Hybrid Image Segmentation Using Fuzzy C-Means and Gravitational Search Algorithm

Emadaldin Mozafari Majd, M.A. As’ari, U.U. Sheikh, S.A.R. Abu-Bakar Computer Vision, Video and Image Processing Research Group (CvviP),

Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Johor Bahru, Malaysia e.mozafarimajd@gmail.com, amir-asari@biomedical.utm.my, usman@fke.utm.my,

syed@fke.utm.my

ABSTRACT

In this paper, we propose a new hybrid approach for image segmentation. The proposed approach exploits spatial fuzzy c-means for clustering image pixels into homogeneous regions. In order to improve the performance of fuzzy c-means to cope with segmentation problems, we employ gravitational search algorithm which is inspired by Newton’s rule of gravity. Gravitational search algorithm is incorporated into fuzzy c-means to take advantage of its ability to find optimum cluster centers which minimizes the fitness function of fuzzy c-means. Experimental results show effectiveness of the proposed method in segmentation different types of images as compared to classical fuzzy c-means.

Keywords: gravitational search algorithm, fuzzy c-means, segmentation, cluster centers

1. INTRODUCTION Image segmentation, which is an advancing research field in computer vision, refers to clustering an image into set of

pixels possessing similar characteristics in terms of intensity values and geometrical features. Much effort has been done to achieve an efficient segmentation algorithm to be able to work in a wide range of images. Efficient algorithms have been used for image segmentation, can be categorized as follows. Thresholding based techniques [1,2] exploit certain values called threshold to discriminate between intensity values of objects in images. Active contours [3,4] employ a set of primary contours which move towards the object boundaries according to the information derived from energy function. Besides, active contours have demonstrated good performance in segmentation of organs in medical images. Graph-based techniques define segmentation problem as a graph G = (V;E) in which each node virefers to a pixel in the image, and the edges in Econnect certain pairs of neighboring pixels. Felzenszwalb proposed an efficient graph based image segmentation in [5]. Boykov and Funkalea [6] suggested graph cut algorithm for N-dimensional image segmentation.

One alternative solution for segmentation problems is to relate the concept of clustering to segmentation. Generally, clustering is assigning a set of ndata points into a smaller number of cclusters. Besides, clustering techniques have been widely applied to the image segmentation in which among them K-means and fuzzy c-means attracted more popularity due to their high performances. Fuzzy c-means has proved to be the most effective clustering technique for image segmentation purposes. The algorithm employs an iterative procedure to minimize the objective function depending on the Euclidean distance of the pixels to the cluster centers. Since correlation of pixels in the immediate neighborhood is high, incorporation of spatial correlation into fuzzy c-means algorithm can improve the results of conventional fuzzy c-means. Hence, spatial fuzzy c-means [7,8]can be used as an alternative in such applications. Despite high performance of fuzzy c-means, it happens that the algorithm is unable to segment precisely. One solution is to incorporate evolutionary algorithms into the problem. Particle swarm optimization [9,10]has been applied to the problem and generated promising results in terms of removing spurious segments.

Gravitational Search Algorithm (GSA) is a heuristic optimization method which was recently developed by Rashedi et al. [11,12]. It is inspired by the Newton’s rule of gravity and laws of motion. The algorithm exploits a group of agents which interact with each other by the gravity force, and laws of motion. Global movement is directed towards agents possessing heavier masses. They represent good solutions and on the other hand, are equivalent with slower motions to satisfy exploitation part of algorithm. Algorithm is converged to an optimum value in an iterative process. Moreover, being computationally effective motivates further application of this algorithm in new research fields.

Fourth International Conference on Digital Image Processing (ICDIP 2012), edited by Mohamed Othman,Sukumar Senthilkumar, Xie Yi, Proc. of SPIE Vol. 8334, 83342V · © 2012 SPIE

CCC code: 0277-786X/12/$18 · doi: 10.1117/12.956460

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In this paper, we incorporate the GSA algorithm into fuzzy c-means algorithm to find better estimation of cluster centers such that objective fuzzy c-means is minimized and expect to achieve higher performance compared to classical fuzzy c-means. The rest of paper is organized as follows. In section II and III, fuzzy c-means and GSA is discussed in detail, respectively. The proposed algorithm is presented in section IV. Experimental results of applying proposed algorithm is illustrated in section V. In section VI, paper is concluded and future research is discussed.

