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Mathematical Methods Applied to Digital Image Processing

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Research Article | Open Access

Volume 2014 |Article ID 974024 | https://doi.org/10.1155/2014/974024

Kanjana Charansiriphaisan, Sirapat Chiewchanwattana, Khamron Sunat, "A Global Multilevel Thresholding Using Differential Evolution Approach", Mathematical Problems in Engineering, vol. 2014, Article ID 974024, 23 pages, 2014. https://doi.org/10.1155/2014/974024

A Global Multilevel Thresholding Using Differential Evolution Approach

Revised23 Jan 2014
Accepted03 Feb 2014
Published20 Mar 2014

Abstract

Otsu’s function measures the properness of threshold values in multilevel image thresholding. Optimal threshold values are necessary for some applications and a global search algorithm is required. Differential evolution (DE) is an algorithm that has been used successfully for solving this problem. Because the difficulty of a problem grows exponentially when the number of thresholds increases, the ordinary DE fails when the number of thresholds is greater than 12. An improved DE, using a new mutation strategy, is proposed to overcome this problem. Experiments were conducted on 20 real images and the number of thresholds varied from 2 to 16. Existing global optimization algorithms were compared with the proposed algorithms, that is, DE, rank-DE, artificial bee colony (ABC), particle swarm optimization (PSO), DPSO, and FODPSO. The experimental results show that the proposed algorithm not only achieves a more successful rate but also yields a lower threshold value distortion than its competitors in the search for optimal threshold values, especially when the number of thresholds is large.

1. Introduction

Thresholding is the simplest and most commonly used method of image segmentation. It can be bilevel or multilevel [1]. Both of these types can be classified into parametric and nonparametric approaches [1]. Surveys of thresholding techniques for image segmentation can be found in [27]. The surveys revealed that Otsu’s method is a commonly used technique [4, 8]. This method finds the optimal thresholds by maximizing the weighted sum of between-class variances (BCV) [9]. The BCV function is also called Otsu's function. However, the solution finding process is an exhaustive search and it is a very time-consuming process because the complexity grows exponentially with the number of thresholds.

Multilevel image thresholding based on Otsu’s function has been used as a benchmark for comparing the capability of evolutionary algorithms (EA). The EA is a nongradient based optimization algorithm. Several algorithms have been widely applied to solve multilevel thresholding. A group of successful works were based on a combination of Otsu’s function with some state-of-the-art algorithms: PSO [10], DE [11], ABC [12], and FOSPSO [13]. Kulkarni and Venayagamoorthy [14] showed that PSO was faster than Otsu’s method in searching the optimal thresholds of multilevel image thresholding. Akay [15] presented a comprehensive comparative study of the ABC and PSO algorithms. The results showed that the ABC algorithm with both the between-class variance and the entropy criterion can be efficiently used in multilevel thresholding. Hammouche et al. [16] focused on solving the image thresholding problem by combining Otsu’s function with metaheuristic techniques, that is, genetic algorithm (GA), PSO, DE, ant colony, simulated annealing, and Tabu search. Their results revealed that DE was the most efficient with respect to the quality of solution. Osuna-Enciso et al. [17] presented an empirical comparative study of the ABC, PSO, and DE algorithms to perform image thresholding using a mixture of Gaussian functions. The results showed that the DE algorithm was superior in performance in minimizing the Hellinger distance and used less evaluations of the Hellinger distance. Ghamisia et al. [18] showed that a global optimal search for optimal threshold values of Otsu’s function was essential for the multilevel segmentation of multispectral and hyperspectral images.

The DE algorithm was selected for multilevel image thresholding. It is simple to implement and produces good results. However, based on our experiments, DE could not reach an optimal solution when it was applied to a very difficult problem. Therefore, a better DE algorithm is required. We noticed that the mechanism of vector selection and the size of the higher ranked population are an important criterion for success.

The contribution of this paper is as follows.

DE with the onlooker and ranking-based mutation operation, named , is proposed to overcome the drawback of the DE algorithm for multilevel image thresholding, especially when the number of thresholds is large. The proposed algorithm homogenizes the onlooker phase of the ABC algorithm and the ranking-based mutation operator of the [19]. The main advantage of the proposed algorithm is that a user can adjust the balancing of the exploitation and exploration capabilities of the algorithm.

To verify the capabilities of the proposed algorithm, experiments to find the optimal solutions in the multilevel image thresholding, when the number of thresholds ranged from two to 16, were set up. It was found that the optimal solutions could be effectively reached using the proposed algorithm.

The remainder of the paper is organized as follows. Section 2 describes the multilevel thresholding problem. Section 3 presents a brief review of the differential evolution algorithm (DE). In Section 4, the proposed new version of the DE algorithm with the onlooker and ranking-based mutation operator algorithm, , is described in detail. Section 5 shows the experimental results of applying the proposed method to multilevel segmentation in different images. Finally, the conclusion of the paper is discussed in Section 6.

2. Multilevel Thresholding Problem Formulation

Otsu’s method [9] is based on the maximization of the between-class variance. Consider a digital image having the size , where is the width and is the height. The pixels of a given picture are represented in gray levels and they are in . The number of pixels at level is denoted by and the total number of pixels by . The gray-level histogram is normalized and regarded as a probability distribution and is written as follows: The total mean of the image can be defined as The multilevel thresholding with respect to the given threshold values , can be performed as follows: where is the coordinate of a pixel and denotes the intensity level of a pixel. The pixels of a given image will be divided into classes in this regard.

The optimal threshold can be determined by maximizing the between-class variance function (BCV), , which can be defined by where represents a specific class in such a way that and are the probability of occurrence and the mean of class , respectively. Equation (4) is also called Otsu’s function. The probabilities of occurrence of classes are defined by The mean of each class can be given by Thus, the -level thresholding problem is transformed to an optimization problem. The process is to search for thresholds that maximize the value , which is generally defined as

3. Differential Evolution Algorithm

The DE algorithm is an evolutionary optimization technique proposed by Storn and Price [11]. The main procedures of DE are briefly described as follows.

3.1. Initialization

The DE algorithm starts with a population of initial solutions, each of dimension , , where the index denotes the solution, or vector, of the population, is the generation, and is the population size. The initial population (at = 0) is randomly generated to be within the search space constrained by the minimum and maximum bounds, and . The vector is initialized as follows: where is a uniformly distributed random real number between 0 and 1, .

3.2. Mutation Operators

The differential mutation operator is one of the three operators of DE. The mutation operator is applied to generate the mutant vector for each target vector in the current population. A mutant vector is generated according to where the randomly chosen indexes, random indexes, are mutually different random integer indices and they are also different from the running index . Further, , and are different so that . is a real and constant factor, , which controls the amplification of the differential variation; is called the base vector, is called the terminal vector, is called the other vector, and is called the difference vector.

There have been many proposed mutation strategies for DE [20, 21]. Each different strategy has different characteristics and is suitable for a set of problems. However, the choice of the best mutation operators for DE is difficult for a specific problem [2224]. The “DE/rand/1/bin” strategy has been widely used in DE literature [2528]. It is more reliable than the strategies based on the best-so-far solution such as “DE/best/1” and “DE/current-to-best/1”. However, “DE/rand/1/bin” has slower convergence. Simply put, it has high exploration but low exploitation abilities.

3.3. Crossover

DE utilizes the crossover operation to generate new solutions by shuffling competing vectors and to increase the diversity of the population. The classical version of the DE (DE/rand/1/bin) uses the binary crossover. It defines the following trial vector: where ( = problem dimension) and

CR is the crossover rate , is the th evaluation of a uniform random number generator with outcome , and is a randomly chosen index that ensures will get at least one parameter from .

3.4. Selection

Selection determines whether the target or the trial vector survives to the next generation. The selection operation is described as where is the objective function to be minimized. Therefore, if the objective of the new trial vector, , is equal to or less than the objective of the old trial vector, , then is set to ; otherwise, the old value is retained.

The pseudocode of basic DE with “DE/rand/1/bin” strategy is shown in Algorithm 1.

 (1)  Generate the initial population randomly (2)  Evaluate the fitness for each individual in the population (3)  while the maximum generation G is not reached do (4)   for to NP do (5)     Select uniform randomly (6) (7)     for to do (8)        if or is equal to then (9) (10)      else (11) (12)      end if (13)    end for (14)  end for (15)  for to NP do (16)      Evaluate the offspring (17)       if is better than or equal to then (18)      Replace with (19)      end if (20)    end for (21) end while

The function returns a uniformly distributed random integer number between 1 and D. is a uniformly distributed random real value of . The word “better” in line 17 means “less than” if the problem requires minimization, see (12) and its explanation, and it means “greater than,” if the problem requires maximization. The best , where is the maximum number of generations, is the solution of the algorithm. The word “best” also depends on the type of problem.

4. The Proposed DE with Onlooker Ranking-Based Mutation Operator

In 2013 Gong and Cai [19] proposed a algorithm. They claimed that probabilistically selecting the vectors and in the mutation operator from the better population can improve the exploitation ability of basic DE. To the best of the authors’ knowledge, may, however, also lead to premature convergence (this will be shown in the experiments). That means that the has too much exploitation ability. Furthermore, it cannot balance between the exploration and the exploitation abilities. In order to balance between the two abilities, we propose DE with the onlooker and ranking-based mutation operator, named . The proposed algorithm is an improvement of the rank-DE by homogenizing the rank-DE with the onlooker phase of ABC algorithm. The detail of the algorithm is described as follows.

