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Article

A Chaotic Electromagnetic Field Optimization Algorithm Based on Fuzzy Entropy for Multilevel Thresholding Color Image Segmentation

College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(4), 398; https://doi.org/10.3390/e21040398
Submission received: 1 April 2019 / Revised: 11 April 2019 / Accepted: 12 April 2019 / Published: 15 April 2019
(This article belongs to the Special Issue Entropy in Image Analysis)

Abstract

:
Multilevel thresholding segmentation of color images is an important technology in various applications which has received more attention in recent years. The process of determining the optimal threshold values in the case of traditional methods is time-consuming. In order to mitigate the above problem, meta-heuristic algorithms have been employed in this field for searching the optima during the past few years. In this paper, an effective technique of Electromagnetic Field Optimization (EFO) algorithm based on a fuzzy entropy criterion is proposed, and in addition, a novel chaotic strategy is embedded into EFO to develop a new algorithm named CEFO. To evaluate the robustness of the proposed algorithm, other competitive algorithms such as Artificial Bee Colony (ABC), Bat Algorithm (BA), Wind Driven Optimization (WDO), and Bird Swarm Algorithm (BSA) are compared using fuzzy entropy as the fitness function. Furthermore, the proposed segmentation method is also compared with the most widely used approaches of Otsu’s variance and Kapur’s entropy to verify its segmentation accuracy and efficiency. Experiments are conducted on ten Berkeley benchmark images and the simulation results are presented in terms of peak signal to noise ratio (PSNR), mean structural similarity (MSSIM), feature similarity (FSIM), and computational time (CPU Time) at different threshold levels of 4, 6, 8, and 10 for each test image. A series of experiments can significantly demonstrate the superior performance of the proposed technique, which can deal with multilevel thresholding color image segmentation excellently.

1. Introduction

Image segmentation is an important technology in image processing, which is a frontier research direction in computer vision, as well as one of the key preprocessing steps in image analysis [1,2]. It has been widely adopted in medicine, agriculture, industrial production, and various other fields. Image segmentation can be defined as the procedure of dividing an image into different regions [3]. In the subsequent research, the relevant regions can be extracted from the segmented image expediently according to specific requirements. Nowadays, the common image segmentation methods include threshold-based, cluster-based, edge-based methods and so on. Thresholding is extensively applied due to its simplicity, efficiency, and robustness. Depending on the number of thresholds, it can be classified as bi-level segmentation and multilevel segmentation [4]. Bi-level thresholding techniques use one threshold to partition an image into two segments; whereas multilevel segmentation determines several thresholds to separate an image into more than two classes. Many thresholding approaches have been proposed by scholars around the world in the past few years, Otsu’s (between-class variance criterion) [5,6] technique pushes the thresholding segmentation to an upsurge and inspires the scholars constantly in this field. Then diverse entropy-based criteria have emerged in the thresholding segmentation study, such as maximum entropy (Kapur’s) [7], minimum cross entropy [8], fuzzy entropy [9], etc.
Gray-scale image thresholding technology is relatively popular and mature. Compared with the segmentation of gray-scale images, color image segmentation plays a more beneficial role in practical applications, which separates an image into several disjoint and homogenous components based on the information of texture, color or histogram [10]. Color image segmentation is more complex and challenging than gray-scale images. Nevertheless, considering that color images contain more characteristics and they are closer to human visual effects [11], the research of color image segmentation is more meaningful. There will appear some problems when a traditional segmentation method is adopted to segment a color image, for example, the computation is massive and accuracy of segmented images cannot be guaranteed [12,13]. In this paper, fuzzy entropy is one of the research objects with high segmentation accuracy. In the fuzzy entropy thresholding technique, each threshold needs to be determined by three fuzzy parameters. Hence the calculation of thresholds is more accurate, at the same time the process is more complicated and the running time of the program will be longer. With the improvement of the threshold level, the computation of fuzzy entropy will exponentially increase for searching the optimal thresholds and the efficiency of segmentation will gradually decrease [14,15,16]. In order to enhance the practicability of fuzzy entropy thresholding technique, this paper combines fuzzy entropy thresholding with intelligent optimization algorithms to improve the performance with respect to accuracy and efficiency.
Meta-heuristic algorithms are utilized to obtain the optimal solution of the problem [17]. Generally, they are inspired by nature and try to handle the problems from mimicking ethology, biology or physics [18]. For instance, Bird Swarm Algorithm (BSA) [19], Firefly Algorithm (FA) [20], and Flower Pollination Algorithm (FPA) [21] are inspired by ethology or biology; Electromagnetic Optimization (EMO) [22], Wind Driven Optimization (WDO) [23], and Gravitational Search Algorithm (GSA) [24] are inspired by physics. At present, a number of scholars have coupled the optimization algorithms with the field of image segmentation in the literature. For instance, Sowjanya et al. [25] combined a WDO algorithm with Otsu’s method for the segmentation of brain MRI images, it has shown the superior performance in the experiment results. Wasim et al. [26] proposed an improved Bee Algorithm (BA) for multilevel image segmentation, whereby they embedded Levy fight into a Bees Algorithm (the Levy Bees Algorithm, LBA), and the results show that LBA is more stable than BA in this field. Rakoth et al. [27] tried to combine Dragonfly Optimization with Self-Adaptive weight (SADFO) and used SADFO for image segmentation experiments with satisfying results. These references confirm the feasibility of applying optimization algorithms to image thresholding segmentation. However, the above experiments all concentrate on gray-scale images and do not extend the experiments to the analysis of color images. Applying meta-heuristic algorithms to the field of multilevel image segmentation can enhance the convergence speed and efficiency [28]. Therefore, in this paper, Electromagnetic Field Optimization algorithm (EFO) [29] is modified and combined with fuzzy entropy thresholding method to eliminate the complex computation, which is used into the multilevel color image segmentation field for searching the best threshold values.
Electromagnetic Field Optimization is a new meta-heuristic algorithm inspired by the electromagnetic theory developed in physics. EFO algorithm has been applied in several applications, for example, Behnam et al. [30] created a method using EFO for hiding sensitive rules simultaneously, which has fewer lost rules than other well-known algorithms. Bouchekara et al. [31] proposed the optimal coordination of directional overcurrent relays based on EFO, and the results show that EFO is better than other optimization algorithms such as Particle Swarm Optimization (PSO) [32], or the Differential Evolution (DE) algorithm [33], etc. This paper embeds a new chaos strategy into standard EFO algorithm according to the specific problem of color image segmentation named as Chaotic Electromagnetic Field Optimization (CEFO). Employing the CEFO algorithm to optimize the fuzzy parameters which determine the optimal thresholds of an image in fuzzy entropy. To the best of our knowledge, this topic has not been investigated yet. The rest of this paper is organized as follows: in Section 2, the concept of EFO algorithm is elaborated. In Section 3, the chaotic strategy in CEFO algorithm is introduced and explained. In Section 4, the problem definitions and formulas of the Otsu’s, Kapur’s entropy, and the fuzzy entropy are illustrated. In Section 5, the experimental environment is reported. In Section 6, the experimental results and discussions are provided and analyzed. Finally, a brief conclusion of this paper and future works are drawn in Section 7.

2. Electromagnetic Field Optimization

Electromagnetic Field Optimization is a novel meta-heuristic intelligent algorithm proposed by Hosein in 2016 [29]. In contrast to the swarm-based meta-heuristic algorithms widely inspired by biology, the EFO algorithm is based on the electromagnetic field principle used in physics. In the EFO algorithm, due to the forces of attraction and repulsion in the electromagnetic field, the electromagnetic particle (EMP) keeps away from the worst solution and moves towards the best solution. In the end, all the electromagnetic particles (EMPs) gather around the optimal solution.
A magnetic field is generated around the electrified iron core, which is made of an electromagnet. An electromagnet has only one polarity and it is contingent on the direction of the electric current. Hence, an electromagnet has two characteristics of attraction or repulsion, electromagnets with the different polarity attract each other, and those with identical polarity repel each other. The intensity of attraction is 5-10% higher than repulsion and the ratio between attraction and repulsion is set as golden ratio [29,31], which can promote the algorithm to explore the optimal solution effectively in the search space. The essence of the optimization problem is to find the pole (maximum or minimum) about the objective function and the corresponding fitness in the prescriptive range [34]. Each potential solution of the problem is represented with an electromagnetic particle composed of a group of electromagnets. The electromagnetic field comprises several electromagnetic particles and it can be defined as a space in 1-D (dimension), 2-D, 3-D, or hyperdimensional space [35]. The number of electromagnets of an electromagnetic particle corresponds to variables of the optimization problem, as well as the dimension of the electromagnetic space. Moreover, all electromagnets of one electromagnetic particle have the same polarity. Therefore, an electromagnetic particle has the same polarity with its electromagnets. The set of electromagnetic particles can be considered in a matrix as:
E M P s = [ P 1 , 1 P 1 , 2 P 1 , d P 2 , 1 P 2 , 2 P 2 , d P n , 1 P n , 2 P n , d ]
where n is the number of electromagnetic particles and j is the number of variables (dimension).
The mechanism of the EFO algorithm can be described as follows:
Step 1:
A certain number of electromagnetic particles are generated randomly in the electromagnetic field, and the fitness of each electromagnetic particle is evaluated by the objective function. Then the electromagnetic particles are sorted on the basis of their fitness.
Step 2:
The electromagnetic field is divided into three regions: positive, negative and neutral. Then all electromagnetic particles are classified into these three groups. The first group consists of the best particles with positive polarity. The second group consists of the worst particles with negative polarity. The third group consists of neutral particles which have a little negative polarity almost near zero. And all electromagnetic particles are located in the corresponding electromagnetic regions.
Step 3:
In each iteration of the algorithm, a new electromagnetic particle ( E M P N e w ) is generated. If the fitness of E M P N e w is better than the original worst particle, the E M P N e w will remain and its fitness and polarity will depend on the list of fitness, furthermore, the original worst particle will be eliminated. If else, the will be eliminated directly. This process continues until the algorithm reaches the maximum number of iterations.
The core of the EFO is the method of generating E M P N e w in each iteration, and each electromagnet in E M P N e w is shaped separately. The main process can be described as follows: three electromagnetic particles are randomly extracted from three electromagnetic regions (one EMP from each region), and then three electromagnets are randomly extracted from three electromagnetic particles obtained just now (one electromagnet from each EMP). Consequently, there are three electromagnets with different polarities. The neutral electromagnet is attracted and repelled by positive and negative electromagnets. Owing to the intensity of attraction is stronger than repulsion and the neutral electromagnet has a slight negative polarity, the neutral electromagnet moves a distance away from the negative electromagnet and approaches towards the positive electromagnet. In other words, each electromagnet in E M P N e w is a result of interaction between attraction and repulsion, which is shown in Figure 1.
Figure 1 shows the process of generating E M P N e w , in this figure, each electromagnetic particle contains three electromagnets for example, and positive, neutral and negative electromagnets are colored as green, blue and red respectively. In accordance with the above mechanism, three electromagnets of E M P N e w is selected from nine original electromagnets, which increases randomness and enhances the strength of the optimization algorithm. Establishing a mathematical model to describe the update mechanism of E M P N e w as below:
D j P j K j = E M P j P j E M P j K j
D j N j K j = E M P j N j E M P j K j
E M P j N e w = E M P j K j + [ ( φ r ) D j P j K j ] ( r D j N j K j )
where j is the number of electromagnets in EMP; E M P j P j is the positive electromagnet; E M P j N j is the negative electromagnet; E M P j K j is the neutral electromagnet; D j P j K j is the distance between positive and neutral electromagnets. D j N j K j is the distance between negative and neutral electromagnets; r is the random value between 0 and 1; φ is the golden ratio of ( 5 + 1 ) / 2 .
In order to preserve the diversity of particles in the electromagnetic field and reduce the probability of falling into local optima [36], randomness is an indispensable part in EFO algorithm. Therefore, the probability of P s _ r a t e about the new position is determined by the selected electromagnet from a positive field, which accelerates the convergence rate and improves the accuracy of the optimum. Additionally, the probability of R _ r a t e is used to replace one electromagnet in E M P N e w with randomly generated electromagnet within the space. The most important feature of EFO algorithm is the high degree of cooperation among particles. Another pivotal characteristic is high randomization, which avoids obtaining the local optimum. Meanwhile, the application of the golden ratio makes EFO more efficient. All of the above strategies lead EFO to a robust optimization algorithm.

