Application of Fuzzy Cellular Neural Network and its Stability to Color Edge Detection

Transcription

1 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science Application of Fuzzy Cellular Neural Network its Stability to Color Edge Detection M. Kalpana P. Balasubramaniam Abstract--- The main objective of this paper is to derive innovative vector-valued color edge detection, which is different from the papers existing in the literature. In this paper, based on the Lyapunov function, the global stability of fuzzy cellular neural network (FCNN) is studied. Moreover, a simple threshold method is used to threshold each (Red), (Green), (Blue) valued functions so FCNN method is employed to each thresholded, valued functions to incur the precise convergent vector-valued color edge image, by single iteration. Experimental results are given to prove that the proposed method is more efficient. Keywords-Component--- Color Images, Edge Detection, Fuzzy Cellular Neural Network, Simple Thresholding C I. INTRODUCTION ELLULAR NEURAL NETWORKS (CNNs) are a parallel computing paradigm similar to neural networks, with the difference that communication is allowed only between neighboring units, that is an array of locally coupled nonlinear electrical circuits (or cells), which are capable of processing a large amount of information in parallel in real time, see Chua Yang in [5], [6]. Its theoretic foundations practical applications include image signal processing, analyzing 3D surfaces, robotic biological visions so on. However, in mathematical modeling of real world problems, uncertainty or vagueness is unavoidable. Fuzzy set theory provides mathematical support to capture these uncertainties associated with human cognitive processes. Each cell in an FCNN contains fuzzy operating abilities, yet, the entire network is governed by cellular computing laws, proposed by Yang in [20], [21]. It is a very useful paradigm for image processing pattern recognition [1], [17], [18]. In image processing, a pixel s value is calculated based on its neighbor pixels. Neural networks like Hopfield networks lacking this property of locality make them unsuitable for image processing applications [13]. However, the FCNN characteristics of its local interconnection make it applicable to image processing VLSI realization. The analysis of stability of neural networks has received much attention [10], [11], [19]. Thresholding is a popular, fast, computationally inexpensive segmentation technique [3]. The fundamental difference between color images gray- M. Kalpana, Junior Research Fellow, Department of Mathematics, Ghigram Rural Institute - Deemed University, Ghigram , Tamilnadu, India. kalpana.nitt@gmail.com P. Balasubramaniam, Professor Head, Department of Mathematics, Ghigram Rural Institute - Deemed University, Ghigram , Tamilnadu, India. balugru@gmail.com level images is that in color image processing, vector-valued image functions are treated instead of the scalar image functions used in gray-level image processing [12]. For a digital color image three vector components are given for one image pixel Edges play important roles in many applications such as image segmentation, boundary detection, object recognition classification, image registration so on. Color edge detection schemes perform extremely better than gray scale approaches, when edges exist at the boundary between regions of different colors with no change in intensity [9]. In graylevel images, a discontinuity in the gray-level function is indicated as an edge, the term color edge has not been clearly defined for color images. Although color-based edge detection includes more information than gray-based detection, it is much less well defined has not received the same attention as gray-based ones [2], [14]. Recently, there are some literatures with respect to color edge detection [4], [7], [8], [15], [22], are failed to produce vector-valued edge detection for the color images. However, best of authors knowledge; there are no results on finding the vector-valued edge detection for the color images in the literature. Motivated by the above discussion, in this paper, a novel approach is developed to find the vector-valued edge detection for the color images by using a simple threshold method to threshold each valued function so FCNN method is applied to each thresholded valued functions obtain the precise convergent vectorvalued color edge image, by single iteration. Finally, the experimental results show that the proposed method performs better than classic methods such as Roberts, Prewitt, Sobel, LOG neural network method such as CNN. II. MODEL DESCRIPTION OF FCNN AND PRELIMINARIES The circuit of a cell in an FCNN is shown in Fig. 1., where the suffices denote input, state output, respectively. Voltages denote input, state output voltages of cell Here, where is a nonlinear function. are two fuzzy logical operations. Also, they may be any expression of fuzzy OR /or fuzzy AND

2 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science State equation of is given by Figure 1: The Circuit of Cell in a FCNN Definition 2.1 The cell on the row column is denoted by Definition 2.2([6]). influence) (the sphere of (1) where are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template, fuzzy feed-forward MAX template, respectively. are elements of feedback template feed-forward template, respectively. denote fuzzy AND fuzzy OR, respectively. denote state, output, input bias of respectively. denote as capacitor resistor, respectively. Output equation of is given by Constraint conditions are given by Parameter assumptions (2) Figure 3: The Neighborhood of Cell defined for respectively. The of radius of a cell in CNN is defined by where is a positive integer number. Definition 2.3 ([16]). The inner cell is the cell which has neighbor cells, where is defined in above. Other cells are called boundary cells. III. GLOBAL STABILITY In this section, we shall see the global stability of FCNN. Definition 3.1 The fuzzy feedback MIN/MAX templates fuzzy feedforward MIN/MAX templates take the constant values respectively; if for all there exists a connection between neurons Thus, in terms of the above definition, (1) can be reformulated as, (5) where Figure 2: Architecture of FCNN