4.1. Ranking Assignment

To perform the maximization, the fitness of each vector is sorted in ascending order (i.e., from worst to best). Then, the rank of the th vector, , is assigned based on its sorted ordering as follows: As a result, the best vector in the current population will obtain the highest ranking, that is, NP.

4.2. Probabilistic Selection

After assigning the ranking for each vector, the selection probability of the th vector is calculated as

4.3. A New Strategy for Base Vector, Terminal Point, and the Other Vector Selections

Definition 1 (a worse population and a better population). Let be a real value and . A population having probability less than is called a worse population and a population having probability greater than or equal to is called a better population.

In the , the base vector and the terminal point were based on their selection probabilities. The other vector in the mutation operator, , is selected randomly as in the original DE algorithm. The vectors with higher rankings (higher selection probabilities) are more likely to be chosen as the base vector or the terminal point in the mutation operator.

Our investigation revealed that if both and vectors of were chosen from better vectors, then the distribution of the target vector may collapse quickly and possibly lead to premature convergence. Accordingly, when the was applied to a very difficult problem, it could not reach the optimal solution.

If the steps of the DE algorithm are compared with the ABC algorithm, the population in the current generation can be considered as the employed bees and the population in the next generation can be considered as the onlooker bees. To follow the concept of ABC, a new vector, , which is called the base vector, chooses a food source with respect to the probability that is computed from the fitness values of the current population. The probability value, , of which is chosen by a base vector can be calculated by using the expression given in (14). After a base source for a new vector is probabilistically chosen, both and are also chosen in the same manner as the terminal point and the other vector selections in the . The target vector is created by a mutation formula of DE. The mutant vector is created after the target vector is crossed with a randomly selected vector, and then the fitness value is computed. As in the ordinary DE, a greedy selection is applied between and . Hence, the new population contains better sources and positive feedback behavior appears. This idea can be expressed as pseudocode, as in Algorithm 2. Since the selection of is the onlooker selection and the selections of and are brought from the , then the algorithm is called onlooker and ranking-based vector selection.

 (1)  Input: The target vector index , the last index of onlooker , and (2)  Output: The selected vector indexes (3)   ; if then ; end if (4)  while //onlooker-like selection (5)      ; if then ; end if (6)  end while (7)  Randomly select terminal vector index (8)  while or or do (9)     Randomly select (10) end while (11) Randomly select the other vector index (12) while or or do (13)    Randomly select (14) end while

The pseudocode of onlooker and ranking-based vector selection is shown in Algorithm 2. The differences between the original ranking-based and onlooker and ranking based selection are highlighted by “”.

The function is added to generalize the algorithm. Its output depends on the parameter . The outcome can be either a constant value of or a value of the uniform random function . The balance of the exploration and exploitation ability can be set by the parameter . And the function is defined by

4.4. The DE with Onlooker-Ranking-Based Mutation Operator

The procedures in Sections 4.1, 4.2, and 4.3 are combined together to create a better DE algorithm. The parameter determines the fraction of the worse population to be eliminated. When there is no worse population; each single vector in the current population will act as the base vector. If , then each single vector having a probability less than is a worse vector and will not be selected as the base vector. If is not a constant or is outside , each single base vector is an onlooker bee. Accordingly, the name of the algorithm is Ranking-Base Differential Evolution . To achieve the global solution, a user can set a proper value for to control the balance of the exploration and exploitation abilities of the algorithm. The pseudocode of is shown in Algorithm 3 and the differences between the and are highlighted by “”.

 (1)  Randomly generate the initial population (2)  Evaluate the fitness for each individual in the population (3)  while the maximum generation G is not reached do (4)     Sort and rank the fitness values of population according to (13) (5)     Calculate the selection probability for each individual according to (14) (6) (7)   for to NP do (8)       Select , , as shown in Algorithm 2 based on the current and (9) (10)      for to do (11)          if or is equal to then (12) (13)          else (14) (15)          end if (16)      end for (17)    end for (18)    for to NP do (19)          Evaluate the offspring (20)          if is better than or equal to then (21)            Replace with (22)          end if (23)    end for (24) end while

5. Experiments and Results

5.1. Experimental Setup

The global multilevel thresholding problem deals with finding optimal thresholds within the range that maximize the BCV function. The dimension of the optimization problem is the number of thresholds, , and the search space is . The parameter of is or is set to be one of 0.0, . The variation of the proposed was implemented and compared with the existing metaheuristics that performed image thresholding, that is, PSO, DPSO, FODPSO, ABC, and several variations of DE algorithms. All the methods were programmed in Matlab R2013a and were run on a personal computer with a 3.4 GHz CPU, 8 GB RAM with Microsoft Windows 7 64-bit operating system. The experiments were conducted on 20 real images. The 19 images, namely, starfish, mountain, cactus, butterfly, circus, snow, palace, flower, wherry, waterfall, bird, police, ostrich, viaduct, fish, houses, mushroom, snow mountain, and snake, were taken from the Berkeley Segmentation Dataset and Benchmark [29]. The last image, namely, Riosanpablo, is a satellite image “New ISS Eyes see Rio San Pablo”, March 1, 2013 (http://visibleearth.nasa.gov/view.php?id=80561). Each image has a unique gray level histogram. These original images and their histograms are depicted in Figure 1. An experiment of an image with a specific number of thresholds is called a “subproblem.” The number of thresholds investigated in the experiments was . Thus, there are subproblems per algorithm. Each subproblem was repeated 50 times and each time is called a run.

To compare with PSO, ABC, and DEs algorithms, the objective function evaluation is computed for , where is population size and is the number of generations. A population of PSO and the DEs calls Otsu’s function one time per generation. The population size in the PSO and DEs algorithms was set to 50. A bee in the ABC calls Otsu’s function two times per generation; therefore their number of food sources were set to a half of the PSO’s size, that is, 25. The stopping criteria were set by the maximum amount of generations . In this experiment, was set to 50, 100, 150, 200, 300, 400, 600, 800, 1000, 1500, 2000, 3000, 4000, 5000, and 6000 when was 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16, respectively. For the PSO, DPSO, and FODPSO algorithms, the parameters were set as per the suggestion in [30] and is shown in Table 1. The other control parameter of the ABC algorithm, limit, was set to 50 [15]. The control parameters and of the DE algorithms were set to 0.5 and 0.9, respectively [31, 32].

 Parameter PSO DPSO FODPSO Population 50 50 50 1.5 1.5 1.5 1.5 1.5 1.5 1.2 1.2 1.2 2 2 2 −2 −2 −2 255 255 255 0 0 0 Min population — 10 10 Max population — 50 50 No. of swarms — 4 4 Min swarms — 2 2 Max swarms — 6 6 Stagnancy — 10 10 Fractional coefficient — — 0.75
5.2. Comparison Strategies and Metrics

To compare the performance of different algorithms, there are three metrics: the convergence rate of algorithms was compared by the average of generations (), a lower means a faster convergence rate; the stability of algorithms was compared by the average of the success rate, (SRHM), a higher SRHM means higher stability; the reliability was compared by the threshold value distortion measure (TVD), a lower TVD means higher reliability. The details of the three metrics are described as follows.

When all 50 runs of an algorithm performing on an image with a specific number of thresholds are terminated, the outcomes will be analyzed. Run ’th is called a successful run if there is a generation of such that and the number of generations (NG) of the successful run is recorded. Thus, the number can be defined by The average of from those successful runs is represented by as follows: The ratio of success rate (SR) for which the algorithm succeeds to reach the VTR for each subproblem is computed as The experiments were conducted on 20 images. The arithmetic mean (AM) of () over the entire set of images with a specific number of thresholds is calculated as where is the total number of images. is shown in Table 3. The worst-case scenario is that there is no successful run for a subproblem; this subproblem is called an “unsuccessful subproblem.” If an algorithm encounters this scenario, the subproblem will be grouped by its number of thresholds and the number of images in the group will be counted and assigned to . These scenarios will be represented by , as shown in Tables 3 and 4.

The average of the success rate over the entire dataset with a specific number of thresholds () is averaged by the Harmonic mean, HM, as follows:

The is very important in measuring the stability of an algorithm and it means the ratio of runs that are achieving the target solution. Because the evolutionary methods are based on stochastic searching algorithms, the solutions are not the same in each run of the algorithm and depend on the search ability of the algorithm. Therefore, the is vital in evaluating the stability of the algorithms. The comparison of the stability gives us valuable information in terms of the ratio representing the success rates (). A higher means better stability of the algorithm.

An algorithm producing means that more than 50 percent of the independent runs of the algorithm cannot reach the global solution. Thus, the algorithm that yields should not be selected to solve the problem. The experiments were conducted for the number of thresholds varying from 2 to 16. These experiments contained the maximum number of thresholds such that the algorithm yields , which is represented by in Table 4. Furthermore, the experiments also contained the maximum number of thresholds that the algorithm can solve; and above this value there was the case such that all 50 runs of some subproblems missed the VTR. This number is represented by . In this case the success rate was zero and the associated was zero too. And the definitions of the two values are presented in (21)

Let be the number of thresholds. The reliability of a solution is measured by threshold value distortion measure () and is computed as where is the threshold value producing the VTR, is the threshold value obtained from the algorithm, and is the indicator function, which is equal to 1 when and is zero otherwise. is zero if the algorithm can reach the VTR in every run. The lower the the more reliable the algorithm is.