3. Proposed Algorithm

One of the essential points in the EFO algorithm is the degree of chaos about the electromagnetic particles in the electromagnetic field; if the degree of chaos is higher, the search power will be stronger. In the literature, the initial position of electromagnetic particles is processed by a chaotic strategy, which disturbs the distribution of particles and increases the unpredictability of the system.
Chaotic phenomena refer to the external complex behavior in a non-linear deterministic system due to the inherent randomness [37]. Almost all meta-heuristic algorithms need to be initialized randomly, and usually it is achieved by using probability distribution, which can advantageous to replace such randomness with chaotic map [38]. Owing to the dynamic behavior of chaos, chaotic maps have been commonly acknowledged in the field of optimization, which can promote algorithms in exploring optima more effective globally in the search space. Table 1 lists some common chaotic maps, which are expressed by mathematical equations.
For instance, logistic chaos is widely used because of its simple expression and good performance, and it is shown in Figure 2. As can be seen, the logistic system has missed certain values. In consideration of the multilevel color image segmentation problem, this paper proposes a new chaotic map as follows:
x n + 1 = r a n d ( ) × sin ( 2 π x n ) + x n
where r a n d ( ) is the random value between 0 and 1.
The new chaotic map is shown in Figure 3 and its distribution is more symmetrical than Logistic chaotic map. Taking advantage of this chaos strategy in EFO, the total performance of the algorithm will be improved and it is known as CEFO. The pseudo-code of the CEFO algorithm is presented in Algorithm 1.
Algorithm 1. Pseudo-code of CEFO algorithm
/* Part 1: Algorithm parameters initialization */
N _ v a r : The number of electromagnets in each electromagnetic particle.
N _ e m p : The number of electromagnetic particles in population.
P s _ r a t e : The probability of changing one electromagnet with a random electromagnet.
R _ r a t e : The probability of selecting electromagnets from the positive field.
P _ f i e l d : The portion of particles belonging to positive.
N _ f i e l d : The portion of particles belonging to negative.
 min = lower boundary; max = upper boundary
/* Part 2: Main loop of the algorithm */
for i = 1 to N _ e m p do
   for j = 1 to N _ v a r do
    position [i, j] = min + rand ( ) (max − min)
   end for
end for
Update position by using the chaotic map of Equation (5)
 fitness = function (position)
while t (current iteration) < max iterations
   Divide the electromagnetic field into three regions
   for i = 1 to N _ v a r do
     if rand (0,1) > P s _ r a t e
      Generate the E M P N e w by Equation (4)
     else
      Generate the E M P N e w from positive particles
     end if
     Check if any particle beyond the search space
   end for
   if rand (0,1) < R _ r a t e
     Change one electromagnet of E M P N e w randomly
   end if
   Compare the fitness of E M P N e w with worst particle
   t = t + 1
end while
 Output the best particle

4. Thresholding Segmentation Methods

The process of multilevel thresholding color image segmentation is to find more than two optimal thresholds to segment three components (red, green, and blue) respectively. In RGB images, each color component consists of P pixels and L number of gray levels. The obtained thresholds are within the range of [0, L 1 ], L is considered as 256 and each gray-level is associated with the histogram representing the frequency of its gray level pixel used by g ( x , y ) .

4.1. Between-Class Variance Thresholding

Between-class variance (Otsu’s) [5] thresholding method can be defined as follows:
Assuming that n 1 thresholds form the threshold vector T = [ t 1 , t 2 , , t n 1 ] to split an image into n classes:
{ C 1 = { ( x , y ) | 0 g ( x , y ) t 1 1 } C 2 = { ( x , y ) | t 1 g ( x , y ) t 2 1 } C n = { ( x , y ) | t n 1 g ( x , y ) L 1 }
Constructing image histogram { f 0 , f 1 , , f L 1 } , where f i is the frequency of gray-level i . Then, the probability of gray-level i can be represented as:
p i = f i i = 0 L 1 f i ,   i = 0 L 1 p i = 1
For every class C k , the cumulative probability ω k and average gray level μ k in every region can be defined as:
ω k = i C k p i ,   μ k = i C k i p i ω k
and Otsu’s function can be expressed as:
σ B 2 = k = 0 K ω k ( μ k μ T ) 2 ,   μ T = i = 0 L 1 i p i
where μ k is the average gray intensity of the image.
Therefore, the optimal threshold vector is as follows:
T * = arg max ( σ B 2 )

4.2. Kapur’s Entropy Thresholding

Kapur’s entropy method maximizes the entropy value of the segmented histogram such that each separated region has more centralized distribution [39]. Extending Kapur’s entropy for multilevel image segmentation problem:
H 1 = i = 0 t 1 1 p i ω 1 ln p i ω 1 ,     ω 1 = i = 0 t 1 1 p i H 2 = i = t 1 t 2 1 p i ω 2 ln p i ω 2 ,     ω 2 = i = t 1 t 2 1 p i H j = i = t j 1 t j 1 p i ω j ln p i ω j ,     ω j = i = t j 1 t j 1 p i H n = i = t n L 1 p i ω n ln p i ω n ,     ω n = i = t n L 1 p i
where H j represents the entropy value of j-th region in the image.
There are n thresholds which can be configured as the n dimensional optimization problem. And the optimal threshold vector is obtained analogously by:
T * = arg max ( i = 0 m H i )

4.3. Fuzzy Entropy Thresholding

In the fuzzy entropy technique, let an original image be D = { ( i , j ) | i = 0 , , M 1 ; j = 0 , , N 1 } , where M and N represent the width and height of an image. Supposed that t 1 and t 2 are two thresholds to divide the original image into 3 parts named as E d , E m , E b [10]. E d consists of pixels of low gray levels; E m is made of pixels with middle gray levels; E b is composed of pixels of high gray levels. Usually, using (13) to calculate the image histogram:
h k = n k M N
where k = 0 , 1 , , 255 ; n k is the number of the k-th pixel in D k ; h k is the histogram of the image at gray-level k , k = 0 255 h k = 1 .
Consider Π 3 = { E d , E m , E b } as an unknown probabilistic partition of D , whose probability distribution can be expressed as:
p d = P ( E d ) ;             p m = P ( E m ) ;           p b = P ( E b )
For each ( i , j ) D , let:
D d = { ( i , j ) | 0 g ( i , j ) t 1 } , D m = { ( i , j ) | t 1 g ( i , j ) t 2 } , D b = { ( i , j ) | t 2 g ( i , j ) L 1 }
Utilizing μ d , μ m , μ b as the membership functions of E d , E m , E b , which is shown in Figure 4 [40]. There are six fuzzy parameters of u 1 , v 1 , w 1 , u 2 , v 2 , w 2 in the membership functions, in other words, t 1 and t 2 are determined by these six parameters. According to the above statement, we can have the probability distribution of three regions expressed as:
p d = k = 0 255 p k p d | k = k = 0 255 p k μ d ( k ) p m = k = 0 255 p k p m | k = k = 0 255 p k μ m ( k ) p b = k = 0 255 p k p b | k = k = 0 255 p k μ b ( k )
where p d | k , p m | k , p b | k are the conditional probability of a pixel partitioned into three classes. Moreover, a pixel of k in an image satisfies the constraint of p d | k + p m | k + p b | k = 1 .
The three membership functions have been shown in Figure 4. And these mathematical formulas are defined as follows:
μ d ( k ) = { 1 k u 1 1 ( k u 1 ) 2 ( w 1 u 1 ) ( v 1 u 1 ) u 1 k v 1 ( k w 1 ) 2 ( w 1 u 1 ) ( w 1 v 1 ) v 1 k w 1 0 k w 1
μ m ( k ) = { 0 k u 1 ( k u 1 ) 2 ( w 1 u 1 ) ( v 1 u 1 ) u 1 k v 1 1 ( k w 1 ) 2 ( w 1 u 1 ) ( w 1 v 1 ) v 1 k w 1 1 w 1 k u 2
μ b ( k ) = { 0 k u 2 ( k u 2 ) 2 ( w 2 u 2 ) ( v 2 u 2 ) u 2 k v 2 1 ( k w 2 ) 2 ( w 2 u 2 ) ( w 21 v 2 ) v 2 k w 2 1 k w 2
where u 1 , v 1 , w 1 , u 2 , v 2 , w 2 should meet the condition of 0 u 1 < v 1 < w 1 < u 2 < v 2 < w 2 255 .
Then, the fuzzy entropy of each part is as follows:
H d = k = 0 255 p k μ d ( k ) p d ln ( p k μ d ( k ) p d ) H m = k = 0 255 p k μ m ( k ) p m ln ( p k μ m ( k ) p m ) H b = k = 0 255 p k μ b ( k ) p b ln ( p k μ b ( k ) p b )
The whole fuzzy entropy function is defined as:
H ( u 1 , v 1 , w 1 , u 2 , v 2 , w 2 ) = H d + H m + H b
Equation (21) is determined by six variables which are called fuzzy parameters. Seeking the optimal group of u 1 , v 1 , w 1 , u 2 , v 2 , w 2 when (21) reach the maximum value. Therefore, the most applicable threshold can be calculated as:
μ d ( t 1 ) = μ m ( t 1 ) = 0.5 μ m ( t 2 ) = μ b ( t 2 ) = 0.5
As is shown in Figure 4, according to the above equation, t 1 and t 2 can be defined by (17)–(19), and the result is as follows:
t 1 = { u 1 + ( w 1 u 1 ) ( v 1 u 1 ) / 2 ( u 1 + w 1 ) / 2 < v 1 < w 1 w 1 ( w 1 u 1 ) ( w 1 v 1 ) / 2 u 1 < v 1 < ( u 1 + w 1 ) / 2 t 2 = { u 2 + ( w 2 u 2 ) ( v 2 u 2 ) / 2 ( u 2 + w 2 ) / 2 < v 2 < w 2 w 2 ( w 1 u 2 ) ( w 2 v 2 ) / 2 u 2 < v 2 < ( u 2 + w 2 ) / 2
Fuzzy entropy thresholding can meet the requirement from single threshold segmentation to multiple thresholds segmentation, and the optimal threshold vector obtained is more precise. However, each threshold should be determined by three parameters in fuzzy entropy thresholding, and thresholds need to be defined by 3n fuzzy parameters [41,42].
With the increase of threshold level gradually, the degree of the computation will be significantly risen, which diminish the speed of the process and the practicability will be reduced. In order to improve the convergence efficiency, it is necessary to use the optimization algorithm for searching the optimal threshold vector. This paper takes advantage of the CEFO to ensure the segmentation accuracy and greatly decrease the execution time. The general flow of fuzzy entropy thresholding based on the CEFO algorithm is presented in Figure 5.