3 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science According to (4), we have immediately. Lemma 3.1 ([19]). Let be two states of system (5), then, The neuron activation function continuous; that is, there exist constants for all Let the matrix be, then we have Lemma 3.2. are Lipschitz such that Lemma 3.2 ([19]). If the spectral radius of the matrix is less than then there exists only one globally stable equilibrium point in (5). Let have be an equilibrium state of (5). In terms of (5), we (6) Theorem 3.1 Suppose the spectral radius of the matrix is less than under assumption Then, the equilibrium state in (6) is globally stable. Proof Since there exists only one equilibrium state in (6). We construct the following Lyapunov function when when Along the solution of (6), we calculate the Dini upper-right differential of as (7) Then, the equilibrium state in (6) is globally stable. Its global stability shows that the state of a neuron will finally find a stable equilibrium point. The last inequality is satisfied when Here, the pixels of color image is converted into neurons of FCNN. Each neuron will change its state iteratively according to (5), until the entire FCNN network converges. IV. FCNN BASED VECTOR-VALUED COLOR EDGE DETECTION ALGORITHM The following algorithm is the novel approach of finding vector-valued edge detection for the color images based on a simple threshold FCNN method, by single iteration. 1. Divide each pixel of input color image into valued functions. However, three separated grayscale images will be obtained. 2. A simple threshold method is used for each pixel in the three separated grayscale images to create corresponding binary images. The threshold images are respectively. 3. The FCNN method is applied to each pixel of Moreover, we acquire edge detection of valued functions. 4. Assign each three edge detection components into an output pixel. 5. Finally, a novel vector-valued color edge image will be obtained.

4 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science V. EXPERIMENTAL STUDIES In this Section, several results of experiments are illustrated. The key point for image processing based on FCNN is to find proper templates to complete different tasks. Example 5.1 According to Theorem 3.1, the original templates for color image are given below Figure 4: The Algorithm Flow Chart Letting assumption we get which satisfies In order to prove the efficiency of the proposed method in color edge detection, four other classic methods (Roberts, Prewitt, Sobel, Log) neural network such as CNN are used for comparisons as references. Fig. 5. represents the original image of size The original image is divided into three components. A simple threshold method is applied to each components to obtain respectively. Simulations of Roberts, Prewitt, Sobel, Log, CNN our proposed method of FCNN are applied to to obtain (a) - edge detection, (b) - edge detection, (c) - edge detection, (d) - vector valued color edge detection, which are represented in each following figures, respectively. Figure 5: Original Image

5 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science Figure 6: Results of Edge Detection based on Roberts Method: (a) - Edge Detection, (b) - Edge Detection, (c) - Edge Figure 9: Results of Edge Detection based on Log Method: (a) - Edge Detection, (b) - Edge Detection, (c) - edge Figure 7: Results of Edge Detection based on Prewitt Method: (a) - Edge Detection, (b) - Edge Detection, (c) - Edge Figure 10: Results of Edge Detection based on CNN Method: (a) - Edge Detection, (b) - Edge Detection, (c) - Edge Figure 8: Results of Edge Detection based on Sobel Method: (a) - Edge Detection, (b) - Edge Detection, (c) - edge Figure 11: Results of Edge Detection based on our proposed FCNN Method: (a) - Edge Detection, (b) - Edge Detection, (c) - Edge Detection, (d) - Vector Valued Color Edge Detection

