Literature DB >> 34293587

Ant colony optimization with Cauchy and greedy Levy mutations for multilevel COVID 19 X-ray image segmentation.

Lei Liu1, Dong Zhao2, Fanhua Yu3, Ali Asghar Heidari4, Chengye Li5, Jinsheng Ouyang6, Huiling Chen7, Majdi Mafarja8, Hamza Turabieh9, Jingye Pan10.   

Abstract

This paper focuses on the study of multilevel COVID-19 X-ray image segmentation based on swarm intelligence optimization to improve the diagnostic level of COVID-19. We present a new ant colony optimization with the Cauchy mutation and the greedy Levy mutation, termed CLACO, for continuous domains. Specifically, the Cauchy mutation is applied to the end phase of ant foraging in CLACO to enhance its searchability and to boost its convergence rate. The greedy Levy mutation is applied to the optimal ant individuals to confer an improved ability to jump out of the local optimum. Furthermore, this paper develops a novel CLACO-based multilevel image segmentation method, termed CLACO-MIS. Using 2D Kapur's entropy as the CLACO fitness function based on 2D histograms consisting of non-local mean filtered images and grayscale images, CLACO-MIS was successfully applied to the segmentation of COVID-19 X-ray images. A comparison of CLACO with some relevant variants and other excellent peers on 30 benchmark functions from IEEE CEC2014 demonstrates the superior performance of CLACO in terms of search capability, and convergence speed as well as ability to jump out of the local optimum. Moreover, CLACO-MIS was shown to have a better segmentation effect and a stronger adaptability at different threshold levels than other methods in performing segmentation experiments of COVID-19 X-ray images. Therefore, CLACO-MIS has great potential to be used for improving the diagnostic level of COVID-19. This research will host a webservice for any question at https://aliasgharheidari.com.
Copyright © 2021 Elsevier Ltd. All rights reserved.

Entities:  

Keywords:  Ant colony optimization; COVID-19; Diagnosis; Image; Meta-heuristic; Swarm-intelligence

Mesh：

Year:  2021        PMID： 34293587      PMCID： PMC8254401          DOI： 10.1016/j.compbiomed.2021.104609

Source DB:  PubMed          Journal:  Comput Biol Med        ISSN： 0010-4825            Impact factor:   4.589


World Health Organization An infectious disease of coronavirus infection Multilevel threshold image segmentation Wilcoxon signed-rank test Friedman test A novel variant of ant colony optimization CLACO-based MIS model Peak Signal to Noise Ratio Structural Similarity Index Feature Similarity Index The pheromone evaporation rate The archive size The algorithm coefficient A self-adapting mutation coefficient A Levy distribution random number A number uniformly distributed between A scaling parameter The Cauchy distribution function A random number The current number of iterations The current number of evaluations The maximum number of evaluations An algorithm coefficient The gray-scale values of pixels . and of image Population size A corresponding filter value The corresponding weight The corresponding standard deviation Local area average values A block-oriented at A block-oriented at A gray image Non-local means image The normalized 2D histogram 2D Kapur's entropy The gray level The segmentation level The Gaussian kernel function The Gaussian function The weight vector The mean vecto The choosing probability The ant individual Problem dimension Search space The best ant

Introduction

In December 2019, an outbreak of a novel coronavirus, named COVID-19 by the World Health Organization (WHO) [1], occurred in Wuhan, China. In March 2020, the WHO declared COVID-19 a world pandemic. As COVID-19 is spreading around the world, healthcare systems are under tremendous pressure. Although the current diagnostic criteria for COVID-19 are a positive nucleic acid test, the false-negative rate for nucleic acid testing is as high as 17%–25.5%. Therefore, imaging-based diagnostic methods are used to screen for many suspected cases and serve as an effective way to stifle the spread of the disease. Since the outbreak, many researchers have conducted numerous studies using X-ray images of the lungs of infected patients [[2], [3], [4]]. Data science [5] has witnessed significant progress during recent years in dealing with a global epidemic. These kinds of studies are an essential step in developing effective healthcare or bio-informatic systems [[6], [7], [8]]. It is well known that multilevel threshold image segmentation (MIS) is a simple and efficient image processing method. To further improve the diagnostic level of COVID-19 [[9], [144], [145]], this paper focuses on the study of MIS and proposes an MIS method with excellent segmentation results. In recent years, MIS has gained much attention as an efficient and straightforward image segmentation method. A wide variety of MIS have been proposed and applied to various complex medical image processing. For instance, Zhao et al. [10] proposed a fast 2D Otsu image segmentation algorithm to segment lung tissue images based on enhanced particle swarm optimization (PSO). Rodríguez-Esparza et al. [12] proposed an excellent method for MIS by using the minimum cross-entropy as a fitness function of Harris hawks optimization, which was tested on medical images of digital mammography. Anter et al. [13] developed a robust optimization approach for automatic segmentation of liver and liver lesions based on fast fuzzy c-means, chaos theoretical, and bio-inspired ant-lion optimizer. Moreover, Abdel-Basset et al. [14] proposed a new hybrid approach by integrating a sticky mode algorithm and whale optimizer according to the threshold segmentation technique of COVID-19 chest X-ray images. Sambandam et al. [15] proposed an adaptive dragonfly optimization approach for performing MIS of medical images. Verma et al. [16] proposed a hybrid MIS method with Fuzzy c-means and the PSO algorithm, which was applied to execute experiments on publicly available real brain datasets. Radha et al. [17] proposed a combination of an intelligent fuzzy level set approach and an improved quantum PSO for magnetic resonance image segmentation. Table 1 compares some related works, which, with relatively weak adaptability, were developed by carrying out a straightforward combination of techniques. Therefore, in the present study, we first propose the CLACO algorithm with excellent performance and then combine it with 2D Kapur's entropy, non-local means, and 2D histogram to achieve a better segmentation effect and a more robust adaptability at different threshold levels.
Table 1

The comparison between some relevant works.

StudyMethodImage typeThe relevant result
Work of this paperCLACO+ 2D Kapur's entropy + non-local means + 2D histogramCOVID-19 X-ray imagesIt obtained a better segmentation effect and more robust adaptability at different threshold levels.
Zhao et al. [10]improved PSO + 2D Otsulung tissue imagesIt not only satisfied the requirement of segmentation precision but also met the requirement of operation speed.
Rodríguez-Esparza et al. [12]MCET-HHO + Minimum cross entropymedical images of digital mammographyIt produced efficient and reliable results in terms of quality, consistency, and accuracy.
Anter et al. [13]CAL OF CM + fuzzy c-meanshepatic lesion segmentation from CT scansIt showed good detection and segmentation performance.
Abdel-Basset et al. [14]HSMA_WOA + Kapur's entropyCOVID-19 chest X-ray imagesIt outperformed SMA under Kapur's entropy for all the metrics.
Sambandam et al. [15]SADFO + Kapur's entropysix medical images of eyes, liver, head and tongueIt effectively optimized the threshold values by exploring the solution space.
Verma et al. [16]hybrid fuzzy c-means and PSOreal brain datasetsIt improved up to 30% for real brain images.
Radha et al. [17]IQPSO + intelligent fuzzy level set methodmagnetic image resonance imagesIt showed a promising significant improvement in the image segmentation process.
The comparison between some relevant works. The MIS method based on the swarm intelligence optimization algorithm is widely adopted. It is also found that the optimum threshold set in the MIS method is highly dependent on the swarm intelligence optimization algorithm. In other words, a swarm intelligence optimization algorithm with high performance can greatly improve the results of MIS. Complex feature spaces often are challenging enough to be dealt with [18,19], especially in the medical area [20] and image quality assessment [132]. Hence, more attention should be paid to the accuracy and efficacy of the procedure used to tackle the model [21,22]. The swarm intelligence optimization algorithm has shown great potential in solving a multitude of practical problems, including but not limited to, detection of feature selection cases [[23], [24], [25]], parameter optimization [[26], [27], [28]], imaging array [128], communication systems [129], and engineering problems [[29], [30], [31]]. Also, we can refer to PID optimization control [[32], [33], [34]], prediction problems in educational field [[35], [36], [37]], the hard maximum satisfiability problem [38,39], foreign fiber in cotton [40,41], medical diagnosis [[42], [43], [44]], scheduling problem [45,46], wind speed prediction [47], bankruptcy prediction [[48], [49], [50]], fault diagnosis of rolling bearings [51,52], and gate resource allocation [53,54]. As far as swarm intelligence optimization algorithms are concerned, a number of related algorithms have been proposed, including PSO [57], Harris hawks optimization (HHO)2 [61], slime mould algorithm (SMA)3 [62], hunger games search (HGS)4 [63], Runge Kutta optimizer (RUN)5 [64], modified SCA (m_SCA) [65], boosted GWO (OBLGWO) [66], opposition-based SCA (OBSCA) [67], A-C parametric WOA (ACWOA) [68], biogeography-based learning PSO (BLPSO) [69], comprehensive learning PSO (CLPSO) [70], moth-flame optimizer with sine cosine mechanisms (SMFO) [71], enhanced comprehensive learning particle swarm optimizer (GCLPSO) [72], enhanced GWO with a new hierarchical structure (IGWO) [73], improved WOA (IWOA) [74], and ant colony optimization (ACO) for continuous domains (ACOR) [75]. Notably, it is well known that ACO [76,77] is an algorithm for solving discrete optimization problems, whereas ACOR can be used to solve optimization problems other than discrete ones. Accordingly, in recent years, a broad array of related studies have been carried out on original optimizers such as PSO [[136], [137], [138]], and its variants [[139], [140], [141], [142]], and more methods based on well-known swarm-based method continuous ACO algorithms. Zhao et al. [78] proposed a modified continuous ACO algorithm by refining the selection scheme and applying horizontal cross-search and vertical cross-search to ACOR. Chen et al. [79] proposed a new elite hybrid continuous ACO method based on the central initialization of the population. Zhao et al. [80] proposed a novel variant of the continuous ACO by applying a stochastic standby strategy and a chaotic reinforcement strategy based on the original ACO. Wu et al. [81] described a multi-modular and continuous ACO algorithm and devised an efficient method as a local search operation. Kumar et al. [82] proposed a novel continuous ACO algorithm by incorporating an interaction scheme between ants based on the Laplace distribution. Chen et al. [83] developed a modified continuous ACO with crossover operators, where there are three crossover methodologies adopted to yield some new sets of probability density functions. Karakonstantis et al. [84] presented a hybrid variant of ACO for continuous domains, which can handle continuous optimization problems either with or without constraints. Although these proposed various variants of the continuous ACO algorithm outperform the original ACOR on specific problems, they suffer from some performance problems and thus need improving. It is possible that, they might fail to obtain satisfactory results when applied to different fields due to their poor search capability, slow convergence rate, and tendency to fall into local optimum in the process of solving specific problems. To tackle these problems and obtain a high-quality threshold set in multi-threshold COVID-19 X-ray image segmentation, this paper proposes a novel variant of ACO with the Cauchy mutation and the greedy Levy mutation for continuous domains (CLACO). Specifically, the Cauchy mutation introduced to the end stage of foraging in CLACO effectively enhances the searchability and convergence speed. The greedy Levy variant applied to the optimal ant individuals confers an improved ability to jump out of the local optimum. The optimal individuals have enhanced experiential information and the ability to learn on their own. In this paper, the variant algorithms consisting of the Cauchy mutation and the greedy Levy mutation were compared to demonstrate the superior performance of CLACO, using 30 test functions from IEEE CEC2014. Then, CLACO was also experimentally compared with ten other similar algorithms. Finally, all experimental results obtained were re-analyzed and compared using the Wilcoxon signed-rank test (WSRT) [85] and the Friedman test (FT) [86]. A careful analysis and comparison of experimental results demonstrate that CLACO is superior in searchability, convergence speed and ability to jump out of the local optimum. Meanwhile, to improve the diagnostic level of COVID-19, a new CLACO-based MIS model (CLACO-MIS), which can obtain high-quality segmentation results, is proposed in this paper and applied to segment X-ray images from patients with COVID-19. In CLACO-MIS, to obtain an optimal threshold set, 2D histograms are adopted composed of non-local mean filtered images and grayscale images and 2D Kapur's entropy is taken as the CLACO fitness function based on the composed 2D histograms. To demonstrate that CLACO-MIS has a superior segmentation effect and stronger adaptability to different threshold levels, based on 9 X-ray images of COVID-19 patients, CLACO-MIS and eight other similar methods were compared in segmentation experiments not only at levels 4, 5, and 6, representing low threshold levels, but also at levels 15, 20, and 25, representing high threshold levels, Finally, we evaluated the COVID-19 X-ray image segmentation results using Peak Signal to Noise Ratio (PSNR) [87], Structural Similarity Index (SSIM) [88], Feature Similarity Index (FSIM) [89] to effectively account for the segmentation results. In addition, we compared the evaluation results using mean and variance together with the WSRT [85] and FT [86]. The comparative results show that, compared with other similar methods, CLACO-MIS has a better segmentation effect and a stronger adaptability to different threshold levels in performing multi-threshold COVID-19 X-ray image segmentation. The other sections of this paper are organized as follows. Section 2 describes the Cauchy mutation and the greedy Levy mutation and proposes a new ACOR (CLACO) version. Section 3 describes non-local means for the 2D histogram, 2D Kapur's entropy, and presents a novel multilevel COVID-19 X-ray image segmentation method based on CLACO. In section 4, some comparative experiments are carried out to demonstrate the performance of CLACO and CLACO-MIS. The conclusions of the whole paper and some future works are presented in section 5. In conclusion, this paper's main innovations and contributions lie in the following aspects. A new enhanced ACOR version combining the Cauchy mutation and the greedy Levy mutation, called CLACO, is first proposed based on the original ACOR. CLACO is compared with many similar methods using IEEE CEC2014, and the core advantages of CLACO are fully demonstrated. Using non-local means, 2D histogram and 2D Kapur's entropy, a CLACO-based MIS method, called CLACO-MIS, is first developed. In a comparative experiment between CLACO-MIS and its peers at several threshold levels, its segmentation effect is well demonstrated using COVID-19 real pathology images. Compared with other peers, CLACO obtains a more significant improvement in benchmark function problems and image segmentation quality.