2. FUZZY C-MEANS Fuzzy c-mean, proposed by Bezdek [13], is an unsupervised learning algorithm which is widely applied to the image

segmentation. SupposeX = fx1; x2; :::; xng, refers to a set of n data points (an image with npixels), c is number of clusters to which data points are to be partitioned. Fuzzy c-mean exploits an iterative process aimed to minimize the objective function as follows:

Jm(U;V ) =cX

i=1

nXk=1

(uik)md2

ik(xk; vi)

(1)

Where uik represents the membership degree of xk in the ith cluster, a value in the interval [0 1], vi is the ith cluster center, m is the weighting exponent on each fuzzy membership and

dik = kxk ¡ vik = (xk ¡ vi)T (xk ¡ vi)

(2)

Where dik is the Euclidean distance between object xk and cluster center vi. A minimization criterion is accomplished when pixels in the local neighborhood of cluster centers possess high membership value and pixels far from the cluster centers possess low membership values. The membership function denotes the probability that a pixel belongs to a specific cluster. Taking the first derivatives of Jm with respect to uik and vi, the membership function and cluster centers, corresponding to the optimum results, are computed as follows;

uik =1

cPj=1

³dik

djk

´ 2m¡1

(3)

vi =

nPk=1

(uik)mxk

nPk=1

(uik)m (4)

The algorithm can be terminated in two steps at each iteration, the first step includes computing the fuzzy membership function. In the second step, the algorithm calculates the values of cluster centers. Since the unknown variables regarding the cluster centers and fuzzy membership arrays cannot be computed directly, at the starting point of the algorithm, the cluster centers are initialized to random values ranged between maximum and minimum intensity level. The algorithm exploits its iterative nature to estimate cluster centers and fuzzy membership values in the desired accuracy. The stopping criterion for algorithm is met when the difference between two cluster centers at two successive iterations is less than a small value of ɛ.

3. GRAVITATIONAL SEARCH ALGORITHM Gravitational Search Algorithm is a new population-based optimization method, proposed by Rashedi et al. [11]. The

algorithm is founded upon the Newton’s rule of gravity and laws of motion in physics. According to the Newton’s law of gravity, each particle attracts every other particle with a force, called gravitational force. GSA initializes a set of agents which cooperate with each other in the search space through gravitational force to find the optimum solution.

GSA can be modeled as a system with N agents in which Xi denotes position of ith agent. Where n is dimension of the problem, and xi

d determines the position of ith agent in the dth dimension. GSA exploits the values of fitness function to calculate the masses as formulated in (6) and (7). Where fi represents the fit and w(t) and b(t) represents the worst and best.

Xi = (x1i ; :::; x

di ; :::; x

ni ); i = 1; 2; :::; N

(5)

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mi(t) =fi(t)¡w(t)

b(t)¡w(t) (6)

Mi(t) =mi(t)

NPj=1

mj(t) (7)

Where Mi(t) and fi(t) represent the mass and the fitness value of the agent i at time t, respectively, b(t)and w(t)corresponds to different definitions depending on the kind of optimization problem. For a minimization problem, b(t)and w(t)are defined as follows:

b(t) = maxj2f1;:::;Ng

fj(t)

(8)

w(t) = minj2f1;:::;Ng

fj(t)

(9)

For a maximization problem, b(t) and w(t) are the opposite. Since the acceleration of the agent is updated in every iteration, the total forces imposed by set of heavier masses, corresponding to good solutions, on the agent is computed based on law of gravity as follows:

Fdij(t) = G(t)

Mi(t)£Mj(t)

Rij(t) + "(xd

j(t)¡xdi (t))

(10)

Fdi (t) =

Xj2kb;j 6=i

rjFdij(t)

(11)

Where G(t) is the gravitational constant at tth iteration, rj is a random number in the interval [0,1], Rij is the Euclidean distance between the two agents i and j, kb refers to those agents possessing the best fitness value and heaviest mass, and ε is a small constant. The gravitational constant, G decreases at each iteration to account for efficient exploitation. Hence, G can be modeled as a function of an initial value G0 and the value of iteration as G(t) = G(G0; t). The acceleration of the agent

calculated according to the law of motion in physics is formulated as adi (t) =

Fdi (t)