5.2.1. The Value to Reach (VTR)

Following the completion of all of the experiments the best values of the between-class variance and the corresponding thresholds were collected and are shown in Table 6. The results are shown image by image and the numbers of thresholds vary from 2 to 16. The between-class variance values in column 3 are used as the VTR values.

5.2.2. Results Produced by Local Search Method

The multithresh function of the Matlab toolbox was conducted on the same images and number of thresholds as the other search methods. The capabilities of solving the optimal solution between a local search and a global search will be discussed here. This is the reason we focused on the global search, that is, the proposed algorithm. Table 2 shows the between-class variances and threshold values of the “mountain” image. These values were the best outcomes of 50 runs produced by the multithresh function in the Matlab R2013a toolbox and by the proposed algorithm. The terminated condition of the multithresh function was set by “MaxFunEvals” = 500000. That is the multithresh function performs more function calls than that of the algorithm. It can be seen from columns 3 and 5 that all the BCVs produced by the algorithm are better than the BCVs produced by the multithresh function; the difference of the BCVs is shown in column 7. The differences in the thresholds from the two algorithms, shown in column 8, tended to be large if the number of thresholds increased.

 Image (1) (2) Produced by multithresh Produced by the O(0.0)R-DE algorithm Objdiff Threshdiff Between-class variance (3) Thresholds (4) Between-class variance (5) Thresholds (6) Mountain 2 2372.886 60, 127 2372.923 61, 128 0.037 2 3 2495.852 32, 76, 130 2496.113 33, 77, 131 0.261 3 4 2551.778 32, 72, 109, 145 2551.955 33, 73, 109, 147 0.177 4 5 2580.154 31, 68, 98, 124, 158 2580.336 32, 69, 99, 125, 159 0.182 5 6 2588.264 35, 75, 98, 118, 152, 175 2596.956 24, 46, 74, 101, 126, 160 8.692 122 7 2598.800 33, 70, 98, 118, 141, 166, 207 2608.807 24, 46, 73, 98, 119, 145, 175 10.007 153 8 2605.785 31, 68, 91, 105, 121, 140, 163, 192 2616.294 24, 46, 73, 97, 115, 135, 160, 191 10.509 70 9 2619.512 27, 52, 75, 95, 111, 129, 148, 168, 197 2622.314 20, 37, 54, 76, 98, 116, 136, 160, 191 2.802 114 10 2620.392 23, 46, 73, 97, 116, 135, 157, 175, 204, 232 2627.194 20, 36, 53, 74, 93, 106, 121, 140, 163, 194 6.802 258 11 2625.275 23, 45, 72, 93, 105, 119, 137, 156, 175, 204, 228 2630.496 20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201 5.221 199 12 2629.641 22, 39, 58, 78, 95, 109, 123, 139, 156, 177, 207, 247 2633.189 18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201 3.548 246 13 2631.372 20, 38, 54, 74, 94, 107, 121, 139, 157, 173, 193, 224, 248 2635.290 18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207 3.918 283 14 2633.865 19, 36, 53, 74, 91, 103, 115, 129, 144, 160, 174, 190, 217, 245 2637.088 17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207 3.223 307 15 2635.190 19, 33, 50, 69, 85, 96, 107, 121, 137, 152, 167, 178, 190, 212, 240 2638.449 17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207 3.259 351 16 2636.357 16, 30, 45, 59, 77, 93, 105, 118, 133, 148, 162, 176, 192, 209, 230, 244 2639.616 17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209 3.259 415
 No.maxGen. FODPSO DPSO PSO ABC Rank-DE DE O(rand)R-DE O(0.9)R-DE O(0.8)R-DE O(0.7)R-DE O(0.6)R-DE O(0.5)R-DE O(0.4)R-DE O(0.3)R-DE O(0.2)R-DE O(0.1)R-DE O(0.0)R-DE 18.337 15.713 13.218 13.202 10.538 14.713 10.117 6.394 6.903 7.636 8.208 9.147 9.261 9.901 10.709 11.631 20.850 Rank 16 15 13 12 9 14 8 1 2 3 4 5 6 7 10 11 17 35.200 36.210 27.479 33.210 19.488 29.049 18.958 10.492 12.296 13.587 14.807 16.969 17.107 18.100 19.546 21.749 44.192 Rank 15 16 12 14 9 13 8 1 2 3 4 5 6 7 10 11 17 45.391 58.683 49.596 63.251 31.060 48.130 29.595 14.733 18.526 20.885 22.815 26.632 26.456 28.058 30.175 33.499 72.100 Rank 12 15 14 16 10 13 8 1 2 3 4 6 5 7 9 11 17 55.664 81.877 87.441 108.131 44.157 69.088 41.283 19.508 24.524 28.097 30.963 36.935 36.748 38.647 41.455 45.631 100.693 Rank 12 14 15 17 10 13 8 1 2 3 4 6 5 7 9 11 16 NA(2) 100.664 141.937 157.328 61.935 99.700 56.405 24.623 32.687 38.167 42.608 51.003 50.281 52.089 55.729 61.748 138.716 Rank 17 13 15 16 11 12 9 1 2 3 4 6 5 7 8 10 14 NA(1) NA(1) 192.267 236.928 86.938 142.096 75.340 29.763 41.691 49.408 56.193 67.619 67.063 69.696 74.389 83.326 184.371 Rank 17 16 14 15 11 12 9 1 2 3 4 6 5 7 8 10 13 NA(7) NA(6) NA(2) NA(1) 117.092 198.986 100.944 33.967 51.573 62.263 74.523 91.836 89.666 95.198 102.552 115.583 244.880 Rank 17 16 15 14 11 12 8 1 2 3 4 6 5 7 9 10 13 NA(11) NA(13) NA(12) NA(3) 151.004 264.253 125.299 41.228 63.742 79.432 90.201 113.530 110.416 114.397 122.555 137.538 299.492 Rank 17 16 15 14 11 12 9 1 2 3 4 6 5 7 8 10 13 NA(11) NA(16) NA(16) NA(7) NA(1) 321.373 147.770 NA(3) 73.564 91.121 105.020 132.968 128.999 132.691 141.825 157.997 356.546 Rank 17 16 15 14 12 10 8 13 1 2 3 6 4 5 7 9 11 NA(16) NA(17) NA(19) NA(9) NA(1) 405.090 185.020 NA(3) NA(2) NA(2) 127.135 162.822 157.184 160.123 175.152 198.522 434.877 Rank 17 16 15 14 12 8 6 13 10 10 1 4 2 3 5 7 9 NA(18) NA(20) NA(20) NA(9) 278.743 535.542 229.965 NA(4) NA(5) 135.266 160.047 205.266 197.386 203.814 222.094 251.535 574.104 Rank 17 16 15 14 12 8 6 13 11 10 1 4 2 3 5 7 9 NA(20) NA(19) NA(20) NA(14) NA(3) NA(3) 275.860 NA(12) NA(5) NA(1) NA(1) NA(1) NA(1) 261.522 278.082 316.468 726.521 Rank 17 16 15 14 12 9 2 13 11 10 6 6 6 1 3 4 5 NA(20) NA(20) NA(20) NA(14) NA(4) NA(4) 330.194 NA(11) NA(8) NA(4) NA(1) NA(2) 296.414 NA(1) 331.626 385.951 889.711 Rank 17 16 15 14 12 9 1 13 11 10 7 8 6 5 2 3 4 NA(20) NA(20) NA(20) NA(19) NA(13) NA(5) 399.228 NA(11) NA(10) NA(8) NA(6) 363.506 362.665 383.788 417.958 481.830 1086.001 Rank 17 16 15 14 13 8 1 12 11 10 9 7 6 5 2 3 4 NA(20) NA(20) NA(20) NA(18) NA(10) NA(8) NA(4) NA(18) NA(11) NA(9) NA(5) NA(1) NA(2) 447.570 491.772 552.856 1374.438 Rank 17 16 15 14 12 9 4 13 11 10 8 6 7 5 1 2 3 Avg for to 16 ,NA( ) 5, NA(146), 38.648 6,NA(152), 58.629 7, NA(149), 85.323 7, NA(94), 102.008 9, NA(32), 88.995 12, NA(20), 193.456 15, NA(4), 144.713 9, NA(62), 22.589 10, NA(41), 36.167 10, NA(24), 52.586 12, NA(13), 66.593 12,NA(4), 106.519 12, NA(3), 119.204 13, NA(1), 143.971 16, NA(0), 167.708 16, NA(0), 190.391 16, NA(0), 436.499 Rank 17 16 15 14 12 9 4 13 11 10 8 7 6 5 1 2 3
 No Multi-thresh FODPSO DPSO PSO ABC Rank-DE DE O(rand)R-DE O(0.9)R-DE O(0.8)R-DE O(0.7)R-DE O(0.6)R-DE O(0.5)R-DE O(0.4)R-DE O(0.3)R-DE O(0.2)R-DE O(0.1)R-DE O(0.0)R-DE 2 SRHM 0 0.977 1 1 1 1 1 1 0.999 1 1 1 1 1 1 1 1 1 Rank 18 17 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 3 SRHM 0.000 0.635 0.972 0.991 0.979 0.998 1.000 0.997 0.986 0.993 0.999 0.997 0.998 0.999 1.000 1.000 1.000 1.000 Rank 18 17 16 13 15 8 1 10 14 12 6 10 8 6 1 1 1 1 4 SRHM 0.000 0.379 0.742 0.934 0.833 0.987 0.995 0.986 0.897 0.945 0.972 0.971 0.986 0.993 0.992 0.991 0.998 0.997 Rank 18 17 16 13 15 7 3 8 14 12 10 11 8 4 5 6 1 2 5 SRHM 0.000 0.185 0.391 0.767 0.706 0.936 0.977 0.960 0.782 0.878 0.874 0.944 0.950 0.959 0.968 0.960 0.985 1.000 Rank 18 17 16 14 15 10 3 5 13 11 12 9 8 7 4 5 2 1 6 SRHM 0.000 0.000 0.187 0.433 0.453 0.863 0.968 0.900 0.601 0.706 0.826 0.878 0.934 0.922 0.944 0.935 0.970 0.999 Rank 18 17 16 15 14 10 3 8 13 12 11 9 6 7 4 5 2 1 7 SRHM 0.000 0.000 0.000 0.153 0.164 0.696 0.884 0.835 0.366 0.561 0.675 0.741 0.840 0.851 0.883 0.914 0.949 0.998 Rank 18 17 16 15 14 10 4 8 13 12 11 9 7 6 5 3 2 1 8 SRHM 0.000 0.000 0.000 0.000 0.000 0.300 0.458 0.530 0.105 0.256 0.351 0.502 0.592 0.628 0.724 0.757 0.838 0.972 Rank 18 17 16 14 14 11 9 7 13 12 10 8 6 5 4 3 2 1 9 SRHM 0.000 0.000 0.000 0.000 0.000 0.495 0.695 0.681 0.083 0.274 0.372 0.548 0.721 0.725 0.769 0.815 0.852 0.999 Rank 18 17 16 14 14 10 7 8 13 12 11 9 6 5 4 3 2 1 10 SRHM 0.000 0.000 0.000 0.000 0.000 0.000 0.381 0.375 0.000 0.073 0.218 0.311 0.409 0.370 0.460 0.578 0.686 0.998 Rank 18 17 16 14 14 12 6 7 13 11 10 9 5 8 4 3 2 1 11 SRHM 0.000 0.000 0.000 0.000 0.000 0.000 0.271 0.267 0.000 0.000 0.000 0.144 0.407 0.316 0.424 0.584 0.704 0.999 Rank 18 17 16 14 14 12 7 8 13 10 10 9 5 6 4 3 2 1 12 SRHM 0.000 0.000 0.000 0.000 0.000 0.090 0.270 0.423 0.000 0.000 0.089 0.177 0.400 0.407 0.501 0.641 0.810 1.000 Rank 18 17 16 14 14 12 8 5 13 11 10 9 7 6 4 3 2 1 13 SRHM 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.151 0.000 0.000 0.000 0.000 0.000 0.000 0.254 0.267 0.381 0.974 Rank 18 17 16 14 14 12 6 5 13 10 10 6 6 6 4 3 2 1 14 SRHM 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.102 0.000 0.000 0.000 0.000 0.000 0.110 0.000 0.253 0.494 0.993 Rank 18 17 16 14 14 12 7 4 13 10 10 7 7 6 5 3 2 1 15 SRHM 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.090 0.000 0.000 0.000 0.000 0.095 0.117 0.149 0.288 0.466 1.000 Rank 18 17 16 14 14 12 8 4 13 10 10 8 7 6 5 3 2 1 16 SRHM 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.110 0.192 0.288 0.977 Rank 18 17 16 14 14 12 6 4 13 10 10 6 6 6 5 3 2 1 Avg. for to 16 ,   ,  SRHM <2, <2, 0.000 3, 5, 0.376 4, 6, 0.443 5, 7, 0.454 5, 7, 0.464 7, 9, 0.665 9, 12, 0.554 9, 15, 0.304 6, 9, 0.263 7, 10, 0.313 7, 10, 0.530 9, 12, 0.420 9, 12, 0.654 9, 12, 0.625 9, 13, 0.619 12, 16, 0.498 12, 16, 0.656 16, 16, 0.994 Rank 18 17 16 15 14 12 8 4 13 11 10 9 6 7 5 3 2 1