Proposed CLACO

Cauchy mutation

Based on the description of ACOR in Appendix C, in this subsection, the Cauchy mutation strategy is briefly illustrated to enhance the search capability of CLACO. It is also found that some relevant experimental results proved that the Cauchy mutation strategy has a powerful search ability in literature [90]. Therefore, based on the above inspiration, the Cauchy mutation is introduced into CLACO to improve the search power of CLACO, which also further enhances the convergence rate. The Cauchy distribution function, in this case, can be represented as Eqs. (1) and (2) [91]. The corresponding density function is defined as follows,where is the scaling parameter [92], is a number uniformly distributed between , and . Although the corresponding density function is similar between Gaussian distribution and Cauchy distribution, differences can be found between them. The main distinguishing feature is that the Cauchy distribution is smaller than the Gaussian distribution in the vertical direction, but it is broader than the Gaussian distribution in the horizontal direction. The search capability of search individuals can be improved by adding some neighbors in every generation because it guarantee that individuals can enhance themselves within the main scope and easily jump out of the local optimum. Therefore, the Cauchy distribution, which can be regarded as an effective mutation operation, is applied to the end of the foraging phase of an ant colony in CLACO while . This process can be illustrated by Eqs. (1) and (2) [91]. At the end of the foraging phase, the Cauchy mutation is introduced to confer a better ability to utilize the prospective space. Therefore, it is possible to improve solution quality with the Cauchy mutation over the whole simulation process. The generated random number is applied to the end of the foraging phase of an ant colony and is shown as Eq. (3) [91].where is a random number sampled from the Cauchy distribution.

Greedy Levy mutation

In the basic ACOR, some optimal individuals guide the entire group in the search direction. However, the best individuals face a lack of experiential information and thus the ability to learn independently. Therefore, it may be difficult for them to be improved and thus they tend to fall into the local optimum. A greedy Levy mutation-based optimal individual strategy is proposed in order to avoid falling into the local optimum and to overcome inefficiency in the later period. As a result, the individuals can jump out of the original optimal position previously searched by the mutation operation, thus preserving the population diversity. The Greedy Levy mutation-based optimal individual methodology is represented as Eq. (4) [93].where is the random number obtained by the Levy distribution, is a self-adapting mutation coefficient, and is the dimension value of the optimal individual position at iteration . Due to the difficulty of integrating the probability density function of the Levy distribution, in the greedy Levy mutation, it has been shown that the Mantegna algorithm can be implemented as an equivalent calculation [94]. It can be calculated by Eq. (5) [93].where , , , , and can be calculated by Eq. (6) [93].where indicates the standard Gamma function. Besides, aiming to obtain a large mutation in the initial stage of ant foraging to perform global interference, and to diminish the mutation in the late-foraging stage to accelerate the local search, the self-adaptive mutation control coefficient can be expressed as given in Eqs. (7)–(9) [93]. where is the current number of iterations, is the current number of evaluations, is the maximum number of evaluations, is an algorithm coefficient, is the difference between the value of the position vector of the current optimal individual and the average of the values of the position vector of each individual in the archive, and is the difference between the maximum and minimum values among the jth position vector values in the archive.

The proposed CLACO

Given that satisfactory results are often not obtained when ACOR was applied to different domains for its poor search capability, slow convergence speed and tendency to fall into local optima in solving specific problems, we propose a new version of ACOR, called CLACO. The introduction of the Cauchy mutation to the end stage of ant foraging in CLACO effectively enhances not only the search capability but also the convergence speed of foraging ants. Aiming to confer an improved ability to jump out of local optima, the greedy Levy mutation is applied to optimal ant individuals, conferring enhanced experiential information and an ability to learn independently. By analyzing the effects of the Cauchy mutation and the greedy Levy mutation on the process of ant foraging, a flowchart of CLACO is presented in Fig. 1 .
Fig. 1

The flowchart of CLACO.

The flowchart of CLACO.

Proposed MIS method

Non-local means for 2D histogram

Non-local means, a novel denoising technique, was developed by Buades et al. [95]. Taking full advantage of the redundant information in the image, this method can preserve the image's detailed features as much as possible while de-noising. Assuming that and are the corresponding gray-scale values of pixels and of image , it is possible to calculate the non-local mean values from Eqs. 10–13 [95] for image . where is a corresponding filter value, is the corresponding weight, is the corresponding standard deviation, and are local area average values, is a block-oriented at , and L(q) is an m × m block-oriented at . An image's 2D histogram can be constructed using a non-local mean image and grayscale image. If we suppose that a gray image has levels and the image size is , the image obtained by non-local means filtering also has levels and the image size is also . As a result, the point can be constructed using the levels and gray values of and . denotes the gray value of the pixel in and denotes the gray value of the corresponding pixel in . Consequently, it is possible to have the number of pixels h(i,j) that appear at this point (s,t), as well. The 2D histogram developed above in this way is normalized by Eq. (14) [96]. We can produce a final 2D histogram shown in Fig. 2 and the corresponding plane view.
Fig. 2

The 2D histogram and the 2D plan view.

The 2D histogram and the 2D plan view.

2D Kapur's entropy

As we have the definition of non-local mean 2D histogram above, it is possible to construct correspondingly the 2D histogram and 2D plane view in Fig. 2, where represents the greyscale image levels, and characterizes levels in the non-local mean image. Based on the 2D histogram presented above, we can construct the corresponding 2D Kapur's entropy. Since the primary alignment of the 2D histogram contains the most image information, the 2D Kapur's entropy is calculated only for the subregions on the central diagonal to make the calculation correct and straightforward. Consequently, through the description above, the 2D Kapur's entropy is expressed as Eq. (15) [80] in the image. Therefore, the threshold set that maximizes is the optimal threshold set, when the 2D Karpur's entropy is considered as the objective function of CLACO.where

The proposed MIS method

The main purpose of threshold-based segmentation is to find a reasonable threshold set to distinguish the target from the background in an image, a compelling image segmentation method. Furthermore, MIS is defined as the process of finding a threshold set in an image, segmenting the image into multiple parts using a threshold set. To achieve better one-threshold image segmentation, Pun [97] presented a maximum entropy-based thresholding algorithm, which regards the histogram of an image as a probability distribution and calculates the maximum entropy to identify an optimal threshold value. Ler, Kapur et al. [98] introduced a modified maximum entropy-based threshold segmentation algorithm, which is simple to compute and yields better segmentation results called Kapur's entropy. It is theoretically easy to extend the maximum entropy-based one-threshold image segmentation algorithm to MIS. Nevertheless, Given repeated calculations that are involved in MIS, its computational efficiency is relatively low. For instance, by applying the exhaustive method to multilevel maximum entropy threshold segmentation, the time complexity is , and it is increasing exponentially, in which denotes the grayscale range of the image and is the number of levels in the segmentation. Moreover, for the 1D histogram-based segmentation method, a serious misclassification phenomenon occurs when the target occupies a relatively small area in an image, rendering the segmentation results vulnerable to noise interference. By contrast, the conventional 2D histogram segmentation method based on the local mean does not consider some detailed information in an image, such as some points, lines, and planes [95]. As a result, the proposed MIS method is based on 2D histograms with non-local means to minimize the above drawbacks, and it employs 2D Kapur's entropy as the fitness function of CLACO. In addition, it is applied to multilevel COVID-19 X-ray image segmentation, and is described in detail in Fig. 3 .
Fig. 3

The flowchart of the CLACO-MIS method.

The flowchart of the CLACO-MIS method.

Experiments and results

In this section, it is mainly to verify the performance of CLACO experimentally and to evaluate the segmentation effect of CLACO-MIS. First, ACOR variants consisting of the Cauchy mutation and the greedy Levy mutation were compared, and then CLACO was compared with ten other similar algorithms on the 30 test functions from IEEE CEC2014. Subsequently, based on 9 COVID-19 X-ray images, a segmentation experiment using CLACO and eight other similar methods was performed not only at levels 4, 5, and 6, representing low threshold levels, but also at levels 15, 20, and 25, representing high threshold levels.

Experiment setup

In the experiments on the IEEE CEC2014 benchmark functions shown in Table 2 , which contains a convincing test suite of unimodal functions, simple multimodal functions, hybrid functions, and composition functions. Its main purpose is to demonstrate that CLACO has a high performance. At first, different ACOR variants, namely, LACO, CACO, and CLACO, are constructed using the Cauchy mutation and the greedy Levy mutation, and the constructed variant algorithms are employed for strategy comparison experiments. Subsequently, the comparison experiments with similar algorithms were mainly carried out by comparing CLACO with GWO [134], MFO [135], PSO, ACOR, SCA [135], WOA [135], OBLGWO, m_SCA, OBSCA, and ACWOA. In the process of conducting the relevant experiments, all the algorithms involved in the comparison were carried out under the same conditions to guarantee the fairness and reliability of the experimental results [133], where the population size was set to 30, and the maximum number of evaluations was uniformly set to 300,000. Furthermore, all algorithms were independently trained 30 times to minimize the influence of random conditions. Notably, both the WSRT and FT were utilized to analyze and compare the obtained experimental results once again. After careful analysis and comparison on a series of experimental results, CLACO was shown to have an enhanced search capability, a faster convergence speed, and a higher convergence accuracy as well as an improved ability to step out of the local optimum.
Table 2

Descriptions of the CEC2014 functions.

ClassIDDescriptionRangeFi∗=Fi(x∗)
Unimodal Functions1Rotated High Conditioned Elliptic Function[−100,100]100
2Rotated Bent Cigar Function[−100,100]200
3Rotated Discus Function[−100,100]300
Simple Multimodal Functions4Shifted and Rotated Rosenbrock's Function[−100,100]400
5Shifted and Rotated Ackley's Function[−100,100]500
6Shifted and Rotated Weierstrass Function[−100,100]600
7Shifted and Rotated Griewank's Function[−100,100]700
8Shifted Rastrigin's Function[−100,100]800
9Shifted and Rotated Rastrigin's Function[−100,100]900
10Shifted Schwefel's Function[−100,100]1000
11Shifted and Rotated Schwefel's Function[−100,100]1100
12Shifted and Rotated Katsuura Function[−100,100]1200
13Shifted and Rotated HappyCat Function[−100,100]1300
14Shifted and Rotated HGBat Function[−100,100]1400
15Shifted and Rotated Expanded Griewank's plus Rosenbrock's Function[−100,100]1500
16Shifted and Rotated Expanded Schaffer's F6 Function[−100,100]1600
Hybrid Functions17Hybrid Function 1 (N = 3)[−100,100]1700
18Hybrid Function 2 (N = 3)[−100,100]1800
19Hybrid Function 3 (N = 4)[−100,100]1900
20Hybrid Function 4 (N = 4)[−100,100]2000
21Hybrid Function 5 (N = 5)[−100,100]2100
22Hybrid Function 6 (N = 5)[−100,100]2200
Composition Functions23Composition Function 1 (N = 5)[−100,100]2300
24Composition Function 2 (N = 3)[−100,100]2400
25Composition Function 3 (N = 3)[−100,100]2500
26Composition Function 4 (N = 5)[−100,100]2600
27Composition Function 5 (N = 5)[−100,100]2700
28Composition Function 6 (N = 5)[−100,100]2800
29Composition Function 7 (N = 3)[−100,100]2900
30Composition Function 8 (N = 3)[−100,100]3000
Descriptions of the CEC2014 functions. Experiments were conducted for segmentation using 9 COVID-19 X-ray images shown in Fig. 4 from a public database collected by Cohen et al. [99]. Their corresponding 2D histograms are represented by A, B, C, D, E, F, G, H, and I, respectively. It is mainly to demonstrate that the proposed CLACO-MIS method can obtain relatively good segmentation results and has a strong adaptability to different threshold levels. Firstly, CLACO and eight other similar methods were used to perform segmentation experiments at levels 4, 5, and 6, representing low threshold levels, and secondly, we also performed segmentation experiments at levels 15, 20, and 25, representing high threshold levels. In the segmentation experiments, the 8 similar comparison methods involved were ACOR-MIS, MVO-MIS, HHO-MIS, SCA-MIS, BLPSO-MIS, IGWO-MIS, IWOA-MIS, and CLPSO-MIS. During the process of performing the experiments, all segmentation experiments were performed in 100 times iterations to assure the fairness as well as the reliability of the experimental results, where the chosen segmentation image size was defined as 512 × 400, and the population size was set to 20 for all the segmentation methods involved in the comparison. To eliminate the experimental randomness, all experiments were run 30 times independently. The experimental results obtained were evaluated using PSNR, SSIM, and FSIM, and a comparative analysis was performed using the mean and variance as well as WSRT and FT. By analyzing and comparing a series of experimental results, the superior segmentation effect of CLACO-MIS and its stronger adaptability to different threshold levels were well demonstrated when CLACO-MIS was compared with other similar methods to segment COVID-19 X-ray images with multi-threshold levels.
Fig. 4

The original COVID-19 X-ray images and the corresponding 2D histograms.

The original COVID-19 X-ray images and the corresponding 2D histograms. In addition, considering that the parameters of CLACO are crucial, we applied a trial-and-error approach to obtain values of the relevant parameters shown in Table 3 . The corresponding parameters of CLACO-MIS also adopted the same values as CLACO. Furthermore, all experiments were performed on a desktop computer with an Intel(R) Core(TM) i7-10750H CPU @2.60 GHz, and all programs were coded on MATLAB 2018B.
Table 3

Some parameter descriptions of CLACO.

Variable nameVariable descriptionVariable value
ξThe pheromone evaporation rate1
kThe archive size10
qThe coefficient0.5
Some parameter descriptions of CLACO.

Benchmark function validation

In this subsection, the performance of CLACO is demonstrated mainly from two aspects. The first is the experiment comparing CLACO with other variants on 30 benchmark functions; and the second is the experiment comparing CLACO with 10 other similar methods on 30 benchmark functions. Through the variance, mean, and WSRT and FT analysis, the obtained experimental results demonstrate that CLACO is superior in search capability, convergence speed, and convergence accuracy as well as inability to avoid local optimum.

The impact of Cauchy and greedy Levy mutations

Table 4 presents the composition of the various versions of ACOR, where CM denotes Cauchy mutation, GLM denotes greedy Levy mutation, ‘1′ means that the corresponding ACOR version has this mutation operation, and ‘0′ means that the corresponding ACOR version does not have this mutation operation. As shown in Table 4, ACOR is the original version, LACO is the version with the greedy Levy mutation, CACO is the version with the Cauchy mutation, and CLACO is the version with both.
Table 4

Various variants with different mutation strategies.