5. Experimental Environment

In order to verify the superiority of the CEFO algorithm in dealing with the multilevel color image segmentation problem, this section will introduce the description of our benchmark images and then select several other algorithms for comparison. The parameters of each algorithm will be described firstly and a series of quality metrics used to evaluate the quality of segmented images will be calculated at the end.

5.1. Benchmark Images

In this experiment, ten images are chosen from the Berkeley segmentation data set, which is shown in Figure 6. It has presented the histogram of three components about every color image. Among these images, Test 1–3 are animal images; Test 4 and 5 are about human; Test 7 and 8 are landmark buildings; Test 6 and 9 are images related to landscape architecture; Test 10 is the normal scenery image.

5.2. Experimental Settings

When applied to solve the problem of multilevel color image segmentation, different meta-heuristic algorithms have different optimization performances due to their strategies and mathematical formulations [43]. Therefore, it is essential to compare the CEFO algorithm with other different algorithms such as EFO, ABC [44], BA [10], BSA [19], WDO [23]. Among these algorithms, ABC, BA, and BSA are proposed from biology; EFO and WDO are inspired from physics. The number of maximum iterations of each algorithm is set to 500, and the initial population is set to 15, with a total of 30 runs per algorithm, other specific parameters are presented in Table 2.
All the algorithms are programmed in Matlab R2016a (The Mathworks Inc., Natick, MA, USA) and implemented on a Windows 7 – 64 bit with 8 GB RAM environment.

5.3. Segmented Image Quality Metrics

To evaluate the quality of segmented images under different algorithms at selected threshold levels, four metrics are selected as follows [45,46]:
  • Peak Signal to Noise Ratio (PSNR)
The index is used to measure the difference between the original image and the segmented image, and a higher value is gained when the segmented image has a better effect. It can be defined as:
P S N R ( x , y ) = 20 log 10 ( 255 M S E ) M S E = 1 M N i = 0 M 1 j = 0 N 1 x ( i , j ) y ( i , j ) 2
where M and N represent the size of the image; x is the original image; y is the segmented image.
  • Mean Structural Similarity (MSSIM)
The index evaluates the overall image quality, which is in the range of [ 1 , 1 ] . The higher value of MSSIM is obtained when it represents the segmented image is more similar to the original image. The MSSIM is the average of every component and SSIM can be calculated as:
S S I M ( x , y ) = ( 2 μ x μ y + c 1 )   ( 2 σ x y + c 2 ) ( μ x 2 + μ y 2 + c 1 )   ( σ x 2 + σ y 2 + c 2 )
  • Feature similarity (FSIM)
The index is in the range of [ 0 , 1 ] , and the segmented image is better when the value is closer to 1. The FSIM can be expressed as:
F S I M ( x , y ) = x Ω S L ( X ) P C m ( x ) x Ω P C m ( x )
  • Computation Time (CPU Time)
The index measures the convergence rate of each algorithm. The algorithm is more efficient when the time is shorter.

6. Results and Discussions

6.1. Comparison of Other Meta-Heuristic Algorithms

Utilizing 6 algorithms based on fuzzy entropy criterion to conduct the experiment on 10 images at the threshold level of 4, 6, 8, and 10 ( K = 4, 6, 8, 10). The results of the optimal threshold vector are presented in Table 3, Table 4 and Table 5 exhibiting each threshold level of three component about every image. And the results of segmented images are presented in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, which take each Test image as a group. Furthermore, the results of the four metrics are shown in Table 6 and Table 7.
Table 6 and Figure 17 and Figure 18 compare the CPU Time and PSNR values while Table 7 and Figure 19 and Figure 20 compare the MSSIM and FSIM values of the segmented images. As can be seen from these tabulated values, all algorithms have lower values of PSNR, MSSIM, and FSIM at lower threshold levels. With the improvement of the threshold level, the values of PSNR, MSSIM, and FSIM increase gradually. Consequently, it can be clearly known that segmentation performance will be improved as the threshold level increases. However, the time of each algorithm will rise equally on the increasing threshold levels indicating the computation of algorithms is more complex on the higher threshold levels. PSNR, MSSIM, and FSIM are used to measure the similarity and qualify among the segmented images. Higher PSNR, MSSIM, and FSIM demonstrate that segmented images have more excellent segmentation performances.
Then, when comparing the differences in CPU time between various algorithms, it can be found that CEFO and EFO are significantly faster than ABC, BA, WDO, and BSA. Moreover, the running time of CEFO has decreased about 12.26% when comparing with EFO, which indicates the modified electromagnetic field optimization algorithm has a faster convergence rate. As for other algorithms, ABC has the longest time of computation due to its slow convergence rate, it needs nearly 30 times as much as CEFO. Afterward, BA, WDO, and BSA have an approximate running time, they are about 15 times longer than CEFO. With the increase of execution time, the practicability of the algorithm will be reduced. For a clearer presentation of convergence speed about these algorithms, the convergence curves are shown in Figure 21.
In terms of PSNR, the chart of all algorithms is shown in Figure 18. It can be seen that CEFO has higher values among these algorithms; ABC has similar values to EFO in some images at a high threshold level. For instance, in Test 1 of K=10, PSNR value of EFO is 24.8085 while ABC is 24.9762, but CEFO is 25.1603. WDO has much lower values in smaller threshold level and BSA has good values in higher threshold level, all in all, PSNR values of BA, WDO, and BSA have different diversification, but they are mediocre on the whole.
Comparing the results of MSSIM and FSIM in Table 7, FSIM is considered to be more authoritative and application and CEFO also performs better than other algorithms in this index. Although EFO can have a banner performance at K = 4, ABC will usually be close to EFO at high levels. BA has better values at some images such as in Test 2, 4, etc. WDO and BSA have higher values than BA at K = 6 in some images but they are all lower than CEFO on the whole.
From what has been mentioned above, the CEFO algorithm has superior performance when searching for the optimal threshold vector in multilevel thresholding color image segmentation.

6.2. Comparison of Other Segmentation Methods

In the last experiment, the superiority of CEFO has been verified. And in this experiment, fuzzy entropy has been regarded as the research objective. To show the performance of fuzzy entropy thresholding in multilevel color image segmentation, Otsu’s and Kapur’s entropy based on color image segmentation are used to be a comparison. Applying the CEFO algorithm to Fuzzy entropy, Otsu’s, and Kapur’s entropy respectively to segment selected 10 Berkeley images in Figure 6. The threshold level is chosen as = 4, 6, 8, and 10, which is used to obtain the corresponding threshold points for each component of the color image. The results of segmented images are in Figure 22, and the corresponding optimal threshold values are in Table 8. Table 9 compares the performance of different thresholding approaches based on parameters of CPU Time, PSNR, MSSIM, and FSIM.
As can be seen in Table 9 and Figure 23, Figure 24, Figure 25 and Figure 26, Otsu’s thresholding has the fastest speed of execution time, fuzzy entropy and Kapur’s entropy are a bit slower than Otsu’s. However, three thresholding segmentation methods are all in 0.5 (seconds) at different threshold levels, which can also indicate CEFO algorithm has a fast convergence rate. In terms of PSNR, it is clear that fuzzy entropy thresholding has higher values in general, the ranking of PSNR among these segmentation methods is Fuzzy > Kapur’s > Otsu’s. As for MSSIM and FSIM, Fuzzy entropy also performs well, which is in advance of two other methods overall. And the ranking of MSSIM and FSIM among three methods is Fuzzy > Kapur’s > Otsu’s. Therefore, fuzzy entropy is better as compared to others showing CEFO based on the fuzzy entropy technique can be applied in the color image segmentation field excellently.

6.3. ANOVA Test

A statistical test known as “the analysis of variance” (ANOVA) has been performed at 5% significance level to evaluate the significant difference between algorithms. In the experiment, CEFO algorithm is regarded as the control group and is compared with EFO, ABC, BA, WDO and BSA algorithms in terms of four measure metrics. The null hypothesis assumes that there is no significant difference between the mean values of 5 selected algorithms, whereas, the alternative hypothesis can be considered as a significant difference between them. Table 10 exhibits the –value of CPU Time, PSNR, MSSIM, and FSIM by the ANOVA test. As can be seen, the -value for CPU Time is less than 0.05, which implies a significant difference between the proposed algorithm and other algorithms and CEFO has a much fast convergence rate. With respect to another three measures, CEFO also has significant difference about BA, WDO and BSA. It can be observed that CEFO algorithm has a better performance.

7. Conclusions and Future Work

In this paper, multilevel thresholding color image segmentation has been considered as an optimization problem in which the fuzzy entropy technique has been presented as the objective function. To achieve efficient segmentation, it is essential for algorithms to search the optimal fuzzy parameters and threshold values. Electromagnetic Field Optimization is a novel meta-heuristic algorithm which use is attempt herein for the first time in this field. Additionally, a new chaotic strategy is proposed and embedded into the EFO algorithm to accelerate the convergence rate and enhance segmentation accuracy. In order to demonstrate the superior performance of the CEFO-based fuzzy entropy technique, a series of experiments have been conducted and results are evaluated in terms of CPU Time, PSNR, MSSIM, and FSIM. On the one hand, the CEFO algorithm is compared with EFO, ABC, BA, WDO, and BSA based on fuzzy entropy for segmenting ten Berkeley benchmark images at different threshold levels (K = 4, 6, 8, and 10). The obtained results illustrate the obvious effect of proposed chaotic strategy and CEFO needs less than 0.35 seconds to find the optimal threshold vector which makes it an effective algorithm to handle the above problem. On the other hand, the fuzzy entropy method is compared with Otsu’s variance and Kapur’s entropy method based on CEFO on the basis of the same experimental environment. The high precision of fuzzy entropy has been validated with four metrics. Although CEFO-fuzzy is not the fastest among the three techniques, its execution time is suitable for practical applications within 0.5 seconds. To sum up, CEFO-based fuzzy entropy is a robust technique in multi-threshold color image segmentation.
In the future, the proposed technique can be applied to solve practical problems such as medical images, satellite images, etc. It is also interesting to modify EFO algorithm in other aspects to improve its performance for higher threshold levels (e.g., K = 15 and 20). Furthermore, the merits of CEFO can be investigated using Tsallis entropy, Renyi’s entropy, and cross entropy for multilevel thresholding.

Author Contributions

S.S. and H.J. contributed to the idea of this paper; J.M. performed the experiments; S.S. and J.M. wrote the paper; H.J. contributed to the revision of this paper.