Figure 2 shows the graph of the TVD of all the images and thresholds. These results are in the same pattern of the results of the “mountain” image in Table 2. That means the ability to search for the optimal solution of the proposed global search algorithm is higher than that of the multithresh function, especially when the number of thresholds is large. This goes to illustrate the difficulty of the problem. The problem with this kind is that it can be multimodal [33] or can be a nearly flat top surface [34]. The multithresh function solves the problem by performing the Nelder-Mead Simplex method [35], which is a local search method that cannot guarantee an optimal solution. Thus, its solutions are inferior to the solution produced by the algorithm using a global search.

5.2.3. Convergence Rate Comparison

The number of generations (NG) is a measure used for the convergence rate comparisons. If the target value, VTR, is achieved in a lesser number of generations (NG), it means a faster convergence rate for the algorithm. Table 3 shows the average of () for each specific number of thresholds. The results of each algorithm are represented in the corresponding column’s name. In each column, the cell containing starts from the row associated with until the row associated with . The cells associated with to the row associated with are filled by . The second last row of the table is filled by the triple:

The algorithm with the highest , that is, the lowest number of unsuccessful subproblems and the lowest average of generation is the winner. The ranking of the algorithms depends on the ordering of (, , ) and (, , ) as follows.First, rank on and . Since both and are numeric, the higher value has the higher rank.Second, rank on and b2. If is 16, then must be NA(0). If is less than 16, then must be NA (number of unsuccessful subproblems) and has the same characteristics as . Perform the order of the two numeric values in reverse; the lower value has the higher rank.Third, rank on c1 and . They are numeric but the lower is better; perform the order of the two numeric values in reverse. The lower value has the higher rank.When the ordering is finished, assign the numeric value of “1” to the object having the highest rank, assign the numeric value of “2” to the first runner up, and so on. These values are represented in the last row of the table.

If the row having must be ranked, it can be done in the same manner as above with some minor modifications. If is less than , the values of the triple pair will be , , . If is greater than , the values of the triple pair will be (, , ). Thus the ranking can now be performed.

From the ranking results, see the second last row of Table 3; the convergence rate can be ranked from best to worst in the following order: , , , , , , , , DE, , , , , ABC, PSO, DPSO, FODPSO.

As can be seen from Table 3, the DE algorithm cannot complete the task when and the algorithm cannot complete the task when . Thus, cannot compete with DE on searching for global multilevel thresholding.

In order for to outperform DE then must be in the range of or set to .

5.2.4. Stability Analysis

The harmonic mean of the success rate () for each specific number of thresholds was computed and is presented in Table 4. The results of each algorithm are represented in the corresponding column’s name. The second row from the bottom shows the harmonic mean of the success rate of each algorithm for all threshold levels.

In each column, the cells containing start from the row associated with to the row associated with . The cells from the row associated with to the row associated with are filled by . The cells from the row associated with to the row associated with are the cells that have ; these cells will be excluded from the comparison. The second last row of the table is filled with the triple:

The algorithm with the highest , the highest , and the highest average success rate is the winner. The ranking of the algorithms depends on the ordering of (a1, , ) and (, b2, ) as follows.First, rank on and a2.Second, rank on and .Third, rank on and .

Because they are numeric the higher value has the higher rank. When the ordering is finished, the numeric value of “1” is assigned to the object having the highest rank, the numeric value of “2” is assigned to the first runner up, and so on.

If the row having must be ranked, it can be done in the same manner as above with some minor modifications. If is less than or equal to , the values of the triple pair will be (, , ). If , then the values of the triple pair will be (, , ). If is greater than , the values of the triple pair will be (, , ). Thus, the ranking can now be performed.

From the ranking results, see Table 4, the success rate can be ranked from best to worst in the following order: , , , , , , , DE, , , , , , ABC, PSO, DPSO, FODPSO, multithresh.

As can be seen from Table 4, the DE algorithm has an until and the algorithm has an until . This result confirms that cannot compete with DE on searching for global multilevel thresholding. If the correct is selected, the proposed algorithm can work very well. For , has a higher rank than DE. The multithresh function cannot compete with any of the other algorithms. It can also be seen that the proposed algorithm has the best stability because its is greater than 0.5 when to 16.

5.2.5. Reliability Comparison

The threshold value distortion a.k.a. for each specific threshold is computed, shown in Table 5 and depicted in Figure 2. The results of each algorithm are illustrated in the corresponding column’s name. The second last row of Table 5 shows the slope or the approximated growth rate, , of the of each algorithm for all threshold levels. The is the slope of the robust linear regression computed by the Matlab function “robustfit.” The lower slope exhibits the better reliability. The of each algorithm is sorted in descending order. From the results, the reliability can be ranked from best to worst in the following order: , , , , , , , DE, , ABC, , , , , DPSO, FODPSO, PSO, multithresh.

