Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 3196958, 24 pages
http://dx.doi.org/10.1155/2016/3196958

Research Article

Improved Glowworm Swarm Optimization Algorithm for Multilevel Color Image Thresholding Problem

Lifang He¹ and Songwei Huang²

¹Department of Electronics and Communication Engineering, Kunming University of Science and Technology, Kunming 650093, China
²Department of Mineral Processing, Kunming University of Science and Technology, Kunming 650093, China

Received 12 May 2016; Revised 22 July 2016; Accepted 27 July 2016

Academic Editor: Masoud Hajarian

Copyright © 2016 Lifang He and Songwei Huang. 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.

The thresholding process finds the proper threshold values by optimizing a criterion, which can be considered as a constrained optimization problem. The computation time of traditional thresholding techniques will increase dramatically for multilevel thresholding. To greatly overcome this problem, swarm intelligence algorithm is widely used to search optimal thresholds. In this paper, an improved glowworm swarm optimization (IGSO) algorithm has been presented to find the optimal multilevel thresholds of color image based on the between-class variance and minimum cross entropy (MCE). The proposed methods are examined on standard set of color test images by using various numbers of threshold values. The results are then compared with those of basic glowworm swarm optimization, adaptive particle swarm optimization (APSO), and self-adaptive differential evolution (SaDE). The simulation results show that the proposed method can find the optimal thresholds accurately and efficiently and is an effective multilevel thresholding method for color image segmentation.

1. Introduction

Image segmentation is to partition an image into multiple segments or regions and extract the meaningful and interested objects, which is the critical step in image processing and image analysis. The goal of image segmentation is to make an image more meaningful and easier to understand and analyze [1, 2]. Nowadays image segmentation has been widely used in many practical applications such as medical imaging [3], object detection [4], optical character recognition (OCR) [5], and remote sensing [6].

By now, several algorithms and techniques have been proposed for image segmentation in the literature and thresholding is one of the simplest but most effective methods in all the image segmentation algorithms. The fundamental principle of thresholding technique is to divide the whole pixel points of an image into several classes by setting different threshold values, so the key to this method is to find the proper threshold values. According to the number of threshold values, thresholding techniques can be separated into two groups: bilevel and multilevel thresholding. Bilevel thresholding is to divide an image into two parts by using one threshold value. If the threshold value is more than one, it will be extended into multilevel thresholding, which can accurately divide an image into several significant parts. Therefore, multilevel thresholding is an effective and famous technique, and it is extensively applied in many fields.

For years, a great number of thresholding techniques have been described in the literature. In 1979, Otsu’s method was presented by Otsu [7], which is one of the best ways of thresholding. However, the computation time is much longer in multilevel threshold problem because it exhaustively searches optimal threshold values by maximizing the between-class variance. In 1985, Tsai [8] proposed a new method to find optimal thresholding values of an input gray-level image by moment-preserving principle, which is called Tsallis entropy method. In 1986, Kittler and Illingworth [9] assumed that the pixel level values of each object in an image are normally distributed. Li and Lee [10] proposed minimum cross entropy thresholding method which selects the optimal thresholding values by minimizing the cross entropy between the original image and its segmented image. The Otsu, Tsallis entropy, and minimum cross entropy methods can be easily extended to multilevel thresholding. For bilevel thresholding, the traditional thresholding algorithms can find optimal threshold quickly and effectively and the image is accurately segmented into two parts. But, for multilevel thresholding, the traditional techniques become very time-consuming because a large number of iterations are needed for computing the optimal threshold values [11–13].

Swarm intelligence algorithms are very popular global optimization schemes, and the techniques imitate the collective behavior of natural or artificial systems that show some intelligence [14]. The algorithms have been widely used in complex optimization problems which show better performances. In order to solve the multilevel thresholding problem, swarm intelligence algorithms have been used to find optimal thresholds over the years, including genetic algorithm (GA) [15, 16], particle swarm optimization (PSO) [17–19], artificial bee colony (ABC) [20–22], differential evolution (DE) [23–25], firefly algorithm (FA) [26, 27], cuckoo search algorithm (CS) [28, 29], wind driven optimization (WDO) [29], and electromagnetism-like optimization (EMO) algorithm [30]. These methods use different basic swarm intelligence algorithms, different improved algorithms, and different objective functions for different types of images, such as between-class variance, Tsallis entropy, Kapur’s entropy, and minimum cross entropy. GA was inspired by the process of natural selection which has been used for multilevel thresholding [31, 32]. The PSO and the improved PSO algorithms were also used to solve the multilevel thresholding problem which mimics the social behavior of bird flock or fish school. The researchers Yin [17] and Maitra and Chatterjee [18] applied PSO algorithm to multilevel thresholding. Thereafter, particle swarm optimization (PSO) and artificial bee colony (ABC) have been adopted to search the optimal multilevel thresholds by Akay using Kapur’s entropy and between-class variance as objective functions [19]. Horng [20], Zhang and Wu [21], and Cuevas et al. [33] proposed a new image segmentation method using artificial bee colony algorithm for multilevel thresholding. And the algorithm is also used for segmentation of SAR image [34, 35] and satellite image [22]. In 2010, a novel multilevel thresholding segmentation method based on differential evolution (DE) algorithm is presented [23]. Firefly algorithm (FA), inspired by the social behavior of firefly swarm, is also used to find several threshold values on a given image [36]. Also, two new swarm intelligence algorithms, cuckoo search algorithm (CS) and wind driven optimization (WDO), using Kapur’s entropy for multilevel thresholding are proposed, and two algorithms can efficiently and accurately search multiple threshold values [29].

Color images include more information than gray images and multilevel color image segmentation is widely used now. Swarm intelligence algorithms are also used for color image multilevel segmentation. Zingaretti et al. [37] proposed the new method which is based on genetic algorithm (GA) for color image segmentation. Raja et al. [38] presented an improved particle swarm optimization (PSO) algorithm for cancer infected breast thermal images by Otsu’s method. Sarkar et al. [39] developed a novel multilevel color image thresholding method based on differential evolution (DE) algorithm and minimum cross entropy, and simulation results show that it is an effective method. Three different quantum inspired metaheuristic techniques, namely, Quantum Inspired Ant Colony Optimization, Quantum Inspired Differential Evolution, and Quantum Inspired Particle Swarm Optimization technique, for multilevel color image thresholding are presented. Simulations and results prove that the Quantum Inspired Ant Colony Optimization method outperforms the other methods [40]. Rajinikanth and Couceiro [41] used the firefly algorithm (FA) for color image segmentation. The evolutionary and swarm-based algorithms of evolution strategy (ES), genetic algorithm (GA), differential evolution (DE) algorithm, adaptive differential evolution algorithm (JADE), particle swarm optimization (PSO) algorithm, artificial bee colony (ABC) algorithm, cuckoo search (CS), and differential search (DS) algorithm are also used for multilevel color image thresholding problem [42].

The glowworm swarm optimization (GSO) is a novel swarm intelligence algorithm for optimization developed by Krishnanand and Ghose in 2005 [43] which mimics the flashing behavior of glowworms. In the algorithm, each glowworm carries a luminescence quantity called luciferin, which is decided by the function value of glowworm’s current location. During the course of movement, glowworm identifies its neighbors based on local-decision domain and selects a neighbor which has a luciferin value higher than its own using a probabilistic mechanism and moves toward it [44–49]. GSO algorithm has been applied for numerous complex optimization problems. Qifang et al. [50] and Horng [51] used GSO algorithm based on Otsu’s method and minimum cross entropy for multilevel threshold image segmentation and the experimental results show that the method has better performance for gray images. In order to improve the performance of the standard GSO algorithm and search the global optimal value efficiently and accurately, the improved glowworm swarm optimization (IGSO) is presented in this paper. Step size is an important parameter in determining the convergence of GSO algorithm, so a new update method of step size is proposed. Furthermore the sensor range is extended to the whole search space and the random movement of the brightest glowworms of firefly algorithm is also introduced. Subsequently the IGSO algorithm using different objective functions is used for multilevel color image thresholding problem, such as between-class variance and minimum cross entropy (MCE). The performance of IGSO algorithm for multilevel color image thresholding is measured in terms of the optimal threshold values, objective values, the peak signal to noise ratio (PSNR), and structural similarity index (SSIM) and then compared with other swarm intelligence algorithms such as adaptive particle swarm optimization (APSO) [52] and self-adaptive differential evolution (SaDE) algorithm [53].