Funding

This research received no external funding.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Schematic diagram of the electromagnetic field. The relationship between the electromagnetic particle (EMP) and electromagnet. The method of generating the new electromagnetic particle
Figure 1. Schematic diagram of the electromagnetic field. The relationship between the electromagnetic particle (EMP) and electromagnet. The method of generating the new electromagnetic particle
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Figure 2. Logistic Chaotic Map.
Figure 2. Logistic Chaotic Map.
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Figure 3. Chaotic Map in this paper.
Figure 3. Chaotic Map in this paper.
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Figure 4. Membership function graph.
Figure 4. Membership function graph.
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Figure 5. Flow chart of fuzzy entropy thresholding method based on the Chaotic Electromagnetic Field Optimization (CEFO) algorithm.
Figure 5. Flow chart of fuzzy entropy thresholding method based on the Chaotic Electromagnetic Field Optimization (CEFO) algorithm.
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Figure 6. Experimental images of Berkeley. Ten classical images and their histograms of three component (red, green, and blue) are exhibited.
Figure 6. Experimental images of Berkeley. Ten classical images and their histograms of three component (red, green, and blue) are exhibited.
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Figure 7. Segmented images of Test 1 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 7. Segmented images of Test 1 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 8. Segmented images of Test 2 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 8. Segmented images of Test 2 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 9. Segmented images of Test 3 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 9. Segmented images of Test 3 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 10. Segmented images of Test 4 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 10. Segmented images of Test 4 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 11. Segmented images of Test 5 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 11. Segmented images of Test 5 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 12. Segmented images of Test 6 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 12. Segmented images of Test 6 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 13. Segmented images of Test 7 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 13. Segmented images of Test 7 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 14. Segmented images of Test 8 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 14. Segmented images of Test 8 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 15. Segmented images of Test 9 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 15. Segmented images of Test 9 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 16. Segmented images of Test 10 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
Figure 16. Segmented images of Test 10 at K = 4, 6, 8, 10 using selected algorithms based on fuzzy entropy.
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Figure 17. Comparison of Computational Time (CPU Time) based on fuzzy entropy.
Figure 17. Comparison of Computational Time (CPU Time) based on fuzzy entropy.
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Figure 18. Comparison of Peak Signal to Noise Ratio (PSNR) based on fuzzy entropy.
Figure 18. Comparison of Peak Signal to Noise Ratio (PSNR) based on fuzzy entropy.
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Figure 19. Comparison of Mean Structural Similarity (MSSIM) based on fuzzy entropy.
Figure 19. Comparison of Mean Structural Similarity (MSSIM) based on fuzzy entropy.
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Figure 20. Comparison of Feature Similarity (FSIM) based on fuzzy entropy.
Figure 20. Comparison of Feature Similarity (FSIM) based on fuzzy entropy.
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Figure 21. Convergence curves of CEFO, EFO, ABC, BA, WDO, and BSA based on fuzzy entropy at K = 10. (Red line represents CEFO; Blue line represents EFO; Cyan line represents ABC; Yellow line represents BA; Magenta line represents WDO; Green ling represents BSA.)
Figure 21. Convergence curves of CEFO, EFO, ABC, BA, WDO, and BSA based on fuzzy entropy at K = 10. (Red line represents CEFO; Blue line represents EFO; Cyan line represents ABC; Yellow line represents BA; Magenta line represents WDO; Green ling represents BSA.)
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Figure 22. Segmented images at K = 4, 6, 8, 10 using CEFO algorithm based on fuzzy entropy, Otsu’s and Kapur’s entropy.
Figure 22. Segmented images at K = 4, 6, 8, 10 using CEFO algorithm based on fuzzy entropy, Otsu’s and Kapur’s entropy.
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Figure 23. Comparison of CPU Time based on fuzzy entropy.
Figure 23. Comparison of CPU Time based on fuzzy entropy.
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Figure 24. Comparison of PSNR based on fuzzy entropy.
Figure 24. Comparison of PSNR based on fuzzy entropy.
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Figure 25. Comparison of MSSIM based on fuzzy entropy.
Figure 25. Comparison of MSSIM based on fuzzy entropy.
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Figure 26. Comparison of FSIM based on fuzzy entropy.
Figure 26. Comparison of FSIM based on fuzzy entropy.
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Table 1. Chaotic maps.
Table 1. Chaotic maps.
NameChaotic Map
Logistic x i + 1 = a x i ( 1 x i )
Sine x i + 1 = a 4 sin ( π x i )
Cubic x i + 1 = a x i ( 1 x i 2 )
Circle x i + 1 = m o d ( x i + b ( a 2 π ) sin ( 2 π x i ) , 1 )
Iterative x i + 1 = sin ( a π x i )
Tent x i + 1 = a ( 1 a ) | x i |
Table 2. Specific values of parameters used in selected algorithms.
Table 2. Specific values of parameters used in selected algorithms.
AlgorithmParameterExplanationValue
EFOPfieldThe portion of particles belonging to the positive field.0.1
NfieldThe portion of particles belonging to the negative field.0.45
PsfieldThe probability of selecting electromagnets directly from the positive field.0.3
RrateThe probability of changing one electromagnet directly from the positive field.0.2
ABClimitThe value of the max trial limit.10
BAriThe rate of pulse emission.[0, 1]
AiThe value of loudness.[1, 2]
WDO a A constant.0.4
gThe constant of gravitation.0.2
RTA coefficient.3
c Coriolis coefficient.0.4
BSAa1, a2The values of indirect and direct effects on the birds’ vigilance behaviors.2
c1, c2The values of the cognitive coefficient and social coefficient.2
FLThe frequency of birds’ flight behaviors.0.6
Table 3. Comparison of optimal threshold values between Chaotic Electromagnetic Field Optimization (CEFO) and Electromagnetic Field Optimization (EFO) at K = 4, 6, 8, 10 based on fuzzy entropy.
Table 3. Comparison of optimal threshold values between Chaotic Electromagnetic Field Optimization (CEFO) and Electromagnetic Field Optimization (EFO) at K = 4, 6, 8, 10 based on fuzzy entropy.
ImageKCEFOEFO
RGBRGB
Test 1456 93 153 18757 88 132 19159 10 157 19761 90 144 18818 74 115 17058 87 148 202
620 59 92 128 167 19926 47 81 114 161 20325 48 76 111 138 19526 60 95 128 162 19819 47 80 119 171 21018 57 85 122 154 212
811 31 58 82 107 132 168 21112 42 64 92 114 149 181 21018 50 73 97 143 168 196 22317 47 71 90 115 138 171 20912 39 67 93 130 170 205 22326 56 78 100 113 162 190 216
1014 37 55 75 90 111 130 159 182 21011 41 61 95 122 146 168 185 208 23714 40 64 80 105 136 161 189 212 23110 26 56 82 102 133 148 167 192 21511 32 59 76 90 107 134 166 198 22224 41 56 72 93 126 156 181 203 229
Test 2410 93 114 17186 143 194 22691 110 137 18980 116 149 18393 113 146 216104 124 170 206
618 41 75 107 147 17787 101 133 153 192 23454 72 97 121 156 19273 83 108 139 176 21994 114 135 162 197 23116 50 85 129 156 199
815 27 44 59 96 127 171 20414 22 43 65 117 144 190 22239 61 82 117 148 176 195 2237 34 55 78 107 135 166 19949 65 76 139 168 193 211 23245 61 82 101 129 154 182 213
1032 55 75 108 144 171 195 215 226 2395 50 65 85 102 121 139 167 197 22831 47 64 87 108 125 158 179 209 2394 24 47 67 102 122 147 180 204 22319 33 53 71 99 119 144 170 195 22419 36 57 80 108 141 154 177 201 230
Test 3444 92 139 188 31 92 139 18827 97 138 185 33 80 126 18137 87 142 20629 106 154 188
632 62 93 128 173 19919 53 90 119 162 20636 71 105 136 183 20832 76 116 150 177 20750 69 101 143 190 21615 46 70 113 153 197
822 43 68 91 112 138 172 208 17 51 73 98 123 148 181 20711 36 58 88 106 141 170 213 19 54 94 125 141 164 190 21425 49 72 96 126 156 192 22315 49 71 95 128 159 187 214
1026 39 55 69 86 110 131 154 175 20742 29 56 79 104 121 141 163 197 2199 25 49 82 109 132 154 176 199 22217 31 45 62 93 124 168 194 212 22914 39 56 76 95 114 148 176 196 22416 13 34 58 82 115 150 180 199 223
Test 4452 84 127 16467 102 141 20449 74 111 15559 88 129 16464 105 153 19961 101 145 203
638 62 97 137 172 19548 71 111 144 175 21343 77 109 147 178 23540 65 89 112 147 18444 67 105 126 167 21139 63 98 134 160 190
813 31 54 92 110 142 175 19835 60 86 114 137 169 200 22624 41 64 94 129 156 193 22415 26 46 68 97 134 160 18918 43 73 100 128 155 190 22523 57 105 137 169 195 221 237
106 25 43 64 84 106 132 152 172 1897 27 39 52 75 99 130 157 190 22115 25 48 78 103 130 157 178 219 2408 18 31 48 74 100 124 153 181 2118 23 39 57 80 104 127 157 188 21510 27 50 87 117 146 171 194 218 234
