INTUITIONISTIC FUZZY SHORTEST PATH ALGORITHM FOR OPTIMAL BOUNDARY EXTRACTION IN IMAGE PROCESSING

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1 International Journal of Pure and Applied Mathematics Volume 114 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu INTUITIONISTIC FUZZY SHORTEST PATH ALGORITHM FOR OPTIMAL BOUNDARY EXTRACTION IN IMAGE PROCESSING R.SOPHIA PORCHELVI 1 AND A. BANUMATHY 2 Abstract. In this paper, a shortest path algorithm is proposed for optimal boundary extraction based on representing edge segments in the form of graph. Here, intuitionistic triangular fuzzy number is assigned to each arc instead of fuzzy number. An illustrative example is given to express the proposed work. 1. Introduction A problem of fundamental importance in image analysis is edge detection. There are so many edge detection methods and operators developed for noiseless and noisy images. Moreover edges characterize object boundaries and are therefore useful for segmentation and other image analysis techniques. Boundaries are linked edges that characterize the shape of an object and they can be found by tracing the connected edges. Edge detection in images and the concept of boundary extraction are addressed here as given [1, 2, 3, 4]. A boundary path can be obtained using Heuristic searching algorithms [5] and moreover, Dynamic Programming is used for getting the optimal path. It gives the global optimum of multistage processes. Optimal boundary can be obtained by an algorithm in [7] by critical path method. Here in this paper, an attempt is made to find the global optimum using an algorithm based on shortest path length using triangular intuitionistic fuzzy number. Consider the edge weight of the graph as uncertain, which means that it is either imprecise or unknown Mathematics Subject Classification. 03F55, 68U10, 90C70, 94A08. Key words and phrases. Intuitionistic fuzzy number, Intuitionstic fuzzy set, Intuitionistic fuzzy shortest path, optimal boundary

2 International 2 Journal R.SOPHIA of Pure PORCHELVI and Applied ANDMathematics A. BANUMATHY The fuzzy shortest path problem was first analyzed by Dubois and Prade [3] using fuzzy number instead of a real number is assigned to each edges. In this paper we propose an intuitionistic fuzzy number instead of fuzzy number. An algorithm is based on the idea that from all the possible paths from source node to destination node, an edge with shortest length is computed and the Euclidean distance is computed for all the paths with the edge of minimum distance is the shortest path for membership and non membership values. 2. Preliminaries Definition 1 (Intuitionistic Fuzzy Set(IFS)). Let X be an universe of discourse, then an intuitionistic Fuzzy Set A in X is given by A = {x, µ A (x), γ A (x)/ x X} where the functions µ A (x) : X [0, 1] and γ A (x) : X [0, 1] determine the degree of membership and non membership of the element x X respectively for every x X, 0 µ A (x) + γ A (x) 1. Definition 2 (Intuitionistic Fuzzy Numbe(IFN)). Let A = {x, µ A (x), γ A (x) /x X} be an IFS, then we call the pair (µ A (x), γ A (x)) an intuitionistic fuzzy number. We denote it by ( a, b, c, l, m, n), where a, b, c F (I), l, m, n F (I), I = [0, 1], 0 c + n 1. Definition 3 (Triangular Intuitionistic Fuzzy Number and its Arithmetic). A triangular fuzzy number A is denoted by A = {(µ A, γ A ) x R}, where µ A and γ A are triangular fuzzy numbers with γ A µ A c. So a triangular intuitionistic fuzzy number A is given by A = ( a, b, c, e, f, g)with ( e, f, g a, b, c c ) i.e., either e band f corf aand g b are membership and non memberswhip fuzzy numbers of A. The addition of two triangular fuzzy numbers are as follows [5] For two Triangular Intuitionistic Fuzzy Numbers A = ( a 1, b 1, c 1 : µ A, e 1, f 1, g 1 : γ A ) and B = ( a 2, b 2, c 2 : µ B, e 2, f 2, g 2 : γ B ) with µ A µ B and γ A γ B, define A + B = ( a 1 + a 2, b 1 + b 2, c 1 + c 2 : min (µ A, µ B ), e 1 + e 2, f 1 + f 2, g 1 + g 2 : max (γ A, γ B )). A boundary can also be viewed as a path through a graph formed by linking the edge elements together. Linkage rules give the procedure for connecting the edge elements. Suppose a graph with node locations x k, k = 1, 2,... is formed from node 1 to node 10. Also, suppose we are given an evaluation function ϕ(x k ), which gives the 166