The remainder of the paper is organized as follows. Section 2 presents the concepts of between-class variance method and minimum cross entropy method. Section 3 gives a detailed description of GSO algorithm and the proposed IGSO algorithm. In Section 4, the numerical experimental results of IGSO, GSO, APSO, and SaDE algorithms for multilevel color image segmentation are shown and discussions are also given. Finally, the conclusion is presented in Section 5.

2. Formulation of the Problem

Swarm intelligence algorithms find the optimal thresholds by maximizing an objective function. In this paper, two commonly used thresholding methods, between-class variance method (Otsu’s) and minimum cross entropy are used as objective functions to find the optimal multilevel thresholds.

2.1. Between-Class Variance Method (Otsu’s Method)

Thresholding based on Otsu’s method is a nonparametric segmentation method that divides the whole image into classes by maximizing the between-class variance.

Assume that an image has pixels and gray levels, and the number of pixels at level is represented by ; then . The occurrence probability of level is defined by

In bilevel thresholding, the optimum threshold divides the image into two classes, and the cumulative probabilities of each class can be described as follows:

The mean levels of two classes are described as follows:

The between-class variance of two classes is defined by (4):

where is the mean levels of whole image:

The optimum threshold is searched exhaustively by maximizing the between-class variance, and the optimal threshold is

Otsu’s method can be extended to multilevel thresholding. Assume that an image is divided into classes; the extended between-class variance of classes is calculated by

The sigma terms are determined using (9) and the mean levels are calculated by (10):

The optimum thresholds are searched by maximizing the between-class variance by

2.2. Minimum Cross Entropy Method

Assume that two probability distributions, and , belong to the same set. The cross entropy between and is defined as follows:

The concept of cross entropy is widely used for optimization problem, and the minimum cross entropy thresholding method selects an optimal threshold that minimizes the cross entropy between the original image and the processed image. If an original image in gray levels can be divided into two segments by threshold and , where   is the number of gray levels, then the cross entropy can be calculated by

where

We can select an optimal threshold by minimizing the cross entropy based on (13):

Since the first item is constant, the expression of the cross entropy can be modified as

where is the zero-moment and is the first-moment of the image histogram.

It is quite straightforward to extend minimum cross entropy thresholding method to multilevel thresholding segmentation. If an image is required to find thresholds , the cross entropy is given by

We can obtain the optimal threshold by minimizing (17). As the swarm intelligence algorithms are usually used to solve maximization problems, we modified (17) as shown below:

All algorithms used in this paper calculate the minimum cross entropy fitness function by (18) for multilevel thresholding segmentation, which can find the optimal thresholds.

3. Glowworm Swarm Optimization Algorithm

3.1. The Standard Glowworm Swarm Optimization Algorithm

The standard GSO algorithm includes the following steps.

Step 1 (parameters’ definition). The key parameters impact the performance of GSO algorithm, that is, , , , , and .

Step 2 (glowworms’ initialization). In the phase, the glowworms are initially distributed randomly in the given fitness function space so that they are well dispersed, which have equal quantity of luciferin and sensor range. Furthermore, the current iteration is set to 1.

Step 3 (luciferin update phase). The luciferin depends on the function value at the current position of the glowworm, so the position of glowworms changes and the luciferin updates accordingly in the each iteration. Each glowworm updates luciferin according to the following equation:

where is the luciferin of glowworm at time , is the luciferin decay constant , represents the luciferin enhancement constant, and is the function value.

Step 4 (movement phase). Each glowworm has a variable local-decision domain, which is bounded by a radial sensor range , and is attracted to brighter glowworms. In the movement phase, glowworms search a neighbor by a probabilistic mechanism that has higher luciferin value and move to it. For each glowworm , the probability equation of moving toward a neighbor can be stated as

where , is the set of neighbors of glowworm ,   is the variable local-decision domain, and represents the Euclidean distance between glowworms and at time . Then, the equation of the glowworm movements is given by

where represents the location of glowworms at time , is the step size, and is the Euclidean norm operator.

Step 5 (local-decision domain update). In the GSO algorithm, the local-decision domain is a dynamic value that is a function with the number of peaks captured. In order to update adaptively the local-decision domain range of each glowworm, the rule is stated as

where is a constant parameter and is a threshold parameter used to control the number of neighbors.

3.2. The Proposed Glowworm Swarm Optimization Algorithm

The size step is a key parameter, which affects the convergence of the GSO algorithm. In GSO algorithm, the step size should be smaller than the -distance in order to find the optimal solutions [54]. However, the lower step size value may lead to a slow convergence speed. So it can be a large value for starting the algorithm and gradually reducing the value of step size with the increase of iteration [54]. Also, the step size is relevant to the size of the search space. For the larger space, step size must be a larger value. In the standard GSO, is a constant. In order to improve the convergence of the GSO algorithm, we present a new step size update method in this section. Considering the search space and dimension of image segmentation, the expression is given by

where is the range of step size, is iteration, is the maximum number of iterations, and , where is the dimension of the search space. When the dimension is a large value and is a low value, the step size will correspondingly increase and algorithm can search more quickly.

The local-decision domain is an important parameter that affects the ability of capturing multiple peaks. When the sensor range of each glowworm extends to the whole search space, all the glowworms move to the global optimal value and ignore the local optimum [54]. To ensure that the GSO algorithm can find the global optimal threshold of image segmentation, the sensor range covers the entire space of image gray histogram and the value is set to in this paper.

In the GSO algorithm, the glowworms with the maximum luciferin remain stationary during the each iteration. The above feature leads to the movements of glowworms that are confined to the interior region of the convex hull [54]. The firefly algorithm can search multiple global solutions simultaneously, and thus it is a very efficient algorithm. However, in the FA algorithm, the brightest fireflies move randomly. So the random movement of the brightest agents is introduced to GSO in this paper.

The pseudocodes of the IGSO algorithm are shown as Pseudocode 1.

Set number of dimensions

Set number of glowworms

Let be the location of glowworm at time

Generate initial population of glowworms    randomly

for = 1 to do

    ;

set maximum iteration number = iter_max

set

While ( < iter_max) do

    for each glowworm do

    ;

     for each glowworm do

     ;

      for each glowworm do

      = select_glowworm

     ;

     for to do

         ;

     end

 ;

Pseudocode 1: The pseudocode of IGSO algorithm.

4. Results and Discussion

In this paper, the improved glowworm swarm optimization (IGSO) algorithm is applied to the multilevel color image thresholding problem. The RGB color model is a simple and effective model of color image which has three basic color components of red, green, and blue, so we should search the optimal threshold values and fitness values for the each component of the color images. The optimal fitness values () of color image is equal to the sum of the optimal fitness values of three components. The work of IGSO algorithm for multilevel color image thresholding problem in this paper is briefly illustrated in Figure 1.

Figure 1: The wok of IGSO algorithm for multilevel color image thresholding problem.

In this section, a large number of experiments are carried out on ten well-known color test images in order to test the performance of the IGSO algorithm for multilevel color image thresholding. Two simple segmentation methods, namely, between-class variance (Otsu) method and minimum cross entropy, are utilized as the fitness functions. All algorithms are implemented in MATLAB Release 2010. Ten well-known color test images and their histograms are shown in Figure 2. All the images are 512 × 512 in size. The population size is set to be 50 and the maximum number of iterations is 100 in all experiments in this paper. The IGSO algorithm is also compared with two efficient optimization algorithms of APSO and SaDE algorithms. Tables 1–3 give the value of important parameters used for IGSO, APSO, and SaDE algorithms, respectively. In order to estimate the quality of segmented image, the two parameters of peak signal to noise ratio (PSNR) and structural similarity (SSIM) index are used. The higher value of PSNR and SSIM shows a better quality of thresholding.

Table 1: The parameters used in the IGSO algorithm.

Parameters
Explanation
Value

Luciferin enhancement constant
0.6

Update rate of decision domain
0.08

Luciferin decay constant
0.4

Sensor range
255

Initialization range for the position of the particles
0, 255

Table 2: The parameters used in the APSO algorithm.

Parameters
Explanation
Value

Maximum of inertia weight
0.9

Minimum of inertia weight
0.1

Acceleration constants
2.0

Acceleration constants
2.0

Initialization range for the position of the particles
0, 255

Table 3: The parameters used in the SaDE algorithm.