Test 5452 98 149 19855 89 164 21577 125 162 20987 122 152 19038 58 106 16280 120 160 194
622 63 93 131 179 22516 69 97 125 158 20415 61 97 131 166 20723 64 90 140 183 21326 75 114 134 169 21120 59 84 132 169 205
816 59 77 94 116 134 162 19920 58 75 98 129 155 185 21419 56 77 115 145 173 196 22714 56 75 104 127 162 191 22537 37 60 76 111 135 175 21417 39 66 97 122 171 205 234
1013 29 57 77 93 107 130 165 188 21414 30 62 78 94 126 152 179 205 23119 44 56 72 89 109 129 159 176 20511 26 50 72 92 115 149 180 201 22416 25 48 64 85 106 127 142 175 20624 41 56 70 96 123 150 177 202 229
Test 6455 97 158 19564 99 153 19046 75 131 17952 100 144 19256 92 123 178 52 88 149 176
643 60 91 118 165 20225 38 76 119 165 20547 69 97 133 155 19044 75 112 150 176 21160 87 109 135 167 19632 51 85 125 172 207
816 42 73 91 129 157 190 22110 34 52 79 108 136 168 20723 39 64 86 120 137 172 20228 45 74 91 120 150 191 22515 26 55 85 111 141 176 20523 38 57 87 119 159 183 214
1010 32 50 77 99 134 162 192 209 22813 36 47 69 107 140 164 185 200 22314 33 60 82 101 130 156 183 208 2288 30 46 72 99 125 145 166 189 22110 28 53 72 100 125 146 172 187 20614 31 51 68 86 115 134 167 190 221
Test 7466 95 139 17625 66 108 15723 88 134 22026 89 132 17943 89 133 17830 94 137 208
617 33 80 120 168 20520 80 108 133 164 18913 59 84 102 128 15120 41 60 92 131 180 21 54 86 111 144 183 31 67 101 138 196 223
820 54 79 106 126 146 169 20913 50 80 102 132 166 197 22319 40 63 88 119 148 200 22814 58 80 98 122 146 167 20718 55 82 105 128 161 193 22818 36 67 91 128 149 200 220
1010 30 41 63 89 110 129 154 178 21614 54 68 82 106 132 169 197 220 24315 41 66 85 105 133 151 190 214 23312 28 59 82 115 146 179 199 215 22712 31 57 76 92 117 141 168 200 22310 34 63 95 120 140 157 185 212 232
Test 8448 73 173 21262 107 141 20743 79 124 21522 83 139 19955 105 149 20137 102 183 234
638 64 101 141 181 22432 61 85 126 155 19638 61 91 126 197 21840 66 96 151 182 21240 66 97 140 175 20934 54 88 117 182 230
815 41 64 85 110 133 175 21344 34 74 110 135 158 179 20325 40 69 104 136 182 207 23131 55 77 93 108 135 178 21725 51 80 117 155 179 206 22730 32 60 103 117 141 179 215
1018 39 58 87 115 137 172 186 210 23021 36 63 79 111 129 150 180 208 22728 47 74 90 116 131 153 178 208 22622 42 69 84 109 133 158 182 208 22739 91 103 127 146 159 176 193 208 22521 35 65 98 119 140 165 184 211 231
Test 9447 74 107 14955 90 132 16742 66 117 14933 67 109 15553 87 123 16545 89 155 211
613 43 62 92 126 16515 50 89 125 160 19519 45 81 115 151 23012 42 59 96 127 16024 33 76 115 143 19028 48 80 116 149 167
814 36 53 75 92 123 152 17311 40 55 77 101 126 151 18326 41 66 91 128 155 200 23311 38 54 81 115 139 166 18717 66 86 105 125 145 172 19728 51 76 104 132 154 172 207
1013 23 41 59 82 106 126 158 181 19811 43 54 72 89 105 134 163 188 21324 47 71 94 118 140 160 177 208 2379 41 60 92 116 131 150 167 185 20313 30 47 67 86 108 131 150 183 21220 43 71 92 115 155 175 190 226 240
Test 10448 76 132 17531 69 117 16953 86 116 20062 101 132 19045 70 123 18652 83 139 193
634 57 90 127 166 19653 90 116 142 180 21442 56 104 136 169 22033 56 87 127 162 19843 76 118 155 190 22542 70 100 133 164 226
819 33 62 91 133 165 195 21420 26 57 90 112 150 195 22012 38 55 88 113 150 186 22934 60 98 128 148 170 194 22117 38 58 87 119 147 187 21610 23 49 79 121 149 170 189
1016 35 54 71 93 119 141 169 196 2319 26 36 51 72 106 132 166 195 22410 26 41 61 77 108 134 160 185 21314 33 56 86 106 129 159 189 212 2358 29 43 75 99 118 141 168 196 22610 33 48 69 90 113 144 169 194 230
Table 4. Comparison of optimal threshold values between Artificial Bee Colony (ABC) and Bat Algorithm (BA) at K = 4, 6, 8, 10 based on fuzzy entropy.
Table 4. Comparison of optimal threshold values between Artificial Bee Colony (ABC) and Bat Algorithm (BA) at K = 4, 6, 8, 10 based on fuzzy entropy.
ImageKABCBA
RGBRGB
Test 1455 103 154 20169 104 156 20740 96 165 21949 78 150 21466 92 128 19578 116 176 213
624 63 109 149 184 22047 74 111 152 183 21839 84 126 154 188 22031 58 81 105 150 19520 45 85 115 165 21229 71 118 154 190 230
824 51 81 113 142 177 211 22620 54 77 108 150 174 200 22315 47 75 107 143 171 207 23712 51 80 98 117 155 193 22513 48 77 109 159 181 205 22533 63 94 110 143 168 203 242
1016 41 59 85 113 142 167 187 211 23615 43 62 88 112 139 167 189 209 23213 39 62 83 113 139 165 192 221 23620 50 82 99 125 149 168 190 211 23115 41 88 107 129 146 158 177 206 23231 51 72 84 99 129 162 191 227 242
Test 2463 108 164 21678 119 159 23172 98 171 224117 150 188 275129 152 182 23188 113 140 189
649 76 112 149 196 23652 79 125 159 189 22356 78 124 148 182 21981 107 143 173 205 23283 97 112 130 212 23383 114 153 178 206 238
838 51 73 110 143 182 221 24037 56 90 122 143 170 195 22231 51 79 109 149 175 207 24635 48 68 89 140 181 210 23651 85 120 156 182 214 242 2505 20 45 82 113 149 177 217
1022 37 57 85 112 142 167 192 223 2376 45 60 80 109 133 160 188 218 24423 40 65 88 115 144 163 188 218 23964 69 75 81 98 120 155 187 210 23030 50 70 92 120 147 171 193 214 23828 58 79 98 135 162 186 206 234 248
Test 3431 103 162 20660 102 151 20944 103 157 20653 99 141 19753 121 164 20323 100 143 191
634 68 103 140 190 22629 66 101 140 186 21731 67 98 138 186 21729 66 116 154 185 21539 72 122 154 181 21045 86 116 143 173 219
818 51 79 111 138 173 205 23228 56 84 108 140 168 194 23120 50 77 103 141 168 199 22220 43 75 97 136 181 205 23625 51 80 108 136 177 203 23112 56 86 110 138 175 202 227
1019 37 63 87 112 137 170 195 222 24422 46 65 92 118 141 165 194 216 23721 52 71 96 110 136 157 196 214 23019 40 64 90 117 147 174 206 227 24436 62 88 105 116 135 162 196 216 23930 51 74 100 113 135 146 160 184 214
Test 4440 90 136 17263 118 170 21063 96 146 20868 110 141 16860 112 161 20849 109 189 234
632 69 114 155 190 21942 66 111 161 201 23735 65 104 146 186 23538 59 116 147 173 19737 78 135 168 193 22340 74 108 137 155 226
821 45 78 107 137 169 206 22732 56 83 120 142 177 212 24126 49 80 116 156 182 213 2338 36 61 92 125 197 222 24126 52 84 112 135 160 198 22826 58 91 121 146 170 198 228
1025 45 60 84 114 140 168 191 217 23226 43 62 86 108 136 161 191 216 24522 38 63 85 113 137 159 193 223 2418 33 64 96 128 156 174 195 217 23438 63 84 105 129 145 168 192 206 23125 41 54 67 96 130 152 186 209 233
Test 5434 94 167 22229 93 159 22075 118 169 23133 107 160 19629 75 134 22572 106 149 208
626 72 97 138 187 22321 73 108 140 188 22620 71 118 153 191 23159 86 122 163 202 22819 62 94 144 184 23452 101 136 176 213 234
816 52 84 117 141 175 202 22913 62 86 117 143 178 203 22818 56 77 103 135 172 200 23427 61 85 117 142 171 201 23889 124 141 157 171 183 198 22556 78 98 126 157 167 199 234
1018 36 68 99 124 147 167 183 206 22812 50 68 88 114 144 169 192 219 23818 41 65 92 111 133 163 190 220 24117 37 54 72 104 138 160 178 191 23114 68 86 104 118 141 168 197 216 23116 43 68 98 128 154 177 198 220 241
Test 6472 116 162 19759 92 142 17760 91 143 19660 121 178 21055 91 135 17459 94 128 166
635 54 103 142 184 22231 66 111 161 192 22836 74 114 158 197 21812 59 96 121 156 20161 110 147 175 204 23344 74 103 131 190 227
816 45 79 113 151 189 210 23113 51 85 119 156 182 212 23421 44 84 120 148 176 206 22413 33 48 60 119 152 190 23641 64 98 119 158 183 203 22211 40 62 125 165 183 213 234
1015 42 60 86 114 144 164 194 214 23111 41 60 81 101 130 155 181 212 23326 46 64 80 110 131 164 185 201 22011 47 75 102 121 134 150 178 204 22714 39 58 84 108 138 161 186 210 23521 38 51 71 90 115 144 172 191 228
Test 7455 97 147 21045 96 148 21641 92 139 22067 96 125 17184 112 163 24424 76 173 219
633 72 111 153 177 21919 75 110 147 181 22725 71 109 148 190 22757 75 92 122 166 21227 87 118 141 185 23421 67 100 140 192 235
824 59 79 102 129 157 188 22129 65 87 113 145 166 195 23124 56 86 113 147 189 211 23441 72 92 116 138 154 169 21222 45 79 104 133 162 200 22614 45 72 111 154 188 212 244
1015 39 68 98 125 142 166 195 220 23713 40 69 87 110 140 170 202 223 24217 44 71 89 113 138 158 193 214 23421 62 94 122 144 163 183 200 217 24243 80 102 121 140 159 175 196 223 23720 50 84 111 131 167 180 205 223 239
Test 8455 86 183 22060 104 153 22533 85 121 22632 78 163 22663 106 148 22254 97 126 192
638 82 126 154 193 22944 74 114 152 181 23035 57 76 105 130 23546 87 148 181 215 23753 78 119 144 184 22727 42 65 90 112 141
814 48 81 105 142 175 212 23924 55 90 116 146 174 208 23629 62 80 110 133 172 208 23628 71 98 124 149 179 204 23324 54 81 104 142 177 211 2369 21 42 84 113 132 192 240
1011 34 57 78 101 133 167 191 216 23417 35 64 85 109 134 155 183 210 23920 40 72 99 116 135 167 186 218 24016 51 79 103 126 144 169 194 224 23918 50 66 83 101 121 147 172 207 22718 47 82 105 127 145 169 194 220 240
Test 9464 110 138 18243 96 149 19452 101 145 22168 104 122 15135 73 109 18238 92 144 208
619 62 101 131 157 19739 74 106 145 183 21230 65 102 135 166 23439 63 90 116 141 18112 51 100 129 161 19840 54 72 101 129 158
821 56 98 134 154 173 194 21416 43 67 99 141 175 197 21622 50 83 115 146 168 205 23938 58 74 104 128 145 175 19840 60 105 141 163 187 213 23615 29 60 96 126 167 210 239
1012 42 63 87 114 141 169 197 223 23815 46 66 86 110 137 160 187 214 23625 45 60 85 111 136 164 193 227 24223 66 88 123 142 154 164 179 199 21614 42 64 84 111 135 159 184 203 22616 32 48 99 135 158 174 202 227 243
Test 10458 104 164 22170 110 161 21377 122 159 21574 134 165 19970 109 164 23878 104 163 193
638 76 125 159 192 22953 79 111 139 176 21848 71 101 135 182 22126 80 137 166 196 23351 96 118 138 169 20942 74 121 155 198 229
831 49 76 110 142 169 202 24032 52 86 117 148 180 206 22629 52 83 114 146 171 195 23432 67 94 119 147 174 199 24047 75 90 103 117 159 187 21935 67 108 136 162 185 215 243