CMGLM
ACOR00
LACO01
CACO10
CLACO11
Various variants with different mutation strategies. Table A1 lists the means and variances obtained by four different versions of ACOR after 30 randomized experiments on 30 test functions, where ‘AVG’ indicates the mean value obtained by the algorithm on the benchmark function and ‘STD’ indicates the algorithm variance obtained. As can be seen from Table A1, the original algorithm ACOR obtained optimal results on only 1 benchmark function, CACO on nine functions, LACO on five functions, and CLACO on 15 functions. Therefore, CACO, LACO, and CLACO all have a certain degree of improvement on ACOR performance, with CLACO being a variant of ACOR with the best performance. In Table 5 , the analysis results of the WSRT are presented, where '+' indicates the number of functions in which CLACO's overall performance is better than that of the other methods among the 30 tested functions, '-' indicates the number of functions in which CLACO's overall performance is worse than that of the other methods, ' = ' indicates that CLACO and the other methods perform equally, ‘Mean’ indicates the overall ranking means obtained by each method, and ‘Rank’ indicates the final ranking result based on the performance of each algorithm. As can be seen from Table 5, CLACO is superior in overall performance to CACO, LACO, and ACOR. Specifically, CLACO performed worse than CACO in only 3 functions, and CLACO was not worse than LACO and ACOR in all functions. On the overall ranking mean, CLACO also obtained the No. 1 result with 1.677, indicating that CLACO is a high-performance variant of ACOR. Besides, the results of the FT analysis given in Fig. 5 also show that CLACO outperforms CACO, LACO, and ACOR. The convergence curves of different variants of ACOR shown in Fig. B1 further demonstrate that, compared with CACO, LACO, and ACOR, CLACO has not only an improved ability to obtain the optimal solution quality and to jump out of the local optimum but also an elevated convergence rate.
Table 5

Comparison results of four variants based on WSRT.

ItemCLACOACORCACOLACO
+/−/ =~21/0/915/3/122/0/28
Mean1.6673.5332.4672.333
Rank1432
Fig. 5

The Friedman ranking of different versions of ACOR.

Comparison results of four variants based on WSRT. The Friedman ranking of different versions of ACOR.

Comparison with some peers

In this section, the core advantages of CLACO are verified by comparing CLACO with ten other similar high-performance algorithms including GWO, MFO, PSO, ACOR, SCA, WOA, OBLGWO, m_SCA, OBSCA, and ACWOA. To make the experimental results more convincing, not only some traditional basic algorithms but also some variant algorithms were included. Moreover, we followed fair evaluation instructions [[100], [101], [102]]. Table A2 gives the means and variances obtained by CLACO and ten peers for different test functions, where CLACO obtained the smallest mean on 20 functions and obtained the minor variance on 14 functions, demonstrating the strong ability of CLACO to obtain high-quality solutions. Table A3 gives the results of further analysis of the obtained experimental results using the WSRT, where CLACO obtained the No. 1 result with a total ranked mean value of 1.800 and performed better than OBLGWO (ranked No. 2) on 25 functions. The p-values obtained by the WSRT are given in Table A4, where only a small number of values are more significant than 0.05, indicating that the results obtained for the experimental analysis are credible. The results of the analysis of the experimental results using Friedman are presented in Fig. 6 , and they further illustrate that CLACO has a more robust performance than the original ACOR and the ten other similar methods. Fig. B2 gives the convergence curves of CLACO and its similar methods, where F2, F8, F12, F28, F29, F30 demonstrated the faster convergence speed and higher convergence accuracy of CLACO, and F1, F17, F21 demonstrated a better ability of CLACO to jump out of the local optimum.
Fig. 6

The Friedman ranking of CLACO and its peers.

The Friedman ranking of CLACO and its peers. Therefore, the experiments comparing CLACO with other ACOR variants on 30 benchmark functions and ten other similar methods fully demonstrate that CLACO has a significantly enhanced search capability over ACOR, a faster convergence speed, a higher convergence accuracy, and a better ability to escape from the local optimum.

Experiment on multilevel COVID-19 X-ray image segmentation

In this subsection, the segmentation effect of CLACO-MIS on the COVID-19 X-ray image is demonstrated not only at the low threshold levels (4, 5, and 6) but also at the high threshold levels (15, 20, and 25). The segmentation methods used for comparison were CLACO-MIS, ACOR-MIS, MVO-MIS, HHO-MIS, SCA-MIS, BLPSO-MIS, IGWO-MIS, IWOA-MIS, and CLPSO-MIS.

Performance evaluation indicators

In order to comprehensively analyze the obtained experimental results, three commonly adopted evaluation methods were applied in this paper, namely Peak Signal to Noise Ratio (PSNR) [87], Structural Similarity Index (SSIM) [88], Feature Similarity Index (FSIM) [89]. Table 6 presents the relevant descriptions, where the larger values of the evaluation results of PSNR, SSIM, and FSIM indicate a better segmentation effect obtained by each method. In addition, we conducted a comparative analysis of the evaluation results obtained using PSNR, FSIM, and SSIM using mean and variance and WSRT [85] and FT [86].
Table 6

The definitions and descriptions of the three evaluation indicators.

IndicatorsFormulationRemark
Peak Signal to Noise Ratio (PSNR)PSNR=20×log10(255RMSE)Assess the difference between the split image and the original image
Structural Similarity Index (SSIM)SSIM=(2μIμSeg+c1)(2σI,Seg+c2)(μI2+μSeg2+c1)(σI2+σSeg2+c2)Finds the similarity between segmented image and uncompressed or distortion-free image.
Feature Similarity Index (FSIM)FSIM=∑I∈ΩSL(X)PCm(X)∑I∈ΩPCm(X)Defines the quality score, which reflects the significance of a local structure.
The definitions and descriptions of the three evaluation indicators.

Experimental result analyses

In the area of image and video processing [103], it is critical to apply efficient methods for quality assessment [[104], [105], [106]]. Tables A5 - A7 present the mean and variance results obtained by all segmentation methods under the three performance evaluation metrics, where CLACO-MIS obtained the highest number of both maximum mean and minimum variance at different threshold levels, which is also a very strong indication that it can obtain better segmentation results in most cases. Tables A8 – A10 give the results of WSRT for further analysis of PSNR, FSIM, and SSIM, and it can be seen that CLACO-MIS is second to none in PSNR, FSIM, and SSIM at all threshold levels for its ranking, and in most cases, the segmentation results of other segmentation methods are not better than CLACO-MIS. Additionally, Table A11 gives the maximum 2D Kapur's entropy obtained by each segmentation model during the segmentation process. CLACO-MIS obtained 44 times the maximum 2D Kapur's entropy in all segmentation cases, ACOR-MIS obtained only seven times the maximum 2D Kapur's entropy, and MVO-MIS obtained only three times the maximum 2D Kapur's entropy. Therefore, the ability of CLACO-MIS to obtain the optimal solution during the segmentation process is unquestionable, and it obtains the maximum 2D Kapur's entropy in most cases, which also indicates that the optimal segmentation threshold set found using CLACO-MIS is also reliable. In addition, Figs. B3 - B5 give the results of further analysis of PSNR, FSIM, and SSIM using FT, where CLACO-MIS also obtained optimal values in all evaluated cases, which further illustrates that CLACO-MIS has better segmentation performance for COVID-19 X-ray images and can obtain high-quality segmentation results. Based on the evaluation results of PSNR, FSIM, and SSIM, Fig. B6, Fig. B7, Fig. B8 give the mean performance of the segmentation results of each segmentation method at each threshold level. By observing the mean performance under each threshold level, we can find that CLACO-MIS can obtain the maximum value in all cases. Moreover, Fig. 7, Fig. 8, Fig. 9 give the mean values obtained by each segmentation method concerning all the threshold levels, where CLACO-MIS can obtain the maximum value under each evaluation metric. Therefore, CLACO-MIS obtains excellent results for COVID-19 X-ray image segmentation. Furthermore, the convergence curves of each segmentation method on the 2D Kapur's entropy are given in Figs. B9 – B10 in the process of searching for the optimal threshold set. From the given convergence curves, it can be seen that the convergence speed of CLACO-MIS is the fastest, and it is not easy to fall into local optimum in the process of convergence, and thus can obtain the maximum 2D Kapur's entropy. In addition, the ability of CLACO-MIS to obtain the optimal threshold set is significantly enhanced at the threshold level 25. Therefore, in addition to obtaining better segmentation effects, CLACO-MIS also has better adaptability to different threshold levels. In other words, as the threshold level increases, the advantages of CLACO-MIS become more apparent.
Fig. B6

Overall average results of PSNR evaluation at each threshold level

Fig. B7

Overall average results of FSIM evaluation at each threshold level

Fig. B8

Overall average results of SSIM evaluation at each threshold level

Fig. 7

Overall average results of PSNR evaluation of all threshold levels.

Fig. 8

Overall average results of FSIM evaluation of all threshold levels.

Fig. 9

Overall average results of SSIM evaluation of all threshold levels.

Overall average results of PSNR evaluation of all threshold levels. Overall average results of FSIM evaluation of all threshold levels. Overall average results of SSIM evaluation of all threshold levels. Furthermore, when the threshold level is 6, Figs. B11 - B19 give the specific optimal segmentation thresholds obtained by all segmentation methods, and Fig. B29 gives the specific segmentation results of all images obtained by all methods. Also, the specific optimal segmentation thresholds obtained by all methods at the threshold level 25 are given in Figs. B20 - B28, and the specific segmentation results obtained by all methods at the threshold level 25 are given in Fig. B30. Based on the observation and analysis of the optimal segmentation thresholds and the segmentation results, it is further demonstrated that CLACO-MIS can obtain better segmentation results and verified that CLACO-MIS has good adaptability to different threshold levels. As exploration and exploration of the method exposes, the experts can also utilize the proposed ACO-based optimizer to tackle multi-faced feature spaces in other families of problems such as works in Refs. [[107], [108], [109]]. For future work, since CLACO is a superior swarm intelligence optimization algorithm, we will consider applying it to more fields, such as face recognition and micro-expression recognition [110,111], 3D deformable shape analysis [112,113], micro-expression spotting [111,114], and service ecosystem [115,116]. Also, we can suggest areas such as Lunar impact crater identification and age estimation [117], large scale network analysis [118], energy storage planning and scheduling [119], medical diagnosis [[120], [121], [122]], structure designs [130], power generation [131], and prediction of brain-behavior [123,124].

Conclusions and future works

This paper presents the first ACO with the Cauchy mutation and the greedy Levy mutation, namely CLACO, for continuous domains. Besides, to improve the diagnostic level of COVID-19, this paper proposes a novel CLACO-based MIS model, namely CLACO-MIS, and applies it to the segmentation of COVID-19 X-ray images and obtains segmentation results with high quality. The Cauchy mutation has been effectively applied to the end phase of ant foraging in CLACO, which has achieved an effective enhancement of the searchability of ants; in other words, the lack of searchability of ACOR has been successfully enhanced as well as the convergence speed has been dramatically boosted. To make CLACO have a solid ability to step out of the local optimum, the greedy Levy mutation is applied to optimal individuals to make the optimal individuals enhance the experiential knowledge and learn by themselves. To provide a robust demonstration of CLACO's performance based on the 30 test functions from IEEE CEC2014, the variant algorithms, which are made up of the Cauchy mutation and the greedy Levy mutation, were first compared experimentally with each other. CLACO was then compared experimentally with ten other peer algorithms. At last, we analyzed and compared all the experimental results obtained again using WSRT and FT, which powerfully illustrate that CLACO has an enhanced search capability, a further improvement in convergence speed, and an improved ability to jump out of the local optimum. In CLACO-MIS, it mainly adopts 2D histograms made up of non-local mean filtered images and grayscale images and takes 2D Kapur's entropy as the CLACO fitness function based on 2D histograms to obtain an optimal threshold set. Based on 9 COVID-19 X-ray images, CLACO-MIS and eight other similar methods at levels 4, 5, and 6, representing low threshold levels, and at levels 15, 20, and 25, representing high threshold levels, have been applied to perform COVID-19 X-ray image segmentation experiments. Then, we evaluated the image segmentation results using PSNR, FSIM, and SSIM to illustrate the effectiveness of the segmentation results effectively, and the comparative analysis results were also obtained by using the mean, variance, and WSRT and FT. Therefore, CLACO-MIS has a better segmentation effect and a stronger adaptability to different threshold levels, compared with other similar methods. However, due to the introduction of the Cauchy mutation and the greedy Levy mutation, the time complexity of CLACO-MIS inevitably increases, and the CPU computation time required for its segmentation has a corresponding extension. Thus, parallel computation is used in the experimental process to improve CPU utilization and reduce CPU computation time. In future, as the CLACO-MIS model is also a segmentation method that can obtain high-quality segmentation results, we will consider using it to segment more pathological images to realize greater value and make our due contribution to the advancement of medical diagnosis technology.