Parameters
Explanation
Value

Differential weight
0.9

CR
Crossover probability
0.1

Initialization range for the position of the particles
0, 255

Figure 2: The ten test images: (a) airplane, (b) couple, (c) flower, (d) girl, (e) monarch, (f) pen, (g) pepper, (h) soccer, (i) test, and (j) yacht and corresponding histograms: (a′) airplane, (b′) couple, (c′) flower, (d′) girl, (e′) monarch, (f′) pen, (g′) pepper, (h′) soccer, (i′) test, and (j′) yacht.

The peak signal to noise ratio (PSNR) is one of the important performance criteria of image segmentation. The expression of PSNR is defined as follows:

where

where is the size of image, is the original image, and is the segmented image.

For a RGB color image, the PSNR value of three basic components is computed independently and the average values of them are considered as the PSNR value of color image.

The structural similarity (SSIM) index [55] is used to measure the similarity between the original image and the segmented image. The SSIM between two images can be stated as

where is the average of , is the average of , is the variance of , is the variance of , is the covariance of and , two variables and stabilize the division with weak denominator, and is the dynamic range of the pixel-values, and . In addition, the SSIM can be extended for color RGB images as shown below:

where and are the channel of the original image and segmented color image, respectively, and is channel number.

4.1. Experiment 1

In this section, the IGSO algorithm is used for ten color test images using Otsu’s method as fitness function and the results are compared with the GSO, APSO, and SaDE algorithms. Tables 4 and 5 present the number of thresholds , corresponding optimal thresholds, and objective values for , and computed by IGSO, APSO, and SaDE algorithms using Otsu’s method. In Table 6, comparison of PSNR (dB) and SSIM values is depicted for each method and number of thresholds, which reveals the quality of segmented images. The segmented images using IGSO algorithm based on Otsu’s method at 2, 4, 6, and 8 levels of thresholding are shown in Figures 3 and 4.

Table 4: Optimal thresholds and objective values obtained by IGSO and GSO algorithms using Otsu’s method.

IGSO
GSO

(×10³)

(×10³)

Airplane

2
120,177
94,166
124,181
5.15456
120,177
94,166
124,181
5.15456

4
83,126,169,199
66,116,164,201
112,150,180,203
5.48484
83,126,169,199
66,116,164,201
112,150,180,203
5.48484

6
68,99,129,162,
187,204
57,98,132,167,
193,209
102,134,156,176,
195,207
5.58695
68,100,130,162,187,204
55,98,133,167,
193,207
102,136,156,
176,196,209
5.58690

8
64,92,114,140,
163,184,198,209
47,80,107,132,159, 183,200,212
56,102,132,152,
170,188,201,211
5.63154
63,93,116,141,
166,186,199,210
47,81,108,134,161,184,202,214
57,104,134,154,
173,190,202,211
5.63124

Couple

2
35,83
30,81
27,76
2.64122
35,83
30,81
27,76
2.64122

4
17,42,71,109
16,39,74,122
17,39,72,117
2.91159
17,42,71,109
16,39,74,122
17,39,72,117
2.91159

6
13,32,52,73,97,131
13,31,51,81,123,
176
11,26,46,77,119,
177
2.99013
15,33,52,75,97,131
13,32,54,81,123,
176
13,27,46,77,119,
178
2.99009

8
12,28,45,62,81,
104,135,204
10,23,37,54,78,
104,138,200
9,22,38,56,80,
108,142,180
3.02731
12,29,43,61,78,
100,131,200
10,24,40,60,73,86,
106,135,201
9.20,34,55,74,102,
133,176
3.02709

Flower

2
101,190
53,129
73,151
10.35564
101,190
53,129
73,151
10.35564

4
42,92,154,209
44,84,132,188
32,68,108,163
11.25529
42,92,154,209
44,84,132,188
32,68,108,163
11.25529

6
41,76,119,160,194,
226
32,56,84,121,162,
206
26,55,86,114,152,
199
11.38993
41,76,118,159,
193,225
32,54,82, 121,163,206
26,55,83,114,
152,198
11.38990

8
35,53,78,109,142,
173,201,229
30,54,77,102,129,
157,189,222
20,38,55,76,96,
118,154,200
11.47485
34,53,76,109,142,
172,200,228
31,54,77,101,138,
162,193,220
24,40,59,72,91,
117,161,209
11.47454

Girl

2
69,124
34,88
31,81
4.34647
69,124
34,88
31,81
4.34647

4
37,69,106,155
25,58,92,129
24,52,81,117
4.84594
37,69,106,155
25,58,92,129
24,52,81,117
4.84594

6
35,62,88,112,140,
179
25,57,89,124,194,
208
23,44,67,94,124,
190
4.93170
34,62,89,114,140,
179
22,57,89,123,194,
208
23,48,67,94,124,190
4.93167

8
27,44,64,83,104,
125,151,187
16,34,52,72,92,
115,138,169
13,29,47,67,88,
110,134,163
5.02842
26,45,65,87,112,
130,160,192
12,31,50,67,88,
110,136,172
12,25,40,65,84,
103,131,167
5.02814

Monarch

2
114,176
79,142
62,122
5.19253
114,176
79,142
62,122
5.19253

4
89,127,162,198
63,90,125,175
55,86,122,164
5.63990
89,127,162,198
63,90,125,175
55,86,122,164
5.63990

6
77,105,129,149,
174,204
45,64,83,103,134,
180
40,60,85,113,144,
177
5.77439
79,105,129,150,
174,206
48,64,83,104,134,
181
40,60,85,113,
141,178
5.77434

8
63,85,109,130,148,
166,191,212
12,45,63,81,98,
119,149,186
38,54,70,86,108,
131,156,184
5.83061
68,90,103,123,145,160,187,208
21,46,63,81,98,
119,149,186
35,49,63,80,101,127,149,182
5.83002

Pen

2
76,151
60,138
63,152
9.77709
76,151
60,138
63,152
9.77709

4
51,84,124,178
44,82,128,183
44,85,125,184
10.58262
51,84,124,178
44,82,128,183
44,85,125,184
10.58262

6
44,69,92,120,157,
195
36,63,86,114,151,
192
32,61,88,115,153,
201
10.78635
44,69,93,120,156,
194
36,63,85,114,152,
192
32,61,88,117,
155,201
10.78632

8
39,59,77,96,120,
149,179,206
29,50,70,89,110,
137,169,199
29,53,74,93,116,
142,177,212
10.86604
36,55,74,93,114,
141,171,202
28,48,67,85,106,
145,176,199
27,49,70,90,110,135,169,209
10.86570

Pepper

2
97,159
78,157
58,127
8.61756
97,159
78,157
58,127
8.61756

4
81,124,156,186
30,82,140,185
25,60,95,146
9.28037
81,124,156,186
30,82,140,185
25,60,95,146
9.28037

6
62,96,121,143,167,
191
20,46,81,125,163,
193
23,50,73,96,129,
172
9.45358
64,97,122,143,168,191
20,49,81,126,163,
194
20,50,73,96,129,171
9.45354

8
53,85,107,127,
145,164,183,199
17,42,69,100,
132,159,182,201
11,32,50,69,87,
108,140,177
9.53018
54,85,108,128,
145,165,184,200
17,40,64,92,122,
153,177,199
11,31,50,69,87,
108,140,179
9.52962

Soccer

2
86,149
71,138
55,126
7.27688
86,149
71,138
55,126
7.27688

4
57,103,144,180
41,83,120,166
38,77,113,166
7.96150
57,103,144,180
41,83,120,166
38,77,113,166
7.96150

6
44,75,109,137,160,
192
30,57,86,112,135,
175
27,56,82,105,133,
177
8.15225
44,75,108,136,
159,192
32,58,87,113,136,
176
27,55,82,105,
133,179
8.15218

8
40,63,86,113,135,
153,171,198
26,48,73,97,114,
133,158,192
23,46,68,86,105,
127,154,192
8.22619
36,58,79,110,
129,147,168,195
24,43,69,95,111,
129,151,186
22,45,67,86,104,126,155,192
8.22581

Test

2
89,148
86,144
82,150
7.43581
89,148
86,144
82,150
7.43581

4
61,92,127,171
57,90,126,165
55,88,127,172
7.94367
61,92,127,171
57,90,126,165
55,88,127,172
7.94367

6
52,78,100,127,161,
193
46,72,95,124,157,
185
40,62,86,114,147,
181
8.10926
51,77,100,127,
160,194
45,71,95,124,158,
185
40,62,86,1154,147,182
8.10922

8
45,68,87,104,129,
157,184,207
43,67,88,107,130,
158,180,198
33,52,69,90,115,
145,176,197
8.17834
40,63,78,99,124,
147,175,204
36,59,85,102,126,
153,172,196
35,51,67,86,106,
138,173,189
8.17792

Yacht

2
71,141
64,132
74,148
9.16339
71,141
64,132
74,148
9.16339

4
56,97,136,176
50,86,125,166
45,85,120,163
9.83344
56,97,136,176
50,86,125,166
45,85,120,163
9.83344

6
46,78,105,133,160,
185
48,81,111,141,
168,189
28,55,86,115,144,
176
9.98335
45,78,105,132,
159,185
48,80,109,140,
168,189
29,55,86,115,
143,176
9.98330

8
37,62,84,107,132,
155,173,190
29,53,74,92,110,
133,155,178
25,49,74,97,118,
139,162,185
10.00658
32,59,75,105,130, 146,162,185
29,51,74,92,110,
133,156,179
24,48,70,91,112,132,154,181
10.00622

Table 5: Optimal thresholds and objective values obtained by APSO and SaDE algorithms using Otsu’s method.

