FUZZY FA: A MODIFIED FIREFLY ALGORITHM

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1 Applied Artificial Intelligence, 28:47 65, 2014 Copyright 2014 Taylor & Francis Group, LLC ISSN: print/ online DOI: / FUZZY FA: A MODIFIED FIREFLY ALGORITHM Tahereh Hassanzadeh and Hamidreza Rashidy Kanan Department of Electrical, Computer and IT Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran The firefly algorithm (FA), which is usually used in optimization problems, is a stochastic, population-based algorithm inspired by the intelligent, collective behavior of fireflies in nature. In the standard FA, each firefly in each neighborhood is compared with other fireflies, and the less-bright firefly moves toward the brighter one (in the maximization optimization). In fact, in the standard FA, firefly movement is based on the local optima, and the global optima do not have any effect on the movement of fireflies. So, the exploration rate of the FA decreases. In this study, we propose a new, fuzzy-based, modified version of the standard FA the fuzzy firefly algorithm (FFA) to increase the exploration and improve the global search of the FA. In the FFA, to improve the speed of finding the global optima, in each iteration the global optima and some brighter fireflies have influence on the movement of fireflies. The effect of each firefly depends on its attractiveness, which is considered as a fuzzy variable in the FFA. In order to evaluate the proposed FFA, seven well-known benchmark functions, including Sphere, Rastrigin, Rosenbrock, Step, Schwefel s P2.22, Ackly, and the Xin-She Yang functions are used in 10-, 20-, and 30-dimensional spaces. Also, to indicate the effectiveness of the proposed FFA, we utilize our approach in a multilevel image-thresholding application, which is one of the most important and challenging issues in image segmentation, and we propose the FFA-Otsu method. The experimental results show that our proposed method can be effective to find the global optima and can improve the global search and the exploration rate of the standard FA. Moreover, the obtained thresholding results show that the proposed FFA-Otsu algorithm is more efficient when compared with other benchmark approaches for image segmentation. INTRODUCTION Collective intelligence is a type of artificial intelligence that is based on collective behavior of agents in decentralized and selforganized systems. These systems are usually organized from simple agents that cooperate locally with other elements and environments. Address correspondence to Hamidreza Rashidy Kanan, Department of Electrical, Computer and IT Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran. h.rashidykanan@qiau.ac.ir

2 48 T. Hassanzadeh and H. R. Kanan However, there is not any centralized mechanism on the agents behaviors. So, nature-inspired metaheuristic algorithms are becoming more powerful algorithms for solving optimization problems (Baeck, Fogel, and Michalewicz 1997; Bonabeau, Dorigo, and Theraulaz 1999; Kennedy, Eberhart, and Shi 2001; Yang 2005; Liu and Passino 2005). There are different methods for solving optimization problems. Particle swarm optimization (PSO; Kennedy and Eberhart 1995), ant colony optimization (ACO; Darigo, Birattari, and Stutzle 2006), artificial fish swarm algorithm (AFSA; Li, Shao, and Qian 2002) and bee colony (Pham et al. 2006) are the most well-known algorithms that have been proposed for optimization. By idealizing some of the flashing characteristics of the fireflies, the firefly algorithm (FA) was recently introduced by Yang at Cambridge University (Yang 2008). This swarm intelligence optimization technique is based on the assumption that the solution of an optimization problem can be shown as a firefly, which glows proportionally to its quality in a considered problem setting. Consequently, each brighter firefly attracts its partners, which makes the search space being explored. Yang used the FA for nonlinear design problems (Yang 2010a) and multimodal optimization problems (Yang 2009) and showed the efficiency of the FA for finding global optima in two-dimensional environments. Yang also introduced a new version of the FA (Levy FA), which combined Levy flight with the search strategy via the firefly for improving the randomization of FA (Yang 2010b). The discrete firefly algorithm was also introduced recently by Sayadi, Ramezanian, and Ghafari-Nasb (2010). The Yang optimization results imply that the FA is potentially more powerful than other existing algorithms such as PSO (Yang 2010a). Some characteristics of the FA that make it suitable for solving optimization problems are that the FA is superior to both PSO and genetic algorithms (GA) in terms of efficiency and convergence (Yang 2009). In this article, we introduce a fuzzy firefly algorithm (FFA) for improving the global search of FA. In the proposed algorithm, to improve the convergence rate of the FA, more than one firefly representing the global optima and some brighter fireflies have an effect on other fireflies movements to search the landscape and find the global best. The degree of effectiveness of each k-best firefly in the movement of other fireflies in each iteration depends on their attractiveness, which is considered as a fuzzy variable in FFA. Image segmentation is one the most difficult and challenging issues in image processing. It is often the first preprocessing step in many image-processing applications. Segmentation is defined as separating objects from background, or objects that have distinct gray levels, from each other. Image thresholding is widely used for image segmentation.