Declaration of competing interest

The authors declare that there is no conflict of interests regarding the publication of article.
Table A1

The mean and variance values gained by relevant variants on IEEE CEC2014

FunItemCLACOCACOLACOACOR
F1AVG3.2250 E+061.6559E + 062.8502 E+066.4907 E+06
STD3.1387 E+067.9411E + 052.0672 E+061.0097 E+07
F2AVG1.3543 E+041.1570E + 041.2641 E+083.1627 E+08
STD1.2715 E+041.2552E + 046.9232 E+081.3099 E+09
F3AVG8.7428 E+025.9578E + 022.8025 E+031.2320 E+04
STD7.3854 E+024.4795E + 023.3751 E+031.5983 E+04
F4AVG4.4999E + 024.6261 E+024.6011 E+024.7870 E+02
STD3.0968E + 014.6270 E+013.5970 E+014.5320 E+01
F5AVG5.2000E + 025.2091E+025.2000E+025.2092E+02
STD1.4892E-035.8613E-026.7377E-035.2908E-02
F6AVG6.1335 E+026.1307 E+026.1592 E+026.1264E + 02
STD5.7492 E+002.9805E + 005.9321 E+003.4019 E+00
F7AVG7.0001E + 027.0001 E+027.0001 E+027.0312 E+02
STD1.1587E-021.7870E-021.5846E-021.2478 E+01
F8AVG8.0673E + 028.3748 E+028.0879 E+028.5970 E+02
STD4.8721E + 007.0867 E+005.0469 E+002.0022 E+01
F9AVG1.0068 E+039.7383E + 021.0526 E+031.0311 E+03
STD3.3213 E+011.8224E + 013.3067 E+015.9325 E+01
F10AVG1.2353 E+032.1862E+031.1939E + 033.0943 E+03
STD1.9363 E+024.2033E+021.6338E + 025.2288 E+02
F11AVG4.1261 E+033.8004E + 034.2291 E+034.2145 E+03
STD7.1379 E+026.9068 E+024.5399E + 021.8687 E+03
F12AVG1.2002E + 031.2013E+031.2002E+031.2024E+03
STD5.9746E-022.6751E-014.9470E-022.7249E-01
F13AVG1.3003E + 031.3005 E+031.3004 E+031.3005 E+03
STD8.2923E-021.2138E-018.0065E-021.5488E-01
F14AVG1.4004E + 031.4007 E+031.4004 E+031.4012 E+03
STD2.2038E-013.0139E-011.8310E-013.2256 E+00
F15AVG1.5103E + 031.5159 E+031.5103 E+031.6409 E+03
STD4.0081 E+002.1711E + 003.9374 E+003.4416 E+02
F16AVG1.6113 E+031.6112E + 031.6114 E+031.6117 E+03
STD4.6112E-014.4951E-017.6039E-013.6573E-01
F17AVG1.8523 E+051.1702E + 053.3839 E+052.3134 E+05
STD1.5170 E+057.9534E + 046.3085 E+057.4030 E+05
F18AVG9.8507 E+031.1328 E+047.9694E + 031.0173 E+04
STD1.0449 E+048.9895 E+038.9481 E+038.2862E + 03
F19AVG1.9137 E+031.9098 E+031.9086E + 031.9256 E+03
STD1.8959 E+012.6922 E+002.6371E + 003.2742 E+01
F20AVG2.9337 E+032.5918E + 035.5125 E+038.7828 E+03
STD9.1804E+025.9689E + 024.8704 E+031.7187 E+04
F21AVG1.0968 E+057.3751E + 041.0965 E+058.3913 E+04
STD8.8399 E+046.2751E + 041.2531 E+059.5793 E+04
F22AVG2.4826E + 032.6282 E+032.5470 E+032.5969 E+03
STD1.7921E + 022.5538 E+021.9402 E+022.3312 E+02
F23AVG2.6143E + 032.6152 E+032.6144 E+032.6175 E+03
STD1.9431E-011.9659E-124.7907E-013.1485 E+00
F24AVG2.6221E + 032.6389 E+032.6238 E+032.6404 E+03
STD1.0423 E+016.8135E + 008.9071 E+008.4687 E+00
F25AVG2.7008E + 032.7067 E+032.7008 E+032.7066 E+03
STD3.9740E-013.2956 E+003.4021E-012.4838 E+00
F26AVG2.7003E + 032.7285 E+032.7004 E+032.7241 E+03
STD1.0667E-017.6358 E+011.0244E-017.2059E+01
F27AVG3.3992E + 033.4280 E+033.4262 E+033.4093 E+03
STD1.5674 E+028.6849E + 011.7446 E+029.1329 E+01
F28AVG3.2587 E+033.8482 E+033.2540E + 033.8560 E+03
STD6.7389 E+011.7913 E+025.9706E + 011.9388 E+02
F29AVG3.1183 E+031.4150 E+063.1169E + 033.6582 E+06
STD2.5634 E+013.2038E+062.1513E + 015.0062 E+06
F30AVG3.8475E + 036.4474 E+033.8544 E+031.0754 E+04
STD2.6490E + 021.0036 E+032.7471 E+026.2312 E+03
Table A2

Comparison results of CLACO and some excellent peers

FunItemCLACOGWOMFOPSOACORSCAWOAOBLGWOm_SCAOBSCAACWOA
F1AVG3.036E + 065.808 E+071.159 E+088.536 E+065.158 E+062.321 E+082.742 E+071.849 E+074.770 E+074.252 E+081.277 E+08
STD2.456E + 063.796 E+071.161 E+082.925 E+065.713 E+065.580 E+071.059 E+077.737 E+062.759 E+071.064 E+085.309 E+07
F2AVG1.119E + 042.284 E+091.284 E+101.440 E+081.878 E+081.717 E+107.023 E+061.480 E+076.955 E+092.418 E+106.188 E+09
STD1.214E + 042.295 E+098.474 E+091.796 E+075.495 E+083.336 E+091.151 E+078.211 E+063.848 E+094.000 E+092.933 E+09
F3AVG8.839E + 023.111 E+041.051 E+059.154 E+029.564 E+033.554 E+043.692 E+049.242 E+032.481 E+045.106 E+045.069 E+04
STD6.470 E+028.956 E+035.982 E+041.175E + 021.216 E+046.026 E+032.677 E+043.333 E+037.957 E+038.727 E+037.977 E+03
F4AVG4.552E + 026.555 E+021.433 E+034.573 E+024.945 E+021.400 E+035.731 E+025.565 E+027.609 E+022.333 E+031.281 E+03
STD3.970 E+019.229 E+018.496 E+023.468E + 016.674 E+012.373 E+024.363 E+014.842 E+011.031 E+025.944 E+023.929 E+02
F5AVG5.200E + 025.209 E+025.203 E+025.209 E+025.209 E+025.209 E+025.203 E+025.210 E+025.206 E+025.210 E+025.207 E+02
STD9.800E-044.807E-021.674E-014.711E-024.875E-025.833E-021.732E-015.601E-021.558E-015.622E-022.045E-01
F6AVG6.142 E+026.138 E+026.228 E+026.218 E+026.132E + 026.337 E+026.352 E+026.192 E+026.215 E+026.324 E+026.346 E+02
STD5.902 E+002.705 E+003.139 E+003.302 E+002.881 E+002.455 E+003.386 E+004.169 E+002.603 E+001.388E + 002.575 E+00
F7AVG7.000E + 027.214 E+027.990 E+027.023 E+027.078 E+028.281 E+027.010 E+027.012 E+027.579 E+029.105 E+027.455 E+02
STD1.043E-021.958 E+016.811 E+011.361E-011.693 E+012.301 E+016.914E-029.422E-023.753 E+014.347 E+012.581 E+01
F8AVG8.073E + 028.771 E+029.436 E+029.736 E+028.624 E+021.038 E+039.771 E+029.245 E+029.379 E+021.064 E+039.911 E+02
STD4.780E + 002.162 E+013.559 E+011.792 E+012.201 E+011.836 E+013.629 E+012.478 E+012.217 E+011.598 E+012.392 E+01
F9AVG1.018 E+039.965E + 021.119 E+031.113 E+031.016 E+031.170 E+031.113 E+031.072 E+031.052 E+031.198 E+031.126 E+03
STD3.264 E+011.927 E+015.523 E+012.971 E+016.235 E+011.976 E+015.208 E+013.315 E+012.689 E+011.715E + 012.134 E+01
F10AVG1.196E + 033.021 E+034.389 E+035.095 E+033.173 E+036.678 E+034.928 E+033.952 E+034.161 E+036.274 E+034.885 E+03
STD1.482E + 024.575 E+021.022 E+036.714 E+024.620 E+026.080 E+029.012 E+027.783 E+026.723 E+023.362 E+021.047 E+03
F11AVG4.256 E+033.824E + 035.136 E+035.933 E+034.851 E+038.018 E+036.243 E+035.085 E+034.713 E+037.336 E+036.138 E+03
STD3.747 E+024.865 E+027.883 E+025.176 E+022.137 E+033.082E + 028.276 E+027.697 E+027.867 E+024.433 E+026.486 E+02
F12AVG1.200E + 031.202 E+031.201 E+031.202 E+031.202 E+031.202 E+031.202 E+031.202 E+031.201 E+031.202 E+031.202 E+03
STD4.637E-021.140 E+002.669E-013.026E-014.616E-013.024E-015.497E-015.424E-013.630E-013.556E-014.680E-01
F13AVG1.300E + 031.300 E+031.302 E+031.300 E+031.300 E+031.303 E+031.300 E+031.301 E+031.301 E+031.304 E+031.302 E+03
STD1.066E-013.810E-011.242 E+006.654E-021.231E-013.738E-011.189E-011.092E-015.461E-014.043E-011.067 E+00
F14AVG1.400 E+031.403 E+031.426 E+031.400 E+031.401 E+031.445 E+031.400E + 031.400 E+031.416 E+031.470 E+031.419 E+03
STD1.895E-014.911 E+002.055 E+011.196E-013.133E-011.091 E+011.385E-011.889E-011.041 E+011.092 E+011.508 E+01
F15AVG1.510E + 031.648 E+031.306 E+051.517 E+031.663 E+033.334 E+031.577 E+031.517 E+032.294 E+031.584 E+041.858 E+03
STD3.108 E+003.117 E+022.563 E+051.079E + 004.544 E+021.720 E+033.135 E+015.842 E+001.242 E+038.689 E+034.358 E+02
F16AVG1.611 E+031.611E + 031.613 E+031.612 E+031.612 E+031.613 E+031.612 E+031.612 E+031.612 E+031.613 E+031.612 E+03
STD4.184E-017.339E-016.048E-013.471E-015.269E-012.519E-014.408E-016.478E-015.979E-011.996E-015.898E-01
F17AVG1.554E + 051.779 E+062.691 E+062.963 E+052.879 E+055.590 E+063.519 E+061.656 E+061.514 E+061.258 E+071.221 E+07
STD9.896E + 041.835 E+064.838 E+061.302 E+055.370 E+052.623 E+062.440 E+061.158 E+069.395 E+056.114 E+068.528 E+06
F18AVG1.002E + 041.027 E+074.257 E+072.100 E+061.013 E+041.610 E+081.484 E+043.356 E+042.122 E+071.747 E+085.337 E+07
STD8.524 E+032.107 E+071.900 E+086.321 E+057.519E + 039.837 E+073.941 E+042.065 E+044.410 E+079.461 E+074.848 E+07
F19AVG1.911E + 031.938 E+031.969 E+031.917 E+031.913 E+031.999 E+031.947 E+031.916 E+031.945 E+032.011 E+032.007 E+03
STD1.343 E+012.832 E+015.062 E+012.171E + 001.107 E+012.330 E+014.620 E+012.175 E+012.056 E+012.409 E+012.777 E+01
F20AVG2.761 E+031.466 E+045.651 E+042.326E + 036.893 E+031.615 E+042.988 E+046.319 E+031.176 E+042.998 E+044.063 E+04
STD6.735 E+025.761 E+032.948 E+046.807E + 017.837 E+035.458 E+031.631 E+042.967 E+032.858 E+039.515 E+031.660 E+04
F21AVG6.879E + 041.437 E+066.577 E+051.156 E+057.994 E+041.322 E+069.813 E+055.627 E+055.401 E+052.367 E+065.185 E+06
STD4.844E + 042.788 E+061.379 E+066.616 E+048.029 E+046.478 E+058.023 E+053.860 E+059.181 E+051.530 E+064.285 E+06
F22AVG2.466E + 032.549 E+032.995 E+032.871 E+032.585 E+032.990 E+032.957 E+032.670 E+032.585 E+033.161 E+033.080 E+03
STD1.509 E+021.934 E+022.577 E+021.843 E+022.658 E+021.103E + 021.982 E+021.849 E+021.706 E+021.650 E+022.669 E+02
F23AVG2.614 E+032.633 E+032.662 E+032.616 E+032.620 E+032.665 E+032.633 E+032.614 E+032.640 E+032.688 E+032.533E + 03
STD1.920E-017.936 E+002.917 E+018.120E-017.911 E+001.252 E+011.068 E+012.168 E+018.011 E+002.114 E+017.451 E+01
F24AVG2.625 E+032.600 E+032.678 E+032.628 E+032.646 E+032.600 E+032.606 E+032.600E + 032.600 E+032.600 E+032.600 E+03
STD1.347 E+016.699E-042.975 E+015.123 E+001.360 E+015.016E-024.542 E+000.000E + 007.328E-043.720E-044.522E-06
F25AVG2.701 E+032.710 E+032.716 E+032.712 E+032.708 E+032.725 E+032.718 E+032.700E + 032.713 E+032.700 E+032.700 E+03
STD2.067E-015.112 E+009.123 E+006.679 E+003.560 E+001.059 E+011.720 E+010.000E + 002.277 E+003.047E-050.000 E+00
F26AVG2.700E + 032.750 E+032.702 E+032.774 E+032.732 E+032.702 E+032.704 E+032.701 E+032.701 E+032.704 E+032.744 E+03
STD9.519E-025.066 E+011.195 E+004.504 E+016.513 E+016.001E-011.818 E+011.662E-014.006E-014.246E-015.000 E+01
F27AVG3.356 E+033.365 E+033.612 E+033.449 E+033.388 E+033.499 E+033.842 E+033.118E + 033.173 E+033.235 E+033.700 E+03
STD1.740 E+021.247 E+021.931 E+022.817 E+021.004 E+023.312 E+023.207 E+023.277 E+021.247 E+023.522E + 013.246 E+02
F28AVG3.228E + 033.912 E+033.938 E+036.858 E+033.839 E+034.868 E+035.065 E+033.588 E+033.995 E+035.509 E+033.947 E+03
STD3.766E + 012.974 E+021.846 E+021.304 E+031.691 E+023.190 E+026.872 E+025.042 E+022.964 E+023.695 E+021.194 E+03
F29AVG3.120E + 031.366 E+063.827 E+064.851 E+041.614 E+061.298 E+076.882 E+063.199 E+061.953 E+061.985 E+072.333 E+07
STD2.925E + 013.309 E+064.157 E+061.052 E+053.692 E+067.052 E+064.256 E+064.245 E+064.241 E+061.089 E+071.819 E+07
F30AVG3.820E + 035.026 E+045.129 E+041.340 E+049.502 E+032.530 E+059.652 E+042.062 E+044.338 E+043.971 E+053.810 E+05
STD2.502E + 023.678 E+043.743 E+045.947 E+035.088 E+038.221 E+045.700 E+041.179 E+042.399 E+041.226 E+053.127 E+05
Table A3

The analysis result by using the WSRT

MethodsCLACOGWOMFOPSOACORSCAWOAOBLGWOm_SCAOBSCAACWOA
+/−/ =~23/4/329/0/123/1/620/0/1028/1/127/1/225/4/128/2/027/3/026/3/1
Mean1.8004.9337.7335.2004.0679.0006.4674.0675.3679.5337.800
Rank1485210726119
Table A4