APSO
SaDE

(×10³)

(×10³)

Airplane

2
120,177
94,166
124,181
5.15456
120,177
94,166
124,181
5.15456

4
83,126,169,199
66,116,164,201
112,150,180,203
5.48484
83,126,169,199
66,116,164,201
112,150,180,203
5.48484

6
71,102,130,164,190,204
59,99,134,169,
194,211
104,136,157,179,
196,207
5.58678
69,101,131,163,
188,204
57,100,133,167,
192,210
102,135,158,177,
196,203
5.58683

8
63,90,110,134,160,
180,197,209
45,79,105,129,
154,180,199,207
52,97,125,150,
168,186,200,210
5.63107
64,91,111,136,
161,182,197,209
46,79,105,130,
156,181,198,208
53,99,127,150,
169,188,201,210
5.63113

Couple

2
35,83
30,81
27,76
2.64122
35,83
30,81
27,76
2.64122

4
17,42,71,109
16,39,74,122
17,39,72,117
2.91159
17,42,71,109
16,39,74,122
17,39,72,117
2.91159

6
16,34,52,76,99,136
15,32,56,84,126,
179
13,32,48,78,122,
178
2.98998
14,33,53,75,98,
133
14,33,55,82,127,
177
14,27,46,79,121,
179
2.99003

8
13,30,48,67,87,
110,142,205
10,24,40,56,82,
113,149,206
10,24,41,64,90,
122,146,187
3.02679
13,30,48,66,85,
109,139,204
10,23,39,55,83,
109,143,202
11,23,40,62,88,
119,147,185
3.02687

Flower

2
101,190
53,129
73,151
10.35564
101,190
53,129
73,151
10.35564

4
42,92,154,209
44,84,132,188
32,68,108,163
11.25529
42,92,154,209
44,84,132,188
32,68,108,163
11.25529

6
42,77,123,163,196,
224
35,59,90,122,165,
208
26,57,86,116,
156,197
11.38983
41,74,117,159,191,225
34,56,86,123,159,
203
27,55,84,112,
150,200
11.38986

8
34,54,79,115,149,
176,203,230
32,55,81,110,139,
167,192,225
24,44,63,81,100,
121,155,202
11.47416
35,54,78,112,146,
176,202,230
31,54,78,106,137,
158,192,226
22,43,60,81,100,
120,153,201
11.47424

Girl

2
69,124
34,88
31,81
4.34647
69,124
34,88
31,81
4.34647

4
37,69,106,155
25,58,92,129
24,52,81,117
4.84594
37,69,106,155
25,58,92,129
24,52,81,117
4.84594

6
37,60,85,108,138,
178
28,59,93,125,196,
210
27,45,69,96,127,
195
4.93159
35,65,89,113,141,
179
24,59,90,127,196,
209
26,46,69,95,126,
192
4.93162

8
29,47,67,89,110,
131,155,190
17,36,58,81,102,
123,130,164
16,34,51,73,92,
116,139,167
5.02802
27,46,66,87,107,
127,156,192
17,35,55,74,95,
119,139,175
15,32,50,71,91,
114,137,165
5.02796

Monarch

2
114,176
79,142
62,122
5.19253
114,176
79,142
62,122
5.19253

4
89,127,162,198
63,90,125,175
55,86,122,164
5.63990
89,127,162,198
63,90,125,175
55,86,122,164
5.63990

6
78,109,133,151,
176,205
46,66,84,105,137,
183
42,61,87,116,
146,180
5.77425
79,104,130,153,
175,205
46,62,82,101,132,
179
40,62,87,114,
145,178
5.77428

8
62,82,104,123,144,
162,189,217
13,49,67,88,100,
123,158,192
38,55,70,90,116,
141,166,186
5.82961
62,84,105,125,145
,164,187,210
17,46,68,84,101,
121,153,191
39,56,73,93,105
,135,160,184
5.82970

Pen

2
76,151
60,138
63,152
9.77709
76,151
60,138
63,152
9.77709

4
51,84,124,178
44,82,128,183
44,85,125,184
10.58262
51,84,124,178
44,82,128,183
44,85,125,184
10.58262

6
46,72,94,121,158,
196
38,66,89,116,153,
193
35,62,91,117,
152,199
10.78625
44,71,91,122,158,197
37,64,87,116,153,192
33,63,88,116,
153,199
10.78627

8
42,64,82,104,126,
158,185,209
30,53,74,95,120,
151,177,202
29,55,79,100,
122,151,183,215
10.86558
40,62,81,101,126,
154,184,208
30,50,71,92,116,
145,174,201
30,55,78,98,
122,148,182,214
10.86549

Pepper

2
97,159
78,157
58,127
8.61756
97,159
78,157
58,127
8.61756

4
81,124,156,186
30,82,140,185
25,60,95,146
9.28037
81,124,156,186
30,82,140,185
25,60,95,146
9.28037

6
61,91,120,141,168,
191
22,47,84,127,166,
194
22,48,69,94,128,
171
9.45348
59,93,121,142,
167,191
24,48,81,126,164,193
20,49,72,96,130,
172
9.45350

8
49,77,102,122,141,162,183,198
19,45,71,107,138,
165,185,203
13,32,54,74,89,
114,147,184
9.52933
53,79,104,124,
143,161,179,198
20,42,71,105,137,
162,187,202
12,35,51,71,90,
111,145,183
9.52941

Soccer

2
86,149
71,138
55,126
7.27688
86,149
71,138
55,126
7.27688

4
57,103,144,180
41,83,120,166
38,77,113,166
7.96150
57,103,144,180
41,83,120,166
38,77,113,166
7.96150

6
44,78,110,139,158,
193
32,60,87,112,136,
175
30,59,84,106,
133,178
8.15211
45,76,111,137,
159,192
31,59,88,114,136,
176
28,58,84,107,
134,178
8.15213

8
46,68,90,119,137,
158,176,199
28,51,75,102,118,
136,163,199
27,51,70,91,110,
134,162,202
8.22550
40,65,92,118,138,
155,174,200
27,50,76,98,116,
136,165,195
25,48,71,89,
109,133,163,198
8.22559

Test

2
89,148
86,144
82,150
7.43581
89,148
86,144
82,150
7.43581

4
61,92,127,171
57,90,126,165
55,88,127,172
7.94367
61,92,127,171
57,90,126,165
55,88,127,172
7.94367

6
51,79,102,129,160,
194
48,73,96,127,158,
185
39,60,83,114,
145,180
8.10924
51,76,99,126,159,
193
45,71,95,123,159,
186
41,63,86,113,
149,182
8.10927

8
48,70,91,110,137,
164,185,210
41,72,89,112,132,
165,186,203
39,56,77,98,120,
154,176,202
8.17762
47,71,89,109,133,
161,187,209
43,69,91,109,133,
163,184,201
37,57,75,96,121
,150,177,197
8.17770

Yacht

2
71,141
64,132
74,148
9.16339
71,141
64,132
74,148
9.16339

4
56,97,136,176
50,86,125,166
45,85,120,163
9.83344
56,97,136,176
50,86,125,166
45,85,120,163
9.83344

6
46,76,103,132,160,
183
49,84,114,141,
169,189
29,57,85,114,
142,176
9.98324
45,77,105,132,
159,185
47,80,110,139,
168,190
27,56,86,116,
145,175
9.98326

8
42,65,90,112,134,
163.181,198
26,45,66,83,101,
126,151,175
25,49,70,95,109,
142,165,189
10.00592
41,66,87,111,135,
160,178,195
25,49,68,86,103,
128,150,177
27,51,758,98,12
0,141,166,188
10.00601

Table 6: Comparison of optimal PSNR (dB) and SSIM values obtained by IGSO, GSO, APSO, and SaDE algorithms using Otsu’s method.