3 Fuzzy FA: A Modified Firefly Algorithm 49 There are two thresholding methods: bi-level thresholding and multilevel thresholding. Bi-level threshholding is a division of the pixels into two groups: one group contains those pixels with gray levels higher than a certain threshold, and the other group includes the rest. Multilevel thresholding is a division of the pixels into several clusters, and the pixels are assigned to a cluster based on their gray level and the specified threshold values. Some objective functions, which are very efficient for the bi-level thresholding purpose, are not suitable for multilevel thresholding because of the exponential growth of computational complexity. Images of gray-level histograms are considered as efficient tools for improvement of the image-segmentation algorithms. The objective is to find the gray-level values, called thresholds, and then segment the pixels according to the thresholds. The simplest way is a bi-level thresholding problem in which only one threshold value is to be found. However, the problem gets more and more complex when we try to achieve segmentation with more details by employing multilevel thresholding in which more than one threshold are to be found. Several algorithms have so far been proposed in the literature to address the issue of histogram thresholding (Otsu 1979; Liao, Chen, and Chung 2011; Kapur, Sahoo, and Wong 1985; Pun 1981; Saha and Udupa 2001; Hu, Hou, and Nowinski 2006; Cheng, Chen, and Li 1998; Cheng and Lui 1997; Brink 1995). Many of them handle the bi-level thresholding; others have considered the multilevel problem. Methods based on optimizing an objective function include maximization of posterior entropy to measure homogeneity of segmented classes (Kapur, Sahoo, and Wong 1985; Pun 1981; Saha and Udupa 2001; Cheng and Lui 1997; Lim and Lee 1990) and maximization of the measure of separability on the basis of between-class variance (Otsu 1979). Several algorithms such as the Kapur (Kapur, Sahoo, and Wong 1985) and Otsu (Otsu 1979) methods were originally developed for bi-level thresholding and later extended to multilevel thresholding. The disadvantage of these methods is that with the increase in the number of thresholds, the computation rate and total runtimes, because of the exhaustive search employed, also increased. In order to solve the problem of optimizationbased approaches, we also introduced the FFA-Otsu method, which is more efficient. We evaluate our proposed FFA by using seven famous benchmark functions (i.e., Sphere, Rastrigin, Rosenbrock, Step, Schwefel s P2.22, Ackly, and Xin-She Yang) in 10-, 20-, and 30-dimensional spaces and comparing the results with the standard FA. The experimental results indicate the better performance of the proposed algorithm to find the global optima and to improve the global search and the exploration

4 50 T. Hassanzadeh and H. R. Kanan rate of the standard FA, especially in high-dimensional and complex environments. Also, the obtained thresholding results show that our proposed FFA-Otsu method is more efficient in comparison with other image-segmentation benchmarks algorithms. In the following, Firefly Algorithm briefly presents the FA. This is followed bysections that decribe the Proposed Fuzzy Firefly Algorithm and Application of the Proposed FFA in Multilevel Thresholding. In Experimental Results, the proposed method is extensively examined under various dimensions and compared with seven benchmark approaches. This section also presents the experimental results of the proposed FFA-Otsu algorithm in image segmentation. The Conclusions section summarizes the article. FIREFLY ALGORITHM Most fireflies produce short and rhythmic flashes, and fireflies use these flashes for communicating and attracting their potential prey. Yang used this behavior of fireflies and introduced the firefly algorithm (FA) in 2008 (Yang 2008). In the FA algorithm, there are three idealized rules: (1) All fireflies are unisex. So, one firefly will be attracted to other fireflies regardless of their sex. (2) Attractiveness is proportional to their brightness. Thus, for any two flashing fireflies, the less bright one will move toward the brighter one. The attractiveness is proportional to the brightness, and they both decrease in attractiveness as their distance increases. If there is no brighter one than a particular firefly, it will move randomly. (3) The brightness of a firefly is affected or determined by the landscape of the objective function. For a maximization problem, the brightness can simply be proportional to the value of the objective function (Yang 2008). In the firefly algorithm, there are two important issues for variation of light intensity and the formulation of the attractiveness. For simplicity, it is assumed that the attractiveness of a firefly is determined by its brightness, which is associated with the objective function of the optimization problem. Because a firefly s attractiveness is proportional to the light intensity seen by adjacent fireflies, we can now define the attractiveness of a firefly by (r ) = 0 e r 2, (1) where 0 is the attractiveness at r = 0 and is the light absorption coefficient at the source. It should be noted that the r i,j, which is described by Equation (2), is the Cartesian distance between any two