The p-values obtained by conducting the WSRT

FunGWOMFOPSOACORSCAWOAOBLGWOm_SCAOBSCAACWOA
F11.734E-061.921E-068.466E-069.368E-021.734E-061.921E-061.921E-061.734E-061.734E-061.734E-06
F21.734E-061.734E-061.734E-068.936E-011.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F31.734E-061.734E-066.143E-012.879E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F41.734E-061.734E-063.820E-013.162E-031.734E-061.734E-063.182E-061.734E-061.734E-061.734E-06
F51.734E-061.921E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F68.130E-013.182E-063.724E-054.165E-011.734E-061.734E-061.114E-032.597E-051.734E-061.734E-06
F71.734E-061.734E-061.734E-063.331E-031.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F81.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F94.114E-032.127E-061.734E-068.130E-011.734E-063.182E-061.150E-049.627E-041.734E-061.734E-06
F101.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F117.157E-041.639E-051.921E-063.086E-011.734E-062.353E-068.188E-051.319E-021.734E-061.734E-06
F126.892E-051.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-06
F133.001E-021.734E-068.972E-021.057E-041.734E-064.449E-051.494E-051.734E-061.734E-061.921E-06
F142.183E-021.734E-061.915E-011.150E-041.734E-061.414E-012.289E-012.603E-061.734E-062.879E-06
F151.025E-051.734E-063.182E-063.112E-051.734E-061.734E-062.225E-041.734E-061.734E-061.734E-06
F162.765E-032.879E-062.603E-063.501E-021.734E-062.603E-068.307E-043.609E-031.734E-061.799E-05
F173.515E-061.639E-053.065E-049.263E-011.734E-061.921E-061.734E-061.921E-061.734E-061.734E-06
F181.114E-038.972E-021.734E-068.290E-011.734E-065.999E-011.150E-041.734E-061.734E-061.734E-06
F191.360E-054.729E-063.112E-053.162E-031.734E-061.639E-058.919E-051.360E-051.734E-061.734E-06
F202.127E-061.734E-061.484E-032.052E-041.734E-061.734E-063.515E-061.734E-061.734E-061.734E-06
F218.466E-061.494E-052.255E-039.426E-011.734E-061.734E-061.734E-064.286E-061.734E-061.734E-06
F221.109E-011.734E-062.127E-067.190E-021.734E-061.734E-062.225E-042.183E-021.734E-062.127E-06
F231.734E-061.734E-061.734E-061.734E-061.734E-061.734E-063.112E-051.734E-061.734E-067.691E-06
F241.734E-062.127E-063.286E-013.405E-051.734E-069.316E-061.734E-061.734E-061.734E-061.734E-06
F257.691E-061.734E-061.734E-061.734E-063.182E-061.484E-031.734E-061.734E-061.734E-061.734E-06
F265.307E-051.734E-062.127E-064.286E-061.734E-065.320E-033.182E-061.734E-061.734E-061.734E-06
F277.189E-013.881E-041.846E-015.170E-018.590E-021.025E-052.415E-031.477E-042.957E-035.706E-04
F281.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.484E-031.734E-061.734E-062.059E-01
F291.734E-061.734E-064.196E-041.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.921E-06
F301.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.734E-061.360E-05
Table A5

The mean and variance values of the PSNR evaluation at various threshold levels

ImageThresholdsItemCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
A4AVG1.959E + 011.934 E+011.880 E+011.665 E+011.654 E+011.720 E+011.880 E+011.713 E+011.792 E+01
STD1.066E + 001.224 E+002.165 E+002.914 E+001.996 E+001.342 E+001.727 E+002.016 E+001.318 E+00
5AVG2.038E + 012.022 E+012.012 E+011.844 E+011.705 E+011.871 E+011.985 E+011.887 E+011.915 E+01
STD1.358 E+001.149 E+001.239 E+001.731 E+002.671 E+008.464E-011.849 E+002.106 E+001.129 E+00
6AVG2.204E + 012.127 E+012.140 E+011.829 E+011.785 E+012.003 E+012.078 E+011.983 E+011.990 E+01
STD9.438E-019.622E-011.259 E+002.739 E+002.252 E+001.626 E+001.652 E+001.872 E+001.106 E+00
15AVG2.782E + 012.740 E+012.615 E+012.569 E+012.369 E+012.542 E+012.587 E+012.524 E+012.590 E+01
STD8.048E-018.418E-011.470 E+001.940 E+002.423 E+002.272 E+001.771 E+002.224 E+001.291 E+00
20AVG2.927E + 012.838 E+012.756 E+012.779 E+012.510 E+012.665 E+012.747 E+012.747 E+012.690 E+01
STD1.229 E+009.606E-011.542 E+002.197 E+001.956 E+001.531 E+001.672 E+001.498 E+001.752 E+00
25AVG3.030E + 013.008 E+012.941 E+012.858 E+012.760 E+012.941 E+012.905 E+012.955 E+012.924 E+01
STD1.584 E+001.276E + 001.838 E+002.334 E+002.214 E+001.382 E+001.649 E+002.028 E+001.422 E+00
B4AVG1.722 E+011.750E + 011.685 E+011.692 E+011.399 E+011.648 E+011.651 E+011.655 E+011.697 E+01
STD1.063 E+009.113E-011.386 E+001.618 E+002.532 E+001.546 E+001.222 E+001.396 E+001.846 E+00
5AVG1.921 E+011.956E + 011.866 E+011.775 E+011.504 E+011.817 E+011.752 E+011.844 E+011.760 E+01
STD8.006E-019.548E-011.232 E+001.856 E+002.622 E+001.531 E+002.022 E+001.609 E+002.134 E+00
6AVG2.088 E+012.122E + 012.003 E+011.878 E+011.699 E+011.914 E+011.907 E+011.901 E+011.916 E+01
STD8.633E-019.209E-011.542 E+001.871 E+002.238 E+001.429 E+002.089 E+001.649 E+001.497 E+00
15AVG2.768E + 012.672 E+012.604 E+012.453 E+012.424 E+012.535 E+012.492 E+012.507 E+012.498 E+01
STD1.113E + 001.640 E+001.830 E+002.704 E+002.177 E+001.515 E+001.637 E+002.217 E+001.816 E+00
20AVG2.938E + 012.848 E+012.772 E+012.711 E+012.530 E+012.702 E+012.677 E+012.735 E+012.702 E+01
STD1.177E + 001.683 E+001.622 E+001.882 E+002.104 E+001.305 E+002.184 E+002.122 E+001.454 E+00
25AVG3.051E + 012.997 E+012.952 E+012.934 E+012.738 E+012.895 E+012.877 E+012.857 E+012.822 E+01
STD1.321 E+001.446 E+001.956 E+001.480 E+001.480 E+001.745 E+001.251E + 002.478 E+001.757 E+00
C4AVG1.946 E+011.929 E+011.950E + 011.785 E+011.763 E+011.801 E+011.924 E+011.895 E+011.812 E+01
STD2.185E-013.951E-015.338E-011.746 E+002.318 E+001.317 E+001.264 E+001.044 E+001.035 E+00
5AVG2.066E + 012.034 E+012.011 E+011.865 E+011.860 E+011.913 E+012.031 E+011.892 E+011.901 E+01
STD4.865E-018.497E-011.348 E+002.090 E+002.254 E+009.773E-011.514 E+001.566 E+001.323 E+00
6AVG2.171 E+012.180E + 012.143 E+012.053 E+011.910 E+012.067 E+012.101 E+011.927 E+012.029 E+01
STD9.943E-011.060 E+001.218 E+001.615 E+002.159 E+001.389 E+001.779 E+001.842 E+001.380 E+00
15AVG2.770E + 012.687 E+012.659 E+012.574 E+012.460 E+012.518 E+012.565 E+012.532 E+012.584 E+01
STD1.562E + 001.671 E+001.913 E+002.378 E+001.657 E+001.613 E+001.917 E+001.845 E+001.964 E+00
20AVG2.960E + 012.862 E+012.859 E+012.789 E+012.584 E+012.788 E+012.692 E+012.772 E+012.751 E+01
STD1.491E + 001.901 E+002.019 E+002.370 E+002.221 E+001.659 E+002.086 E+002.099 E+001.940 E+00
25AVG3.057E + 013.046 E+012.934 E+012.917 E+012.798 E+012.991 E+012.901 E+012.891 E+012.931 E+01
STD1.879 E+001.527 E+002.385 E+002.418 E+002.705 E+001.189E + 002.132 E+002.402 E+001.741 E+00
D4AVG1.894E + 011.876 E+011.803 E+011.645 E+011.511 E+011.684 E+011.773 E+011.649 E+011.722 E+01
STD1.612E + 001.618 E+002.316 E+002.330 E+002.630 E+001.983 E+002.352 E+002.261 E+001.940 E+00
5AVG2.015 E+012.021E + 011.931 E+011.734 E+011.776 E+011.821 E+011.883 E+011.829 E+011.802 E+01
STD1.237 E+001.087E + 001.887 E+002.943 E+001.948 E+002.202 E+002.293 E+001.725 E+002.513 E+00
6AVG2.127E + 012.104 E+012.069 E+011.809 E+011.696 E+011.947 E+011.942 E+011.890 E+011.941 E+01
STD8.270E-011.222 E+001.676 E+002.840 E+002.845 E+001.575 E+002.115 E+002.212 E+001.811 E+00
15AVG2.719E + 012.707 E+012.647 E+012.480 E+012.282 E+012.552 E+012.548 E+012.592 E+012.560 E+01
STD1.446 E+001.364E + 001.610 E+003.439 E+002.459 E+001.790 E+001.858 E+002.340 E+001.487 E+00
20AVG2.937E + 012.824 E+012.816 E+012.678 E+012.474 E+012.754 E+012.748 E+012.794 E+012.696 E+01
STD1.376E + 001.457 E+001.910 E+002.784 E+002.584 E+001.470 E+001.883 E+001.998 E+002.259 E+00
25AVG3.058E + 012.985 E+012.925 E+012.963 E+012.749 E+012.889 E+012.878 E+012.864 E+012.906 E+01
STD1.726 E+001.730 E+001.650 E+001.952 E+002.209 E+001.587E + 001.630 E+002.624 E+001.612 E+00
E4AVG1.882E + 011.864 E+011.833 E+011.641 E+011.533 E+011.720 E+011.760 E+011.677 E+011.721 E+01
STD1.177 E+001.087E + 001.306 E+002.316 E+002.221 E+001.546 E+001.547 E+001.810 E+001.504 E+00
5AVG2.047E + 012.029 E+011.948 E+011.741 E+011.756 E+011.885 E+011.906 E+011.802 E+011.806 E+01
STD1.437 E+001.170E + 001.971 E+002.466 E+002.180 E+001.863 E+001.559 E+001.451 E+001.825 E+00
6AVG2.164 E+012.172E + 012.140 E+011.860 E+011.825 E+011.844 E+011.959 E+011.893 E+011.906 E+01
STD1.081 E+001.019E + 001.091 E+002.727 E+001.867 E+001.947 E+001.636 E+001.752 E+001.669 E+00
15AVG2.745E + 012.710 E+012.669 E+012.587 E+012.427 E+012.464 E+012.480 E+012.532 E+012.521 E+01
STD1.182E + 001.408 E+001.621 E+001.763 E+001.653 E+001.419 E+001.574 E+001.818 E+001.709 E+00
20AVG2.891E + 012.857 E+012.833 E+012.651 E+012.552 E+012.737 E+012.672 E+012.633 E+012.704 E+01
STD1.696 E+001.460 E+001.717 E+002.345 E+002.489 E+001.416E + 001.676 E+002.367 E+001.945 E+00
25AVG3.048E + 012.979 E+012.869 E+012.921 E+012.751 E+012.906 E+012.884 E+012.911 E+012.901 E+01
STD1.425 E+001.091E + 001.896 E+001.840 E+002.103 E+001.424 E+001.697 E+001.882 E+001.428 E+00
F4AVG1.888 E+011.887 E+011.866 E+011.722 E+011.619 E+011.739 E+011.937E + 011.741 E+011.737 E+01
STD1.098 E+009.216E-011.299 E+002.048 E+002.426 E+001.172 E+001.293 E+002.108 E+001.432 E+00
5AVG1.986E + 011.954 E+011.964 E+011.844 E+011.708 E+011.850 E+011.975 E+011.863 E+011.847 E+01
STD9.483E-018.464E-011.409 E+002.573 E+002.496 E+001.381 E+001.407 E+001.951 E+001.196 E+00
6AVG2.097E + 012.069 E+012.066 E+011.982 E+011.880 E+011.978 E+012.029 E+011.945 E+011.966 E+01
STD7.260E-016.737E-011.936 E+002.457 E+002.486 E+008.686E-011.917 E+001.679 E+001.564 E+00
15AVG2.702E + 012.653 E+012.565 E+012.500 E+012.379 E+012.485 E+012.536 E+012.493 E+012.507 E+01
STD1.268E + 001.279 E+002.150 E+002.621 E+002.571 E+001.945 E+001.519 E+002.216 E+002.044 E+00
20AVG2.879E + 012.818 E+012.779 E+012.699 E+012.568 E+012.751 E+012.653 E+012.710 E+012.675 E+01
STD1.375 E+001.690 E+001.895 E+002.702 E+002.168 E+001.373E + 002.488 E+002.426 E+001.725 E+00
25AVG2.963 E+012.977E + 012.878 E+012.884 E+012.735 E+012.904 E+012.805 E+012.871 E+012.915 E+01
STD2.037 E+001.839 E+001.672 E+002.262 E+001.954 E+002.342 E+001.766 E+001.912 E+001.434E + 00
G4AVG2.021 E+012.039E + 011.996 E+011.888 E+011.736 E+011.736 E+011.993 E+011.896 E+011.823 E+01
STD9.290E-012.820E-011.545 E+002.107 E+001.770 E+001.870 E+008.923E-011.401 E+001.716 E+00
5AVG2.150 E+012.155E + 012.065 E+011.859 E+011.822 E+011.937 E+012.093 E+011.926 E+011.870 E+01
STD5.570E-017.762E-012.008 E+002.386 E+002.033 E+002.017 E+001.345 E+002.189 E+001.907 E+00
6AVG2.220 E+012.278E + 012.229 E+012.113 E+011.823 E+011.951 E+012.050 E+011.982 E+011.977 E+01
STD1.407 E+006.127E-011.454 E+001.893 E+002.739 E+001.529 E+001.845 E+002.303 E+002.437 E+00
15AVG2.731E + 012.671 E+012.703 E+012.624 E+012.295 E+012.483 E+012.523 E+012.536 E+012.546 E+01
STD1.699 E+001.484E + 001.614 E+002.480 E+002.535 E+001.565 E+001.919 E+002.071 E+001.642 E+00
20AVG2.943E + 012.915 E+012.826 E+012.803 E+012.596 E+012.754 E+012.750 E+012.796 E+012.824 E+01
STD1.267 E+001.348 E+002.094 E+003.132 E+002.137 E+001.894 E+001.883 E+002.051 E+001.193E + 00
25AVG3.017E + 012.945 E+012.957 E+012.972 E+012.835 E+012.958 E+012.970 E+012.974 E+012.871 E+01
STD1.524 E+001.658 E+001.544 E+001.995 E+001.673 E+001.506E + 001.551 E+001.564 E+001.851 E+00
H4AVG1.965 E+012.004E + 011.909 E+011.607 E+011.628 E+011.791 E+011.873 E+011.776 E+011.741 E+01
STD1.265 E+007.748E-011.296 E+002.344 E+002.147 E+001.720 E+001.298 E+001.660 E+002.063 E+00
5AVG2.109 E+012.138E + 012.030 E+011.800 E+011.773 E+011.879 E+012.001 E+011.882 E+011.956 E+01
STD9.367E-016.617E-011.443 E+002.559 E+001.806 E+001.420 E+001.862 E+001.814 E+001.211 E+00
6AVG2.250E + 012.219 E+012.125 E+011.934 E+011.816 E+011.982 E+012.102 E+011.939 E+011.981 E+01
STD6.691E-011.205 E+001.802 E+002.291 E+002.343 E+001.806 E+001.456 E+002.459 E+001.655 E+00
15AVG2.767E + 012.680 E+012.648 E+012.609 E+012.473 E+012.527 E+012.455 E+012.567 E+012.543 E+01
STD1.197E + 001.652 E+002.067 E+001.661 E+002.376 E+001.760 E+002.201 E+001.430 E+001.820 E+00
20AVG2.966E + 012.794 E+012.779 E+012.712 E+012.551 E+012.760 E+012.675 E+012.758 E+012.774 E+01
STD1.164E + 001.727 E+002.127 E+002.066 E+002.286 E+001.482 E+002.341 E+001.914 E+001.310 E+00
25AVG3.049E + 013.022 E+013.006 E+012.946 E+012.798 E+012.890 E+012.918 E+012.981 E+012.941 E+01
STD1.596 E+001.134 E+001.557 E+002.082 E+001.489 E+001.660 E+001.880 E+001.625 E+001.104E + 00
I4AVG1.939E + 011.896 E+011.848 E+011.642 E+011.650 E+011.684 E+011.778 E+011.686 E+011.635 E+01
STD1.246E + 001.626 E+001.631 E+002.342 E+002.119 E+002.412 E+001.735 E+001.923 E+002.238 E+00
5AVG2.032E + 012.005 E+012.010 E+011.797 E+011.728 E+011.797 E+011.922 E+011.831 E+011.855 E+01
STD1.291 E+001.252E + 001.573 E+001.860 E+002.271 E+002.005 E+001.977 E+002.164 E+002.073 E+00
6AVG2.211E + 012.166 E+012.145 E+011.854 E+011.866 E+011.966 E+012.019 E+011.933 E+011.945 E+01
STD1.156 E+001.052E + 001.438 E+002.518 E+001.873 E+001.888 E+001.810 E+002.027 E+002.008 E+00
15AVG2.795E + 012.674 E+012.680 E+012.517 E+012.303 E+012.518 E+012.535 E+012.601 E+012.532 E+01
STD1.385E + 001.749 E+002.299 E+002.489 E+001.990 E+001.789 E+001.875 E+001.722 E+001.853 E+00
20AVG2.948E + 012.886 E+012.774 E+012.742 E+012.597 E+012.724 E+012.748 E+012.778 E+012.730 E+01
STD1.548 E+001.779 E+002.356 E+002.923 E+002.847 E+001.420E + 002.098 E+001.497 E+001.763 E+00
25AVG3.096E + 012.984 E+012.930 E+012.907 E+012.812 E+012.918 E+012.928 E+012.948 E+012.857 E+01
STD1.544 E+001.078E + 002.200 E+002.300 E+001.992 E+001.677 E+001.612 E+001.877 E+001.685 E+00
Table A6