Image

IGSO
GSO
APSO
SaDE

PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM

Airplane
2
10.0144
0.7740
10.0144
0.7740
10.0144
0.7740
10.0144
0.7740

4
10.0188
0.8246
10.0188
0.8246
10.0188
0.8246
10.0188
0.8246

6
10.0300
0.8575
10.0263
0.8398
10.0211
0.8346
10.0243
0.8369

8
10.3589
0.9134
10.3524
0.9029
10.3439
0.8958
10.3481
0.9016

Couple
2
27.0382
0.8025
27.0382
0.8025
27.0382
0.8025
27.0382
0.8025

4
27.0470
0.8830
27.0470
0.8830
27.0470
0.8830
27.0470
0.8830

6
27.0579
0.9143
27.0518
0.9093
27.0492
0.8976
27.0507
0.9036

8
27.0602
0.9427
27.0559
0.9362
27.0517
0.9294
27.0526
0.9316

Flower
2
19.3175
0.8460
19.3175
0.8460
19.3175
0.8460
19.3175
0.8460

4
19.3280
0.9040
19.3280
0.9040
19.3280
0.9040
19.3280
0.9040

6
19.3296
0.9372
19.3278
0.9340
19.3283
0.9207
19.3289
0.9222

8
20.3152
0.9604
20.2937
0.9549
20.2859
0.9508
20.2924
0.9526

Girl
2
24.0610
0.7112
24.0610
0.7112
24.0610
0.7112
24.0610
0.7112

4
24.0734
0.8011
24.0734
0.8011
24.0734
0.8011
24.0734
0.8011

6
24.0798
0.8584
24.0766
0.8546
24.0739
0.8512
24.0748
0.8527

8
24.0824
0.9094
24.0802
0.8982
24.0767
0.8927
24.0789
0.8936

Monarch
2
19.4727
0.7658
19.4727
0.7658
19.4727
0.7658
19.4727
0.7658

4
19.4767
0.8589
19.4767
0.8589
19.4767
0.8589
19.4767
0.8589

6
19.4785
0.9155
19.4778
0.9116
19.4771
0.9059
19.4776
0.9081

8
19.4941
0.9454
19.4879
0.9387
19.4811
0.9255
19.4918
0.9312

Pen
2
19.8912
0.8420
19.8912
0.8420
19.8912
0.8420
19.8912
0.8420

4
19.8935
0.8791
19.8935
0.8791
19.8935
0.8791
19.8935
0.8791

6
19.8983
0.9330
19.8961
0.9238
19.8938
0.9170
19.8947
0.9215

8
19.9014
0.9629
19.9001
0.9581
19.8979
0.9543
19.8990
0.9566

Pepper
2
18.1771
0.8046
18.1771
0.8046
18.1771
0.8046
18.1771
0.8046

4
18.1789
0.8384
18.1789
0.8384
18.1789
0.8384
18.1789
0.8384

6
18.1856
0.9004
18.1825
0.8976
18.1796
0.8880
18.1811
0.8916

8
18.1891
0.9476
18.1836
0.9302
18.1801
0.9164
18.1822
0.9251

Soccer
2
20.3329
0.8088
20.3329
0.8088
20.3329
0.8088
20.3329
0.8088

4
20.3368
0.8570
20.3368
0.8570
20.3368
0.8570
20.3368
0.8570

6
20.3466
0.9023
20.3421
0.8931
20.3389
0.8822
20.3405
0.8886

8
20.3474
0.9529
20.3437
0.9493
20.3410
0.9330
20.3422
0.9375

Test
2
18.5023
0.7611
18.5023
0.7611
18.5023
0.7611
18.5023
0.7611

4
18.5081
0.8508
18.5081
0.8508
18.5081
0.8508
18.5081
0.8508

6
18.5106
0.9017
18.5091
0.8959
18.5086
0.8883
18.5089
0.8913

8
18.5144
0.9446
18.5120
0.9274
18.5095
0.9122
18.5107
0.9208

Yacht
2
18.5073
0.8133
18.5073
0.8133
18.5073
0.8133
18.5073
0.8133

4
18.5083
0.8503
18.5083
0.8503
18.5083
0.8503
18.5083
0.8503

6
18.5090
0.9035
18.5086
0.8910
18.5082
0.8865
18.5085
0.8894

8
18.5164
0.9542
18.5122
0.9389
18.5097
0.9120
18.5106
0.9274

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Figure 3: For = 2, 4, 6, and 8, images (a)–(d) for airplane, (e)–(h) for couple, (i)–(l) for flower, (m)–(p) for girl, and (q)–(t) for monarch, using IGSO algorithm based on Otsu.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Figure 4: For = 2, 4, 6, and 8, images (a)–(d) for pen, (e)–(h) for pepper, (i)–(l) for soccer, (m)–(p) for test, and (q)–(t) for yacht, using IGSO algorithm based on Otsu.

4.2. Experiment 2

In the second part of the experiments, we find optimal threshold values by maximizing the modified equation (18). The number of thresholds, the optimal thresholds, and the optimal objective values for IGSO, GSO, APSO, and SaDE algorithms are listed in Tables 7 and 8. The parameter values of PSNR (dB) and SSIM value are shown in Table 9. Figures 5 and 6 give the multilevel thresholding segmented results at 2, 4, 6, and 8 levels of thresholding using IGSO algorithm based on minimum cross entropy for ten test color images, respectively.

Table 7: Optimal thresholds and objective values obtained by IGSO and GSO algorithms using MCE.

IGSO
GSO

(×10²)

(×10²)

Airplane

2
109,168
72,153
120,180
28.75021
109,168
72,153
120,180
28.75021

4
73,117,162,197
53,100,146,191
109,147,178,203
28.7628
73,117,162,197
53,100,146,191
109,147,178,203
28.7628

6
65,95,124,155,183,
204
28,63,100,136,
177,205
57,108,144,169,
191,207
28.76704
64,95,124,155,183,
203
29,63,100,136,
176,205
56,107,143,169,
190,207
28.76703

8
59,85,105,127,152,
176,194,207
24,53,82,108,137,
168,193,210
55,96,121,146,163,
181,198,208
28.76901
59,85,105,128,153, 176,194,208
24,53,81,108,
136,167,192,209
52,95,121,145,
163,181,198,209
28.76899

Couple

2
26,78
21,74
22,71
3.97893
26,78
21,74
22,71
3.97893

4
12,30,57,93
8,22,42,81
8,21,40,78
4.02494
12,30,57,93
8,22,42,81
8,21,40,78
4.02494

6
10,29,52,82,126,209
9,22,40,68,111,232
7,21,39,72,137,192
4.02990
10,30,53,84,134,
210
7,21,39,71,115,
235
8,21,40,74,145,
196
4.02989

8
6,15,28,44,62,85,
118,178
4,11,20,35,52,78,
120,232
6,13,26,41,69,
109,176,218
4.03718
6,15,28,45,64,87,
120,183
5,11,21,35,52,80,
122,223
5,12,22,39,67,
108,173,217
4.03715

Flower

2
41,119
44,111
53,114
12.34354
41,119
44,111
53,114
12.34354

4
39,79,135,199
27,52,93,159
23,56,98,157
12.39484
39,79,135,199
27,52,93,159
23,56,98,157
12.39484

6
26,47,75,118,166,
213
22,40,59,85,129,
184
6,34,63,99,142,
189
12.40942
26,48,75,118,167,
213
22,40,60,86,128,
184
8,33,62,99,141,
190
12.40941

8
20,32,46,59,83,
125,171,215
21,38,57,79,105,
134,171,210
7,28,45,66,89,115,
150,197
12.41612
23,35,48,62,84,127,172,216
21,39,53,76,102,
133,171,212
6,27,47,67,90,
115,153,203
12.41609

Girl

2
41,90
22,72
26,80
7.65874
41,90
22,72
26,80
7.65874

4
33,65,102,149
11,31,62,109
11,28,56,101
7.70152
33,65,102,149
11,31,62,109
11,28,56,101
7.70152

