. Han et al [4] have proposed memory learning framework for image retrieval application. Their method uses a knowledge memory model for storing the semantic information by accumulating user-provided data during query interactions. The semantic relationship is predicted by applying a learning strategy among images according to the memorized knowledge. Image queries are finally performed based on a seamless combination of low-level features and learned semantics. Similarly, a unified framework has also been proposed, which uses textual keywords and visual features for image retrieval applications [6]. The framework has built a set of statistical models using visual features of a small set of manually labeled images that represents semantic concepts. This semantic concept has been used for propagating keywords to unlabeled images. These models are updated regularly when more images implicitly labeled by users become available through relevance feedback. This process performs the activity of accumulation and memorization of knowledge learned from relevance feedback from users. The efficiency of relevance feedback for keyword query is improved by an entropy-based active learning strategy. An integrated patch model has been suggested by Xu and Zang [13], which is a generative model for image categorization based on feature selection. Feature selection strategy is categorized into three steps for extracting representative feature and also to remove the noise feature. Initially, salient patches present in images are detected,

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clustered and keyword vocabulary is constructed. Later, the region of dominance and the salient entropy measure is calculated for reducing the non-common noises of salient patches. Using visual keywords, categories of the images are described by an integrated patch model. The textual keywords appearing in web pages are recently being used for identifying unsuitable, offensive and pornographic web sites [5]. In this framework, the web pages are categorized as continuous text pages, discrete text pages and image pages using decision tree with respect to their contents. These pages are handled by respective classifiers. Statistical and semantic features are used for identifying pornographic nature of continuous text web pages. Similarly, Bayesian classifier and object’s contour of images are being used to identify the pornographic content of discrete and image web pages respectively. Another method has been recently proposed for reducing the gap between the extracted features of the systems and the user’s query [14]. The semantics of the image is discovered from web pages for semantic-based image retrieval using self-organizing maps. The semantics of images is described based on the environmental text, which is surrounded by the images. Text mining process has been applied by adopting Self Organizing Map learning algorithm as a kernel. Some of the implicit semantic information is also discovered after the text mining process. A semantic relevance measure is used for retrieval. Recently, the keyword is being extracted from the HTML pages and low-level features are extracted from the images present in HTML documents for retrieval applications [12]. Various low-level features have been extracted and integrated with the high-level features for relevant image retrieval. However, the association or importance of the keywords with the HTML page or with the image is not measured. Based on the above discussion, it is observed that none of the methods dynamically measuring the association of textual keywords with image information for capturing the semantics for retrieval. It may be noticed that the importance of query content has not been captured by any of these methods. In addition, the advantages of combined textual based query have also been not addressed. In this paper, we propose a technique for integration of keywords with image information for retrieval applications. In the proposed technique, we have used a large number of HTML documents containing text and images from Internet. Using a crawler, the HTML documents are fetched. The content of the HTML documents is segregated into text, image and HTML tags. From the text, keywords are extracted. These keywords are considered as relevant keywords for representing high-level semantics of the images contained in the same HTML document. In the following section of the paper, we present the method of measuring the relevancy of keyword in HTML document. Experimental result is provided in Section 3 and in the last section, we conclude the paper.

2 Weighted Matrix for Measuring the Relevancy of Keywords with Images in HTML Documents The semantics of image can be described by a collection of one or more keywords. Let I be a set of images and K be a set of keywords. Assignment of a keyword to an image may be treated as a labeling predicate written as follows:

Weighted Matrix for Associating High-Level Features with Images in Web Documents

l : K × I → {True , False

}

321

(1)

Thus, a keyword k ∈ K can be assigned to an image i ∈ I if l (k , i ) = True . Since a keyword can be assigned to multiple images and multiple keywords may be assigned to one image, given a subset of keywords K ′ ⊆ K , K ′ can be used to define a subset of images from I as follows. I C KD ( K ′ ) = {i

i ∈ I and ∀ k ∈ K ′ , l (k , i ) = T rue }

(2)