The mean and variance values of the FSIM evaluation at various threshold levels

ImageThresholdsItemCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
A4AVG8.484E-018.399E-018.217E-017.569E-017.419E-017.662E-018.208E-017.658E-017.932E-01
STD4.071E-024.537E-026.682E-028.780E-026.248E-024.307E-025.978E-026.630E-024.044E-02
5AVG8.725E-018.691E-018.625E-018.144E-017.604E-018.172E-018.522E-018.212E-018.279E-01
STD4.302E-023.778E-023.726E-025.096E-028.170E-022.666E-025.614E-026.654E-023.805E-02
6AVG9.173E-018.989E-018.996E-018.058E-017.815E-018.521E-018.712E-018.433E-018.491E-01
STD2.276E-022.665E-023.439E-027.868E-026.517E-025.117E-024.571E-025.894E-023.268E-02
15AVG9.796E-019.761E-019.566E-019.434E-019.121E-019.402E-019.471E-019.356E-019.499E-01
STD6.867E-037.505E-032.093E-022.943E-025.060E-024.040E-022.928E-023.977E-021.906E-02
20AVG9.821E-019.767E-019.630E-019.603E-019.291E-019.509E-019.600E-019.616E-019.527E-01
STD7.883E-038.161E-032.033E-022.462E-023.250E-022.261E-022.065E-021.625E-022.370E-02
25AVG9.815E-019.817E-019.733E-019.645E-019.560E-019.739E-019.719E-019.727E-019.720E-01
STD1.224E-028.472E-031.744E-022.730E-022.906E-021.147E-021.372E-022.372E-021.311E-02
B4AVG7.304E-017.341E-017.269E-017.056E-016.816E-016.851E-017.168E-016.933E-017.090E-01
STD1.717E-022.114E-022.471E-023.072E-022.313E-023.681E-022.245E-023.181E-023.749E-02
5AVG7.621E-017.686E-017.424E-017.288E-016.965E-017.084E-017.432E-017.297E-017.175E-01
STD1.959E-021.715E-023.614E-022.701E-023.098E-023.230E-022.398E-023.387E-023.998E-02
6AVG7.907E-017.906E-017.770E-017.509E-017.127E-017.373E-017.539E-017.494E-017.400E-01
STD1.966E-022.033E-022.537E-023.058E-023.686E-023.413E-023.600E-023.169E-022.854E-02
15AVG9.147E-018.994E-018.834E-018.608E-018.469E-018.603E-018.579E-018.621E-018.553E-01
STD2.068E-023.261E-022.985E-024.478E-023.488E-023.207E-023.055E-023.873E-023.973E-02
20AVG9.347E-019.173E-019.086E-019.021E-018.617E-018.904E-018.896E-019.032E-018.924E-01
STD1.718E-022.878E-022.916E-023.526E-023.845E-022.413E-023.560E-023.439E-022.435E-02
25AVG9.442E-019.345E-019.303E-019.274E-018.940E-019.143E-019.184E-019.163E-019.088E-01
STD1.671E-021.976E-022.368E-022.088E-022.531E-023.170E-021.862E-023.534E-022.674E-02
C4AVG7.927E-017.915E-017.896E-017.527E-017.406E-017.560E-017.669E-017.685E-017.497E-01
STD1.707E-021.899E-022.273E-024.502E-024.853E-023.451E-022.745E-023.888E-023.850E-02
5AVG8.031E-017.948E-017.874E-017.728E-017.701E-017.548E-017.761E-017.630E-017.688E-01
STD1.751E-022.609E-022.551E-024.132E-024.657E-023.328E-024.263E-023.881E-023.373E-02
6AVG8.140E-018.104E-018.164E-017.973E-017.674E-017.871E-018.013E-017.812E-017.686E-01
STD1.952E-021.635E-023.041E-022.979E-024.847E-024.058E-023.874E-024.152E-024.052E-02
15AVG9.241E-019.109E-019.052E-018.878E-018.597E-018.866E-018.806E-018.840E-018.874E-01
STD1.766E-022.679E-023.065E-023.790E-023.903E-023.565E-024.313E-023.505E-023.281E-02
20AVG9.414E-019.286E-019.290E-019.129E-018.921E-019.176E-019.053E-019.182E-019.156E-01
STD1.850E-022.117E-022.887E-023.654E-023.351E-022.214E-023.816E-022.398E-022.632E-02
25AVG9.485E-019.482E-019.329E-019.310E-019.162E-019.337E-019.288E-019.311E-019.298E-01
STD1.822E-021.602E-022.636E-022.196E-022.942E-021.721E-022.739E-022.809E-022.289E-02
D4AVG7.525E-017.442E-017.339E-016.835E-016.764E-017.035E-017.187E-016.915E-017.047E-01
STD4.031E-024.374E-025.586E-025.068E-024.858E-024.097E-026.282E-025.934E-024.023E-02
5AVG7.805E-017.826E-017.600E-017.175E-017.232E-017.373E-017.586E-017.366E-017.337E-01
STD4.024E-023.551E-025.260E-026.389E-024.454E-024.459E-025.487E-023.950E-025.784E-02
6AVG8.176E-018.093E-018.033E-017.448E-017.093E-017.664E-017.703E-017.611E-017.687E-01
STD2.553E-023.514E-024.225E-025.582E-024.971E-023.838E-025.468E-024.787E-023.954E-02
15AVG9.419E-019.360E-019.271E-018.915E-018.437E-019.004E-019.036E-019.087E-019.063E-01
STD2.209E-022.309E-022.720E-026.149E-026.141E-023.409E-023.398E-024.728E-022.954E-02
20AVG9.622E-019.448E-019.404E-019.147E-018.811E-019.297E-019.307E-019.361E-019.209E-01
STD1.570E-022.204E-022.698E-024.710E-025.056E-022.392E-022.846E-023.000E-023.923E-02
25AVG9.674E-019.599E-019.495E-019.534E-019.250E-019.458E-019.455E-019.406E-019.449E-01
STD1.646E-021.867E-022.195E-022.462E-023.501E-022.115E-022.340E-023.736E-022.319E-02
E4AVG7.509E-017.527E-017.363E-017.236E-017.298E-017.140E-017.389E-017.196E-017.319E-01
STD1.638E-028.176E-033.199E-023.562E-023.568E-022.728E-022.658E-024.396E-023.186E-02
5AVG7.682E-017.634E-017.541E-017.302E-017.454E-017.409E-017.541E-017.359E-017.325E-01
STD2.885E-022.996E-023.786E-024.391E-024.073E-023.299E-022.831E-023.392E-022.917E-02
6AVG7.960E-017.849E-017.901E-017.459E-017.438E-017.435E-017.631E-017.482E-017.398E-01
STD2.664E-022.965E-022.580E-024.265E-023.332E-023.604E-023.406E-023.958E-023.033E-02
15AVG9.132E-019.057E-019.025E-018.822E-018.528E-018.583E-018.619E-018.719E-018.681E-01
STD1.994E-022.618E-022.484E-023.668E-023.706E-022.589E-022.952E-023.516E-022.897E-02
20AVG9.323E-019.249E-019.206E-018.908E-018.713E-018.986E-018.920E-018.878E-019.005E-01
STD2.678E-022.498E-022.877E-024.332E-024.224E-022.437E-023.108E-023.665E-023.518E-02
25AVG9.457E-019.379E-019.192E-019.276E-019.014E-019.239E-019.187E-019.286E-019.240E-01
STD1.852E-021.845E-022.999E-022.909E-023.174E-022.001E-022.802E-022.556E-022.578E-02
F4AVG8.658E-018.628E-018.564E-018.139E-017.546E-018.149E-018.702E-018.151E-018.133E-01
STD3.356E-022.939E-024.224E-025.517E-029.472E-023.748E-024.110E-026.037E-024.909E-02
5AVG8.891E-018.802E-018.807E-018.414E-017.864E-018.422E-018.804E-018.413E-018.442E-01
STD2.643E-022.336E-023.907E-027.118E-029.087E-024.978E-023.969E-026.807E-023.541E-02
6AVG9.163E-019.088E-019.035E-018.629E-018.349E-018.803E-018.878E-018.633E-018.729E-01
STD1.521E-021.773E-024.176E-027.072E-027.613E-022.809E-024.609E-026.057E-024.293E-02
15AVG9.756E-019.697E-019.553E-019.430E-019.233E-019.402E-019.506E-019.432E-019.453E-01
STD9.831E-031.347E-022.670E-023.596E-025.248E-024.076E-022.144E-023.164E-023.037E-02
20AVG9.805E-019.759E-019.707E-019.598E-019.488E-019.695E-019.565E-019.606E-019.612E-01
STD8.329E-031.193E-021.644E-022.401E-023.092E-021.310E-022.879E-022.868E-021.937E-02
25AVG9.799E-019.811E-019.753E-019.729E-019.641E-019.717E-019.697E-019.732E-019.771E-01
STD1.464E-021.157E-021.310E-021.716E-021.855E-022.292E-021.572E-021.539E-029.268E-03
G4AVG7.425E-017.473E-017.452E-017.412E-017.133E-016.998E-017.496E-017.334E-017.204E-01
STD2.017E-028.573E-032.868E-024.026E-024.258E-023.840E-021.947E-023.483E-023.584E-02
5AVG7.752E-017.798E-017.658E-017.356E-017.281E-017.433E-017.746E-017.440E-017.225E-01
STD1.434E-021.345E-024.173E-024.934E-024.262E-024.318E-023.190E-024.987E-024.293E-02
6AVG7.971E-018.076E-018.078E-017.978E-017.400E-017.469E-017.724E-017.578E-017.622E-01
STD3.026E-021.881E-023.372E-023.680E-025.366E-023.454E-024.621E-025.408E-025.453E-02
15AVG9.106E-018.967E-019.049E-018.885E-018.166E-018.580E-018.661E-018.691E-018.679E-01
STD3.696E-023.037E-023.204E-025.442E-025.883E-022.907E-024.153E-024.591E-023.835E-02
20AVG9.413E-019.347E-019.185E-019.123E-018.757E-019.003E-019.055E-019.117E-019.158E-01
STD2.141E-022.378E-024.171E-025.903E-024.763E-024.165E-023.069E-024.290E-022.369E-02
25AVG9.424E-019.322E-019.334E-019.361E-019.200E-019.298E-019.351E-019.380E-019.183E-01
STD2.319E-022.643E-022.403E-022.991E-022.823E-022.623E-022.663E-022.327E-023.018E-02
H4AVG7.315E-017.356E-017.245E-016.779E-016.897E-016.962E-017.276E-017.002E-016.949E-01
STD2.676E-021.101E-022.159E-024.609E-023.437E-023.651E-022.443E-023.268E-022.455E-02
5AVG7.637E-017.684E-017.437E-017.093E-017.140E-017.167E-017.475E-017.287E-017.369E-01
STD2.170E-021.729E-023.891E-025.248E-023.871E-023.275E-024.263E-023.701E-023.239E-02
6AVG7.972E-017.840E-017.666E-017.539E-017.078E-017.513E-017.696E-017.506E-017.503E-01
STD1.662E-023.440E-024.952E-023.856E-025.289E-022.513E-024.233E-024.327E-023.598E-02
15AVG9.149E-018.973E-018.996E-018.843E-018.531E-018.609E-018.532E-018.756E-018.676E-01
STD2.300E-023.201E-022.693E-023.266E-025.105E-023.644E-023.234E-022.591E-023.294E-02
20AVG9.392E-019.097E-019.136E-018.990E-018.709E-018.995E-018.859E-019.060E-019.068E-01
STD2.048E-022.905E-023.073E-023.189E-024.083E-022.815E-024.348E-023.205E-022.326E-02
25AVG9.471E-019.417E-019.387E-019.279E-019.043E-019.178E-019.252E-019.341E-019.263E-01
STD1.832E-021.555E-022.301E-022.814E-022.545E-022.839E-022.551E-022.524E-021.789E-02
I4AVG7.804E-017.745E-017.527E-017.212E-017.278E-017.322E-017.413E-017.226E-017.117E-01
STD3.384E-023.757E-025.209E-024.575E-024.768E-024.700E-024.474E-024.094E-024.171E-02
5AVG7.993E-017.954E-017.989E-017.618E-017.492E-017.533E-017.732E-017.603E-017.593E-01
STD3.520E-023.654E-023.957E-024.441E-024.804E-025.200E-024.933E-024.346E-024.531E-02
6AVG8.361E-018.243E-018.283E-017.671E-017.722E-017.844E-017.954E-017.828E-017.794E-01
STD2.900E-022.807E-023.832E-025.075E-024.925E-024.401E-024.499E-024.824E-024.694E-02
15AVG9.459E-019.269E-019.242E-018.997E-018.607E-018.874E-018.969E-019.086E-018.986E-01
STD2.070E-023.257E-024.346E-024.479E-023.288E-023.954E-023.761E-023.389E-023.439E-02
20AVG9.581E-019.481E-019.302E-019.285E-019.017E-019.248E-019.285E-019.344E-019.232E-01
STD1.959E-022.559E-023.974E-024.531E-024.816E-022.282E-023.593E-022.351E-023.511E-02
25AVG9.682E-019.576E-019.468E-019.453E-019.300E-019.447E-019.471E-019.505E-019.355E-01
STD1.382E-021.346E-023.021E-022.823E-023.441E-022.193E-022.377E-022.272E-022.518E-02
Table A7