1027 47 69 94 120 143 164 185 212 24226 43 66 95 117 142 165 192 221 239 22 39 62 89 110 135 157 185 214 23833 66 88 111 125 143 156 180 201 23911 29 53 84 114 136 166 193 220 23413 39 58 105 132 166 183 207 230 247
Table 5. Comparison of optimal threshold values between Wind Driven Optimization (WDO) and Bird Swarm Algorithm (BSA) at K = 4, 6, 8, 10 based on fuzzy entropy.
Table 5. Comparison of optimal threshold values between Wind Driven Optimization (WDO) and Bird Swarm Algorithm (BSA) at K = 4, 6, 8, 10 based on fuzzy entropy.
ImageKWDOBSA
RGBRGB
Test 1468 107 150 19265 103 156 19877 115 164 19654 119 190 24312 57 174 24324 93 175 238
644 74 109 134 172 20553 84 120 149 175 20561 95 131 154 186 21822 90 100 165 239 2556 30 78 158 214 23935 58 93 153 186 227
847 72 101 121 144 168 188 21148 75 104 129 149 170 192 21346 68 91 110 131 156 174 2127 57 108 118 129 158 216 2519 55 68 97 181 209 241 2554 32 78 123 152 179 196 247
1045 67 93 115 131 142 161 176 194 21051 65 82 98 114 133 152 171 195 21725 45 65 80 95 113 139 164 194 22312 27 46 70 102 136 153 178 198 22817 37 49 72 95 132 155 194 213 22612 33 63 99 132 161 189 212 228 243
Test 24123 146 174 202128 154 183 21698 131 170 2087 39 194 23224 112 145 22771 105 128 183
6100 117 138 160 185 205109 129 153 171 195 22587 103 128 164 181 21417 51 79 129 191 2303 18 55 104 145 22326 70 90 142 217 240
878 93 104 118 134 146 169 20093 111 131 147 165 182 202 22862 74 92 111 131 154 179 21117 52 83 123 181 216 230 24718 73 92 119 152 190 225 25259 90 102 121 142 198 224 240
1070 82 95 107 122 134 150 168 189 20980 90 101 112 120 130 142 157 170 19070 83 103 118 137 147 166 182 200 2181 20 75 79 82 87 118 142 179 21116 38 73 114 130 163 196 248 253 25522 73 90 123 137 148 179 205 215 239
Test 3448 104 158 19968 121 163 19677 119 161 20139 94 147 21442 117 173 20124 107 208 221
644 80 113 148 176 21247 77 105 141 180 21456 83 114 136 168 21162 96 128 176 199 22316 42 63 76 122 21912 34 67 126 183 212
837 67 92 112 135 163 189 21738 59 82 102 129 156 188 21346 70 91 117 140 162 190 2131 37 51 77 115 159 209 23719 60 102 149 177 203 223 25120 57 73 95 125 159 188 221
1036 69 100 120 142 156 172 190 204 22330 54 73 96 119 137 158 178 198 22224 49 67 83 104 120 143 170 192 2148 77 109 140 164 179 202 207 231 2541 25 82 98 109 164 178 196 230 25517 56 80 97 119 146 172 216 229 251
Test 4454 94 132 16971 117 154 20078 109 147 20746 89 145 21418 95 184 24436 77 130 213
649 72 101 128 156 18358 87 109 143 181 21251 89 115 152 194 22335 52 85 128 167 19341 70 93 146 203 23118 54 89 128 169 221
840 67 94 120 139 156 176 19348 73 93 118 138 166 196 22545 76 106 129 153 176 205 2291 3 30 81 120 181 218 23726 56 92 133 185 205 224 24423 42 91 133 163 204 223 236
1041 59 83 103 124 141 151 167 185 19844 64 81 100 115 134 155 175 199 22343 67 86 103 123 141 159 176 198 22713 26 41 66 92 119 147 190 226 24623 54 74 99 123 145 167 190 216 24015 26 59 95 113 130 156 183 212 238
Test 5486 123 161 19888 123 165 20378 118 159 1998 55 142 20382 107 159 19822 57 114 226
671 90 116 154 188 21878 103 127 157 186 21165 101 133 162 190 22242 106 138 203 227 25311 41 110 152 204 24875 133 164 194 214 237
833 67 88 114 131 158 183 21259 76 97 114 135 164 192 22258 78 97 121 142 168 192 22711 33 86 110 175 195 211 2473 63 123 177 197 216 239 25526 54 85 123 150 178 211 230
1057 77 100 116 132 148 166 181 201 22336 52 70 88 109 128 146 160 183 20164 87 101 115 128 144 161 179 196 21732 46 62 94 121 149 171 188 209 2388 21 50 87 104 138 148 166 192 2299 36 88 110 123 142 163 182 216 249
Test 6459 104 157 19769 106 147 19359 100 130 17725 127 153 24616 78 149 22428 98 172 231
654 81 108 141 175 20255 83 115 151 177 20941 71 104 137 163 19936 100 122 148 178 20429 87 116 142 163 19813 28 69 91 137 228
845 74 100 123 146 167 191 21555 81 105 131 151 171 191 21338 67 92 122 142 169 186 2118 52 79 123 158 181 210 24311 50 109 132 169 204 219 23420 63 104 130 137 168 207 246
1037 53 73 90 110 131 147 170 197 22247 69 89 105 128 143 162 179 197 21936 61 83 99 118 137 157 175 199 21623 42 60 77 108 132 152 182 218 2356 33 61 108 126 158 180 207 228 24514 71 105 126 149 173 208 219 230 244
Test 7474 110 145 17685 107 145 18475 100 136 18228 95 224 25543 70 135 23546 134 198 247
663 84 114 143 170 20122 76 100 132 169 22148 79 107 130 151 1917 30 49 115 157 19430 71 107 144 183 2424 55 137 166 205 251
841 68 97 120 139 165 184 21643 76 97 119 140 164 181 22034 57 82 109 129 151 200 23316 37 65 85 108 141 169 22213 44 70 118 144 157 177 23613 57 95 117 145 183 214 230
1053 70 89 109 122 141 159 176 195 21725 44 66 81 103 123 142 158 180 22755 80 105 123 133 144 156 164 184 23414 35 57 75 88 120 141 154 173 21411 38 58 76 100 126 149 172 189 24017 37 74 100 124 155 175 210 227 242
Test 8469 124 160 20762 118 152 20744 92 122 18948 83 156 21322 81 125 21661 98 114 212
650 85 114 157 184 21354 85 116 149 176 21638 71 103 128 192 2216 31 73 108 214 25449 103 122 161 190 23514 55 109 163 195 234
841 61 86 111 136 161 188 22041 67 88 110 130 157 193 22438 68 91 105 123 140 184 23221 41 67 76 107 134 163 23121 47 88 119 159 194 221 24212 37 67 116 137 160 184 235
1034 47 68 88 112 132 157 180 214 23645 70 89 108 125 137 152 164 178 20628 42 55 76 104 132 167 194 210 23015 38 76 102 125 160 178 209 224 24310 33 46 73 101 138 162 204 227 24233 47 68 93 114 132 142 153 177 213
Test 9470 104 138 16071 106 142 17658 107 149 22547 86 157 23443 92 125 21937 92 147 223
657 82 110 133 156 17761 92 116 137 163 18743 74 102 126 155 20641 93 139 167 213 22113 79 118 156 166 20511 71 112 155 175 239
850 72 97 120 138 155 172 19054 75 93 110 131 151 173 19437 62 91 121 143 165 205 2193 19 32 70 111 155 188 22711 35 76 122 141 181 216 23023 74 94 113 147 189 214 241
1049 64 79 92 105 118 137 158 173 19150 69 90 108 121 139 155 170 187 20438 67 85 98 113 129 145 164 196 2227 35 69 99 115 136 157 184 210 2515 17 37 91 113 158 202 229 247 25225 57 90 112 141 155 174 196 216 232
Test 10462 89 136 17970 103 136 19170 111 156 19132 132 173 21737 90 134 21570 129 166 192
652 93 129 155 187 22654 88 116 148 184 21650 80 111 141 171 20751 85 123 155 187 21814 33 61 117 187 23741 67 108 139 182 219
835 66 89 107 136 164 188 21949 72 96 117 141 167 198 22347 67 89 117 143 166 193 22421 53 63 86 123 185 213 23619 56 78 112 143 186 238 25311 44 70 123 154 173 193 215
1041 67 90 114 131 153 174 200 215 23241 62 80 96 109 128 139 157 189 22741 61 81 102 121 136 158 172 199 23219 37 65 91 104 134 165 194 223 24629 50 74 98 114 137 161 190 226 2406 28 57 105 132 157 180 199 224 239
Table 6. Comparison of CPU Time (in seconds) and PSNR computed by CEFO, EFO, ABC, BA, WDO, and BSA using fuzzy entropy. The bold numbers are the best values in the relevant index.
Table 6. Comparison of CPU Time (in seconds) and PSNR computed by CEFO, EFO, ABC, BA, WDO, and BSA using fuzzy entropy. The bold numbers are the best values in the relevant index.
ImageKComputational Time (CPU Time)Peak Signal to Noise Ratio (PSNR)
CEFOEFOABCBAWDOBSACEFOEFOABCBAWDOBSA
Test 140.175410.203795.137592.452882.493672.9713819.086918.530417.802017.321215.957616.8238
60.216230.266176.639712.981023.045033.3342322.496621.085120.882619.487219.252119.3995
80.262190.304167.403983.542153.532033.9665723.931523.315323.007923.090221.255321.9558
100.332840.357698.548344.059834.120604.6100825.160324.808524.976223.562323.517424.2976
Test 240.181520.197455.480452.624912.550213.0576818.501116.638417.921714.995614.662116.8609
60.230140.251406.528062.988643.116443.6757621.418620.841021.426117.996216.197522.5361
80.267450.301717.392663.545453.540034.0092423.474523.563324.057418.365219.037122.6106
100.354150.364938.322704.077064.178214.7133827.196026.536226.553724.767019.962022.7476
Test 340.193540.209525.429622.540042.323543.0583617.926618.070617.755915.079517.825916.1882
60.244060.277246.516393.072123.148873.2104121.459520.972421.055420.646520.909618.7586
80.265720.301957.432543.612013.647064.0583623.525722.755722.983222.703123.072921.4050
100.311040.357568.497564.055374.270574.9792925.110724.179624.624323.907624.863021.4820
Test 440.206540.222865.223642.535702.510942.8916718.579118.556218.453817.859418.105018.4835
60.256130.270556.529433.002673.053363.4038821.996221.575221.472821.093120.441921.3494
80.295650.317347.348613.539053.740834.0859223.657422.951322.509122.495022.123921.8930
100.342260.430228.482344.192344.200274.7786724.821224.708124.393724.577423.112424.5733
Test 540.200170.225255.276452.538572.487182.9830418.023917.624216.808916.387017.351615.8329
60.254850.260246.894153.069933.062723.5314820.347920.432920.054720.263019.166818.4271
80.294240.321017.551793.862213.687224.0933222.166121.812722.169321.520621.500119.8220
100.344800.351748.371594.095324.243254.9587524.013823.746223.870423.493622.239923.5245
Test 640.194720.208415.121372.493692.539053.0806618.109517.649117.650917.785717.729316.8704
60.242630.267456.576613.070783.064473.5830321.483120.628721.011720.225420.243119.4964
80.308640.320497.356413.574643.507184.1288523.229023.325622.777321.812221.413520.8800
100.337250.357088.549004.037514.157834.8669424.768124.473724.525924.650923.179322.5603
Test 740.206780.230055.225562.531312.447273.0095218.873918.503918.027716.412816.486316.0152
60.235620.258096.624903.071993.216643.4159122.144321.621320.076219.980620.824818.3119
80.298070.302837.414923.560563.691534.0513423.783923.193223.238422.041524.367522.0538
100.342170.362738.588244.209104.367514.7435924.743324.033524.324423.062923.862724.2057
Test 840.214500.223785.337522.527272.337842.9335318.949517.491116.639916.577418.261417.6314
60.250680.280806.435233.039053.088653.4436821.410920.579121.893320.766122.174719.2986