I If we do not restrict the set of images by keyword, i.e. if K ′ = φ , C KD (φ) = I meaning that all the images in the database form one set. Thus, if images could be correctly labeled by a set of K keywords, retrieval based on a subset K ′ ⊆ K of keywords would retrieve only relevant images resulting in 100% recall and precision. The main problem with such a keyword-based retrieval is that, it is not feasible to manually annotate each image with keywords. As is evident from Eq. 2, if a user can specify the keywords that describe the required set of images correctly and if the relevant database images are also labeled with the same set of keywords, we would achieve 100% recall and precision. However, due to the subjective nature of annotations, such high accuracy is not achieved in practice. Now it would be appropriate and suitable that the more high-level features of images could be used for improving the semantics of the images and is shown below. Let us consider an HTML page that contains N images and M keywords and can be written as

K =

{k 1 , k 2 , …

kM

}

(3)

I = {i1 , i 2 , … i N }

and

(4)

We consider that all the keywords in one HTML page are equally important and relevant to all the images in the same HTML page. The strength of the keyword k j in the

HTML document, denoted by stg (k

) , is the number of occurrence of keyword

j

supporting k j . Thus stg

(k ) = {k j

j

|k

j

∈ K

}j

= 1K M

(5)

An association between k j and h is an expression of the form k j ↔ h , where k j is a keyword

(j

= 1… M

) in an HTML page h. This can be denoted by

stg (k

j

⎛ stg (k j ) ↔ h )= ⎜ ⎜ Max (stg (k ⎝

⎞ ⎟ ⎟ j )) ⎠

(6) j = 1… M

Based on Eq. (6), association of a set of keywords K can be denoted by stg

(K , h ) =

∩ stg (k , h )

k∈K

(7)

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The above expression denotes the association of set of keywords K to each image i n belonging to HTML page h. This implies that we consider the frequency of the keyword having maximum frequency in a set of keywords to capture the association of set keywords to an image in the HTML page. Let us consider that an HTML page consists of five paragraphs and set of images. The keywords extracted from each paragraph can be written as

p 1 = {k 11 , k 12 , … k 1 m } p2 = p

3

=

{k 21

, k 22 , … k 2 m

{k

,k

31

32

,… k

3 m

(8)

}

(9)

}

(10)

p 4 = {k 41 , k 42 , … k 4 m }

(11)

p 5 = {k 51 , k 52 , … k 5 m }

(12)

The semantics of the images present in HTML page is captured using the textual information available such as name, optional text and caption of the images. This can be written as I = {name (i ), option _ t (i ), caption (i )}

(13)

Where i=1,2,....N. Using Eq. (8),(9),(10),(11) and (12), the importance of keyword with the HTML page can be measured and could be used for retrieving the images. Table 1. Weighted Matrix between keyword and image high-level features

HTML Page

K1 K2 K3 K4 K5

Name (i)

Option_t(i)

Caption(i)

1

0

1

1

1

1

0

1

0

1

0

1

1

1

1

In the above table, K i is the keywords extracted from paragraph i. While extracting the keywords the stop words are removed and the stemming operation is carried out. The high-level semantic information of the images is extracted from the above table. Say for example, the caption text is matching with most of the keywords and thus the corresponding keyword has more association with the HTML page and captures the semantics of the images. Similarly, the combination of K i is also possible with logical OR, AND, etc operators for improving retrieval performance.

Weighted Matrix for Associating High-Level Features with Images in Web Documents

323

3 Experimental Results We have developed our own Internet crawler for crawling HTML documents along with the images present and it is shown in Fig.1.