5 Fuzzy FA: A Modified Firefly Algorithm 51 fireflies i and j at x i and x j, where, x i and x j are the spatial coordinates of the fireflies iand j, respectively. r i, j = xi x j = d ( ) 2 xi, k x j, k (2) The movement of a firefly i, which is attracted to another moreattractive firefly j, is determined by ( X i = x i + 0 e r ij 2 ( ) xj x i + rand 1 ), (3) 2 where the second term is the attraction and the third term is randomization including randomization parameter and the random number generator rand which numbers are uniformly distributed in interval [0,1]. For most cases of implementations, 0 = 1 and [0, 1] (Yang 2010a) The parameter characterizes the variation of the attractiveness and its value is important to determine the speed of the convergence and how the FA behaves. In the most applications, it typically varies from 0.01 to 100 (Yang 2010a). k=1 THE PROPOSED FUZZY FIREFLY ALGORITHM (FFA) In this study, a new FFA is proposed. In the standard FA, in each iteration the brighter firefly (local optima) exerts its influence over other fireflies and attracts them toward itself in maximization optimization. In fact, in the standard FA, fireflies move regardless of the global optima, which can increase the number of iterations to find the global best and decrease exploration of the firefly algorithm. In this study, to eliminate the weaknesses of the standard FA and to improve the collective movement of fireflies, we propose a fuzzy-based modified version of FA in which some fireflies in each iteration can influence others and their movements. It should be mentioned that, in the standard FA, only one firefly in each iteration can affect others and attract its neighbors. The attractiveness of each firefly depends on its brightness. The level of attractiveness of each k-best firefly is represented as a fuzzy variable. In the proposed algorithm, the k-brighter fireflies in each iteration are selected to be candidates, where k is a user-set parameter and it depends on the complexity and the population size of the problem. The numbers of alternatives are possible for the attractiveness membership function. In this research, we use the Cauchy function as a membership

6 52 T. Hassanzadeh and H. R. Kanan function of the proposed algorithm. Let h be one of the k-brighter fireflies in each iteration, and f (p g ) refers to the fitness of the local optima (a firefly that is brighter than the compared one) in each iteration. We compute the attractiveness (h) of firefly h as (h) = 1 ( f(p ), (4) h) f(p g ) where f(p h ) is the fitness function of k-brighter fireflies. To avoid dependence on the scale of the fitness function, we set = f ( ) p g, (5) l where l is a user-specified parameter. For a fixed f(p h ), the larger the value of l, the smaller the attractiveness (h). In the FFA, we use Cartesian distance to compute the distance of fireflies (same as the standard FA), but for movement of the firefly, Equation (3) is replaced by ( X i = x i + 0 e r ij 2 ( ) k xj x i + (h) 0 e r ih 2 (xh x i ) h=1 ) ( rand 1 ), (6) 2 where x i is the spatial coordinate of the less brighter firefly, second term is due to the attraction of the brighter firefly, and the third term is a fuzzy variable that shows the level of attractiveness of k-brighter fireflies (in all n fireflies) in the movement of fireflies. APPLICATION OF THE PROPOSED FFA IN MULTILEVEL THRESHOLDING In this section, to show the capability of the proposed algorithm, the combination of the proposed FFA and the Otsu method (Otsu 1979) has been used for image segmentation. Otsu introduced a method that maximizes the between-class variance to separate the segmented classes as far apart as possible in order to find the optimal threshold for segmentation. When partial background is mistakenly taken as a target, variance of intracluster will lessen; therefore, the largest Otsu means the least probability of a mistake occurrence. This between-class variance measure is called Otsu s multithreshold measure and is used as the objective function in the FFA. It is formulated as follows: Assume the image size is M N and the number of its gray levels is L ranging from 0 to (L 1). The number of pixels with gray level i is