80.319940.359807.366673.570583.466084.1433723.067722.439823.061922.375923.484621.3875
100.337250.368888.489884.082544.166084.8944325.435725.372023.968923.787325.459824.9694
Test 940.193240.235585.283172.550342.322813.1190416.456815.977616.552516.145016.199418.1708
60.237890.273846.589093.063633.081473.3235318.467518.674919.263918.569818.417519.1084
80.270570.297387.320343.582043.532994.0408321.009820.531220.600320.390819.959721.0047
100.331510.363618.345544.153944.263644.6002722.366623.065524.448622.463020.835822.9661
Test 1040.190390.232475.457312.582812.425823.1071818.581718.486018.102517.935818.451717.3690
60.236560.257626.467013.039823.124203.4578320.893820.845020.818720.759220.568519.7354
80.263900.293607.348703.590423.547113.9562522.624122.334622.382722.233022.646622.6210
100.333040.343018.491224.125454.297274.7636425.364024.891624.855924.729923.308524.8469
Table 7. Comparison of MSSIM and FSIM computed by CEFO, EFO, ABC, BA, WDO, and BSA using fuzzy entropy. The bold numbers are the best values in the relevant index.
Table 7. Comparison of MSSIM and FSIM computed by CEFO, EFO, ABC, BA, WDO, and BSA using fuzzy entropy. The bold numbers are the best values in the relevant index.
ImageKMean Structural Similarity (MSSIM)Feature Similarity (FSIM)
CEFOEFOABCBAWDOBSACEFOEFOABCBAWDOBSA
Test 140.973680.963120.961920.942010.940170.956420.758820.748800.742210.729400.724850.69178
60.986310.982800.982440.957690.970510.973050.850390.839380.811250.766480.806940.75111
80.991990.989850.989340.989290.982990.984420.886180.876830.869150.868600.854350.83095
100.993610.992510.993290.989590.988780.992150.910980.902820.909270.883840.894800.89012
Test 240.960130.953060.954840.930350.925880.958140.729720.689090.692470.648070.645400.65013
60.984590.974950.977460.937220.946640.981100.813150.737860.747090.702360.675940.73024
80.990980.989080.983430.966780.971160.987650.862100.846950.856250.710720.743000.83126
100.996310.993890.995450.991940.977090.988990.928760.924970.914650.878800.761620.84512
Test 340.973730.972930.970770.901060.964890.958820.719190.722850.718870.684560.708710.65393
60.988620.986230.987670.985230.981450.978320.831440.811850.825590.805440.827050.74121
80.993190.991430.991960.991370.992740.988060.885100.872950.878660.872920.874570.84056
100.995130.993930.994470.992820.994670.984720.914710.886830.912140.901640.909960.86459
Test 440.975040.973820.973290.970050.969990.971440.759660.754930.753450.753290.728730.73845
60.988170.987360.986340.984890.982560.987950.858250.847360.847060.826800.804020.84761
80.992210.990990.991610.990250.987860.986840.897380.885820.893470.884080.846680.86167
100.994850.994520.994460.993080.990110.993850.922180.918790.912270.904670.865750.91742
Test 540.976530.973150.968860.960220.966750.962670.767950.762960.749130.756390.757090.72777
60.986160.985130.985170.987320.977100.976270.828390.827380.819410.821740.800320.78086
80.991310.989550.990270.982580.987180.981580.862820.854100.858430.852360.855590.82891
100.994700.994350.994290.992530.988640.992860.893130.884080.888470.885200.883490.87589
Test 640.970600.965790.964060.959310.966330.956800.773660.741820.738940.752630.744990.71969
60.985910.984940.984170.978460.980730.977180.835680.828930.833660.799840.815450.80549
80.991980.991650.989320.987020.984480.982180.884460.881680.866190.856450.843590.82360
100.994240.993300.993460.992990.989790.987150.903430.897370.900760.895850.879330.85478
Test 740.975960.972790.973490.956290.951680.957060.751910.720870.731960.698860.722290.62681
60.988620.988400.980610.980130.982780.970280.849830.839050.794820.797730.825470.70824
80.992600.991160.991290.989740.990200.987920.885820.875680.876490.865370.876620.83777
100.994070.992590.993200.990840.990530.992680.906590.892690.901240.872110.880490.89262
Test 840.983540.974690.961640.970040.982760.979900.790860.767870.753290.782190.774830.78629
60.991110.989090.990310.988550.990550.982730.847160.839150.836300.826780.836810.79247
80.993810.992270.991030.992170.992660.990660.866160.846810.858920.840040.857030.80136
100.998470.996000.995060.993630.996060.995540.888320.882980.882880.871880.877180.86633
Test 940.971620.970390.969420.968220.965870.970860.812900.795450.804530.788910.791350.80267
60.982810.982090.981730.980960.979840.979060.867510.857100.852100.844830.855140.83386
80.986340.988260.984860.985710.986010.987270.884140.895370.906780.878920.884680.87814
100.992550.992360.992220.991190.988450.990310.923260.917150.917320.913800.896380.90067
Test 1040.978600.976800.972290.969560.974240.970190.789550.781690.743770.744450.754730.78499
60.986060.955430.985500.983450.983260.982650.818370.821210.808520.806240.794440.80907
80.992070.991770.991290.989940.989410.991090.861640.860150.853350.841450.835160.84721
100.995320.995000.994410.993840.991050.994230.893750.887110.879180.872360.845160.88534
Table 8. Comparison of optimal threshold values of CEFO at K = 4, 6, 8, and 10 using fuzzy entropy, Otsu’s and Kapur’s entropy.
Table 8. Comparison of optimal threshold values of CEFO at K = 4, 6, 8, and 10 using fuzzy entropy, Otsu’s and Kapur’s entropy.
ImageKFuzzyOtsuKapur
RGBRGBRGB
Test 1456 93 153 18757 88 132 19159 10 157 19752 86 130 190 60 80 100 17350 89 133 22364 103 145 18875 117 156 19967 109 150 194
620 59 92 128 167 19926 47 81 114 161 20325 48 76 111 138 19552 74 105 161 177 21768 80 101 124 167 19010 55 85 105 155 21251 84 116 150 184 212 55 87 117 147 178 21256 87 115 150 187 221
811 31 58 82 107 132 168 21112 42 64 92 114 149 181 21018 50 73 97 143 168 196 22325 36 69 90 124 152 169 231 42 59 70 79 115 158 202 23847 59 68 79 107 120 177 19738 60 81 104 132 158 185 21349 74 99 122 147 173 199 22416 56 87 117 143 172 200 227
1014 37 55 75 90 111 130 159 182 21011 41 61 95 122 146 168 185 208 23714 40 64 80 105 136 161 189 212 23145 51 67 107 119 129 165 199 222 23741 70 72 86 91 112 156 192 210 23035 50 82 100 101 115 139 148 178 19712 38 58 81 106 133 161 184 203 22745 61 82 103 124 146 168 189 210 22817 52 74 94 116 137 163 185 212 238
Test 2410 93 114 17186 143 194 22691 110 137 18939 92 144 20636 79 116 15820 45 86 14176 120 164 20889 130 171 21068 111 150 194
618 41 75 107 147 17787 101 133 153 192 23454 72 97 121 156 19222 59 88 121 183 19729 61 90 129 165 17918 23 36 64 95 23320 58 96 135 173 21319 58 99 138 177 21541 74 106 137 165 199
815 27 44 59 96 127 171 20414 22 43 65 117 144 190 22239 61 82 117 148 176 195 22326 47 103 127 151 181 222 2355 27 61 84 102 120 132 17013 45 55 61 104 140 180 18616 48 79 109 140 170 200 22817 52 84 114 147 175 201 22815 41 67 92 117 143 169 200
1032 55 75 108 144 171 195 215 226 2395 50 65 85 102 121 139 167 197 22831 47 64 87 108 125 158 179 209 23918 23 47 56 99 115 143 184 194 23722 71 76 78 99 108 138 158 196 21218 43 56 97 129 145 162 234 235 23713 35 56 81 104 130 155 179 205 23015 40 65 92 112 134 159 183 206 22913 37 58 81 102 123 143 163 190 209
Test 3444 92 139 188 31 92 139 18827 97 138 185 63 113 115 17093 99 136 19290 113 150 19249 98 145 20225 85 133 19258 122 157 202
632 62 93 128 173 19919 53 90 119 162 20636 71 105 136 183 20848 78 103 126 153 19036 71 114 168 185 20318 30 101 133 159 20130 70 102 127 166 20020 65 99 143 181 20826 80 104 135 168 202
822 43 68 91 112 138 172 208 17 51 73 98 123 148 181 20711 36 58 88 106 141 170 213 15 68 93 126 130 160 177 2257 40 95 123 146 172 186 20918 80 114 137 167 197 206 23913 33 60 87 125 157 185 21429 52 73 97 124 161 187 22212 42 61 86 108 137 167 205
1026 39 55 69 86 110 131 154 175 20742 29 56 79 104 121 141 163 197 2199 25 49 82 109 132 154 176 199 2221 37 64 71 99 122 151 176 212 23113 51 74 75 104 137 170 197 215 24651 82 106 111 130 168 181 203 233 24111 32 51 73 100 132 154 173 199 22314 37 55 75 93 109 123 149 181 21216 44 66 99 130 150 166 187 208 225
Test 4452 84 127 16467 102 141 20449 74 111 15536 86 170 19666 107 139 22941 83 91 14655 94 133 17368 124 178 21840 79 126 203
638 62 97 137 172 19548 71 111 144 175 21343 77 109 147 178 23533 63 93 127 163 21536 66 112 138 168 23122 48 83 135 183 20651 86 121 155 190 22446 80 116 149 183 21927 55 83 118 155 203
813 31 54 92 110 142 175 19835 60 86 114 137 169 200 22624 41 64 94 129 156 193 22433 49 87 98 142 193 231 2516 25 60 85 118 156 173 21819 37 39 41 47 67 103 15034 58 87 116 145 173 201 23135 61 86 112 139 165 192 22326 54 79 106 129 155 182 204
106 25 43 64 84 106 132 152 172 1897 27 39 52 75 99 130 157 190 22115 25 48 78 103 130 157 178 219 24039 48 73 105 135 152 156 167 197 22526 39 48 70 106 136 158 191 199 2209 28 49 66 71 94 131 159 170 18632 56 78 99 123 145 168 190 211 23526 48 69 90 112 133 157 180 204 22921 43 63 80 101 121 145 165 183 204
Test 5452 98 149 19855 89 164 21577 125 162 20995 147 188 198 59 135 142 212 81 116 164 20374 110 145 194 78 120 160 20522 88 147 218
622 63 93 131 179 22516 69 97 125 158 20415 61 97 131 166 20792 132 165 192 233 24378 116 119 138 187 229 68 125 134 158 190 24866 92 118 145 180 216 67 98 126 160 191 22322 62 99 142 180 219
816 59 77 94 116 134 162 19920 58 75 98 129 155 185 21419 56 77 115 145 173 196 22789 111 120 148 188 207 240 246 62 96 152 160 186 194 199 21517 76 100 131 141 169 197 23165 90 115 140 162 184 204 22564 88 112 135 158 182 206 22718 59 89 119 146 172 197 221
1013 29 57 77 93 107 130 165 188 21414 30 62 78 94 126 152 179 205 23119 44 56 72 89 109 129 159 176 20536 66 73 82 104 120 126 147 181 21060 104 127 170 175 181 185 190 206 221 35 58 87 97 135 174 175 189 214 22254 71 87 106 127 145 168 189 208 22562 81 100 120 143 163 183 201 218 23322 52 81 108 133 154 176 197 217 235
Test 6455 97 158 19564 99 153 19046 75 131 17954 104 163 22269 112 141 214 37 76 121 19160 109 155 20166 112 152 19958 110 156 213
