A Logistic Chaotic Barnacles Mating Optimizer With Masi Entropy for Color Image Multilevel Thresholding Segmentation

Barnacles mating optimizer (BMO) is an evolutionary algorithm that simulates the mating and reproductive behavior of barnacle population. In this article, an improved Barnacles mating optimizer based on logistic model and chaotic map (LCBMO) was proposed to produce the high-quality optimal result. Firstly, the logistic model is introduced into the native BMO to realize the automatic conversion parameters. This strategy maintains a proper relationship between exploitation and exploration. Then, the chaotic map is integrated to enhance the exploitation capability of the algorithm. After that, six variants based on LCBMO are compared to find the best algorithm on benchmark functions. Moreover, to the knowledge of the authors, there is no previous study on this algorithm for multilevel color image segmentation. LCBMO takes Masi entropy as the objective function to find the optimal threshold. By comparing different thresholds, different types of images, different optimization algorithms, and different objective functions, our proposed technique is reliable and promising in solving color image multilevel thresholding segmentation. Wilcoxon rank-sum test and Friedman test also prove that the simulation results are statistically significant.


I. INTRODUCTION
With the emergence of computer technology, image processing has been widely used in many fields. Image segmentation is one of the classical topics in image processing [1]. It divides the original image into significative and multiple sub-regions according to intensity, color, texture and other attributes of the image [2]. Image segmentation is often the pretreatment stage of higher-class processing such as: image analysis, object recognition, and computer vision. Consequently, the performance of higher-class processing system depends on the accuracy of the segmentation technique adopted [3]. Researchers have proposed many kinds of segmentation, including edge detection, histogram based thresholding, region, feature clustering, and neural networks [4]- [6]. Histogram based thresholding is a The associate editor coordinating the review of this manuscript and approving it for publication was Seyedali Mirjalili . simple and the most commonly used image segmentation approach [7], [8]. Thresholding methods can be divided into two categories: bi-level thresholding and multi-level thresholding. Bi-level thresholding means that the target and background can be clearly distinguished by a single threshold value. Multi-level thresholding denotes that the given image can be segmented into various classes by multiple threshold values [9]- [11].
In recent years, the methods to determine the optimal threshold for a given image can be divided into two categories: parametric and non-parametric methods [12]. In the parametric techniques, it is assumed that the probability density function of each class is known. The common parameter methods generally follow a certain distribution of probability density, such as Gauss distribution [13], Poisson distribution [14], [15], generalized Gaussian distribution [16], and so on. This methods differs from the actual situation to some extent. In addition, the segmentation is affected when classes are highly overlapped. Therefore, the parametric approaches are not ideal choices in this case. For non-parametric methods, the probability density function is usually unknown, and the threshold is generally searched by optimizing the objective function [17]. The classical non-parametric methods are mainly as follows: Otsu proposed a method to maximize the variance between classes at first [18]. Then the methods based on information entropy theory are proposed, which are categories to measure homogeneity. Among them, the most representative entropy approaches for image segmentation include: Minimum Cross entropy [19], Kapur entropy [20], Renyi entropy [21], Tsallis entropy [22], and Masi entropy [23]. They can be easily extended to multi-level thresholding.