7 Fuzzy FA: A Modified Firefly Algorithm 53 n i. Thus, the number of image pixels is: L 1 n = n i = M N (7) i=0 So, the normalized histogram and the probability distribution are p i = n i n, L 1 p i = 1 (8) i=0 The image is divided into two classes with the standard threshold t. The class c 1 includes the pixels i t, and the class c 2 includes the pixels i t. Cumulative probabilities of c 1 and c 2 are 1 = 2 = i=0 t p i = (t), (9) i=0 L 1 i=t+1 p i = 1 (t) (10) The mean level and the variances of class c 1 and c 2 are t ip i t ip i 1 = = 1 (t), (11) 2 = 2 1 = L 1 i=t+1 t i=0 L = i=t ip i 2 = i=0 L 1 i=t+1 ip i 1 (t), (12) (i 1 ) 2 p i 1, (13) (i 2 ) 2 p i 2 (14) Based on the above equations, the intercluster and the intracluster variances are as follows, respectively: 2 = , 2 (15) 2 B = 1 ( 1 T ) ( 2 T ) 2, (16) where, the T is the mean intensity of whole original image. The best threshold value T should satisfy the following condition (the is our

8 54 T. Hassanzadeh and H. R. Kanan fitness function that must be optimized): [ 2 B = max 2 ] (17) When the image has a complex target area, the Otsu method should be extended to two or more thresholds. In the proposed method, we used Equation (17) as a fitness function to find optimal thresholds. The proposed method can be summarized as follows: Step 1: Initialize fireflies and set parameters. Step 2: Compute the variance of each segment of the image according to threshold value and compare the computed value with other fireflies values. Step 3: Move firefly to a new position according to the updating mechanism of fireflies position and fitness in FFA. Step 4: Check whether the iteration number has reached the maximum value. If so, deduction ends, otherwise, jump to Step 2. Step 5: Finally, segment the image by optimal thresholds. For evaluation of the segmentation results, we use uniformity measure U = 1 2 c c i=0 i R i (f i j ) 2 /N /(f max f min ) 2, (17) where c is the number of thresholds, R j is the jth segmented region, N is the total number of pixels in the given image, f i is a gray level of pixel i, j is a mean gray level of pixels in the jth region, and f max and f min are the maximum and minimum gray levels of pixels in the given image, respectively. The value of this uniformity measure U should be a positive fraction, in other words, it should be between 0 and 1. A large value of U indicates that there is better uniformity in the segmented image, representing a better quality of thresholding. Conversely, a smaller value of U indicates worse quality of the thresholding procedure. EXPERIMENTAL RESULTS In this part, at first we evaluate the proposed FFA algorithm according to the seven benchmark functions and then show the results of the proposed image-segmentation algorithm.

9 Experimental Results of FFA Fuzzy FA: A Modified Firefly Algorithm 55 In this section, at first the parameters of the FA are studied and tried in order to set the proper parameters to improve the results. One of the most important parameters of FA is which is important for controlling the distance of the random walks. So, it should be linked with scales of the problem variables. In the standard FA, is set in interval [0,1], and it is recommended to set 0.2 (Yang 2008). Considering that, the nature of the firefly algorithm is exploitive. So, increasing can improve the exploration rate of the fireflies as well as the random motion of particles. In order to find the optimal value of, several experiments were performed. For example, we evaluated the effect of different values of this parameter to find the global optima of the Ackly function in 10-dimensional spaces. For this experiment, the number of fireflies is set as 60; also, we set = 1 and 0 = 1. Experiments were repeated 30 times, and the best, mean, and standard deviations have been shown in Table 1. As Table 1 shows, tuning in interval [0,1] cannot be a very effective way to find global optima of function in high-dimensional spaces. The results indicate that by increasing this parameter to 20, the results are improved, and then they decrease again. It shows that the random walk of fireflies shouldn t be so large. In general, we can say that by increasing to 20, we could improve the motions of individual fireflies. Following this section, seven functions, all of which are standard test functions, are utilized for evaluation of the proposed FFA algorithm. This test functions, together with their search space ranges, are presented in Table 2. It should be mentioned that the global optimum solution of the above functions is 0. In this study, for evaluation of the proposed algorithm, we execute the standard FA and the proposed FFA 30 times and the largest iteration numbers are set as 30 for the seven functions. TABLE 1 The Best, Mean, and Standard Deviation Values of the Ackly Function for Different Values of Parameter Best Mean St. D e e e