6
23,43,66,96,131,
185
10,29,56,86,122,
181
12,28,55,96,145,
167
7.70996
22,43,66,98,132,
186
10,30,58,87,123,
182
11,27,55,95,145,
167
7.70995

8
17,34,48,65,85,
107,130,169
5,14,27,40,59,84,
113,149
16,47,53,132,146,
171,207,242
7.71557
19,38,50,67,87,
110,133,168
7,16,30,42,62,88,
114,156
13,45,52,131,147,
168,205,239
7.71555

Monarch

2
108,172
75,135
58,113
14.98290
108,172
75,135
58,113
14.98290

4
79,115,147,186
61,86,119,169
38,61,91,141
15.00562
79,115,147,186
61,86,119,169
38,61,91,141
15.00562

6
62,85,110,137,165,
200
44,63,81,100,130,
177
36,50,66,88,119,
162
15.01305
62,85,111,137,165,
198
45,63,81,102,130,
177
35,50,67,89,121,163
15.01304

8
59,79,97,115,134,
151,175,204
36,49,64,80,96,
117,146,185
33,45,57,71,88,112,
140,173
15.01603
62,83,100,117,
137,153,178,206
33,47,61,78,95,
115,143,182
35,49,60,71,86,
111,141,174
15.01600

Pen

2
61,133
48,125
49,136
13.20604
61,133
48,125
49,136
13.20604

4
45,76,112,169
34,66,101,157
29,63,102,166
13.25261
45,76,112,169
34,66,101,157
29,63,102,166
13.25261

6
35,56,79,106,143,
187
26,47,70,95,133,
183
22,44,69,95,128,
182
13.26480
34,56,79,106,142,
188
26,47,70,95,134,
183
22,43,69,95,128,
183
13.26479

8
30,46,63,80,100,
127,161,199
22,38,56,73,92,
117,151,191
16,29,48,69,90,
113,148,198
13.27007
30,47,67,85,103,
131,162,201
24,41,60,78,93,
121,155,204
17,33,52,70,93,
115,151,200
13.27003

Pepper

2
87,153
22,104
21,74
16.40133
87,153
22,104
21,74
16.40133

4
62,102,137,174
10,39,85,156
15,52,83,133
16.45113
62,102,137,174
10,39,85,156
15,52,83,133
16.45113

6
51,84,111,135,162,
189
8,33,63,102,149,
188
7,29,54,80,116,170
16.46417
52,82,111,135,162,
188
8,33,62,102,148,
187
6,28,55,81,117,
172
16.46416

8
44,69,94,115,133,
153,174,194
6,23,42,65,97,134,
167,195
6,26,47,67,89,115,
151,189
16.46928
44,70,95,116,134,
154,175,195
6,27,45,68,94,
120,170,197
6,23,42,67,93,
118,153,192
16.46926

Soccer

2
64,126
42,97
42,113
13.11371
64,126
42,97
42,113
13.11371

4
43,79,123,165
30,63,105,155
25,60,99,150
13.15363
43,79,123,165
30,63,105,155
25,60,99,150
13.15363

6
37,61,92,126,155,
188
22,41,67,96,124,
167
20,45,75,103,140,
191
13.16536
38,62,92,127,155,
188
22,42,67,96,126,
166
20,46,75,105,142,
194
13.16535

8
32,48,68,93,120,
143,163,193
21,36,55,77,99,
118,138,175
13,24,41,62,84,
103,128,171
13.17071
32,48,68,94,121,
144,164,194
21,37,59,80,101,
120,139,178
16,30,43,60,79,
103,125,174
13.17068

Test

2
80,137
83,140
67,132
16.22951
80,137
83,140
67,132
16.22951

4
55,88,122,167
40,70,97,143
39,65,100,155
16.25759
55,88,122,167
40,70,97,143
39,65,100,155
16.25759

6
43,70,93,121,156,
190
37,64,90,116,149,
182
29,48,68,91,125,
169
16.26698
43,70,94,120,155,
190
38,66,91,118,150,
182
31,49,67,92,126,
169
16.26697

8
36,58,74,92,108,
131,162,194
31,53,70,88,107,
129,159,186
27,45,61,79,99,
125,154,183
16.27108
37,59,76,93,110,
134,163,196
31,55,72,92,111,
134,163,187
27,43,58,74,95,
121,152,181
16.27105

Yacht

2
60,133
55,125
53,127
15.87338
60,133
55,125
53,127
15.87338

4
44,80,119,166
28,56,92,139
25,55,99,154
15.91058
44,80,119,166
28,56,92,139
25,55,99,154
15.91058

6
30,53,80,108,140,
177
24,48,73,96,127,
168
21,40,65,92,123,
162
15.92188
30,52,80,107,140,
177
25,47,73,96,127,
167
20,40,65,91,121,
162
15.92187

8
25,41,62,84,107,
133,159,185
21,36,55,76,95,
118,146,174
16,29,48,70,93,
118,144,175
15.92650
25,45,63,84,108,
130,154,187
21,39,58,82,96,
119,147,174
16,30,49,73,95,
122,144,177
15.92647

Table 8: Optimal thresholds and objective values obtained by APSO and SaDE algorithms using MCE.

APSO
SaDE

(×10²)

(×10²)

Airplane

2
109,168
72,153
120,180
28.75021
109,168
72,153
120,180
28.75021

4
73,117,162,197
53,100,146,191
109,147,178,203
28.7628
73,117,162,197
53,100,146,191
109,147,178,203
28.7628

6
63,95,123,156,185,
205
28,65,102,137,177,
208
54,109,144,169,
193,209
28.76701
64,95,123,155,
183,203
29,64,100,136,176,
203
58,107,143,169,
190,206
28.76702

8
58,82,101,123,147,
172,192,206
26,57,87,111,139,
171,194,209
48,94,120,144,162,
180,197,208
28.76895
59,84,103,126,
149,174,193,205
24,53,85,110,139,
170,194,210
53,95,119,145,
161,180,198,209
28.76897

Couple

2
26,78
21,74
22,71
3.97893
26,78
21,74
22,71
3.97893

4
12,30,57,93
8,22,42,81
8,21,40,78
4.02494
12,30,57,93
8,22,42,81
8,21,40,78
4.02494

6
10,30,56,85,127,
207
11,24,42,71,114,
235
10,22,42,76,138,
194
4.02987
9,29,54,83,127,
211
8,21,40,71,110,
234
8,20,42,75,137,
193
4.02988

8
7,20,31,46,63,89,
120,178
8,15,24,29,50,81,
125,234
8,6,14,29,3941,73,
106,175,214
4.03710
8,19,28,44,64,87,
116,180
7,12,24,41,55,82,
123,228
8,16,29,44,66,
105,174,221
4.03712

Flower

2
41,119
44,111
53,114
12.34354
41,119
44,111
53,114
12.34354

4
39,79,135,199
27,52,93,159
23,56,98,157
12.39484
39,79,135,199
27,52,93,159
23,56,98,157
12.39484

6
28,48,78,119,168,
214
25,41,59,87,131,
186
8,35,64,106,146,
191
12.40939
27,47,77,119,167,
213
22,41,61,87,128,
184
6,34,63,99,146,
192
12.40940

8
15,30,45,57,81,123,
179,218
25,39,58,82,106,
136,173,216
9,31,44,68,87,108,147,198
12.41604
18,29,44,58,82,
124,171,216
22,41,58,77,105,
136,166,205
6,27,45,65,91,
118,154,199
12.41606

Girl

2
41,90
22,72
26,80
7.65874
41,90
22,72
26,80
7.65874

4
33,65,102,149
11,31,62,109
11,28,56,101
7.70152
33,65,102,149
11,31,62,109
11,28,56,101
7.70152

6
22,46,67,100,134,
186
10,33,58,90,124,
183
14,26,57,100,146,
168
7.70993
23,44,68,98,132,
185
10,29,59,89,122,
182
11,26,55,95,144,
166
7.70994

8
21,39,55,67,88,103,
135,172
6,19,35,43,56,87,
110,154
12,44,50,134,141,
167,204,237
7.71551
18,36,52,71,90,
110,132,167
8,22,30,42,61,85,
117,153
11,45,51,128,
152,173,209,239
7.71553

Monarch

2
108,172
75,135
58,113
14.98290
108,172
75,135
58,113
14.98290

4
79,115,147,186
61,86,119,169
38,61,91,141
15.00562
79,115,147,186
61,86,119,169
38,61,91,141
15.00562