Among them, a novel generalized entropy measurement called Masi entropy has attracted increasing attention in the past few years. Furthermore, Tsallis and Renyi entropies are two different generalizations along two different paths. Furthermore, Tsallis entropy is generalized to non-extensive systems, while Renyi entropy is quasi-linear devices. However, Masi entropy is extended to non-extensive systems and non-linear devices, including Tsallis entropy and Renyi entropy [24], [25]. A publication for multilevel thresholding segmentation of color satellite images based on Masi entropy has been proposed by Shubham in 2019. Simulation results show that the proposed method is effective and has better segmentation performance than Kapur, Renyi and Tsallis entropy [26]. Although the exhaustive search is effective in image segmentation, it cannot find the optimal threshold, and the complexity increases exponentially with the number of thresholds. [27], [28]. In order to speed up this process, one option is to replace some classical exhaustive searches based on meta-heuristic search algorithms.
Sulaiman proposed a novel bio-inspired algorithm called Barnacles mating optimizer (BMO) in 2020 [29]. Obviously, the BMO algorithm simulates the intelligent behavior of barnacles in nature, including selection process and reproduction. It can be seen from the lecture that the BMO algorithm has outstanding convergence ability, fast convergence speed and excellent search ability. But according to the no free lunch theorem, it can be seen that no independent algorithm can solve all optimization problems [30]. Therefore, the BMO algorithm need to be improved. The logistic regression model is a common improvement strategy and widely used in various optimization methods. In 2018, Qasim et al. applied logistic regression model for optimization of feature selection. The results showed that the proposed method can obtain a great classification performance with few features [31]. In 2019, by using logistic regression prediction model, Ghazvini et al. solved the problem of the variables affecting tuberculosis [32]. Therefore, this article chooses logistic regression model to improve BMO. Meanwhile, chaotic map is an excellent mathematical strategies, which can improve the performance of meta-heuristic algorithm in avoiding local optimization. Chaotic map can provide random behavior without random component [33]. Accordingly, scholars have added chaotic map to the FIGURE 1. Selection of mating process of ten barnacles [69]. optimization algorithm to improve the ability of algorithms. J. Alikhani Koupaei et al. proposed a new optimization algorithm based on chaotic maps. Experimental results proved that the modified algorithm was competitive in multi/unimodal objective functions [34]. A. Naanaa embedded spatiotemporal map into chaos optimization algorithms to improve its convergence and efficiency [35]. Yang et al. proposed chaos optimization algorithms based on chaotic maps to achieve the high efficiency, which improve the convergence speed and accuracy [36]. Chuang et al. combined chaotic maps with s binary particle swarm optimization, which sped up search process the algorithm [37]. Motivated by these successful applications of the strategies, the authors introduce logistic model and chaotic map into BMO algorithm to increase the diversity of algorithm and prevent skipping over the optimal solutions. In addition, it also better balances the exploration and exploitation trends.
Image segmentation based on histogram and global threshold is most commonly used to determine threshold value. Masi entropy is a bi-level threshold method based on the gray level and its histogram. And Masi entropy objective functions can be maximized by LCBMO to find the optimum threshold value. Furthermore, the provided image is segmented into unique classes. In this article, a series of experiments are conducted, and the experimental results are analyzed and discussed in details. The performance of image segmentation is measured in terms of objective function values, peak signal-to-noise ratio (PSNR) [38], [39], structural similarity index (SSIM) [40]- [42], feature similarity (FSIM) [43], [44], Wilcoxon rank-sum test [45], [46], and Friedman test [47]. In order to compare various algorithms more intuitively, the convergence curve based on objective function values are drawn. The experimental results confirm that the proposed Barnacles mating optimizer based on logistic model and chaotic map can be effectively used for multilevel thresholding.
The remainder of this article is organized as follows: Section II discusses related studies. Section III outlines some preliminaries. Section IV gives the proposed BMO based on logistic model and chaotic map for multilevel thresholding color image segmentation. The benchmark functions experiments are presented in Section V. Other simulation experiments and results analysis are described in Section VI. Finally, Section VII concludes the work and suggests some directions for future studies.