10 56 T. Hassanzadeh and H. R. Kanan TABLE 2 Seven Standard Test Functions Used for Evaluation of the Proposed FFA Test Function Search Space Name of the Function f 0 (x) = D xi 2 [ 100, 100] D Sphere i=1 f 1 (x) = D ( x 2 i=1 10 cos (2 x i) + 10 ) [ 5.12, 5.12] D Rastrigin i=1 f 2 (x) = D ( 100 ( ) x i+1 xi (xi 1) 2) [ 50, 50] D Rosenbrock i=1 f 3 (x) = D ( xi ) 2 [ 100, 100] D Step i=1 f 4 (x) = D xi D i=1 xi [ 10, 10] D Schwefel s P2.22 i=1 1 D D f 5 (x) = 20 + e 20e 0.2 n x 2 1 i 2 cos(2 x i) i=1 e i=1 ( [ 32, 32] D Ackly D [ ( )] ) f 6 (x) = xi exp sin x 2 i [ 2,2 ] D Xin-She Yang i=1 We test the mentioned functions in 10, 20-, and 30-dimensional spaces. The population size for 10-D is 60, 20-D is 80, and 30-D is 100. Selection of appropriate k is one of the most important issues in FFA; k must be selected based on the population size and the problem complexity. The algorithm computation rate will be increased by increasing the k; therefore, it must be selected carefully. In this article, for the proposed algorithm based on the population size, k is determined to be 3 and l is set as 2.6 for evaluation. Also, we set = 1, = 0.2 and = 1 for standard FA as well as FFA; but for FFA we set = 20. Experiments were repeated 30 times and the best, mean, variance, standard deviation and the worst case that is obtained from the standard run in 10, 20-, and 30-dimensional spaces on Sphere, Rastrigin, Rosenbrock, Step, Schwefel s P2.22, Ackly and Xin-She Yang functions are shown in Table 3. As Table 3 indicates, the results of the FFA in all cases are better than FA in 10-, 20-, and 30-dimensional spaces, and it shows the capability of the proposed algorithm to find optimal solutions. In the proposed algorithm, we improved FA using k number of the brighter fireflies for searching in the function landscape for finding the best solution. Especially in high-dimensional spaces, the proposed method can be more effective. For example, we execute FA and FFA for finding the global minimum of Ackly, Rosenbrock, Step, and Rastrigin functions in 20- and 30- dimensional spaces for 30 iterations and show the best result of each iteration in Figure 1. As shown in Figure 1, results show that FFA considerably improves the performance and increases the exploration of FA. The main reasons are

11 Fuzzy FA: A Modified Firefly Algorithm 57 TABLE 3 Comparison of FA and FFA on Sphere, Rastrigin, Rosenbrock, Step, Schwefel s P2.22, Ackly, and Xin-She Yang Functions Functions Dim Criteria FA FFA Sphere 10 Best e e-007 Ave-StD e+004 ± e ± Worst e Best e e-009 Ave-StD e+004 ± e e-004 ± e-004 Worst e Best e e-008 Ave-StD e+004 ± e ± Worst e Rastrigin 10 Best e-008 Ave-StD ± e-004 ± e-004 Worst e Best e-008 Ave-StD ± e-004 ± e-004 Worst Best e-008 Ace-StD ± e-004 ± e-004 Worst e-004 Rosenbrock 10 Best e e-006 Ave-StD e+008 ± e ± Worst e Best e e-010 Ave-StD e+008 ± e ± Worst e Best e e-007 Ave-StD e+009 ± e ± Worst e Step 10 Best Ave-StD e+004 ± e Worst Best Ave-StD e+004 ± e Worst Best Ave-StD ± e Worst Schwefel s P Best e-005 Ave-StD ± ± Worst e Best e-004 Ave-StD e+004 ± e ± Worst e Best e e-005 Ave-StD e+008 ± e ± Worst e Ackly 10 Best e-006 Ave-StD ± ± Worst (Continued)