Fig. 1. The Crawler with Various Components for Weighted Matrix Construction and Retrieval

The Web documents from various domains such as sports, news, cinema, etc have been crawled. Approximately, 10,000 HTML documents with 30,000 images have been extracted and preprocessed for retrieval. The web crawler provides HTML pages along with the image content. The text in each page is separated as normal text and image related text. While extracting the keywords, the stop words are removed and the stemming operation is carried out. Once the keywords and image related text are extracted, the weighted matrix is constructed and it is stored in the repository. For each page, this matrix is constructed and stored in the repository along with a relation between the document and image present. While making query based on textual keywords, search will be carried out in the repository and the final ranking of retrieval is based on the weighted value. The images with higher weights are ranked first and will tip the retrieval result. In our system, the precision of retrieval is used as the performance measure. Initially, we have measured the precision of retrieval using keywords and HCPH and ICICM for measuring the effect of combining textual features and low-level features

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and the precision is measured. Fig 2 shows the average precision (P) in percentage for 10, 20, 50 and 100 nearest neighbours. In Fig. 2, H and K represents the retrieval results using the combination of Haar Wavelet based low-level features and Keywords. D and K refer the combination of Debauches Wavelet based low-level features and Keywords. We determined the precision for each of these images and computed their average precision. From the results shown in Fig. 2, it is observed that the performance of the combined method is quite encouraging. The precision of retrieval of combination of keyword with ICICM is found to be high compared to other feature based combinations. However, the combination of keyword and image information achieves higher average precision.

Perceived Precision

100 80

Keyword

ICICM

H and K

D and K

HCPH and K

ICICM and K

K and image text

60 40 20 0 10

20

50

100

No. of Nearest Neighbors Fig. 2. Comparison of Precision of Retrieval of the Features used for Retrieval

4 Conclusion We have studied the important role of textual keywords in describing high-level semantics of an image. We felt that, to define the high-level semantics of images on the World Wide Web, HTML documents can be effectively used. Using a crawler, the HTML documents along with their images are fetched from the Internet. Keywords are extracted from the HTML documents after eliminating HTML tags, the stop words and common words. The frequency of occurrence of words in the HTML documents is computed to assign weights to the keywords. During retrieval using only keywords as query, the final ranking is based on the frequency of occurrence. In addition, the image name is also treated as a keyword with high value of frequency of occurrence to improve the precision of retrieval. Combining both high-level and image information achieves high average precision of retrieval.

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325

References 1. Cai, D., Yu, S., Wen, L.R., Ma, W.Y.: VIPs: a Vision based Page Segmentation Algorithm. Microsoft Technical Report, MSR-TR-2003-79 (2003) 2. Cai, D., He, X., Ma, W.-Y., Wen, J.-R., Zhang, H.: Organizing WWW Images based on the Analysis of Page Layout and Web Link Structure. In: Int. Conf. on Multimedia Expo., pp. 113–116 (2004) 3. Gevers, T., Stokman, H.M.G.: Robust Histogram Construction from Color Invariants for Object Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(1), 113–118 (2004) 4. Han, J., Ngan, K.N., Li, M., Zhang, H.-J.: A Memory Learning Framework for Effective Image Retrieval. IEEE Transaction on Image Processing 14(4), 511–524 (2005) 5. Hu, W., Wu, O., Chen, Z., Fu, Z., Maybank, S.: Recognition of Pornographic Web Pages by Classifying Texts and Images. IEEE Transactions on Pattern Analysis And Machine Intelligence 29(6), 1019–1034 (2007) 6. Jing, F., Li, M., Zhang, H.-J., Zhang, B.: A Unified Framework for Image Retrieval Using Keyword and Visual Features. IEEE Transaction on Image Processing 14(7), 979–989 (2005) 7. Palm, C.: Color Texture Classification by Integrative Co-Occurrence Matrices. Pattern Recognition 37(5), 965–976 (2004) 8. TASI (2009), http://www.tasi.ac.uk/resources/searchengines.html 9. Tollari, S., Glotin, H., Le, J., Maitre, J.L.: Enhancement of Textual Images Classification using Segmented Visual Contents for Image Search Engine. Multimedia Tools and Applications, Special Issue on Metadata and Adaptability in Web-based Information Systems 25(3), 405–417 (2005) 10. Vadivel, A., Sural, S., Majumdar, A.K.: An Integrated Color and Intensity Co-Occurrence Matrix. Pattern Recognition Letters, Elsevier Science 28(8), 974–983 (2007) 11. Vadivel, A., Sural, S., Majumdar, A.K.: Robust Histogram Generation from the HSV Color Space based on Visual Perception. International Journal on Signals and Imaging Systems Engineering 1(3/4), 245–254 (2008) 12. Vadivel, A., Sural, S., Majumdar, A.K.: Image Retrieval from Web using Multiple Features. Online Information Review, Emerald 33(6), 1169–1188 (2009) 13. Xu, F., Zhang, Y.-J.: Integrated patch model: A Generative Model for Image Categorization based on Feature Selection. Pattern Recognition Letters 28, 1581–1591 (2007) 14. Yang, H.-C., Lee, C.-H.: Image Semantics Discovery from Web Pages for Semantic-based Image Retrieval using Self-organizing maps. Expert Systems with Applications 34, 266– 279 (2008) 15. Zhao, R., Grosky, W.I.: Narrowing the Semantic Gap—Improved Text-Based Web Document Retrieval using Visual Features. IEEE Transactions on Multimedia 4(2), 189–200 (2002)