II. LITERATURE REVIEW
In 2015, A.K. Bhandari et al. proposed satellite image segmentation model based on modified artificial bee colony algorithm, in which the Kapur, Tsallis and Otsu functions are used to determine the threshold [48]. And in 2016, Mozaffari et al. introduced an inclined planes system optimization algorithm to solve the problems in different fields of science and engineering [49]. The convergence heterogeneous particle swarm was utilized to find the best thresholds in literature, which has the better stability and convergence in 2017 [50]. Oliva et al. combines cross entropy with crow search algorithm for image segmentation to reduce computational complexity in the same year [51]. H. N. Liang et al. applied modified grasshopper algorithm in image segmentation technology, which showed excellent results [52]. Furthermore, cuckoo search algorithm based on minimum cross entropy is proposed to make the method more practical and uncomplicated [53]. In 2018, S. Kotte presented an improved differential search algorithm for gray scale images to increase its computational efficiency and accuracy of segmentation [54]. In 2019, H. S. Gill exploits minimize cross entropy as the objective function, and uses teaching-learningbased optimization algorithm to select multilevel threshold values. The experimental outcomes indicate the proposed method has an advantage of efficiency and robustness [55]. S. J. Mousavirad published the human mental to search the optimal threshold to increase segmentation efficiency [56]. Bohat studied a new heuristic for multilevel thresholding of images, and combined whale optimization algorithm. Meanwhile, the results demonstrate that the proposed algorithm is superior to the other algorithm [57]. A novel beta differential evolution algorithm-based fast multilevel thresholding is applied for color image segmentation in 2020. Then the performance is proved to be superior to other methods in image segmentation such as artificial bee colony, particle swarm optimization and differential evolution [58]. And a competitive swarm algorithm was applied in image segmentation guided based opposite fuzzy entropy to improve the segmentation accuracy in the same year [59]. Furthermore, a benchmark of recent population-based metaheuristic algorithms was proposed for high-dimensional multi-level maximum variance threshold selection, which has attract much attention [60]. D. Oliva combined the thresholding techniques and the evolutionary Bayesian network algorithm to generate the accurate class even in complex condition [61]. E. R. Esparza represented an efficient harris hawks method used into the image segmentation so as to produce the efficient and reliable results [62].
These algorithms are successfully applied to multilevel thresholding and reduce the computational complexity, which inspire further research by scholars.

A. MULTILEVEL THRESHOLDING
Threshold segmentation processes the digital image histogram. We use an algorithm as the segmentation criterion, and the threshold that satisfies the criterion function is called the optimal segmentation threshold. By comparing with the optimal threshold, the image is divided into target region and background region. The image threshold method can be summarized into two categories: bi-level thresholding segmentation and multilevel thresholding segmentation. Bi-level thresholding segmentation cannot completely extract the target at a particular image segmentation, so we need multilevel thresholding to divide the whole image into multiple regions. Multilevel thresholding segmentation can highlight the features among image regions.
For a n-bit gray image, the gray level of the image is L = 2 n and the gray level interval is {0, 1, . . . , L − 1}. n i denotes the number of pixels whose gray level is i. N denotes the total number of pixels. p i denotes the probability density of ith the gray value. They are defined as follows: Suppose there are K thresholds of t 1 , t 2 , . . . , t k . They divide the gray level of a given image into K + 1 classes: The selection of threshold is very critical, and it is related to the quality of the segmentation results. In this article, Masi entropy method are adopted.

B. MASI ENTROPY
According to Tsallis and Renyi entropy, Masi proposed a novel generalized entropic measure by introducing the 213134 VOLUME 8, 2020 concept of conventional thermodynamic entropies in 2005 [23]. Masi entropy segment color images by utilizing thorough probability function, and its detailed definition is as follows: Eq. (4) is proposed to express the probabilities of class occurrence ω j , 0 ≤ j ≤ k. Based on the non-extensivity of Tsallis entropy the additivity of Renyi entropy, Eqs. (5) and (6) for calculating Masi entropy are proposed, where E j stands for Masi entropy. r ≤ 0, r = 1, In this article, the power parameter r is set to 1.18 through experiments [24]. Masi entropy method obtains the optimal threshold values according to maximizing the total entropy. The optimal threshold is represented by Eq. (7).
Compared with the histogram of grey scale image, the RGB image is more complex. In RGB space, every color pixel of the image is composed of red, green and blue [63], [64]. In this article, three channel components of R, G and B are extracted at first. Then, each channel is calculated by Masi entropy, and the objective function is maximized to find the optimal threshold for the corresponding channel [65]. The RGB channel components are divided by the optimal threshold and then merged to form the ultimate segmented image.