 No Multi-thresh FODPSO DPSO PSO ABC Rank-DE DE O(rand)R-DE O(0.9)R-DE O(0.8)R-DE O(0.7)R-DE O(0.6)R-DE O(0.5)R-DE O(0.4)R-DE O(0.3)R-DE O(0.2)R-DE O(0.1)R-DE O(0.0)R-DE 2 TVD 74.153 2.217 0.000 0.000 0.000 0.000 0.000 0.000 0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Rank 18 17 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 3 TVD 163.333 41.753 2.968 7.424 1.510 0.150 0.000 0.270 1.004 0.519 0.075 0.246 0.171 0.075 0.000 0.000 0.000 0.000 Rank 18 17 15 16 14 8 1 11 13 12 6 10 9 6 1 1 1 1 4 TVD 257.083 98.848 34.770 36.891 8.716 7.340 1.428 2.184 25.499 13.609 7.318 7.360 7.398 4.303 1.672 0.728 0.130 0.257 Rank 18 17 15 16 12 9 4 6 14 13 8 10 11 7 5 3 1 2 5 TVD 516.582 148.344 97.216 62.319 21.565 5.373 1.864 3.241 20.441 10.064 10.422 4.584 3.904 3.289 2.441 3.385 1.195 0.000 Rank 18 17 16 15 14 10 3 5 13 11 12 9 8 6 4 7 2 1 6 TVD 534.226 219.437 187.620 166.775 45.728 19.536 2.528 7.761 89.938 43.666 30.923 13.010 6.524 6.023 4.136 6.129 2.421 0.050 Rank 18 17 16 15 13 10 3 8 14 12 11 9 7 5 4 6 2 1 7 TVD 599.250 265.029 242.194 285.862 66.259 58.877 27.976 35.481 125.040 75.011 49.851 44.610 28.039 29.841 26.075 17.950 4.776 0.150 Rank 18 16 15 17 12 11 5 8 14 13 10 9 6 7 4 3 2 1 8 TVD 475.458 382.076 380.124 309.635 72.763 120.734 90.287 89.406 218.266 143.225 130.898 99.991 76.353 82.886 57.251 57.665 26.822 2.103 Rank 18 17 16 15 5 11 9 8 14 13 12 10 6 7 3 4 2 1 9 TVD 862.458 394.744 375.229 410.710 113.892 82.099 44.407 46.539 250.470 154.662 96.417 68.197 44.223 35.333 26.627 23.406 12.757 0.780 Rank 18 16 15 17 12 10 7 8 14 13 11 9 6 5 4 3 2 1 10 TVD 847.703 458.536 445.765 498.328 117.664 84.541 43.991 48.474 249.147 136.935 93.935 72.288 43.883 44.110 36.700 27.978 20.213 0.164 Rank 18 16 15 17 12 10 6 8 14 13 11 9 5 7 4 3 2 1 11 TVD 977.976 536.258 528.329 531.211 137.941 87.267 44.944 44.973 296.513 152.195 104.092 75.115 44.461 47.564 34.944 26.231 18.782 0.083 Rank 18 17 15 16 12 10 6 7 14 13 11 9 5 8 4 3 2 1 12 TVD 1070.300 592.750 589.931 605.280 157.169 174.580 109.138 101.282 388.491 258.312 196.098 150.932 101.112 82.365 79.966 51.281 23.429 0.000 Rank 18 16 15 17 10 11 8 7 14 13 12 9 6 5 4 3 2 1 13 TVD 1142.808 567.442 579.495 711.382 183.451 204.665 147.251 123.438 466.508 299.040 216.656 178.394 114.244 114.491 96.104 81.076 52.578 1.975 Rank 18 15 16 17 10 11 8 7 14 13 12 9 5 6 4 3 2 1 14 TVD 1148.641 632.482 660.908 770.319 205.714 218.408 141.431 116.716 524.783 325.936 241.997 177.121 113.426 104.073 88.593 65.908 39.647 0.564 Rank 18 15 16 17 10 11 8 7 14 13 12 9 6 5 4 3 2 1 15 TVD 1245.529 707.835 722.126 816.158 256.477 307.873 219.556 182.464 605.623 428.370 343.439 261.522 174.700 166.309 138.710 100.296 62.940 0.000 Rank 18 15 16 17 9 11 8 7 14 13 12 10 6 5 4 3 2 1 16 TVD 1182.989 768.714 835.747 828.004 249.042 279.753 211.614 195.676 605.790 416.711 311.314 251.408 174.118 167.316 133.922 113.776 76.661 2.539 Rank 18 15 17 16 9 11 8 7 14 13 12 10 6 5 4 3 2 1 Avg. for  to 16 TVD 739.899 387.764 378.828 402.687 109.193 110.080 72.428 66.527 257.838 163.884 122.229 93.652 62.170 59.199 48.476 38.387 22.823 0.578 Growth rate ( ) 86.86 55.25 61.13 67.98 19.25 21.76 15.61 13.48 48.43 32.25 23.91 18.93 12.52 11.86 9.99 7.69 4.94 0.09 Rank 18 15 16 17 10 11 8 7 14 13 12 9 6 5 4 3 2 1
 Image Between-class variance (VTR) Thresholds Starfish 2 2546.885 85, 157 3 2779.925 68, 119, 177 4 2865.707 60, 101, 138, 187 5 2912.859 52, 86, 117, 150, 194 6 2941.728 47, 77, 105, 132, 162, 201 7 2960.158 44, 71, 95, 118, 142, 170, 206 8 2972.356 43, 68, 90, 110, 131, 153, 180, 212 9 2981.138 38, 58, 78, 97, 116, 136, 157, 183, 214 10 2988.206 37, 56, 75, 93, 110, 128, 146, 167, 192, 219 11 2993.348 35, 53, 71, 88, 103, 119, 135, 152, 172, 196, 221 12 2997.352 34, 51, 68, 84, 98, 112, 127, 142, 158, 177, 200, 223 13 3000.480 33, 48, 64, 79, 93, 106, 119, 133, 147, 162, 181, 203, 225 14 3003.076 32, 47, 62, 76, 89, 101, 114, 127, 140, 154, 169, 187, 207, 227 15 3005.235 30, 43, 56, 70, 83, 95, 107, 119, 131, 143, 156, 171, 189, 209, 228 16 3007.060 30, 42, 55, 68, 80, 92, 103, 114, 126, 138, 150, 163, 178, 195, 213, 230 Mountain 2 2372.923 61, 128 3 2496.113 33, 77, 131 4 2551.955 33, 73, 109, 147 5 2580.336 32, 69, 99, 125, 159 6 2596.956 24, 46, 74, 101, 126, 160 7 2608.807 24, 46, 73, 98, 119, 145, 175 8 2616.294 24, 46, 73, 97, 115, 135, 160, 191 9 2622.314 20, 37, 54, 76, 98, 116, 136, 160, 191 10 2627.194 20, 36, 53, 74, 93, 106, 121, 140, 163, 194 11 2630.496 20, 36, 53, 74, 93, 105, 118, 134, 152, 172, 201 12 2633.189 18, 31, 45, 59, 76, 93, 105, 118, 134, 152, 172, 201 13 2635.290 18, 31, 45, 59, 76, 92, 103, 115, 129, 144, 161, 179, 207 14 2637.088 17, 29, 42, 56, 72, 86, 96, 106, 117, 130, 145, 161, 179, 207 15 2638.449 17, 28, 39, 50, 61, 75, 88, 97, 107, 118, 131, 146, 162, 179, 207 16 2639.616 17, 27, 38, 49, 60, 74, 86, 95, 104, 113, 123, 135, 149, 164, 181, 209 Cactus 2 1816.448 73, 151 3 1970.112 64, 106, 173 4 2042.275 55, 87, 125, 187 5 2080.884 49, 76, 102, 138, 196 6 2104.147 46, 70, 93, 119, 157, 208 7 2119.670 44, 65, 85, 105, 131, 168, 215 8 2129.358 42, 60, 77, 94, 113, 138, 174, 218 9 2136.162 41, 58, 74, 89, 105, 125, 152, 186, 224 10 2141.579 40, 56, 71, 85, 99, 115, 135, 161, 193, 228 11 2145.397 39, 53, 66, 79, 91, 104, 119, 139, 165, 196, 229 12 2148.282 38, 51, 64, 76, 88, 100, 114, 131, 152, 176, 204, 233 13 2150.740 37, 49, 61, 72, 83, 94, 105, 118, 135, 155, 178, 205, 233 14 2152.626 36, 47, 58, 68, 78, 88, 98, 109, 122, 138, 158, 181, 207, 234 15 2154.162 36, 46, 56, 66, 76, 85, 94, 104, 115, 128, 144, 164, 187, 212, 237 16 2155.471 36, 46, 56, 66, 75, 84, 93, 102, 112, 124, 138, 155, 174, 195, 217, 239 Butterfly 2 3873.222 84, 155 3 3990.855 81, 144, 199 4 4051.357 77, 121, 167, 207 5 4092.929 60, 89, 129, 172, 209 6 4119.448 59, 87, 122, 161, 193, 221 7 4135.937 53, 74, 98, 128, 165, 195, 222 8 4148.556 53, 73, 96, 123, 156, 183, 203, 228 9 4156.528 52, 72, 94, 118, 144, 170, 190, 207, 231 10 4163.794 48, 63, 80, 99, 121, 147, 172, 191, 208, 231 11 4168.489 47, 62, 79, 98, 118, 141, 164, 183, 198, 213, 234 12 4172.517 46, 59, 73, 89, 104, 122, 144, 167, 185, 199, 214, 235 13 4175.487 46, 59, 72, 87, 102, 118, 137, 157, 175, 189, 202, 216, 236 14 4177.893 44, 55, 66, 79, 93, 106, 121, 141, 161, 178, 191, 203, 217, 237 15 4179.908 44, 55, 66, 78, 92, 105, 119, 137, 156, 173, 186, 