Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 974024, 23 pages

http://dx.doi.org/10.1155/2014/974024

Research Article

A Global Multilevel Thresholding Using Differential Evolution Approach

Kanjana Charansiriphaisan, Sirapat Chiewchanwattana, and Khamron Sunat

Department of Computer Science, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received 3 October 2013; Revised 23 January 2014; Accepted 3 February 2014; Published 20 March 2014

Academic Editor: Yi-Hung Liu

Copyright © 2014 Kanjana Charansiriphaisan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [2–7]. 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 [22–24]. The “DE/rand/1/bin” strategy has been widely used in DE literature [25–28]. 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

Algorithm 1: The DE algorithm with “DE/rand/1/bin” strategy.

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

Algorithm 2: Onlooker and ranking-based vector selection for DE.

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

Algorithm 3: DE with onlooker and ranking-based mutation.

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.

(a)

(b)

Figure 1: The test images and corresponding histograms.

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].

Table 1: Essential parameters of the PSO, DPSO, and FODPSO taken from [30].

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.

Table 2: The best values of the between-class variance and thresholds of the “mountain” image produced by the multithresh function of the Matlab toolbox and by the proposed O(0.0)R-DE algorithm. The number of thresholds varies from 2 to 16.

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

Table 3: The average of mean number of generation () and ranks of the methods.

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

Table 4: The average success rate (SRHM) and ranks of the methods.

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.

Figure 2: The threshold value distortion (TVD) of algorithms versus number of thresholds.

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.

Table 5: The average of threshold value distortion (TVD) and ranks of the methods.

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

Table 6: The between-class variance criterion and the best thresholds for test images.

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.

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