12 58 T. Hassanzadeh and H. R. Kanan TABLE 3 Continued Functions Dim Criteria FA FFA 20 Best e-005 Ave-StD ± ± Worst Best e-007 Ave-StD ± ± Worst Xin-She Yang 10 Best e-004 Ave-StD ± e-004 ± e-005 Worst e Best e e-008 Ave-StD e-005 ± e e-008 ± e-008 Worst e e Best e e-012 Ave-StD e-007 ± e e-012 ± e-012 Worst e e-011 that the proposed algorithm uses more than one firefly to move other fireflies. Therefore, during execution of the algorithm, the capability of the proposed FFA in global searching is improved and could find optimal solution efficiently and more quickly than the standard FA. Experimental Results of Multilevel Thresholding In this part, we present the results of the proposed FFA-Otsu imagesegmentation algorithm. In our experiments, we set 0 = 1 and = 20. Rand is a random number generator uniformly distributed in [0 1]. The parameter, which characterizes the variation of the attractiveness and whose value is important in determining the speed of the convergence and how the FA behaves, is set as 1. Population size is set to 50. The fireflies are initialized as a gray value ranging in intervals of [0 255], and the largest truncated generation is 50. The dimension of input vectors depends on the number of threshold levels that we use for segmentation. In this article, we use different threshold levels from 2 to 5. We evaluate the proposed algorithm on some benchmark images including: Lena, Peppers, Bird, Cameraman, and Gold hill. These original test images and their histograms are shown in Figure 2. As mentioned before, to evaluate the proposed FFA-Otsu segmentation method, we used uniformity measure. The obtained thresholds, uniformity measure, and the objective value on the above images for 2 to 5 threshold levels are shown in Table 4. The uniformity values of the proposed method are compared with the results of FA-Otsu, Gaussian smoothing (Lim and Lee 1990),

13 Fuzzy FA: A Modified Firefly Algorithm 59 FIGURE 1 Comparison of the best results obtained from 30 performances of the standard FA and the proposed FFA on different benchmark functions: (a) and (b), Ackly; (c) and (d), Rosenbrock; (e) and (f), Step; (g) and (h), Rastrigin functions in 20- and 30-dimensional space for 30 iterations. symmetry-duality method (Yin and Chen 1993), improved GA-based algorithm (Yin 1999), the hybrid cooperative-comprehensive learningbased PSO algorithm (HCOCLPSO; Madhubanti and Amitava 2008), and a new social and momentum component adaptive PSO algorithm (SMCAPSO; Akhilesh, Amitava, and Patrick 2011) on the Lena and Peppers images and are shown in Table 5. As is shown, the obtained

14 60 T. Hassanzadeh and H. R. Kanan FIGURE 2 The test images and the corresponding histograms: (a) Lena, (b) Peppers, (c) Bird, (d) Cameraman, and (e) Gold hill. uniformity measures of the proposed method are better than other methods, and it proves the proposed FFA-Otsu method is very effective in image segmentation; in all cases, the uniformity measures are in the upper 98%. In Table 6, we show the obtained objective value on the Lena and Peppers images and compare them with HCOCLPSO (Madhubanti and Amitava 2008) and GA-learning Kapur (Yin 1999) algorithms. As Table 6 shows, the results of the proposed method are better than the two other methods in many cases. Only in two cases for three-level thresholding, TABLE 4 Algorithm Obtained Thresholds and Fitness Values for Test Images for the Proposed FFA-Otsu Benchmark Image c Threshold Uniformity Objective Value Lena ( ) 2 63, , 99, , 74, 97, , 68, 105, 132, Peppers ( ) 2 70, , 124, , 111, 134, , 83, 116, 165, Bird ( ) 2 97, , 108, , 115, 158, , 77, 130, 164, Camera ( ) 2 64, , 121, , 95, 136, , 46, 96, 137, Gold hill ( ) 2 88, , 140, , 94, 153, , 87, 94, 138,