197, 208, 221, 239 16 4181.559 42, 52, 61, 71, 83, 95, 107, 121, 139, 157, 173, 186, 197, 208, 221, 239 Circus 2 1651.257 118, 172 3 1760.512 105, 150, 187 4 1817.487 93, 132, 165, 195 5 1850.243 87, 122, 152, 177, 203 6 1870.083 82, 113, 141, 164, 185, 208 7 1883.966 77, 104, 129, 151, 171, 190, 212 8 1893.450 73, 98, 122, 143, 161, 178, 195, 216 9 1900.592 70, 92, 114, 134, 152, 168, 183, 199, 219 10 1905.992 68, 89, 109, 128, 145, 160, 174, 188, 203, 222 11 1909.887 66, 86, 105, 123, 139, 153, 166, 179, 192, 206, 225 12 1913.079 63, 81, 98, 114, 130, 145, 158, 170, 182, 194, 208, 226 13 1915.676 62, 79, 95, 111, 126, 140, 152, 164, 175, 186, 197, 210, 228 14 1917.756 61, 78, 94, 109, 123, 136, 148, 159, 170, 180, 190, 201, 214, 231 15 1919.524 59, 74, 88, 102, 115, 128, 140, 151, 161, 171, 181, 191, 202, 214, 231 16 1920.958 58, 73, 87, 100, 113, 126, 138, 149, 159, 169, 178, 187, 196, 206, 218, 234 Snow 2 5261.705 80, 169 3 5624.289 71, 139, 207 4 5729.116 50, 92, 144, 208 5 5785.138 49, 91, 140, 192, 231 6 5819.333 45, 81, 111, 148, 194, 232 7 5835.770 43, 76, 101, 127, 159, 196, 232 8 5850.190 30, 55, 83, 107, 133, 163, 197, 233 9 5862.437 30, 55, 82, 106, 129, 157, 185, 211, 237 10 5870.284 29, 52, 75, 94, 111, 132, 159, 186, 212, 237 11 5875.817 29, 52, 75, 94, 111, 132, 158, 183, 206, 227, 244 12 5880.335 29, 52, 74, 93, 109, 127, 149, 169, 188, 209, 228, 244 13 5884.513 23, 39, 57, 76, 94, 110, 128, 150, 170, 188, 209, 228, 244 14 5887.355 22, 37, 54, 70, 84, 98, 112, 129, 151, 170, 188, 209, 228, 244 15 5889.953 21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 175, 192, 211, 229, 245 16 5891.998 21, 36, 53, 69, 83, 97, 110, 124, 141, 159, 174, 189, 207, 222, 236, 248 Palace 2 2623.440 99, 165 3 2791.488 84, 132, 186 4 2860.98 70, 103, 143, 191 5 2908.165 69, 101, 138, 177, 218 6 2934.330 64, 89, 117, 147, 181, 220 7 2953.745 54, 75, 99, 126, 153, 183, 220 8 2966.250 50, 69, 89, 111, 134, 158, 185, 221 9 2974.719 47, 65, 81, 100, 120, 141, 163, 188, 222 10 2981.875 47, 64, 80, 98, 117, 137, 157, 178, 199, 226 11 2986.898 45, 61, 75, 90, 106, 123, 141, 159, 179, 200, 227 12 2990.579 43, 58, 69, 82, 96, 111, 126, 143, 160, 180, 201, 227 13 2993.809 43, 58, 69, 81, 95, 110, 125, 141, 157, 173, 190, 208, 231 14 2996.112 43, 57, 68, 80, 94, 108, 123, 138, 153, 167, 182, 199, 218, 239 15 2998.213 42, 56, 66, 77, 88, 100, 113, 126, 140, 154, 167, 182, 199, 218, 239 16 2999.918 42, 55, 65, 75, 86, 98, 110, 123, 136, 149, 162, 176, 191, 205, 222, 241 Flower 2 1489.281 61, 130 3 1627.897 39, 77, 141 4 1685.956 36, 67, 105, 160 5 1715.220 28, 49, 75, 111, 164 6 1736.423 27, 47, 71, 102, 143, 192 7 1752.512 26, 43, 61, 83, 111, 151, 199 8 1761.968 25, 42, 58, 77, 99, 126, 161, 204 9 1768.354 24, 40, 54, 70, 88, 109, 135, 167, 207 10 1772.903 23, 37, 48, 60, 75, 92, 112, 138, 169, 208 11 1776.470 23, 36, 47, 58, 72, 87, 104, 124, 149, 177, 211 12 1779.106 22, 34, 44, 54, 65, 78, 92, 108, 128, 152, 179, 212 13 1781.251 20, 30, 39, 48, 58, 70, 83, 96, 112, 131, 154, 180, 213 14 1783.049 19, 29, 38, 47, 56, 66, 78, 90, 104, 120, 139, 161, 185, 215 15 1784.478 19, 29, 38, 46, 54, 63, 74, 85, 97, 111, 128, 147, 168, 190, 218 16 1785.641 19, 28, 37, 45, 53, 62, 72, 83, 94, 106, 120, 136, 155, 175, 196, 222 Wherry 2 3313.161 108, 189 3 3543.272 102, 161, 218 4 3599.924 83, 121, 163, 218 5 3634.708 81, 118, 152, 184, 224 6 3656.048 72, 103, 130, 156, 186, 225 7 3668.313 68, 94, 120, 139, 161, 189, 226 8 3678.216 60, 83, 110, 132, 152, 175, 198, 230 9 3686.001 56, 77, 100, 122, 138, 156, 179, 201, 231 10 3691.730 56, 76, 98, 120, 136, 153, 174, 194, 219, 243 11 3696.416 55, 74, 94, 114, 130, 142, 158, 178, 197, 222, 245 12 3699.866 54, 72, 91, 110, 126, 138, 151, 168, 185, 202, 225, 246 13 3702.619 52, 68, 83, 100, 117, 130, 140, 153, 169, 185, 202, 225, 246 14 3705.033 52, 68, 83, 100, 117, 130, 140, 152, 167, 182, 197, 215, 235, 249 15 3706.937 51, 66, 79, 94, 110, 123, 133, 142, 154, 169, 184, 199, 216, 235, 249 16 3708.484 49, 63, 75, 89, 104, 118, 129, 138, 147, 159, 173, 186, 200, 217, 235, 249 Waterfall 2 4512.801 88, 170 3 4646.137 72, 115, 182 4 4711.019 67, 103, 150, 204 5 4752.267 58, 87, 119, 164, 212 6 4777.063 53, 78, 104, 134, 176, 217 7 4793.510 48, 70, 92, 116, 147, 186, 221 8 4805.756 46, 66, 86, 107, 131, 163, 199, 226 9 4814.097 43, 61, 79, 97, 117, 141, 173, 205, 228 10 4820.724 40, 57, 73, 90, 108, 128, 152, 182, 210, 230 11 4825.739 38, 54, 69, 84, 100, 117, 137, 162, 191, 215, 232 12 4829.689 37, 53, 67, 81, 96, 111, 128, 148, 173, 198, 218, 233 13 4832.826 35, 50, 63, 76, 89, 103, 117, 133, 153, 177, 201, 220, 234 14 4835.371 32, 46, 58, 70, 82, 95, 108, 122, 138, 158, 182, 204, 221, 234 15 4837.537 32, 46, 57, 68, 80, 92, 104, 117, 131, 148, 169, 191, 210, 224, 236 16 4839.226 30, 43, 54, 64, 74, 85, 96, 108, 120, 134, 151, 172, 193, 211, 225, 236 Bird 2 901.450 71, 122 3 975.230 64, 111, 140 4 1027.509 61, 104, 131, 164 5 1051.482 54, 93, 119, 138, 169 6 1067.992 47, 82, 108, 127, 142, 172 7 1077.304 40, 70, 94, 113, 129, 143, 173 8 1086.005 39, 69, 93, 112, 128, 141, 159, 192 9 1091.341 37, 65, 88, 105, 119, 131, 142, 160, 193 10 1095.355 37, 64, 86, 103, 117, 129, 139, 149, 167, 200 11 1098.475 33, 56, 77, 93, 107, 119, 130, 140, 150, 169, 202 12 1100.749 33, 56, 77, 93, 107, 119, 129, 138, 146, 158, 178, 209 13 1102.639 31, 52, 71, 87, 100, 111, 121, 130, 139, 147, 159, 179, 210 14 1104.132 29, 48, 66, 82, 95, 107, 118, 127, 134, 141, 149, 161, 181, 211 15 1105.446 28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 142, 150, 162, 182, 212 16 1106.500 28, 45, 62, 77, 90, 101, 111, 120, 128, 135, 141, 148, 157, 171, 191, 219 Police 2 3647.353 74, 150 3 3844.314 70, 135, 192 4 3966.225 63, 112, 158, 209 5 4013.875 61, 104, 140, 174, 214 6 4047.198 32, 67, 106, 141, 175, 214 7 4067.996 32, 67, 104, 133, 161, 186, 219 8 4084.933 29, 52, 78, 106, 134, 162, 187, 219 9 4094.705 29, 52, 78, 103, 125, 147, 169, 190, 220 10 4101.702 29, 52, 78, 102, 123, 143, 165, 184, 203, 228 11 4108.018 28, 49, 71, 90, 107, 126, 146, 166, 185, 204, 229 12 4112.304 28, 49, 71, 89, 105, 122, 139, 157, 174, 189, 207, 231 13 4115.165 28, 46, 62, 78, 92, 106, 123, 140, 158, 174, 189, 207, 231 14 4117.926 28, 46, 62, 78, 91, 105, 120, 135, 151, 166, 180, 193, 210, 232 15 4120.045 28, 46, 62, 78, 91, 103, 116, 129, 143, 158, 172, 185, 197, 214, 235 