Author Index

Abdul Hameed, W. 241 Abdul Samath, J. 289 Ambika Gayathri, P. 289 Anand, S. 65 Anantachar, M. 221 Arora, Ashish 19 Ayisha Begum, A. 289

Mary Shanthi Rani, M. 105 Meda, Venkatesh 221 Modak, Nilkanta 187 Mondal, Tanmoy 121 Monoth, Thomas 258 Mourya, Gajendra Kumar 121 Nair, Madhu S. 113 Natarajan, G. 159

Babu Anto, P. 258 Bagavandas, M. 241 Balasubramaniam, P. 35 Balasubramanian, K. 129 Banerjee, Durga 174 Biswas, Animesh 47, 187 Biswas, Papun 79 Bose, S. 249 Chakraborti, Debjani Easwaramoorthy, D. Gopalan, Sasi

Pal, Bijay Baran 79, 174 Pal, P.C. 145 Palaniappan, N. 213 Palaniappan, R. 271 Pandian, P. 159 Paramanathan, P. 166 Park, Jong Yeoul 1 Prabakaran, K. 204 Prabakaran, R. 232 Prasad, Bhagwati 197

79 89

113

Hiregoudar, Sharanakumar Hote, Yogesh V. 19 Jaganathan, P. 281 Jaya Christa, S.T. 137 Jeeva Sathya Theesar, S. Jeong, Jae Ug 1

221

35

Kalavathi, P. 152 Kalidass, Mathiyalagan 27 Kannan, A. 249 Kathirvalavakumar, T. 271 Katiyar, Kuldip 197 Kim, Seoksoo 266, 297 Kumar, Surendra 73 Kumaresan, Nallasamy 57 Lim, Ji-hoon

266

Majumder, Debasish Mandal, Dinbandhu

47 145

Rajasekaran, P. 232 Rajinikannan, M. 281 Ramakrishnan, K. 11 Ramappa, K.T. 221 Rastogi, Mahima 19 Ratnavelu, Kuru 57 Ray, G. 11 Sahu, Subhasis 47 Saravanasankar, S. 65 Sebastian, Souriar 113 Sekar, S. 204 Sen, Shyamal 174 Shanmugavadivu, P. 129, 318 Sheela, C. 113 Shreshta, Bijay 221 Shyamala, P. 312 Somasundaram, K. 105, 152, 213 Song, Jae-gu 297 Su, Si 97 Subbaraj, P. 65 Subramanian, S. Siva 249

328

Author Index

Sukavanam, N. 73 Sumathy, P. 318 Surendiran, B. 303

Vadivel, A. 303 Venkatesh, P. 137 Vijayakumar, P. 249

Thanigaiselvan, R.

232

Udhaykumar, R. Uthayakumar, R.

221 89, 166

Wong, Bernardine R. Xu, Runping

97

57