15 Fuzzy FA: A Modified Firefly Algorithm 61 TABLE 5 Comparison of Uniformity for FFA-Otsu, FA-Otsu, HCOCLPSO, GA Learning-Kapur, Gaussian Smoothing and Symmetry Duality Algorithms Uniformity GA Gaussian Symmetry FFA- FA- Learning- Smoothing Duality Image c Otsu Otsu SMCAPSO HCOCLPSO Kapur Method Method Lena ( ) Peppers ( ) the objective values of the proposed method are lower than the two other methods. Also, in Table 7, we compare the obtained thresholds of the proposed algorithm for Lena and Peppers images in 2 to 5 threshold levels with three other methods. Another important goal in our research is to decrease the time complexity of the Otsu methods. As mentioned earlier, by increasing the number of thresholds, the time and computational complexity will be increased with the Otsu method. To overcome this problem, a recursive Otsu algorithm has been proposed in Liao, Chen, and Chung (2011). In this method, the time complexity of the Otsu method is partially decreased. In our proposed method, we could reduce the time complexity of Otsu and recursive Otsu algorithms. The mean runtimes of the conventional and recursive Otsu methods and the proposed TABLE 6 Comparison of Objective Values for FFA-Otsu, HCOCLPSO, GA Learning-Kapur Algorithms Objective Values Image c FFA-Otsu HCOCLPSO GA Learning-Kapur Lena ( ) Peppers ( )

16 62 T. Hassanzadeh and H. R. Kanan TABLE 7 Comparison of the Obtained Optimal Thresholds for FFA-Otsu, HCOCLPSO, GA Learning-Kapur Algorithms Optimal Thresholds GA Learning- Image c FFA-Otsu FA-Otsu HCOCLPSO Kapur Lena 2 63, , , , 176 ( ) 3 53, 99, , 118, , 125, , 131, , 74, 97, , 98, 161, , 114, 149, , 105, 143, , 68, 105, 25, 74, 123, 69, 104, 137, 74, 103, 133, 132, , , , 195 Peppers 2 70, , , , 150 ( ) 3 57, 124, , 129, , 113, , 119, , 111, 134, , 78, 124, , 80, 126, , 90, 132, , 83, 116, 22, 82, 116, 43, 78, 118, 56, 88, 121, 165, , , , 194 FFA-Otsu algorithm were calculated and are listed in Table 8. As indicated in Table 8, the mean runtimes of the conventional Otsu method and the recursive Otsu algorithm increase significantly when the threshold level c increases; but the runtime of the proposed method is independent of the number of threshold levels. As a subjective evaluation, we show the obtained segmented images in Figure 3. As shown, the obtained thresholds could cluster the pixels properly. In the segmented images, different clusters displayed with different colors. The pixels located into different clusters according to their colors. One of the criteria of a good segmentation algorithm is that the obtained segmented image would be more similar to the original image when the number of threshold levels is increased. As shown in Figure 3, the obtained segmented images become much more similar to the original images with increase in the number of threshold levels. TABLE 8 Comparison of the Mean Runtimes for Three Methods with Classes c = 4, 5, and 6 Mean Runtime (s) The Proposed Recursive Image c FFA-Otsu Method Otsu* Otsu* Lena Peppers The values with * are from (Huang and Wang 2009).

17 Fuzzy FA: A Modified Firefly Algorithm 63 FIGURE 3 The obtained segmented images based on the proposed FFA-Otsu method with different threshold levels. CONCLUSION In this study, to improve the global search and the exploration rate of the standard firefly algorithm (FA), we have presented a fuzzy-based modified version of the firefly algorithm (FFA). In this research, we used more than one firefly representing the global optima and some brighter fireflies in each iteration to influence other fireflies movements, depending on their degrees of brightness, which is considered as a fuzzy variable to improve the collective motion of fireflies. Also, we investigated the proper value of parameter to improve the random movement of fireflies. Our experimental results on seven standard benchmark functions including Sphere, Rastrigin, Rosenbrock, Step, Schwefel s P2.22, Ackly, and Xin-She Yang functions in 20- and 30- dimensional spaces indicate that our proposed algorithm improves performance of FA especially in high-dimensional spaces. Then, we combined the FFA with the Otsu method and proposed the FFA-Otsu algorithm for image segmentation. The different objective and subjective

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