Mi(t). Moreover, the next velocity of the

agent is a percentage of its current velocity added to its current acceleration. Therefore, the next position and the next velocity of the agent can be calculated as follows:

vdi (t + 1) = ri ¢ vd

i (t) + adi (t)

(12)

xdi (t + 1) = xd

i (t) + vdi (t + 1)

(13)

GSA undergoes an iterative procedure till the algorithm satisfies the stop criteria. Furthermore, GSA provides us with promising results compared to other evolutionary algorithm such as PSO and CFO.

4. PROPOSED METHOD The main contribution of the proposed method is utilizing GSA algorithm to find optimal cluster centers such that better

performance of segmentation can be achieved. This goal can be accomplished by introducing a group of cluster centers, represented as agents, to GSA algorithm. Dimensions of agents determine the number of clusters. On the other hand, fuzzy c-means objective function plays the fitness function role in the GSA algorithm. The agents move through the search space via the algorithm rules and this process continues till the convergence criteria are met. The segmentation algorithm can be summarized in the following steps:

1 Fix c, 2 ≤ c < n, fix m, m >1 and initialize cluster centers {vi}ci=1. 6 Calculate M and a for each agent.

2 Calculate the fuzzy membership matrix U using (3). 7 Update velocity and position for each agents. 3 Initialize a group of agents as cluster centers. 8 Repeat Steps 3-10 till stoppage criteria are met. 4 Evaluate the fitness fuzzy c-means function for each agent. 9 Calculate Xie-Beni index and Partition entropy

index using the parameters obtained from the algorithm.

5 Update the G, best and worst of the population.

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5. EXPERIMENTAL RESULTS We apply the proposed method to a variety of images with different complexities. Here, the performance of the

proposed method can be tested under different number of clusters, different number of agent population and different initialization parameters. To initialize the algorithm, we set the parameters m = 2, ɛGSA = 0.0001, G0=100 and α=0.9. The population of initial agents N, is considered to be 35 which are randomly placed within an interval between lowest and highest intensity values in the image. Furthermore, the gravitational constant, Gis devised as a linear function, which is

decreasing slightly, due to better performance in our problemG = G0(1¡®t

T).The proposed method offers promising

segmentation results as compared with fuzzy c-means in terms of the value of validity functions and visual performance as shown in Table 1. In this paper, we employed two validity functions namely, Xie-Beni [14],

Vxb =

cPi=1

nPk=1

(uik)md2ik(xk; vi)

n ¤ (mini6=ld2il)

and Partition-Entropy [15], Vpe =

¡cP

i=1

nPk=1

uik loguik

n. The experiments showed that in

small number of iteration the fuzzy c-means can yield better results in terms of validity indexes. However, in the large number of iteration the proposed method can outperform the classical fuzzy c-means. The high performance of proposed method can be observed in terms of parameters represented in Table 1. The algorithm was tested under different number of iterations. The fuzzy c-means algorithm converges faster than the proposed algorithm, and generates better results in lower number of iterations. However, the results show that in higher number of iterations, the proposed algorithm can segment more efficiently. The results are obtained by running the entire algorithm 10 times on the images, and then computing the average value. The segmentation results of 3 images are demonstrated in Fig. 1-3.Visual performance shows the high capability of the proposed in segmentation as compared to conventional fuzzy c-means.

Table 1 Parameters of Algorithm

Images and number of clusters Validity Indexes

Fuzzy c-means Fuzzy c-means-GSA Vxb Vpe Vxb Vpe

Peppers, c=4 0.0719 0.1791 0.0723 0.1789 Brain, c=4 0.1125 0.0849 0.1081 0.0842

Cameraman, c=3 0.0759 0.1079 0.0755 0.1076

6. CONCLUSION We presented a new approach that utilizes the ability of GSA for optimization of fuzzy c-means algorithm in image

segmentation. The proposed method can overcome to the sensitivity of fuzzy c-means to initialization. Moreover, utilizing a large number of agents can effectively improve the performance of the proposed algorithm. Hence, the results outperform the fuzzy c-means in terms of robustness to initialization, value of fitness fuzzy c-means and visual performance.