643 60 91 118 165 20225 38 76 119 165 20547 69 97 133 155 19049 60 78 135 146 23952 64 81 132 190 23328 57 96 130 201 22745 79 112 147 181 21454 91 126 155 185 21738 78 119 155 188 222
816 42 73 91 129 157 190 22110 34 52 79 108 136 168 20723 39 64 86 120 137 172 20245 50 81 113 133 174 184 19720 37 51 76 112 141 168 2053 11 39 75 110 120 187 21135 60 86 112 139 166 194 22244 68 91 116 138 163 189 21732 58 86 112 138 164 196 226
1010 32 50 77 99 134 162 192 209 22813 36 47 69 107 140 164 185 200 22314 33 60 82 101 130 156 183 208 22841 49 70 99 111 112 171 191 237 24542 60 61 85 102 148 186 248 24913 24 50 87 104 131 137 156 178 22911 36 59 82 109 135 161 187 209 23238 58 79 100 119 140 165 189 208 22815 37 60 86 110 133 158 186 211 232
Test 7466 95 139 17625 66 108 15723 88 134 22085 132 159 183 66 104 171 190 73 104 155 17575 115 155 19381 119 157 19547 88 131 180
617 33 80 120 168 20520 80 108 133 164 18913 59 84 102 128 15177 106 144 188 212 224 51 65 76 110 152 20045 49 79 90 131 15463 96 128 160 194 22547 77 106 134 163 19517 47 79 112 144 180
820 54 79 106 126 146 169 20913 50 80 102 132 166 197 22319 40 63 88 119 148 200 22848 67 83 123 141 182 224 234 79 100 140 150 165 183 214 2546 45 49 80 107 131 165 20538 61 84 110 144 169 195 22360 88 116 142 168 196 216 23712 31 56 82 111 133 154 180
1010 30 41 63 89 110 129 154 178 21614 54 68 82 106 132 169 197 220 24315 41 66 85 105 133 151 190 214 23338 63 76 101 128 131 157 164 185 18627 41 83 91 103 119 157 184 219 22614 55 72 83 86 115 124 157 164 21441 61 82 101 120 142 167 194 216 23746 69 92 114 138 158 176 196 216 23917 42 67 84 111 130 144 161 179 196
Test 8448 73 173 21262 107 141 20743 79 124 215115 174 190 22592 120 136 156 46 100 139 15059 116 149 20283 122 182 21855 96 130 164
638 64 101 141 181 22432 61 85 126 155 19638 61 91 126 197 21854 146 170 197 219 232 98 119 143 156 167 21212 42 50 121 142 18242 81 118 148 179 20945 83 116 147 182 21825 49 72 101 131 164
815 41 64 85 110 133 175 21344 34 74 110 135 158 179 20325 40 69 104 136 182 207 23165 72 140 154 178 188 218 23969 111 141 150 170 218 24528 29 72 99 116 128 188 20037 61 90 120 144 171 200 22646 73 101 126 152 182 204 22520 43 61 80 99 121 143 164
1018 39 58 87 115 137 172 186 210 23021 36 63 79 111 129 150 180 208 22728 47 74 90 116 131 153 178 208 22628 57 59 113 123 163 168 191 213 21439 70 72 102 128 154 169 183 191 1969 21 27 77 79 128 168 185 187 19735 58 80 100 121 142 165 187 207 230 36 47 71 91 116 139 164 183 214 228 20 37 53 70 89 107 126 143 164 182
Test 9447 74 107 14955 90 132 16742 66 117 14940 93 131 211 80 137 195 20529 81 119 17276 123 167 21277 129 180 22743 90 140 194
613 43 62 92 126 16515 50 89 125 160 19519 45 81 115 151 23057 73 109 164 194 23064 105 136 166 201 23115 61 91 114 123 16562 97 132 167 203 23662 95 128 162 195 23131 63 95 126 158 194
814 36 53 75 92 123 152 17311 40 55 77 101 126 151 18326 41 66 91 128 155 200 23345 62 101 143 153 191 200 22945 92 128 151 156 193 199 21712 65 78 106 118 135 169 21455 83 110 138 162 188 213 23853 80 108 137 164 191 215 23825 47 70 97 119 142 171 199
1013 23 41 59 82 106 126 158 181 19811 43 54 72 89 105 134 163 188 21324 47 71 94 118 140 160 177 208 23730 48 87 101 131 159 164 220 227 25330 50 59 64 107 112 136 166 189 22812 20 47 80 93 106 122 132 150 16547 68 92 115 140 160 182 201 219 23938 55 74 95 118 142 164 187 212 23819 36 54 72 91 111 133 156 176 194
Test 10448 76 132 17531 69 117 16953 86 116 20057 112 164 206 63 108 179 202 47 95 172 21649 91 133 182 63 111 157 20659 99 148 192
634 57 90 127 166 19653 90 116 142 180 21442 56 104 136 169 22018 53 72 129 133 155 55 85 150 177 181 23138 75 127 175 203 22930 67 106 136 177 217 46 79 111 141 170 207044 78 114 152 190 225
819 33 62 91 133 165 195 21420 26 57 90 112 150 195 22012 38 55 88 113 150 186 22934 71 116 118 139 141 159 19915 42 60 103 149 155 170 2106 45 70 127 139 200 219 24822 52 79 109 133 161 189 21939 67 95 124 153 180 206 23040 66 91 116 142 167 194 224
1016 35 54 71 93 119 141 169 196 2319 26 36 51 72 106 132 166 195 22410 26 41 61 77 108 134 160 185 21313 28 49 58 95 105 127 145 151 18426 65 103 145 169 178 183 193 207 23919 49 63 107 161 187 191 226 242 24921 42 67 95 121 148 173 197 217 232 34 55 78 101 122 166 186 207 23129 47 70 92 117 139 161 183 204 226
Table 9. Comparison of CPU Time, PSNR, MSSIM, and FSIM computed by CEFO at K = 4, 6, 8, 10 using fuzzy entropy, Otsu’s and Kapur’s. The bold numbers are the best values in the relevant index.
Table 9. Comparison of CPU Time, PSNR, MSSIM, and FSIM computed by CEFO at K = 4, 6, 8, 10 using fuzzy entropy, Otsu’s and Kapur’s. The bold numbers are the best values in the relevant index.
ImageKCPU TimePSNRMSSIMFSIM
FuzzyOtsu’sKapurFuzzyOtsu’sKapurFuzzyOtsu’sKapurFuzzyOtsu’sKapur
Test 140.175410.119190.1806519.086916.037816.03780.973680.958660.941990.758820.756990.73718
60.216230.125060.2411722.496619.052919.05290.986310.967350.970920.850390.798780.81115
80.262190.134430.2586623.931521.369221.97680.991990.981350.985580.886180.834300.86369
100.332840.156290.2728725.160322.807123.80190.993610.988150.990600.910980.857220.89851
Test 240.181520.106030.2496718.501118.466518.48540.960130.958590.959390.729720.723490.71672
60.230140.110120.2558721.418620.985621.27950.984590.980130.980490.813150.806870.80330
80.267450.126660.2654223.474523.247323.24360.990980.989260.989580.862100.857250.86029
100.354150.145910.3035427.196024.660526.82560.996310.992860.995840.928760.879110.92489
Test 340.193540.114620.3973217.926617.534017.77980.973730.967670.971580.719190.704890.71365
60.244060.120740.4504721.459520.405421.31850.988620.983740.987930.831440.792140.82536
80.265720.133480.4828523.525721.571022.73840.993190.987260.991940.885100.828990.87669
100.311040.162540.2115425.110723.383524.36020.995130.992270.994170.914710.882180.92049
Test 440.206540.092020.2186618.579118.456918.49890.975040.943140.974970.759660.743680.74512
60.256130.126000.2285121.996221.716222.01910.988170.986120.987240.858250.849350.85897
80.295650.153380.2410723.657423.006823.58910.992210.990730.992390.897380.884830.89628
100.342260.166490.2701624.821224.551924.79340.994850.993080.994600.922180.912350.92176
Test 540.200170.105690.2325418.023916.479717.27790.976530.961710.970620.767950.742410.76655
60.234850.127500.2418620.347917.962020.06470.986160.970500.983000.828390.813790.82815
80.294240.152110.2725822.166119.699321.48060.991310.978740.986200.862820.833960.87859
100.344800.161730.2977524.013821.918123.17570.994700.988890.990590.893130.854750.89169
Test 640.194720.102860.2496418.109517.851217.97590.970600.970410.966220.773660.763190.74541
60.242630.123250.2823721.483120.406020.68240.985910.981860.981650.835680.815300.82596
80.308640.153390.2941323.229022.135422.98590.991980.988550.989490.884460.854080.87426
100.337250.183920.3209824.768122.889425.02960.994240.989750.993500.903430.868980.90198
Test 740.206780.104020.2491218.873917.870417.82170.975960.964640.964710.751910.722970.74535
60.235620.109560.2553122.144320.325121.62100.988620.978720.986090.849830.793390.84398
80.298070.129370.2736723.783920.771923.19490.992600.979160.990480.885820.811060.87668
100.342170.142260.2755524.743323.184324.36860.994070.990970.993630.906590.850530.90773
Test 840.214500.106170.2342218.949519.054118.94150.983540.983710.984010.790860.790820.79759
60.250680.128960.2613121.410921.141121.44390.991110.986870.990320.847160.802990.81085
80.319940.141360.2851723.067723.284223.59140.993810.993750.995760.866160.857710.87685
100.337250.187270.2992625.435722.605625.40620.998470.991650.996810.888320.829700.89139
Test 940.193240.110920.2359516.456816.359317.15800.971620.970640.969850.812900.805960.81157
60.237890.129990.1795818.467518.010919.08430.982810.982050.983940.867510.864520.89018
80.270570.142290.2284721.009821.004521.35670.986340.986290.986990.884140.870840.90497
100.331510.181280.2545222.366622.354723.15070.992550.992510.994280.923260.912620.93015
Test 1040.190390.104660.2051118.581717.960718.07650.978600.972680.979340.781690.765860.77935
60.236560.112610.2404620.893820.892521.00270.986060.985260.985860.821210.829070.83085
80.263900.122000.2483422.624121.669123.18040.992070.988820.991190.860150.833650.85931
100.333040.126770.2717925.364023.856725.08860.995320.992890.994280.887110.881950.88555
Table 10. Comparison of p-values between CEFO and other algorithms based on Fuzzy entropy.
Table 10. Comparison of p-values between CEFO and other algorithms based on Fuzzy entropy.
Dependent VariableProposed AlgorithmAlgorithmsp-ValueDependent VariableProposed AlgorithmAlgorithmsp-Value
CPU TimeCEFOEFO0.038791(*)MSSIMCEFOEFO0.201224
ABC3.76E-50(*) ABC0.157924
BA1.08E-46(*) BA0.006606(*)
WDO1.23E-43(*) WDO0.004560(*)
BSA1.95E-47(*) BSA0.006092(*)
PSNRCEFOEFO0.457477FSIMCEFOEFO0.364938
ABC0.466803 ABC0.297925
BA0.037665(*) BA0.034212(*)
WDO0.021986(*) WDO0.016480(*)
BSA0.020720(*) BSA0.003728(*)

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Song, S.; Jia, H.; Ma, J. A Chaotic Electromagnetic Field Optimization Algorithm Based on Fuzzy Entropy for Multilevel Thresholding Color Image Segmentation. Entropy 2019, 21, 398. https://doi.org/10.3390/e21040398

AMA Style

Song S, Jia H, Ma J. A Chaotic Electromagnetic Field Optimization Algorithm Based on Fuzzy Entropy for Multilevel Thresholding Color Image Segmentation. Entropy. 2019; 21(4):398. https://doi.org/10.3390/e21040398

Chicago/Turabian Style

Song, Suhang, Heming Jia, and Jun Ma. 2019. "A Chaotic Electromagnetic Field Optimization Algorithm Based on Fuzzy Entropy for Multilevel Thresholding Color Image Segmentation" Entropy 21, no. 4: 398. https://doi.org/10.3390/e21040398

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