16 4121.861 28, 46, 62, 78, 91, 103, 116, 128, 142, 156, 170, 182, 193, 206, 223, 240 Ostrich 2 1073.452 75, 135 3 1139.260 69, 101, 149 4 1178.650 65, 92, 125, 176 5 1203.749 56, 78, 100, 131, 179 6 1218.643 47, 65, 85, 103, 133, 181 7 1228.925 47, 64, 83, 100, 122, 152, 192 8 1236.023 45, 59, 75, 90, 104, 125, 155, 194 9 1240.756 40, 52, 65, 80, 94, 107, 128, 157, 195 10 1244.909 40, 52, 64, 78, 91, 103, 119, 141, 168, 201 11 1247.879 40, 51, 62, 75, 87, 97, 108, 125, 148, 174, 205 12 1250.313 37, 48, 57, 68, 80, 91, 101, 112, 128, 150, 175, 206 13 1252.298 31, 44, 53, 63, 75, 86, 95, 105, 117, 134, 154, 178, 207 14 1253.956 29, 42, 50, 58, 68, 79, 89, 98, 108, 120, 137, 157, 181, 209 15 1255.366 29, 42, 50, 58, 67, 77, 86, 94, 102, 111, 124, 141, 160, 183, 210 16 1256.490 28, 41, 49, 57, 66, 76, 85, 93, 101, 110, 122, 137, 155, 174, 196, 219 Viaduct 2 7920.458 77, 180 3 8117.991 54, 109, 193 4 8203.807 42, 84, 131, 203 5 8246.806 35, 68, 103, 146, 210 6 8272.775 31, 59, 88, 120, 160, 216 7 8287.714 28, 53, 77, 103, 132, 169, 220 8 8298.322 27, 51, 75, 100, 128, 164, 212, 246 9 8308.240 24, 45, 66, 88, 112, 139, 172, 216, 247 10 8315.133 22, 41, 60, 79, 99, 121, 146, 178, 219, 247 11 8320.032 20, 37, 54, 71, 89, 108, 128, 152, 183, 221, 248 12 8323.799 20, 36, 52, 68, 84, 101, 119, 139, 162, 190, 224, 248 13 8326.793 18, 33, 48, 62, 77, 92, 108, 125, 144, 167, 195, 226, 248 14 8329.119 17, 31, 44, 57, 70, 84, 98, 113, 129, 148, 170, 197, 227, 248 15 8330.983 17, 31, 44, 57, 70, 84, 98, 113, 129, 147, 169, 195, 224, 243, 251 16 8332.744 16, 29, 42, 54, 66, 78, 91, 104, 118, 133, 151, 172, 196, 224, 243, 251 Fish 2 3593.389 64, 148 3 3870.456 44, 104, 177 4 3972.731 34, 81, 127, 188 5 4024.885 27, 63, 101, 139, 194 6 4054.836 24, 56, 90, 123, 156, 205 7 4075.236 22, 49, 78, 107, 135, 168, 213 8 4088.300 20, 44, 68, 93, 118, 142, 173, 216 9 4097.101 19, 40, 62, 85, 107, 128, 149, 178, 218 10 4103.194 18, 38, 58, 78, 98, 118, 137, 158, 185, 222 11 4107.954 16, 34, 51, 69, 88, 107, 125, 143, 163, 190, 224 12 4111.678 15, 31, 47, 64, 82, 100, 117, 133, 149, 169, 195, 227 13 4114.783 14, 28, 43, 58, 74, 90, 106, 121, 136, 152, 172, 198, 228 14 4116.954 14, 28, 43, 58, 73, 88, 103, 117, 131, 145, 160, 179, 203, 231 15 4118.931 13, 25, 38, 51, 64, 78, 92, 106, 120, 133, 147, 162, 181, 205, 232 16 4120.522 13, 25, 37, 49, 62, 75, 89, 102, 115, 127, 139, 152, 168, 188, 211, 235 Houses 2 2543.788 56, 116 3 2627.230 53, 105, 150 4 2663.698 42, 69, 110, 152 5 2694.902 39, 65, 96, 130, 158 6 2708.655 39, 65, 95, 127, 150, 169 7 2720.263 35, 55, 74, 98, 128, 151, 170 8 2726.676 35, 55, 74, 98, 127, 148, 164, 184 9 2732.853 32, 48, 65, 80, 101, 128, 148, 164, 184 10 2736.510 30, 45, 60, 74, 87, 106, 129, 149, 164, 184 11 2739.566 30, 45, 60, 73, 86, 105, 128, 145, 156, 168, 187 12 2742.148 29, 42, 55, 68, 79, 94, 112, 130, 145, 156, 168, 187 13 2744.071 28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 156, 168, 187 14 2745.542 28, 39, 51, 63, 74, 84, 98, 115, 131, 145, 155, 165, 176, 193 15 2746.817 27, 37, 47, 58, 68, 77, 86, 100, 116, 131, 145, 155, 165, 176, 193 16 2747.991 27, 37, 47, 57, 67, 76, 85, 98, 113, 127, 139, 147, 156, 165, 176, 193 Mushroom 2 1988.328 76, 145 3 2153.037 65, 110, 174 4 2237.441 59, 93, 135, 193 5 2277.427 52, 78, 106, 145, 199 6 2301.629 48, 71, 94, 122, 157, 205 7 2317.538 46, 67, 87, 110, 138, 171, 213 8 2328.526 44, 62, 79, 97, 118, 145, 177, 216 9 2335.812 42, 58, 74, 90, 107, 127, 152, 181, 218 10 2340.981 41, 56, 70, 84, 99, 116, 136, 159, 186, 220 11 2345.160 39, 52, 65, 78, 92, 107, 124, 144, 166, 191, 223 12 2348.413 38, 51, 63, 75, 87, 100, 115, 132, 152, 174, 199, 227 13 2350.988 38, 50, 62, 74, 85, 97, 110, 125, 142, 160, 180, 203, 229 14 2353.119 37, 48, 59, 69, 79, 89, 100, 113, 127, 144, 162, 181, 204, 230 15 2354.748 36, 46, 56, 66, 75, 84, 94, 105, 117, 130, 146, 163, 182, 205, 230 16 2356.113 36, 46, 56, 65, 74, 83, 92, 102, 113, 125, 139, 154, 169, 187, 208, 232 Snow mountain 2 1912.613 85, 149 3 2135.274 79, 137, 197 4 2234.669 70, 119, 154, 204 5 2288.799 55, 96, 129, 160, 206 6 2317.224 52, 90, 118, 142, 167, 209 7 2333.078 50, 85, 111, 132, 153, 174, 212 8 2344.629 45, 76, 100, 120, 139, 158, 177, 214 9 2352.951 44, 74, 97, 116, 133, 150, 167, 187, 220 10 2359.175 41, 68, 90, 108, 124, 140, 156, 172, 193, 225 11 2364.304 38, 62, 84, 102, 117, 132, 146, 160, 175, 196, 227 12 2367.971 34, 56, 78, 96, 111, 125, 139, 152, 165, 179, 199, 228 13 2371.208 33, 53, 73, 90, 104, 117, 130, 143, 155, 167, 180, 200, 229 14 2373.567 32, 52, 72, 89, 103, 115, 127, 139, 150, 161, 172, 185, 205, 232 15 2375.586 31, 49, 68, 84, 98, 109, 120, 131, 142, 153, 164, 175, 188, 208, 233 16 2377.255 26, 42, 59, 75, 89, 101, 111, 121, 132, 143, 154, 165, 176, 189, 209, 234 Snake 2 1118.615 87, 134 3 1231.320 76, 114, 154 4 1286.555 69, 101, 129, 166 5 1317.027 63, 91, 115, 140, 175 6 1336.172 59, 84, 105, 126, 149, 182 7 1348.933 55, 78, 97, 115, 133, 155, 187 8 1357.665 52, 73, 91, 107, 123, 140, 161, 192 9 1364.159 50, 70, 87, 102, 116, 131, 148, 169, 198 10 1368.955 47, 65, 81, 95, 108, 121, 135, 151, 172, 200 11 1372.626 46, 63, 78, 91, 103, 115, 127, 140, 156, 176, 203 12 1375.513 45, 61, 75, 88, 99, 110, 121, 133, 146, 162, 182, 208 13 1377.847 43, 58, 72, 84, 95, 106, 116, 127, 138, 151, 166, 185, 211 14 1379.736 42, 56, 69, 81, 92, 102, 112, 122, 132, 143, 155, 170, 189, 214 15 1381.282 41, 54, 66, 77, 87, 97, 106, 115, 124, 134, 145, 157, 172, 191, 215 16 1382.603 40, 52, 63, 74, 84, 93, 102, 111, 120, 129, 139, 150, 162, 177, 195, 219 Riosanpablo 2 2667.020 95, 160 3 2818.660 75, 121, 177 4 2892.439 68, 102, 143, 189 5 2931.654 62, 89, 121, 158, 197 6 2957.018 58, 82, 107, 138, 171, 204 7 2973.269 53, 74, 95, 119, 147, 177, 207 8 2984.972 50, 69, 87, 108, 133, 159, 185, 212 9 2993.359 48, 66, 83, 101, 122, 145, 168, 191, 215 10 2999.615 46, 63, 78, 93, 110, 130, 151, 173, 195, 217 11 3004.374 45, 61, 75, 89, 104, 121, 140, 160, 180, 200, 220 12 3008.082 44, 59, 72, 84, 97, 112, 129, 147, 166, 185, 203, 222 13 3011.101 43, 57, 69, 81, 93, 107, 122, 138, 155, 172, 189, 206, 224 14 3013.465 41, 54, 66, 77, 88, 100, 113, 128, 144, 160, 176, 192, 208, 225 15 3015.423 40, 53, 64, 74, 84, 95, 107, 121, 135, 150, 165, 180, 195, 210, 226 16 3017.081 39, 51, 62, 72, 82, 92, 103, 115, 128, 142, 156, 170, 184, 198, 212, 227