6
64,86,112,141,166,202
42,65,82,101,133,179
34,47,69,92,123,
164
15.01302
61,85,112,139,
165,199
44,63,83,102,130,
175
38,50,68,89,119,
163
15.01303

8
55,73,92,111,136,
154,177,207
39,54,66,86,99,120,
148,190
36,49,63,75,92,116,
146,175
15.01597
61,77,93,110,132,
148,176,200
33,53,62,83,97,
124,143,182
39,47,59,74,90,
118,143,170
15.01599

Pen

2
61,133
48,125
49,136
13.20604
61,133
48,125
49,136
13.20604

4
45,76,112,169
34,66,101,157
29,63,102,166
13.25261
45,76,112,169
34,66,101,157
29,63,102,166
13.25261

6
38,57,82,109,144,
189
28,48,73,98,136,
184
21,42,66,98,124,
180
13.26477
35,55,77,104,141,
187
26,50,71,96,134,
184
22,43,72,97,128,
183
13.26478

8
33,50,64,88,106,
135,168,201
26,41,59,75,102,
120,152,196
19,30,56,73,94,
119,151,200
13.27000
31,48,67,85,105,
132,165,202
24,42,60,78,97,
123,155,192
17,32,54,76,96,
115,154,201
13.27001

Pepper

2
87,153
22,104
21,74
16.40133
87,153
22,104
21,74
16.40133

4
62,102,137,174
10,39,85,156
15,52,83,133
16.45113
62,102,137,174
10,39,85,156
15,52,83,133
16.45113

6
49,82,108,133,164,
191
11,34,62,100,152,
185
6,26,55,84,115,
166
16.46414
51,82,113,135,
163,190
9,35,65,104,149,
189
8,30,56,81,116,
172
16.46415

8
48,72,99,118,136,
157,178,197
9,25,44,69,101,126,
170,201
10,28,45,64,84,110,
160,189,196
16.46921
44,73,98,113,130,
156,177,198
5,22,45,68,94,
132,161,194
6,25,42,65,83,
120,170,192
16.46923

Soccer

2
64,126
42,97
42,113
13.11371
64,126
42,97
42,113
13.11371

4
43,79,123,165
30,63,105,155
25,60,99,150
13.15363
43,79,123,165
30,63,105,155
25,60,99,150
13.15363

6
39,62,94,127,160,
191
22,45,72,97,125,
169
21,48,79,104,141,
193
13.16533
38,62,91,125,156,
187
23,43,67,96,126,
168
22,46,77,104,140,
192
13.16534

8
35,55,71,96,124,
145,166,198
19,38,59,79,106,
121,145,177
16,30,50,65,87,
105,133,173
13.17064
33,49,72,95,125,
146,165,196
21,39,60,82,103,
123,142,176
15,29,44,67,86,
106,123,169
13.17066

Test

2
80,137
83,140
67,132
16.22951
80,137
83,140
67,132
16.22951

4
55,88,122,167
40,70,97,143
39,65,100,155
16.25759
55,88,122,167
40,70,97,143
39,65,100,155
16.25759

6
41,70,90,124,158,
190
37,65,95,119,152,
185
33,50,69,92,126,
169
16.26695
43,70,95,123,157,
192
38,66,90,116,151,
182
30,51,68,91,127,
170
16.26696

8
38,64,78,98,110,
134,164,198
37,60,72,90,104,
127,155,188
30,46,63,83,106,
132,159,184
16.27101
39,62,77,95,113,
128,165,199
35,56,75,91,112,
133,162,190
28,48,63,82,104,
128,157,185
16.27103

Yacht

2
60,133
55,125
53,127
15.87338
60,133
55,125
53,127
15.87338

4
44,80,119,166
28,56,92,139
25,55,99,154
15.91058
44,80,119,166
28,56,92,139
25,55,99,154
15.91058

6
31,51,83,112,141,
179
26,51,77,97,128,
170
21,43,68,94,126,
164
15.92185
30,52,82,108,141,
178
25,48,74,95,126,
167
20,41,64,90,122,
162
15.92186

8
30,43,66,95,114,
135,162,187
23,42,63,83,102,
122,148,178
22,31,52,78,99,
125,150,180
15.92641
28,48,70,89,110,
137,162,186
22,40,53,80,98,
121,145,171
18,34,53,74,92,
124,148,179
15.92644

Table 9: Comparison of optimal PSNR (dB) and SSIM values obtained by IGSO, GSO, APSO, and SaDE algorithms using MCE.

Image

IGSO
GSO
APSO
SaDE

PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM

Airplane
2
10.0178
0.7896
10.0178
0.7896
10.0178
0.7896
10.0178
0.7896

4
10.0216
0.8394
10.0216
0.8394
10.0216
0.8394
10.0216
0.8394

6
10.0381
0.8794
10.0311
0.8708
10.0232
0.8617
10.0258
0.8659

8
10.0641
0.9308
10.0592
0.9248
10.0516
0.9101
10.0548
0.9159

Couple
2
27.0372
0.8270
27.0372
0.8270
27.0372
0.8270
27.0372
0.8270

4
27.0478
0.8860
27.0478
0.8860
27.0478
0.8860
27.0478
0.8860

6
27.0568
0.9128
25.0526
0.9032
27.0499
0.8929
27.0512
0.8973

8
27.0613
0.9460
25.0588
0.9343
27.0523
0.9290
27.0562
0.9311

Flower
2
19.3352
0.8696
19.3352
0.8696
19.3352
0.8696
19.3352
0.8696

4
19.3369
0.9037
19.3369
0.9037
19.3369
0.9037
19.3369
0.9037

6
19.3385
0.9385
19.3376
0.9243
19.3370
0.9159
19.3373
0.9213

8
19.3439
0.9679
19. 3417
0.9538
19.3389
0.9337
19.3402
0.9446

Girl
2
24.0684
0.7274
24.0684
0.7274
24.0684
0.7274
24.0684
0.7274

4
24.0725
0.8369
24.0725
0.8369
24.0725
0.8369
24.0725
0.8369

6
24.0761
0.8740
24.0749
0.8649
27.0732
0.8568
24.0740
0.8614

8
24.0844
0.9192
24.0824
0.9069
24.0771
0.8903
24.0792
0.8971

Monarch
2
19.4791
0.7605
19.4791
0.7605
19.4791
0.7605
19.4791
0.7605

4
19.4851
0.8648
19.4851
0.8648
19.4851
0.8648
19.4851
0.8648

6
19.4867
0.9208
19.4862
0.9121
19.4850
0.9032
19.4855
0.9085

8
19.4939
0.9524
19.4902
0.9408
19,4879
0.9264
19.4893
0.9336

Pen
2
19.8970
0.8281
19.8970
0.8281
19.8970
0.8281
19.8970
0.8281

4
19.8992
0.8878
19.8992
0.8878
19.8992
0.8878
19.8992
0.8878

6
19.8996
0.9317
19.8993
0.9244
19.8987
0.9117
19.8991
0.9184

8
19.9052
0.9637
19.9021
0.9546
19.8994
0.9396
19.9010
0.9441

Pepper
2
18.1823
0.8189
18.1823
0.8189
18.1823
0.8189
18.1823
0.8189

4
18.1849
0.8460
18.1849
0.8460
18.1849
0.8460
18.1849
0.8460

6
18.1867
0.9021
18.1860
0.8967
18.1848
0.8815
18.1853
0.8913

8
18.1897
0.9483
18.1883
0.9377
18.1860
0.9239
18.1875
0.9314

Soccer
2
20.3415
0.8016
20.3415
0.8016
20.3415
0.8016
20.3415
0.8016

4
20.3378
0.8512
20.3378
0.8512
20.3378
0.8512
20.3378
0.8512

6
20.3471
0.9071
20.3442
0.8992
20.3496
0.8869
20.3411
0.8983

8
20.3499
0.9560
20.3475
0.9477
20.3420
0.9298
20.3448
0.9329

Test
2
18.5027
0.7658
18.5027
0.7658
18.5027
0.7658
18.5027
0.7658

4
18.5155
0.8560
18.5155
0.8560
18.5155
0.8560
18.5155
0.8560

6
18.5157
0.9021
18.5154
0.8938
18.5153
0.8802
18.5154
0.8859

8
18.5168
0.9533
18.5161
0.9486
18.5156
0.9372
18.5158
0.9400

Yacht
2
18.5135
0.8195
18.5135
0.8195
18.5135
0.8195
18.5135
0.8195

4
18.5172
0.8611
18.5172
0.8611
18.5172
0.8611
18.5172
0.8611

6
18.5188
0.9150
18.5179
0.9021
18.5174
0.8851
18.5177
0.8960

8
18.5203
0.9608
18.5186
0.9454
18.5178
0.9287
18.5182
0.9342

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Figure 5: For =2, 4, 6, and 8, images (a)–(d) for airplane, (e)–(h) for couple, (i)–(l) for flower, (m)–(p) for girl, and (q)–(t) for monarch, using IGSO algorithm based on minimum cross entropy.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Figure 6: For = 2, 4, 6, and 8, images (a)–(d) for pen, (e)–(h) for pepper, (i)–(l) for soccer, (m)–(p) for test, and (q)–(t) for yacht, using IGSO algorithm based on minimum cross entropy.