C. BARNACLES MATING OPTIMIZER
Barnacles mating optimizer (BMO) [29] is a novel bioinspired optimization algorithm inspired by the mating process of barnacles. Barnacles live in water and are famous for their long penises [66]. According to initialization, selection, and reproduction, simulation optimization process is realized. The mathematical model is described in details as follows.
In the initialization process, the barnacle population can be expressed in the following matrix.
where N is the number of barnacle population, n is the number of control variables. In the next selection process, the parents to be mated are randomly selected from the population. The mathematical forms are proposed in Eq. (9) and (10).
where barnacle_d represents the Dad of the offspring, barnacle_m represents the Mum of the offspring.
In the reproduction process, BMO mainly produces the offspring based on Hardy-Weinberg principle [67], [68]. The interesting fact is that the penis length of the barnacle (pl) plays an important role in determining the exploitation and exploration of BMO algorithm. When pl is equal to 7, it can see from Fig. 1 that barnacle #1 can only mate with one of the barnacles #2-#7. Then, the exploitation process will be occurred. In this case, Eq. (11) is proposed to produce new offspring from parents.
where p is a random number drawn from the standard normal distribution between [0, 1], q = (1 − p), x N barnacle_d and x N barnacle_m are the variables of Dad and Mum of barnacles respectively which are selected in Eq. (9) and (10). Furthermore, p and q represent the percentage of genotype of Dad and Mum in the new generation. The new offspring is produced based on genotype frequencies p and q of parents. If barnacle #1 mates with barnacle #8-#10, the offspring is proceeded by sperm cast process. Then, the exploration process will be occurred. In this case, Eq. (12) is proposed to produce new offspring from parents. (12) where rand() is the random number between [0, 1]. It can be noted that Eq. (12) shows the new offspring is produced only based on Mum. Generally, the positions of barnacles are updated in each iteration by Eq. (11) or Eq. (12) to find the best position (the best solution).

D. LOGISTIC MODEL
The adaptive parameter allows the algorithm to smoothly transit between exploration and exploitation. Therefore, it is important to choose a suitable conversion model. The logistic model and its mathematical expression are given as following [70]. How the conversion parameter accords with the change law of logistic model will be introduced in Section III.  where t is the number of iteration, and λ is the initial decay rate. By solving differential Eq. (13), logistic function (14) is obtained.
It can be seen from (7) that P(t) = P min when t = 0, while P(t) = P max , t → ∞.

E. CHAOTIC MAP
Chaotic map is one of the best mathematical strategies to improve the performance of the metaheuristic algorithm in terms of local optima avoidance. Chaotic map can provide random behavior without the need for random component [33]. The mathematical modulation of six different chaotic maps are as following. Fig. 2 visualizes the chaotic behavior. The initial value may have a significant effect on the fluctuation patterns of some chaotic maps. Fig. 2 is drawn based on the initial value of 0.7 [71], [72].
The Chebyshev map is formulated as [73]: The equation of the Gauss/mouse map is defined as follows [74]: VOLUME 8, 2020  The Logistic map is defined as [75]: The Singer chaotic map equation is expressed as [76]: (18) The Sinusoidal map is represented by the following equation [77]: The family of Tent map can be represented as [78]:

IV. PROPOSED METHOD A. IMPROVED BARNACLES MATING OPTIMIZER (LCBMO)
Metaheuristic algorithms all have two important stages in the search level: exploration and exploitation. The balance between these two capabilities directly affects the performance of the algorithm. In the native BMO algorithm, low search accuracy and limited production capacity are the main drawbacks. In order to improve the competence of BMO algorithm to handle optimization problems, two strategies regarding logistic model and chaotic map are introduced. The pattern and mechanism of improvement will be described in details.
In the original BMO algorithm, pl can be set to 50%-70% of the total population size by repeated experiments, which is beneficial to balance the exploitation and exploration. More exploration processes will occur when the value of pl is small. On the contrary, more exploitation processes occur when the value of pl is large. Finally, the authors set pl to a constant value (70% of the population of barnacles). In view of this, the logistic model is used to improve pl to realize the adaptive transformation of parameters. The parameter is improved by the following equation.
It can be seen from (7) that pl(t) = pl min when t = 0, while pl(t) = pl max , t → ∞. The logistic model makes BMO algorithm to perform high exploration in the initial stage and more exploitation in the final stage of search. It can be regarded as a proper strategy to balance the two stages.
In addition, in order to avoid local optimal values, the chaotic map is used to improve the position updating equation of barnacles. Eq. (12) is replaced by the following form.
x n_new i = m × x n barnacle_m (22) where m is a chaotic vector obtained based on six chaotic maps. The chaotic vector can provide random behavior without the need for random component. The purpose of introducing this strategy is to make solutions search in space as widely, randomly and globally as possible. The exploitation efficiency is the primary beneficiary. Finally, the improved version of BMO is called LCBMO whose pseudocode is provided in Algorithm 1.
The If selection of Dad and Mum = pl 8: For each variable 9: Generate offspring using Eq. (11) 10: End for 11: Else if selection of Dad and Mumpl 12: For each variable 13: Generate offspring using Eq. (22)  The process of finding the threshold by Masi entropy is actually to find the optimal solution. However, they have high computational complexity when dealing with multiple thresholds. In order to achieve efficiency, it is entirely possible to use LCBMO algorithm to deal with this. The basic steps are described as follows: Firstly, we input selected color images and calculate the components of the histogram. Next, the number of search agents and iterations are initialized, and the fitness of initial population is calculated. The dynamic pl value is used to determine the position update mode of barnacles. Individual with high fitness value is preserved. Repeat this process until the maximum number of iterations is completed. The best position represents the optimal threshold values of segmentation. The flowchart is provided in Fig. 3.

V. BENCHMARK FUNCTIONS EXPERIMENT
In this section, 23 standard functions are used to evaluate the optimization improvement of LCBMO algorithm. These benchmark functions are divided into three groups: unimodal (f 1 − f 7 ), multimodal (f 8 − f 13 ) and fixed-dimension multimodal (f 13 − f 23 ). Furthermore, the relevant composition, dimension, range limitation and optimal position of 23 functions can be found in [49]. Meanwhile, all the experimental series are carried out on MATLAB R2016b, and the computer is configured as AMD A8-7410 APU with AMD Radeon R5 Graphics @2.20 GHz, using Microsoft Windows 7 system. For the experiment, the most traditional and improved BMO algorithm for global optimization are adopted. And the population size is set to 30 while the number of iterations is set to 500. Moreover, all experiments are conducted 30 times.
In LCBMO, the logistic model can make the algorithm be highly explored in the initial stage and developed more in the later search period. Compared with the traditional BMO, it has excellent exploration and exploitation. For the chaotic map, putting it into BMO as a strategy can greatly improve the convergence and high efficiency. The LCBMOs are divided into 6 different types. LCBMO-1 to LCBMO-6 all introduce the logistic model, but utilize Chebyshev, Gauss/mouse, Logistic, Singer, Sinusoidal, and Tent maps, respectively. The performance of algorithms is evaluated according to the mean value and standard deviation (Std). The stability of each model is evaluated by Std value. Meanwhile, the best results has been highlighted in boldface in Table 1. It can be found from the Table 1 that the LCBMO-1, LCBMO-2, LCBMO-3, LCBMO-4, LCBMO-5 and LCBMO-6 models show much better results compared to BMO on unimodal benchmark functions. In other words, Chebyshev, Gauss/mouse, Logistic, Singer, Sinusoidal and Tent chaotic maps have successfully improved the performance of the BMO algorithm. For multimodal benchmark functions, it can be seen that the LCBMO-2 model shows the best value and great stability in most cases. Although BMO, LCBMO-1 and LCBMO-6 models show competitive result in some cases, this result still proves that the optimal solution obtained by the proposed method is high-quality. In addition, as for fixed-dimension multimodal benchmark functions, compared to other hybrid model, LCBMO-2 model can keep the population diversity in the later iteration. Therefore, the ability to avoid local optimization has enhanced. Moreover, it can be found from the Table 1 that LCBMO-2 shows the lowest value of Std, which indicates better stability. Thus, it can be said that the VOLUME 8, 2020 proposed method in this article is more effective than BMO in 23 benchmark functions, so this article combines LCBMO with multilevel thresholding segmentation method to improve the image segmentation accuracy.