The mean and variance values of the SSIM evaluation at various threshold levels

ImageThresholdsItemCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
A4AVG5.796E-015.663E-015.608E-014.961E-015.172E-015.072E-015.659E-015.088E-015.335E-01
STD4.528E-025.057E-027.063E-029.947E-026.419E-024.163E-026.636E-026.404E-024.814E-02
5AVG6.155E-016.090E-016.115E-015.696E-015.352E-015.647E-016.137E-015.735E-015.807E-01
STD5.617E-024.662E-024.746E-025.972E-027.978E-022.197E-026.225E-027.262E-024.314E-02
6AVG6.863E-016.570E-016.647E-015.782E-015.603E-016.162E-016.562E-016.141E-016.138E-01
STD3.966E-024.054E-025.153E-028.314E-026.436E-024.538E-025.849E-026.649E-023.670E-02
15AVG8.762E-018.664E-018.343E-018.187E-017.555E-018.049E-018.195E-018.062E-018.226E-01
STD1.893E-022.207E-023.788E-024.815E-027.299E-026.319E-024.832E-025.557E-023.605E-02
20AVG9.011E-018.816E-018.624E-018.656E-017.991E-018.410E-018.559E-018.585E-018.434E-01
STD2.566E-022.266E-023.535E-024.847E-024.918E-023.708E-024.225E-023.830E-023.968E-02
25AVG9.142E-019.107E-018.957E-018.782E-018.585E-018.927E-018.889E-018.987E-018.930E-01
STD2.689E-022.259E-023.674E-025.062E-025.034E-022.928E-023.142E-024.204E-022.726E-02
B4AVG7.230E-017.174E-017.071E-016.582E-016.728E-016.466E-017.044E-016.491E-016.692E-01
STD1.372E-023.120E-024.095E-026.336E-022.787E-025.317E-024.830E-026.668E-024.629E-02
5AVG7.336E-017.312E-017.067E-016.635E-016.804E-016.767E-017.211E-016.853E-016.748E-01
STD2.722E-022.669E-025.249E-026.539E-024.270E-023.930E-024.536E-024.210E-025.938E-02
6AVG7.519E-017.437E-017.271E-017.059E-016.957E-016.999E-017.216E-017.077E-017.017E-01
STD2.523E-022.501E-024.632E-025.296E-024.281E-024.334E-023.649E-024.581E-023.764E-02
15AVG8.458E-018.338E-018.276E-018.137E-017.985E-018.144E-018.105E-018.119E-018.092E-01
STD1.770E-022.726E-022.372E-023.345E-022.693E-022.125E-022.246E-022.931E-022.647E-02
20AVG8.780E-018.647E-018.581E-018.529E-018.189E-018.443E-018.415E-018.524E-018.483E-01
STD1.455E-022.291E-022.254E-023.217E-022.956E-021.799E-022.733E-022.806E-021.583E-02
25AVG8.943E-018.879E-018.848E-018.834E-018.548E-018.692E-018.693E-018.759E-018.661E-01
STD1.554E-021.705E-022.028E-021.831E-021.875E-022.508E-021.787E-022.473E-022.065E-02
C4AVG5.590E-015.601E-015.653E-015.610E-016.098E-015.402E-015.926E-015.751E-015.400E-01
STD2.209E-022.734E-023.682E-027.324E-027.882E-024.050E-026.724E-027.196E-025.305E-02
5AVG5.981E-015.844E-015.824E-015.958E-016.774E-015.924E-016.395E-016.043E-015.915E-01
STD6.864E-026.518E-026.394E-028.674E-028.309E-025.615E-028.772E-029.490E-027.428E-02
6AVG6.490E-016.487E-016.415E-016.948E-016.621E-016.362E-016.882E-016.370E-016.206E-01
STD6.716E-027.715E-027.716E-028.273E-029.748E-027.084E-029.830E-028.089E-027.834E-02
15AVG8.594E-018.413E-018.294E-018.095E-018.077E-018.293E-018.064E-018.145E-018.452E-01
STD4.242E-024.601E-026.406E-027.351E-025.452E-024.961E-026.945E-026.723E-022.580E-02
20AVG8.887E-018.824E-018.792E-018.683E-018.273E-018.658E-018.498E-018.562E-018.640E-01
STD3.642E-022.496E-024.172E-024.535E-027.426E-022.953E-025.400E-025.458E-024.444E-02
25AVG9.094E-019.096E-018.855E-018.762E-018.788E-018.940E-018.840E-018.852E-018.886E-01
STD1.650E-021.669E-024.991E-026.192E-022.243E-021.496E-024.476E-023.913E-022.743E-02
D4AVG6.478E-016.306E-016.110E-015.671E-015.984E-015.970E-016.029E-015.723E-015.890E-01
STD3.083E-023.865E-025.790E-025.883E-025.091E-023.438E-026.445E-026.001E-025.731E-02
5AVG6.476E-016.547E-016.348E-015.996E-016.271E-016.135E-016.268E-016.228E-016.098E-01
STD3.659E-023.160E-024.402E-027.121E-024.150E-025.291E-025.216E-023.875E-026.036E-02
6AVG6.683E-016.643E-016.644E-016.220E-015.996E-016.467E-016.521E-016.322E-016.533E-01
STD2.154E-022.751E-024.061E-026.193E-027.406E-023.811E-024.891E-024.803E-023.850E-02
15AVG8.227E-018.224E-018.124E-017.861E-017.351E-017.910E-017.929E-018.071E-017.987E-01
STD3.256E-023.188E-023.576E-026.074E-025.063E-023.701E-023.787E-024.489E-022.896E-02
20AVG8.729E-018.491E-018.507E-018.302E-017.879E-018.361E-018.342E-018.446E-018.256E-01
STD2.412E-022.881E-023.494E-024.884E-025.270E-022.662E-023.715E-023.846E-023.715E-02
25AVG8.917E-018.776E-018.704E-018.757E-018.336E-018.627E-018.610E-018.606E-018.669E-01
STD2.508E-023.076E-022.656E-023.358E-024.206E-022.495E-022.733E-024.740E-022.476E-02
E4AVG7.141E-017.183E-016.861E-016.350E-016.822E-016.508E-016.966E-016.329E-016.759E-01
STD2.629E-021.030E-024.882E-029.758E-024.561E-024.708E-024.835E-026.287E-024.541E-02
5AVG7.153E-017.132E-016.852E-016.633E-016.870E-016.736E-017.064E-016.650E-016.726E-01
STD2.488E-023.285E-025.622E-026.566E-025.835E-024.834E-023.577E-025.293E-024.611E-02
6AVG7.332E-017.161E-017.169E-016.653E-016.977E-016.789E-016.943E-016.820E-016.746E-01
STD2.265E-022.472E-023.812E-026.230E-023.810E-024.952E-024.741E-025.362E-023.776E-02
15AVG8.316E-018.223E-018.291E-018.120E-017.877E-017.936E-017.918E-018.065E-017.962E-01
STD1.915E-022.265E-022.577E-023.064E-023.554E-022.763E-022.281E-022.737E-022.520E-02
20AVG8.631E-018.523E-018.558E-018.276E-018.109E-018.354E-018.215E-018.226E-018.366E-01
STD2.300E-022.515E-022.468E-024.128E-023.617E-022.150E-022.688E-023.141E-023.073E-02
25AVG8.827E-018.749E-018.618E-018.682E-018.462E-018.641E-018.581E-018.703E-018.645E-01
STD2.001E-021.633E-022.504E-022.516E-022.441E-021.734E-022.513E-022.579E-022.029E-02
F4AVG5.989E-015.947E-015.852E-015.238E-014.737E-015.320E-016.174E-015.320E-015.311E-01
STD5.321E-024.582E-026.759E-021.019E-011.185E-015.514E-026.424E-021.014E-016.284E-02
5AVG6.436E-016.276E-016.353E-015.812E-015.189E-015.834E-016.445E-015.915E-015.868E-01
STD4.536E-024.126E-026.403E-021.156E-011.226E-016.580E-026.089E-029.012E-025.493E-02
6AVG6.956E-016.826E-016.830E-016.513E-016.029E-016.469E-016.696E-016.294E-016.410E-01
STD3.267E-023.167E-027.975E-021.012E-011.170E-013.659E-028.091E-028.011E-026.637E-02
15AVG8.890E-018.742E-018.475E-018.285E-017.817E-018.208E-018.386E-018.244E-018.248E-01
STD2.902E-023.402E-025.781E-026.982E-028.999E-026.227E-024.123E-026.469E-026.215E-02
20AVG9.159E-019.036E-018.936E-018.706E-018.401E-018.884E-018.583E-018.734E-018.713E-01
STD2.550E-023.074E-023.859E-025.363E-025.898E-022.661E-026.141E-025.356E-023.886E-02
25AVG9.225E-019.251E-019.090E-019.051E-018.762E-019.054E-018.925E-019.046E-019.132E-01
STD3.523E-022.948E-023.284E-024.464E-024.314E-024.783E-023.621E-023.720E-022.603E-02
G4AVG6.838E-016.855E-016.889E-016.830E-016.612E-016.330E-016.980E-016.667E-016.638E-01
STD2.726E-021.331E-024.409E-024.790E-025.015E-026.228E-021.629E-024.499E-025.970E-02
5AVG7.068E-017.069E-016.895E-016.738E-016.741E-016.790E-017.102E-016.847E-016.652E-01
STD1.417E-021.968E-026.051E-027.466E-026.784E-024.350E-022.465E-025.105E-025.463E-02
6AVG7.160E-017.277E-017.341E-017.258E-016.851E-016.866E-017.114E-016.896E-016.982E-01
STD3.055E-021.509E-022.552E-024.079E-025.850E-024.186E-024.294E-025.919E-026.789E-02
15AVG8.259E-018.169E-018.302E-018.240E-017.596E-017.921E-017.977E-018.050E-017.989E-01
STD3.323E-022.500E-023.065E-024.799E-025.194E-022.177E-023.778E-023.919E-023.318E-02
20AVG8.695E-018.610E-018.530E-018.521E-018.166E-018.314E-018.385E-018.450E-018.472E-01
STD2.181E-022.371E-023.693E-025.474E-023.889E-023.448E-023.308E-024.053E-022.190E-02
25AVG8.788E-018.693E-018.729E-018.767E-018.605E-018.682E-018.720E-018.733E-018.586E-01
STD2.469E-022.439E-022.467E-023.082E-022.696E-022.377E-022.545E-022.672E-022.242E-02
H4AVG6.793E-016.814E-016.513E-015.590E-016.306E-016.351E-016.756E-016.246E-016.161E-01
STD6.370E-024.418E-027.277E-021.300E-018.977E-026.506E-026.472E-029.002E-029.942E-02
5AVG7.001E-017.050E-016.800E-016.331E-016.637E-016.626E-016.970E-016.595E-016.923E-01
STD3.398E-022.647E-025.287E-029.680E-026.926E-025.341E-027.544E-025.700E-024.166E-02
6AVG7.286E-017.176E-017.096E-016.719E-016.684E-016.797E-017.240E-016.636E-016.952E-01
STD1.966E-023.061E-025.393E-028.638E-025.783E-026.634E-023.065E-028.346E-024.821E-02
15AVG8.387E-018.284E-018.268E-018.227E-017.983E-018.018E-017.904E-018.105E-018.050E-01
STD2.238E-023.021E-023.721E-022.954E-024.531E-022.816E-023.902E-022.576E-023.795E-02
20AVG8.747E-018.472E-018.508E-018.518E-018.217E-018.400E-018.367E-018.537E-018.494E-01
STD1.969E-022.833E-023.522E-022.461E-023.396E-022.328E-023.455E-022.563E-022.184E-02
25AVG8.905E-018.852E-018.812E-018.776E-018.532E-018.654E-018.726E-018.788E-018.709E-01
STD1.907E-021.901E-022.445E-022.730E-022.386E-022.431E-022.936E-022.262E-021.739E-02
I4AVG6.609E-016.595E-016.401E-015.885E-016.204E-016.076E-016.202E-015.918E-015.929E-01
STD2.431E-023.656E-024.673E-025.856E-025.380E-026.375E-024.865E-024.164E-025.040E-02
5AVG6.600E-016.661E-016.647E-016.273E-016.319E-016.403E-016.421E-016.224E-016.345E-01
STD3.058E-024.017E-024.217E-024.204E-025.574E-024.966E-024.030E-024.037E-023.924E-02
6AVG6.891E-016.813E-016.860E-016.380E-016.521E-016.639E-016.610E-016.500E-016.587E-01
STD2.129E-022.072E-023.610E-024.788E-025.264E-024.005E-024.099E-024.800E-023.943E-02
15AVG8.333E-018.081E-018.174E-017.933E-017.552E-017.800E-017.849E-017.978E-017.930E-01
STD2.914E-023.989E-024.142E-025.041E-022.735E-023.376E-023.982E-023.473E-023.440E-02
20AVG8.670E-018.539E-018.406E-018.367E-018.043E-018.269E-018.325E-018.377E-018.255E-01
STD2.699E-023.323E-024.237E-025.000E-025.009E-022.436E-023.818E-022.682E-023.236E-02
25AVG8.895E-018.732E-018.720E-018.602E-018.427E-018.603E-018.626E-018.705E-018.498E-01
STD2.489E-021.857E-023.109E-023.922E-023.538E-022.816E-022.949E-022.976E-023.029E-02
Table A8