7. ACKNOWLEDGEMENT The authors would like to express their gratitude to Universiti Teknologi Malaysia (UTM) and the Ministry of Higher

Education (MOHE), Malaysia for supporting this research work under the Research Grant No. Q.J130000.7123.02J25.

8. REFERENCES [1] R. Kohler, “A Segmentation System Based on Thresholding” Comp. Graphics and Image Proc., vol. 15, no. 4, pp. 319-338, 1981 [2] W. B. Tao, J. W. Tian, and J. Liu, “Image Segmentation by Three-Level Thresholding Based on Maximum Fuzzy Entropy and

Genetic Algorithm” Pattern Recognition Lettrs, vol. 24, no. 16, pp. 3069-3078, 2003. [3] D. Mumford, and J. Shah, “Optimal Approximation by Piecewise Smooth Functions and Associated Variational Problems”

Commun. Pure Appl. Math. vol. 42, no. 5, pp. 577-685, 2009. [4] A. Mansouri, A.Mitiche, and C. Vazquez, “ Multiregion Competition: A Level Set Extension of Region Competition to Multiple

Image Partitioning” Comput. Vision Image Understanding, vol. 101, no. 3, pp. 137-150, 2006. [5] P. F. Felzenszwalb, and D. H. Huttenlocher, “Efficient Graph-Based Image Segmentation” Int. Journal of Computer Vision, vol.

59, no. 2, pp. 167-181, 2004. [6] Y. Boykov, G. F. Lea, “Graph Cuts and Efficient N-D Image Segmentation” Int. Journal of Computer Vision, vol. 70, no. 2, pp.

109-131, 2006.

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[7] K. Chuang, H. Tzeng, S. Chen, J. Wu, and T. Chen, “Fuzzy C-Means Clustering with Spatial Informtion for Image Segmentation” Comput. Med. Imag. Graph., vol. 30, pp. 9–15, 2006.

[8] M. N. Ahmed, S. M. Yamany, N. Mohamed, A. A. Farag, and T. Moriarty, “A Modified Fuzzy C-Means Algorithm for Bias Field Estimation and Segmentation of MRI Data” IEEE Trans. Med. Imag., vol. 21, pp. 193-199, 2002.

[9] S. Das, A. Abraham, and A. Konar, “Spatial Information Based Image Segmentation Using a Modified Particle Swarm Optimization” Proc. 6th Int. Conf. Intelligent Systems Design and applications., pp. 477–485, 2006.

[10] Z. Xian-Cheng, “Image Segmentation Based on Modified Particle Swarm Optimization and Fuzzy c-means Clustering Algorithm” Proc. 2nd Int. Conf. Intelligneet Computation Technology and Applications., pp. 477–485, 2009.

[11] E. Rashedi, H. Nezamabadi-pour, and S. Saryazdi, “GSA: A Gravitational Search Algorithm” Information Sciences, vol.179, no. 13, pp. 2232-2248, 2009.

[12] E. Rashedi, H. Nezamabadi-pour, and S. Saryazdi, “Filter Modeling Using Gravitational Search Algorithm” Engineering Applications of Artificial Intelegence, vol. ??, pp. ??, 2010.

[13] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function” New York: Plenum, 1981. [14] X. L. Xie, G. A. Beni, “Validity Measure for Fuzzy Clustering” IEEE Pattern Anal. Mach. Intell., vol. 13, pp. 841-847, 1991. [15] J. C. Bezdek, “Mathematical Models for Systematic and Taxonomy” 8th Int. Conf. Numerical Taxonomy, pp. 143-146, 1975.

(a)

(b)

(c)

Fig. 1. (a) Original pepper image, segmentation using (b) fuzzy c-means, (c) hybrid fuzzy c-means and GSA

(a)

(b)

(c)

Fig. 2. (a) Original cameraman image, segmentation using (b) fuzzy c-means, (c) hybrid fuzzy c-means and GSA

(a)

(b)

(c)

Fig. 3. (a) Original Brain MRI image, segmentation using (b) fuzzy c-means algorithm, (c) hybrid fuzzy c-means and GSA

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