VI. COLOR IMAGES SEGMENTATION EXPERIMENT A. PREPARED WORKS 1) EXPERIMENTAL SETUP
All the experimental series were carried out on MATLAB R2016b, and the computer was configured as AMD A8-7410 APU with AMD Radeon R5 Graphics @2.20 GHz, using Microsoft Windows 7 system.

2) COMPARED ALGORITHMS
After the thresholding segmentation method is extended from two-level thresholding to multi-level thresholding, its computational complexity increases exponentially. Therefore, a large number of optimization algorithms are applied in multithreshold segmentation. In order to prove the superiority of the modified algorithm, two sets of 10 meta-heuristic algorithms which have been proposed and widely applied to multithreshold segmentation are selected for comparison experiments, including MABC [48], CSA [51], GOA [52], CS [53], EO [79], MPA [80], IDSA [54], TLBO [55], WOA-TH [57], and BDE [58]. These comparison algorithms have different search strategies and mathematical formulas and are representative algorithms for multithreshold. The maximum of iterations for all algorithms is 500 and the population size is 30. We follow the same parameters in the original articles. The main parameters of various algorithms are shown in Table 2.

3) COLOR IMAGE DATABASE
In this article, two sets of twelve color images are selected from the Berkeley university database and NASA landsat image dataset for performance analysis. The satellite images can be downloaded from the website [81]. Fig. 4 shows the original test images and the corresponding histograms for each of color channels (red, green, and blue). All images are in JPG format.
The experiments use the control variable method, in which each algorithm runs each image 30 times separately. The number of threshold K includes: 4, 8, 12, and 16.

B. EVALUATION METRICS 1) PEAK SIGNAL TO NOISE RATIO
The peak signal to noise ratio (PSNR) is an objective image quality evaluation algorithm based on pixel error. A higher PSNR value indicates that the quality of the distorted test image is better and closer to the original reference image. However, it is based on the error between corresponding pixels and does not take into account the visual characteristics of human eyes. Its calculation formula is as follows: where L represents the grayscale range of the image. For 8bit grayscale image, L = 255. MSE is the mean square error between the original image and the processed image.
where M × N is the size of the image, R(m, n) represents the gray value of coordinates at the reference image (m, n), and I (m, n) represents the gray value of coordinates at the distorted image (m, n).

2) STRUCTURAL SIMILARITY INDEX
It is an objective image quality evaluation algorithm based on structural similarity. It measures the image similarity from brightness, contrast and structure. SSIM value range is [0, 1]. If the value is closer to 1, the image distortion is smaller. It is defined as follows where U R and U I are the average gray values of the original image R and the segmented image I . σ 2 R and σ 2 I represent the variance of image R and image I respectively. σ RI is the covariance of image R and image I . C 1 = (0.01L) 2 , C 2 = (0.03L) 2 .They are constants that are used to maintain stability.

3) FEATURE SIMILARITY INDEX
On the basis of SSIM, researchers have proposed a new image quality assessment metric based on underlying features, namely feature similarity algorithm (FSIM). Researchers use two complementary features of phase congruency (PC) and gradient magnitude (GM) to calculate FSIM.
where is the pixel field of the entire image, S L (x) represents the similarity value of each position x,and PC m (x) denotes the phase consistency measure. where S PC (x) is the similarity measure of phase consistency, S G (x) represents the similarity measure of gradient magnitude, and α, β are both constants.
where T 1 and T 2 are positive constants that increase stability.

4) WILCOXON RANK-SUM TEST
Wilcoxon rank-sum test is used to compare the two samples. The p value returned represents the probability whether two independent samples are identical, and the h value returned represents the result of hypothesis test. The null hypothesis H 0 represents the statement of no difference. At significance level 5%, it generally believe that if p <0.05(or h = 1) means rejection of the null hypothesis, if p >0.05 (or h = 0) means that H 0 cannot be rejected at the 5% level. VOLUME 8, 2020

5) FRIEDMAN TEST
Non-parametric Friedman test is applied to estimate which algorithms have significant differences. This multiple comparison can be used for comparisons between more than two algorithms and ranks the each algorithm separately.