4.3. Comparison of the Segmented Performance

In this paper, the results of IGSO algorithm based on Otsu and MCE have been compared with basic GSO algorithm and two other well-known optimization algorithms. For the Otsu fitness function, it can be clearly seen from Tables 4 and 5 that the improved GSO algorithm offers superior optimal thresholds and objective values in comparison with GSO, APSO, and SaDE algorithms when the number of thresholds is more than 4. From Tables 7 and 8, it is easy to deduce that the optimal thresholds and objective values of multilevel color image thresholding segmentation using minimum cross entropy for all the test images are better than GSO, APSO, and SaDE algorithms.

The parameters of peak signal to noise ratio (PSNR) and structural similarity (SSIM) index are introduced to evaluate the quality of segmented images. Tables 6 and 9 show that PSNR and SSIM values obtained with IGSO algorithm are higher than those produced by the other algorithms. The PSNR and SSIM values obtained by IGSO and GSO algorithms using Otsu’s and MCE methods are presented in Table 10. For the same color test image and number of thresholds , the IGSO algorithm based on the minimum cross entropy method has higher PSNR and SSIM values than Otsu’s method. After the analysis of results, it is found that IGSO algorithm based on the minimum cross entropy has better quality of segmented image than the other algorithms for each color test image.

Table 10: Comparison of optimal PSNR (dB) and SSIM values obtained by IGSO and GSO algorithms using OTSU and MCE.

Image

PSNR (Otsu)
PSNR (MCE)
SSIM (Otsu)
SSIM (MCE)

IGSO
GSO
IGSO
GSO
IGSO
GSO
IGSO
GSO

Airplane
2
10.0144
10.0144
10.0178
10.0178
0.7740
0.7740
0.7896
0.7896

4
10.0188
10.0188
10.0216
10.0216
0.8246
0.8246
0.8394
0.8394

6
10.0300
10.0263
10.0381
10.0311
0.8575
0.8398
0.8794
0.8708

8
10.3589
10.3524
10.3641
10.0592
0.9134
0.9029
0.9308
0.9248

Couple
2
27.0382
27.0382
27.0372
27.0372
0.8025
0.8025
0.8270
0.8270

4
27.0470
27.0470
27.0478
27.0478
0.8830
0.8830
0.8860
0.8860

6
27.0579
27.0518
27.0568
25.0526
0.9143
0.9093
0.9128
0.9032

8
27.0602
27.0559
27.0613
25.0588
0.9427
0.9362
0.9460
0.9343

Flower
2
19.3175
19.3175
19.3352
19.3352
0.8460
0.8460
0.8696
0.8696

4
19.3280
19.3280
19.3369
19.3369
0.9040
0.9040
0.9037
0.9037

6
19.3286
19.3278
19.3385
19.3376
0.9372
0.9340
0.9385
0.9243

8
20.3152
20.2937
20.3439
19. 3417
0.9604
0.9549
0.9679
0.9538

Girl
2
24.0610
24.0610
24.0684
24.0684
0.7112
0.7112
0.7274
0.7274

4
24.0734
24.0734
24.0725
24.0725
0.8011
0.8011
0.8369
0.8369

6
24.0758
24.0766
24.0761
24.0749
0.8584
0.8546
0.8740
0.8649

8
24.0824
24.0802
24.0844
24.0824
0.9094
0.8982
0.9192
0.9069

Monarch
2
19.4727
19.4727
19.4791
19.4791
0.7658
0.7658
0.7605
0.7605

4
19.4767
19.4767
19.4851
19.4851
0.8589
0.8589
0.8648
0.8648

6
19.4785
19.4778
19.4867
19.4862
0.9155
0.9116
0.9208
0.9121

8
19.4941
19.4879
19.4939
19.4902
0.9454
0.9387
0.9524
0.9408

Pen
2
19.8912
19.8912
19.8970
19.8970
0.8420
0.8420
0.8281
0.8281

4
19.8935
19.8935
19.8992
19.8992
0.8791
0.8791
0.8878
0.8878

6
19.8983
19.8961
19.8994
19.8993
0.9330
0.9238
0.9317
0.9244

8
19.9014
19.9001
19.9052
19.9021
0.9629
0.9581
0.9637
0.9546

Pepper
2
18.1771
18.1771
18.1823
18.1823
0.8046
0.8046
0.8189
0.8189

4
18.1789
18.1789
18.1849
18.1849
0.8384
0.8384
0.8460
0.8460

6
18.1856
18.1825
18.1857
18.1860
0.9004
0.8976
0.9021
0.8967

8
18.1891
18.1836
18.1897
18.1883
0.9476
0.9302
0.9483
0.9377

Soccer
2
20.3329
20.3329
20.3415
20.3415
0.8088
0.8088
0.8016
0.8016

4
20.3368
20.3368
20.3378
20.3378
0.8570
0.8570
0.8512
0.8512

6
20.3466
20.3421
20.3471
20.3442
0.9023
0.8931
0.9071
0.8992

8
20.3474
20.3437
20.3499
20.3475
0.9529
0.9493
0.9560
0.9477

Test
2
18.5023
18.5023
18.5027
18.5027
0.7611
0.7611
0.7658
0.7658

4
18.5081
18.5081
18.5155
18.5155
0.8508
0.8508
0.8560
0.8560

6
18.5106
18.5091
18.5157
18.5154
0.9017
0.8959
0.9021
0.8938

8
18.5144
18.5120
18.5168
18.5161
0.9446
0.9274
0.9533
0.9486

Yacht
2
18.5073
18.5073
18.5135
18.5135
0.8133
0.8133
0.8195
0.8195

4
18.5083
18.5083
18.5172
18.5172
0.8503
0.8503
0.8611
0.8611

6
18.5090
18.5086
18.5175
18.5179
0.9035
0.8910
0.9150
0.9021

8
18.5164
18.5122
18.5203
18.5186
0.9542
0.9389
0.9608
0.9454

Moreover, the convergence curves for IGSO, GSO, APSO, and SaDE algorithms of ten test color images using Otsu’s method have been shown in Figure 7 for 8-level thresholding. It can be clearly observed from these figures that the IGSO algorithm needs less iterations to obtain optimal segmentation results and has better convergence property than GSO, APSO, and SaDE algorithms for multilevel color image thresholding segmentation.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 7: Convergence curves of IGSO, GSO, APSO, and SaDE algorithms for using Otsu’s method.

5. Conclusions

In this paper, a new multilevel color image thresholding segmentation method based on improved glowworm swarm optimization (IGSO) algorithm is presented. The presented method utilized Otsu’s method and minimum cross entropy as the fitness functions criteria and is tested on ten color test images for 2, 4, 6, and 8. The results obtained by IGSO algorithm are compared with those obtained by GSO, APSO, and SaDE algorithms and the segmentation performance of these methods was evaluated in terms of the optimal threshold values, optimal objective values, PSNR values, and SSIM values. Finally, the convergence curves using minimum cross entropy are drawn of ten test color images for . The comparative results show that the improved GSO algorithm is superior to the GSO, APSO, and SaDE algorithms for multilevel color image thresholding problem. Furthermore, the quality of segmented image of the IGSO algorithm using minimum cross entropy outperforms the IGSO algorithm using Otsu’s method in terms of PSNR and SSIM. The further work will include introduction of the other fitness criteria for multilevel thresholding segmentation. In addition, the IGSO algorithm is also applied to the other complex problems.

Competing Interests

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

Acknowledgments

This work was supported by the National Nature Science Foundation of China (no. 51204077) and the Nature Science Foundation of Kunming University of Science and Technology (no. 2014-9-x-8).

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