3 International INTUITIONISTIC Journal of Pure FUZZY andshortest Applied Mathematics PATH ALGORITHM Special Issue3 value of the path from 1 to 10 constrained to go through the node x k. The boundary path from 1 to 10 is obtained through graph search algorithms. Such a path need not be optimal one. By using the Bellman s principle of optimality in Dynamic programming, the optimal boundary path can be obtained as given in [1]. A Martelli [5] studied that the optimization problem discussed here can be posed as a shortest path problem on a graph, instead of solving it using dynamic programming. Here in this paper, after converting the edge map into a forward connected graph of N stages as dealt in [8], the shortest path can be obtained by the procedure given in the next section. 3. An Intuitionistic Fuzzy Shortest Path Length Procedure [8] In this paper the arc length of a graph is considered to be an intuitionisitc fuzzy number, namely triangular intuitionistic fuzzy umber. The shortest path length procedure is based on the Chuang and Kung [9] method. Step: 1 Compute all the possible path lengths L i from i = 1, 2, 3,..., n. Where L i = ( a i, b i, c i, l i, m i, n i ). Step: 2 Initialize L min = ( a, b, c, l, m, n) = L 1 = ( a i, b i, c i, l i, m i, n i ). Step: 3 Initialize i = 2. Step: 4 (a) Compute for membership values b, if b a i b = bb i aa i (b + b i ) (a + a i ), if b a i a = min(a, a i) c = min(c, b i) (b) Compute ( l, m, n) for non membership values m, if m l i m = mm i ll i (m + m i ) (l + l i ), if m l i l = min(l, l i) n = min(n, m i) Step: 5 Set L min = ( a, b, c, l, m, n) as calculated in Step

4 International 4 Journal R.SOPHIA of Pure PORCHELVI and Applied ANDMathematics A. BANUMATHY Step: 6 i = i + 1. Step: 7 if i n, go to Step Example. Consider a 3 5 array of edges whose gradient magnitudes g and tangential contour directions θ are shown in Figure 1. Figure 1. Gradient magnitude and contour directions Applying the linkage rules and letting α = 4/π, β = 0 as in [1]. We obtain the graph as shown in Figure 2. Consider the graph with Figure 2 the triangular intuitionistic fuzzy arc lengths as shown below. The arc lengths are assumed to be E 12 = ( 4, 6, 8, 9, 11, 16) E 46 = ( 2, 2, 3, 5, 8, 12) E 13 = ( 0, 0, 2, 3, 5, 10) E 57 = ( 1, 2, 3, 4, 7, 15) E 14 = ( 2, 3, 4, 4, 7, 13) E 67 = ( 2, 3, 4, 8, 10, 14) E 32 = ( 5, 7, 9, 10, 14, 17) E 89 = ( 6, 7, 8, 10, 12, 17) E 28 = ( 2, 4, 6, 7, 10, 12) E 910 = ( 4, 5, 6, 8, 13, 18) E 45 = ( 0, 0, 1, 4, 7, 79) E 710 = ( 5, 6, 7, 10, 11, 17) 168

5 International INTUITIONISTIC Journal of Pure FUZZY andshortest Applied Mathematics PATH ALGORITHM Special Issue5 The possible paths and its length are as follows: P 1 : L 1 = ( 16, 22, 28, 34, 46, 63) P 2 : L 2 = ( 17, 23, 31, 38, 54, 74) P 3 : L 3 = ( 8, 11, 15, 22, 32, 54) P 4 : L 4 = ( 11, 14, 18, 27, 36, 56) Using the above procedure for the graph shown in Figure 2, we get an Intuitionistic Shortest Path Length L min = ( 8, 11.83, 11, 22, 29.81, 32 ). 4. To Compute the Shortest Path By combining an intuitionistic fuzzy shortest length method with similarity measure, an algorithm is as follows. An algorithm for intuitionistic fuzzy shortest path [6]. Step: 1 Find out all the possible paths from source node s to destination node d and compute the corresponding path lengths L i, i = 1, 2, 3,..., n.. Step: 2 Compute L min by using an intuitionistic fuzzy shortes path length procedure. Step: 3 Find the Euclidean distance d i for i = 1, 2, 3,..., n between all the possible path and L min. Step: 4 Decide the shortest path with the path having lowest Euclidean distance. On executing the above algorithm on the example graph of Figure 2. The first two steps have been already calculated in Section 3.1. The next step, we compute the similarity degree S (L min, L i )between L min and L i for i = 1, 2, 3,..., n by means of similarity measure S (A, B) = m k=1 [1 A(x k ) B(x k ) ] m. In order to compute the accuracy in similarity degree, we should let a generic element U denoted by µ i (i = 1, 2, 3,..., n). Step: 1 From Section 3.1, we have four possible path lengths in Figure 2 as L 1, L 2, L 3 and L 4. Step: 2 The minimum path length is calculated as L min = ( 8, 11.83, 11, 22, 29.81, 32). 169