C. BERKELEY IMAGES SEGMENTATION EXPERIMENT
This subsection analyzes the results provided by Masi entropy implementations based on CSA, GOA, CS, TLBO, EO, MPA, and LCBMO-2, after being applied to segment the 6 Berkeley images (image 1-6). Fig. 5 represents segmented images into four classes using LCBMO-2 algorithm and the fitted histogram with the thresholds for the segmented images. The Berkeley images are segmented using Eq. (6) and the best threshold values found by the LCBMO-2. Fig. 5 visually shows the search capabilities of LCBMO-2 in K-dimensional search space. Table 3-5 report PSNR, SSIM, and FSIM from the evaluation of the segmented images, respectively. From the    Table 4 that LCBMO-2 based method outperform the other algorithms again, which shows the segmentation accuracy of proposed algorithm is satisfied. On comparing the FSIM values, which are given in Table 5, it can be observed that the values increase as the number of the thresholds increase. And the proposed method gives the highest values, accounting for 75% of the total results. These results indicate the precise search ability of LCBMO-2 based method, which is suitable for color Berkeley images segmentation.
As the stochastic nature of metaheuristic algorithms, the experiments are conducted over 30 runs. Then the average fitness values at K = 16 are presented in Table 6. It can be seen from the tables above that the LCBMO-2 based method gives all the best values. In order to verify the stability of proposed algorithm, the results of the fitness function values at K = 16 obtained for 30 runs is plotted as boxplots. A narrower boxplot indicates better stability. The boxplots obtained by all algorithms are shown in Fig. 6. From the figure it is found that LCBMO-2 based method gives narrower boxplots as compared to other algorithms, which shows the better consistency and stability of proposed algorithm. VOLUME 8, 2020

D. SATELLITE IMAGES SEGMENTATION EXPERIMENT
With the progress of earth observation technology and the deepening of understanding of earth resources and environment, the requirements for the quality and quantity of high-resolution remote sensing data are constantly increasing. The main features of high-resolution satellite images  include: rich texture information corresponding to objects, large imaging spectrum, and short revisit time. Therefore, the segmentation and evaluation of satellite images is a challenging work.
This subsection analyzes the results provided by Masi entropy implementations based on MABC, IDSA, WOA_TH, BDE and LCBMO-2, after being applied to segment the 6 satellite images (image 7-12). The segmented images (image 7, image 8, and image 10) obtained by Masi entropy with different thresholds levels are given in Fig. 7. Besides, the corresponding threshold values are given in Table 7 and Appendix Table 2. From the segmentation results we can find that the images with higher levels (such as K = 8, 12, and 16) contain more information than the others.
The PSNR, SSIM, and FSIM values obtained by all algorithms using Masi entropy techniques are reported in Tables 8-10. In terms of PSNR values, the proposed algorithm gives the highest values, accounting for 79.2% of the total results. Besides, the proposed algorithm gives the highest SSIM and FSIM values, accounting for 75% of the total results. Taking Image 12 (at K = 12) as an example, WOA_TH algorithm achieves the highest SSIM value of 0.9491. Our proposed algorithm ranks second and is not much different from the results obtained by WOA_TH. The average fitness values of Masi entropy functions are presented in Table 11. It can be seen from the table above that the LCBMO-2 based method gives the best values in general. Moreover, in order to reflect the performance of LCBMO-2 more intuitively, the convergence curves of Masi entropy functions (for K = 16) are shown in Fig. 8. It can be found that the proposed algorithm outperforms other algorithms in general. In other words, the LCBMO-2 based method gives higher position curves using Masi entropy technique. It is further proved that the two strategies   (logistic model and chaotic map) can improve the search accuracy and production capacity of the native BMO algorithm, and the LCBMO-2 algorithm can use the search space more effectively to complete the optimization task of image segmentation. For visual analysis, the results of the running time (in second) based on each algorithm are represented as stacked bar diagrams in Fig. 9. It can be seen that the running time is sorted as follows: BDE > MABC > LCBMO-2 > WOA_TH > BDE. Although our proposed algorithm is not the champion algorithm in terms of running time, it is not too bad. The improved strategy used slightly increases the computational cost of the algorithm. In general, the running time of the proposed algorithm (LCBMO-2) is acceptable.
In order to statistically prove the superior performance of the proposed algorithm, Wilcoxon rank-sum test and Friedman test are used to evaluate the significant difference among algorithms. The p values of Wilcoxon rank-sum test are given in Table 12. For example, the proposed algorithm gives better