6 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science The extracted vector-valued color edge detection performed by the proposed method is more precise than the above classic operators CNN method. applied to each valued functions to obtain the continuous accurate novel vector-valued color edge image, by single iteration with less time complexity. Figure 12: Results Detected by Different Algorithms: (a) the Original Image, (b) Roberts, (c) Prewitt, (d) Sobel, (e) Log, (f) CNN, (g) the Proposed FCNN Method In Figs , the size of the original image is with full color luminance change. The size of the original image in Fig. 14. is with various color luminance change. Experimental results show that the proposed approach is stable effective. In all the existing literatures [4], [7], [8], [15], [22], the edges for the color images are in binary color. However, our novel approach provides precise color edge. Moreover, the proposed method can be applied for the real-time application task that the color edge image is required in a short time. Remark 5.1 The color edge detection based on the proposed method for given various input images are well obtained by single iteration with less time complexity than the classic operators CNN method through MATLAB R2008a. Moreover, some obvious edges of background in the images are missed when performed Roberts, Prewitt, Sobel Log. In addition, the neural network method such as CNN has less accurate lines, when compared to our proposed FCNN method. VI. CONCLUSIONS AND FUTURE WORK In this paper, a novel approach of finding the vector-valued edge detection for the color images has been discussed. Based on the Lyapunov function, the global stability of FCNN has been studied. However, a simple threshold method has used to threshold each valued functions, according to human vision system. Moreover, the FCNN method has Figure 13: Results of Color Edge Detection: (a) the Original Image, (b) Roberts, (c) Prewitt, (d) Sobel, (e) Log, (f) CNN, (g) the Proposed FCNN Method Figure 14: Results of Color Edge Detection: (a) the Original Image, (b) Roberts, (c) Prewitt, (d) Sobel, (e) Log, (f) CNN, (g) the Proposed FCNN Method

7 Proceedings of the International Conference on Applied Mathematics Theoretical Computer Science Experimental results have shown that the proposed method performs better than classic methods such as Roberts, Prewitt, Sobel, Log, neural network method such as CNN. Moreover, the authors are expecting to report more results along with medical images in the near future. ACKNOWLEDGMENT The work done by Miss. M. Kalpana is supported by the Government of India, Ministry of Science Technology, Department of Science Technology, under Grant No. DST/INSPIRE Fellowship/2010/[293]/dt. 18/03/2011. REFERENCES [1] P. Balasubramaniam M. Kalpana, CT Liver Segmentation based on Fuzzy Cellular Neural Networks its Stability, International Conference on Affective Computing Intelligent Interaction (ICACII 2012), [2] Taipei, Taiwan, Lecture Notes in Information Technology, ISBN: , Vol. 10, Pp , [3] J. 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Kalpana was born in She received her B.Sc (Mathematics) degree from Arungarai Amman College of Arts Science, Chinnadharapuram affiliated to Bharathidasan University, Trichy, Tamilnadu, India in She received her M.Sc (Operations Research Computer Applications) degree from National Institute of Technology, Trichy in Moreover, she received University 4 th Rank in Undergraduate Course (Bharathidasan University) (2006), Outsting Student Award (2009) Gold Medal (2009) in Post graduate (NIT, Trichy). She was awarded Master of Philosophy in the year 2010 in the field of Mathematics from Ghigram Rural Institute-Deemed University, Tamilnadu, India. Currently, she is pursuing PhD (Mathematics) degree with specialized area of Studies on Stability Analysis for Dynamical System of Neural Networks from Ghigram Rural Institute-Deemed University, Tamilnadu, India. Her recent research interest includes Neural Networks, Stability Analysis of Dynamical Systems Image Processing. P. Balasubramaniam post graduated from the Department of Mathematics of Gobi Arts College affiliated to Bharathiar University, Coimbatore in the year He was awarded Master of Philosophy in the year 1990 Doctor of Philosophy (Ph.D) in 1994 in the field of Mathematics with specialized area of Control Theory from the Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu, India. Soon after his completion of Ph.D degree, he served as Lecturer in Mathematics in Kumaraguru College of Technology also in Kongu Engineering College for three years. Since Feb he served as Lecturer in Mathematics for four years Reader in Mathematics for five years in the Ghigram Rural University, Ghigram, Tamilnadu, India. He is rendering his services as a Professor Head, Department of Mathematics, Ghigram Rural University, Ghigram, Tamilnadu, India from Nov till date. He was selected as a Visiting Research Professor during the year for promoting research in the field of control theory neural networks at Pusan National University, Pusan, South Korea. Recently, he has worked as a Visiting Professor for six months from 12 th Sept to 11 th March 2012 for promoting research in the field of control theory neural networks at University of Malaya, Kuala Lumpur Malaysia. He has 18 years of experience in teaching research. He has published 115 research papers in various SCI journals holding impact factors. He has also published research articles in national journals international conference proceedings. He is also serving as a reviewer for few SCI journals member of the editorial board of Journal of Computer Science. Unique recognition was bestowed on him through a Tamilnadu Scientist Award (TANSA)-2005 for the discipline of Mathematical Sciences from Tamilnadu State Council for Science Technology. His research interest includes the areas of Control Theory, Stochastic Differential Equations, Soft Computing, Neural Networks Cryptography.