6 International 6 Journal R.SOPHIA of Pure PORCHELVI and Applied ANDMathematics A. BANUMATHY Step: 3 Euclidean distance between all path lengths P i (i = 1, 2, 3, 4) and L min d (p 1, L min ) = ( (16 8) 2 + ( ) 2 + (28 11) 2, ) (34 22) 2 + ( ) 2 + (63 32) 2 = ( 21.36, 36.97) d (p 2, L min ) = ( (17 8) 2 + ( ) 2 + (31 11) 2, ) (38 22) 2 + ( ) 2 + (74 32) 2 = ( 24.61, 56.04) d (p 3, L min ) = ( (8 8) 2 + ( ) 2 + (15 11) 2, ) (22 22) 2 + ( ) 2 + (54 32) 2 = ( 4.09, 22.11) d (p 4, L min ) = ( (11 8) 2 + ( ) 2 + (18 11) 2, ) (27 22) 2 + ( ) 2 + (56 32) 2 = ( 7.92, 25.28) Step: 4 Decide the shortest path with the path having lowest Euclidean distance for member and non member by examining the Euclidean distance d between L min and d i for i = 1, 2, 3,..., n. From the above calculations we can see that path p 3 : has the least Euclidean distance for membership and non membership. The shortest path from source node 1 to destination node 10 is The above path can also obtained by means of a C program developed for this algorithm. The output is given below. L min = ( 8, 11.83, 11, 22, 29.81, 32 ) The shortest path from source node to destination node is p

7 International INTUITIONISTIC Journal of Pure FUZZY andshortest Applied Mathematics PATH ALGORITHM Special Issue7 5. Conclusion In this paper, an attempt is made to find the optimal boundary path with triangular intuitionistic fuzzy number. A C program is also developed for this algorithm to find the optimal boundary in image processing. An illustrative example is discussed to show the efficiency of the algorithm as well as the program. This algorithm provided the better output for different types of graphs and nework. References [1] Alberti Martelli, An Application of Heuristic Search Methods to Edge and Contour Detection, Graphics and Image Processing, 19(2), p (1976). [2] K. Attanassov, Intuitionistic Fuzzy Set, Fuzzy Sets and System, volume 20, No. 1(1986), [3] D. Dubois and H. Prade, Fuzzy Sets and System, Academic Press, New York, [4] Hamdy, A. Taha, Operations Research: An Introduction, Eighth Edition, Prentice Hall of India Private LTd., Delhi(2006). [5] Jain, A.K. Fundamentals of Digital Image Processing, Prentice Hall of India, Pvt. Ltd., a. New Delhi (1997). [6] A. Kiran Yadav and B. Ranjit Biswas, On Searching Fuzzy Shortest Path in a Netweok, International Journal of Recent Trends in Engineeering, Vol 2, No. 3, November [7] V. Lakshmana Gomaathi Nayagam, G. Venkateshwari and Geetha Sivaraman, Randing of Intuitionistic Fuzzy Numbers, 2008 IEEE International Conference on Fuzzy Systems (FUZZ 2008). [8] Montanari. U, On the optimal Detection of Curves in Noisy Pictures, Commum, ACM 14, p (1971). [9] A. Nagoor Gani and M. Mohammed Jabarulla, On Searching Intuitionistic Fuzzy Shortest Path in a Network, Applied Mathematical Sciences,Volu.e 4, No. 6 (2010), [10] Tzung-Nan Chuang and Jung-Yuan Kung, A new algorithm for the discrete fuzzy shortest path problem in a network, Applied Mathematics and Computation 174 (2006) PG and Research Department of Mathematics, A.D.M. College for Women (Autonomous), Nagapattinam , Tamil Nadu, India. address: sophiaporchelvi@gmail.com 2 Assistant Professor, Dept. of Mathematics, A.V.C. College of Engineering, Mayiladuthurai, Tamil Nadu, India address: banugiri81@gmail.com 171