We can see from these results that has a higher approximated growth rate than DE. with and are still better than DE. produced the best result with a very flat slope and a very low -intercept. The multithresh function yielded a higher growth rate of solution distortion and a higher -intercept than the other algorithms. The higher growth rate of solution distortion means the quality of solution drops very fast if the number of thresholds increases. The higher -intercept means that the solution distortion at the lowest number of thresholds is high. Thus, the global optimization algorithm is required for solving the multilevel thresholding.

6. Conclusions

The differential evolution with onlooker ranking-based mutation operator algorithm was proposed and applied to the multilevel image thresholding problem. The objective of this proposed algorithm was to increase the ability for adjusting the balance of the exploitation and exploration abilities. Its concept is a combination of the ranking-difference evolution and onlooker selection of the ABC algorithm. The experiments compared the proposed algorithm with six existing algorithms: PSO, DPSO, FODPSO, ABC, DE, and on 20 real images of the Berkeley Segmentation Dataset and Benchmark and a satellite image. The stability analysis, convergence speed, and the reliability were measured. The results signified that the proposed algorithm is more efficient than the six tested algorithms. The onlooker ranking-based mutation operator is able to enhance the performance of the proposed algorithm. The not only obtained more stability analysis, but it also achieved faster convergence rates to reach the target BCV, if a proper value of is set.

For future work based on this paper, the proposed algorithm has one parameter to be set by a user; the mechanism to automatically adapt this parameter is not presented but is required.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.

References

1. W. K. Pratt, Digital Image Processing, John Wiley & Sons, New York, NY, USA, 1978.
2. J. S. Weszka, “A survey of threshold selection techniques,” Computer Graphics and Image Processing, vol. 7, no. 2, pp. 259–265, 1978. View at: Google Scholar
3. K. S. Fu and J. K. Mui, “A survey on image segmentation,” Pattern Recognition, vol. 13, no. 1, pp. 3–16, 1981. View at: Google Scholar | MathSciNet
4. P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding techniques,” Computer Vision, Graphics and Image Processing, vol. 41, no. 2, pp. 233–260, 1988. View at: Google Scholar
5. N. R. Pal and S. K. Pal, “A review on image segmentation techniques,” Pattern Recognition, vol. 26, no. 9, pp. 1277–1294, 1993. View at: Publisher Site | Google Scholar
6. A. T. Abak, U. Baris, and B. Sankur, “The performance evaluation of thresholding algorithms for optical character recognition,” in Proceedings of the 4th International Conference on Document Analysis and Recognition, pp. 697–700, Ulm, Germany, August 1997. View at: Google Scholar
7. M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” Journal of Electronic Imaging, vol. 13, no. 1, pp. 146–168, 2004. View at: Publisher Site | Google Scholar
8. S. U. Lee and S. Yoon Chung, “A comparative performance study of several global thresholding techniques for segmentation,” Computer Vision, Graphics and Image Processing, vol. 52, no. 2, pp. 171–190, 1990. View at: Google Scholar
9. N. Otsu, “A threshold selection method from gray level histograms,” IEEE Transactions on Systems, Man and Cybernetics, vol. 9, no. 1, pp. 62–66, 1979. View at: Google Scholar
10. J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks (ICNN ’95), vol. 4, pp. 1942–1948, Perth, Australia, December 1995. View at: Google Scholar
11. R. Storn and K. Price, “Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces,” Tech. Rep. TR-95-012, International Computer Sciences Institute, Berkeley, Calif, USA, 1995. View at: Google Scholar
12. D. Karaboga, “An idea based on honey bee swarm for numerical optimization,” Tech. Rep. TR06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005. View at: Google Scholar
13. M. S. Couceiro, R. P. Rocha, N. M. F. Ferreira, and J. A. T. Machado, “Introducing the fractional-order Darwinian PSO,” Signal Image and Video Processing, vol. 6, no. 3, pp. 343–350, 2012. View at: Publisher Site | Google Scholar
14. R. V. Kulkarni and G. K. Venayagamoorthy, “Bio-inspired algorithms for autonomous deployment and localization of sensor nodes,” IEEE Transactions on Systems, Man and Cybernetics C, vol. 40, no. 6, pp. 663–675, 2010. View at: Publisher Site | Google Scholar
15. B. Akay, “A study on particle swarm optimization and artificial bee colony algorithms for multilevel thresholding,” Applied Soft Computing Journal, vol. 3, no. 6, pp. 3066–3091, 2013. View at: Publisher Site | Google Scholar
16. K. Hammouche, M. Diaf, and P. Siarry, “A comparative study of various meta-heuristic techniques applied to the multilevel thresholding problem,” Engineering Applications of Artificial Intelligence, vol. 23, no. 5, pp. 676–688, 2010. View at: Publisher Site | Google Scholar
17. V. Osuna-Enciso, E. Cuevas, and H. Sossa, “A comparison of nature inspired algorithms for multi-threshold image segmentation,” Expert Systems with Applications, vol. 40, no. 4, pp. 1213–1219, 2013. View at: Google Scholar
18. P. Ghamisia, M. S. Couceiro, F. Martins, and J. A. Benediktsson, “Multilevel image segmentation based on Fractional-Order Darwinian particle swarm optimization,” IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 1, pp. 1–44, 2013. View at: Google Scholar
19. W. Gong and Z. Cai, “Differential evolution with ranking-based mutation operators,” IEEE Transactions on Cybernetics, vol. 43, no. 6, pp. 2066–2081, 2013. View at: Google Scholar
20. K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer, Berlin, Germany, 2005. View at: MathSciNet
21. R. Storn and K. Price, Home Page of Differential Evolution, International Computer Science Institute, Berkeley, Calif, USA, 2010.
22. A. K. Qin and P. N. Suganthan, “Self-adaptive differential evolution algorithm for numerical optimization,” in Proceedings of the IEEE Congress on Evolutionary Computation (CEC '05), pp. 1785–1791, September 2005. View at: Google Scholar
23. E. Mezura-Montes, J. Velázquez-Reyes, and C. A. Coello Coello, “A comparative study of differential evolution variants for global optimization,” in Proceedings of the 8th Annual Genetic and Evolutionary Computation Conference, pp. 485–492, July 2006. View at: Google Scholar
24. A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Transactions on Evolutionary Computation, vol. 13, no. 2, pp. 398–417, 2009. View at: Publisher Site | Google Scholar
25. S. Das, A. Abraham, and A. Konar, “Automatic clustering using an improved differential evolution algorithm,” IEEE Transactions on Systems, Man, and Cybernetics A, vol. 38, no. 1, pp. 218–237, 2008. View at: Publisher Site | Google Scholar
26. J. Brest, S. Greiner, B. Bošković, M. Mernik, and V. Zumer, “Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems,” IEEE Transactions on Evolutionary Computation, vol. 10, no. 6, pp. 646–657, 2006. View at: Publisher Site | Google Scholar
27. N. Noman and H. Iba, “Accelerating differential evolution using an adaptive local search,” IEEE Transactions on Evolutionary Computation, vol. 12, no. 1, pp. 107–125, 2008. View at: Publisher Site | Google Scholar
28. R. S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama, “Opposition-based differential evolution,” IEEE Transactions on Evolutionary Computation, vol. 12, no. 1, pp. 64–79, 2008. View at: Publisher Site | Google Scholar
29. D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,” in Proceedings of the 8th International Conference on Computer Vision, vol. 2, pp. 416–423, July 2001. View at: Google Scholar
30. P. Ghamisi, M. S. Couceiro, J. A. Benediktsson, and N. M. F. Ferreira, “An efficient method for segmentation of image based on fractional calculus and natural selection,” Expert Systems with Applications, vol. 39, pp. 12407–12417, 2012. View at: Google Scholar
31. R. Gamperle, S. D. Muller, and A. Koumoutsakos, “A parameter study for differential evolution,” in Proceedings of the Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation, vol. 10, pp. 293–298, 2002. View at: Google Scholar
32. M. Saraswat, K. V. Arya, and H. Sharma, “Leukocyte segmentation in tissue images using differential evolution algorithm,” Swarm and Evolutionary Computation, vol. 11, pp. 46–54, 2013. View at: Google Scholar
33. J. Kittler and J. Illingworth, “On threshold selection using clustering criteria,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 5, pp. 652–655, 1985. View at: Google Scholar
34. R. Guo and S. M. Pandit, “Automatic threshold selection based on histogram modes and a discriminant criterion,” Machine Vision and Applications, vol. 10, no. 5-6, pp. 331–338, 1998. View at: Google Scholar
35. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM Journal on Optimization, vol. 9, no. 1, pp. 112–147, 1998. View at: Google Scholar | MathSciNet

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