E. DIFFERENT OBJECTIVE FUNCTIONS EXPERIMENT
It can be seen from the above experimental results that LCBMO-2 based method is superior to other compared algorithms using Masi entropy. In order to obtain a simple and VOLUME 8, 2020

VII. CONCLUSION AND FUTURE WORK
In this article, the Barnacles mating optimizer algorithm based on logistic model and chaotic map for multilevel thresholding color image segmentation is proposed. Among many thresholding segmentation methods, Masi entropy method is adopted. The proposed algorithm is used to find the optimal threshold for color images. Meanwhile, 10 algorithms are selected for comparison. Objective function value, PSNR, SSIM, FSIM, Wilcoxon rank-sum test, and Friedman test are used to evaluate the segmentation quality. Firstly, by the convergence curve and boxplot at K = 16, it can be seen that LCBMO-2 algorithm can find larger objective function value more times. Then, in terms of PSNR, SSIM, FSIM, the value obtained by the LCBMO-2 algorithm is larger than other algorithms in most cases. It concludes that the segmentation performance based on LCBMO-2 algorithm is superior. Furthermore, the results of Wilcoxon rank-sum test and Friedman test demonstrate that LCBMO-2 is significantly different from other algorithms, and the improvement is effective. To sum up, a variety of experiments fully proves that LCBMO-2 algorithm has higher search accuracy and convergence speed, stronger robustness, and the overall performance of the algorithm is enhanced.
However, like other optimization algorithms, LCBMO has certain limitations. The computational complexity needs to be reduced. Runtime is important for real-world problems. The distributed island model can organize population into small independent groups (islands) and make the algorithm run in parallel. We believe that it is a potentially effective strategy to reduce the complexity. In the future, the relevant research directions are given as follows: (1) Extend the algorithm to multi-objective problem for obtaining superior segmentation effect.
(2) Explore to introduce LCBMO-2 algorithm in other fields, such as machine learning and data mining.

APPENDIX
See Tables 17 and 18. GANG ZHENG received the M.S. degree in agricultural electrification and automation from Northeast Forestry University, China, in 2008. He is currently a Lecturer with Northeast Forestry University. His research interests include intelligent detection, information fusion, and image processing.
KANGJIAN SUN was born in Jinzhou, China, in 1996. He received the B.E. degree in electrical engineering and automation from Northeast Forestry University, Harbin, China, in 2019, where he is currently pursuing the M.E. degree in control engineering from the College of Mechanical and Electrical Engineering. His research interests include swarm optimization algorithm, image segmentation, and feature selection.
ZICHAO JIANG was born in Qiqihar, China, in 1995. He is currently pursuing the M.S. degree in control theory and control engineering from Northeast Forestry University, China. His research interests include image segmentation and swarm intelligence algorithm.
YAO LI was born in Yichun, China, in 1997. She is currently pursuing the M.S. degree in control theory and control engineering from Northeast Forestry University, China. Her research interests include image segmentation and swarm intelligence optimization.
HEMING JIA (Member, IEEE) received the Ph.D. degree in system engineering from Harbin Engineering University, China, in 2012. He is currently a Professor with Sanming University. His research interests include nonlinear control theory and application, image segmentation, and swarm optimization algorithm. VOLUME 8, 2020