The analytical results of PSNR evaluation gained by using the WSRT

ThresholdsItemCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
4+/−/ =~1/0/83/0/68/0/19/0/09/0/06/0/39/0/08/0/1
Mean1.5561.8893.1117.4448.6676.4444.0005.8896.000
Rank123897456
5+/−/ =~0/0/93/0/69/0/09/0/09/0/06/0/39/0/09/0/0
Mean1.4441.8893.1118.0008.7785.8894.0005.7786.111
Rank123896457
6+/−/ =~1/1/74/0/59/0/09/0/09/0/07/0/29/0/09/0/0
Mean1.5561.6672.7787.0008.8895.6674.4447.0006.000
Rank123795476
15+/−/ =~3/0/66/0/39/0/09/0/09/0/09/0/09/0/09/0/0
Mean1.0002.2222.7785.8898.8897.0006.3335.6675.222
Rank123698754
20+/−/ =~5/0/47/0/28/0/19/0/09/0/09/0/09/0/09/0/0
Mean1.0002.0003.2225.6679.0005.7787.0005.2226.111
Rank123596847
25+/−/ =~1/0/86/0/37/0/29/0/06/0/38/0/15/0/48/0/1
Mean1.1112.4444.7784.8899.0005.1116.4445.1116.111
Rank123495857
Table A9

The analytical results of FSIM evaluation gained by using the WSRT

ThresholdsItemCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
4+/−/ =~0/0/93/0/68/0/19/0/09/0/04/0/58/0/19/0/0
Mean1.7781.7783.3337.2228.1116.7783.2226.2226.556
Rank114897356
5+/−/ =~0/0/93/0/69/0/09/0/09/0/04/0/59/0/09/0/0
Mean1.4441.8893.1117.2227.8896.8893.5566.3336.667
Rank123897456
6+/−/ =~1/0/83/0/68/0/19/0/09/0/08/0/19/0/09/0/0
Mean1.4442.4442.3336.3338.6676.3334.0006.5566.889
Rank132595478
15+/−/ =~4/0/58/0/19/0/09/0/09/0/09/0/09/0/09/0/0
Mean1.0002.2222.7785.2229.0007.2226.5565.3335.667
Rank123498756
20+/−/ =~5/0/47/0/28/0/19/0/09/0/09/0/09/0/09/0/0
Mean1.0002.2222.8896.3339.0006.2226.7784.8895.667
Rank123796845
25+/−/ =~1/0/86/0/38/0/19/0/08/0/18/0/17/0/28/0/1
Mean1.2222.2224.3335.0008.8895.8896.3334.5566.556
Rank123596748
Table A10

The analytical results of SSIM evaluation gained by using the WSRT

ThresholdsItemCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
4+/−/ =~1/0/84/1/47/0/27/1/19/0/03/2/47/0/28/0/1
Mean2.3332.4443.4447.6675.4447.1112.8896.7786.889
Rank124958367
5+/−/ =~0/0/92/0/78/0/17/1/17/0/21/1/77/0/27/0/2
Mean2.1112.7784.2228.0005.8896.5562.6676.1116.667
Rank134957268
6+/−/ =~2/0/72/1/66/1/28/0/18/0/14/0/58/0/16/0/3
Mean1.6673.0002.6676.2227.4446.4444.0007.0006.556
Rank132596487
15+/−/ =~1/0/84/0/58/0/19/0/09/0/09/0/08/0/19/0/0
Mean1.1112.5562.6675.2228.7786.8896.8895.4445.444
Rank123497755
20+/−/ =~5/0/46/0/38/0/19/0/09/0/09/0/09/0/09/0/0
Mean1.0002.6673.0004.7789.0006.2227.2225.0006.111
Rank123497856
25+/−/ =~1/0/85/0/46/0/39/0/07/0/28/0/15/0/48/0/1
Mean1.2222.2224.1115.3338.7786.0006.5564.7786.000
Rank123596846
Table A11

The maximum 2D Kapur's entropy obtained by each method

ImageThresholdsCLACO-MISACOR-MISMVO-MISHHO-MISSCA-MISBLPSO-MISIGWO-MISIWOA-MISCLPSO-MIS
A43.872E + 013.871 E+013.867 E+013.813 E+013.802 E+013.749 E+013.851 E+013.842 E+013.796 E+01
54.417E + 014.417 E+014.413 E+014.316 E+014.216 E+014.274 E+014.391 E+014.361 E+014.251 E+01
64.933E + 014.930 E+014.926 E+014.871 E+014.647 E+014.792 E+014.887 E+014.841 E+014.805 E+01
158.775E + 018.751 E+018.634 E+018.559 E+017.675 E+018.081 E+018.254 E+018.268 E+018.233 E+01
201.028E + 021.020 E+021.005 E+029.757 E+018.820 E+019.494 E+019.569 E+019.714 E+019.503 E+01
251.135 E+021.137E + 021.098 E+021.128 E+021.039 E+021.042 E+021.070 E+021.075 E+021.084 E+02
B43.865E + 013.864 E+013.862 E+013.846 E+013.762 E+013.567 E+013.855 E+013.810 E+013.759 E+01
54.431E + 014.430 E+014.426 E+014.399 E+014.221 E+014.189 E+014.423 E+014.321 E+014.271 E+01
64.958 E+014.964E + 014.942 E+014.839 E+014.632 E+014.638 E+014.911 E+014.825 E+014.743 E+01
158.630E + 018.601 E+018.525 E+018.372 E+017.764 E+017.842 E+018.241 E+018.274 E+018.108 E+01
201.006E + 029.966 E+019.838 E+019.810 E+018.751 E+019.208 E+019.616 E+019.597 E+019.633 E+01
251.129E + 021.127 E+021.104 E+021.098 E+029.739 E+011.010 E+021.055 E+021.077 E+021.023 E+02
C43.852E + 013.852 E+013.852 E+013.834 E+013.765 E+013.707 E+013.843 E+013.844 E+013.778 E+01
54.441E + 014.440 E+014.426 E+014.333 E+014.274 E+014.272 E+014.427 E+014.400 E+014.322 E+01
64.989E + 014.978 E+014.969 E+014.874 E+014.610 E+014.768 E+014.899 E+014.855 E+014.819 E+01
158.657E + 018.582 E+018.538 E+018.534 E+017.757 E+018.216 E+018.379 E+018.280 E+018.370 E+01
201.008E + 021.007 E+029.858 E+019.994 E+018.893 E+019.576 E+019.614 E+019.788 E+019.522 E+01
251.131E + 021.126 E+021.104 E+021.118 E+029.940 E+011.050 E+021.052 E+021.072 E+021.069 E+02
D43.885 E+013.902E + 013.876 E+013.882 E+013.738 E+013.726 E+013.865 E+013.855 E+013.784 E+01
54.492E + 014.488 E+014.474 E+014.427 E+014.300 E+014.402 E+014.422 E+014.398 E+014.415 E+01
65.036E + 015.033 E+014.994 E+014.913 E+014.704 E+014.750 E+014.932 E+014.982 E+014.877 E+01
158.722E + 018.689 E+018.485 E+018.464 E+017.853 E+018.119 E+018.202 E+018.303 E+018.249 E+01
201.017E + 021.003 E+029.961 E+019.715 E+018.864 E+019.314 E+019.553 E+019.707 E+019.525 E+01
251.124 E+021.129E + 021.092 E+021.101 E+029.887 E+011.036 E+021.040 E+021.093 E+021.049 E+02
E43.865E + 013.865 E+013.863 E+013.830 E+013.768 E+013.667 E+013.862 E+013.793 E+013.802 E+01
54.441E + 014.440 E+014.431 E+014.362 E+014.130 E+014.339 E+014.427 E+014.325 E+014.255 E+01
64.983E + 014.978 E+014.950 E+014.785 E+014.550 E+014.685 E+014.884 E+014.853 E+014.744 E+01
158.665 E+018.685E + 018.558 E+018.411 E+017.637 E+018.089 E+018.119 E+018.292 E+018.142 E+01
201.015E + 021.008 E+029.929 E+011.000 E+028.995 E+019.157 E+019.487 E+019.547 E+019.727 E+01
251.123E + 021.104 E+021.107 E+021.085 E+021.018 E+021.010 E+021.048 E+021.065 E+021.057 E+02
F43.840 E+013.840 E+013.841E + 013.784 E+013.776 E+013.713 E+013.833 E+013.793 E+013.729 E+01
54.401E + 014.401 E+014.397 E+014.266 E+014.200 E+014.259 E+014.368 E+014.343 E+014.258 E+01
64.935E + 014.931 E+014.920 E+014.825 E+014.788 E+014.773 E+014.873 E+014.798 E+014.780 E+01
158.629E + 018.578 E+018.366 E+018.381 E+017.716 E+018.164 E+018.158 E+018.177 E+018.120 E+01
201.010E + 029.946 E+019.867 E+019.667 E+018.795 E+019.335 E+019.390 E+019.553 E+019.386 E+01
251.104E + 021.100 E+021.092 E+021.090 E+029.695 E+011.018 E+021.037 E+021.058 E+021.036 E+02
G43.853E + 013.852 E+013.852 E+013.840 E+013.799 E+013.684 E+013.849 E+013.838 E+013.807 E+01
54.419E + 014.419 E+014.404 E+014.388 E+014.216 E+014.209 E+014.382 E+014.337 E+014.305 E+01
64.965E + 014.960 E+014.952 E+014.857 E+014.699 E+014.733 E+014.876 E+014.851 E+014.781 E+01
158.654E + 018.628 E+018.516 E+018.422 E+017.655 E+018.056 E+018.173 E+018.288 E+018.189 E+01
201.012E + 029.994 E+019.914 E+019.842 E+018.960 E+019.466 E+019.586 E+019.686 E+019.504 E+01
251.113 E+021.126E + 021.088 E+021.101 E+029.681 E+011.039 E+021.055 E+021.078 E+021.070 E+02
H43.872E + 013.872 E+013.869 E+013.853 E+013.739 E+013.690 E+013.868 E+013.854 E+013.751 E+01
54.456E + 014.456 E+014.445 E+014.374 E+014.210 E+014.213 E+014.427 E+014.407 E+014.401 E+01
64.984E + 014.981 E+014.978 E+014.898 E+014.706 E+014.739 E+014.958 E+014.908 E+014.835 E+01
158.639E + 018.604 E+018.468 E+018.366 E+017.692 E+017.984 E+018.077 E+018.267 E+018.172 E+01
201.005E + 029.956 E+011.003 E+029.869 E+018.698 E+019.315 E+019.450 E+019.597 E+019.416 E+01
251.109 E+021.100 E+021.112E + 021.105 E+029.802 E+011.038 E+021.061 E+021.075 E+021.054 E+02
I43.883E + 013.881 E+013.824 E+013.765 E+013.713 E+013.736 E+013.849 E+013.765 E+013.774 E+01
54.418 E+014.411 E+014.421E + 014.316 E+014.246 E+014.276 E+014.344 E+014.293 E+014.313 E+01
64.958E + 014.949 E+014.935 E+014.830 E+014.677 E+014.760 E+014.893 E+014.865 E+014.801 E+01
158.695 E+018.736E + 018.550 E+018.484 E+017.611 E+018.014 E+018.233 E+018.229 E+018.123 E+01
201.010E + 029.974 E+011.003 E+029.938 E+018.744 E+019.282 E+019.474 E+019.648 E+019.595 E+01
251.127E + 021.111 E+021.109 E+021.095 E+029.719 E+011.033 E+021.084 E+021.090 E+021.041 E+02
  5 in total

1.  Multi-Threshold Image Segmentation of Maize Diseases Based on Elite Comprehensive Particle Swarm Optimization and Otsu.

Authors:  Chengcheng Chen; Xianchang Wang; Ali Asghar Heidari; Helong Yu; Huiling Chen
Journal:  Front Plant Sci       Date:  2021-12-13       Impact factor: 5.753

2.  Detection of COVID-19 severity using blood gas analysis parameters and Harris hawks optimized extreme learning machine.

Authors:  Jiao Hu; Zhengyuan Han; Ali Asghar Heidari; Yeqi Shou; Hua Ye; Liangxing Wang; Xiaoying Huang; Huiling Chen; Yanfan Chen; Peiliang Wu
Journal:  Comput Biol Med       Date:  2021-12-24       Impact factor: 4.589

3.  Machine learning for early discrimination between transient and persistent acute kidney injury in critically ill patients with sepsis.

Authors:  Xiao-Qin Luo; Ping Yan; Ning-Ya Zhang; Bei Luo; Mei Wang; Ying-Hao Deng; Ting Wu; Xi Wu; Qian Liu; Hong-Shen Wang; Lin Wang; Yi-Xin Kang; Shao-Bin Duan
Journal:  Sci Rep       Date:  2021-10-12       Impact factor: 4.379

4.  Multilevel thresholding satellite image segmentation using chaotic coronavirus optimization algorithm with hybrid fitness function.

Authors:  Khalid M Hosny; Asmaa M Khalid; Hanaa M Hamza; Seyedali Mirjalili
Journal:  Neural Comput Appl       Date:  2022-09-23       Impact factor: 5.102

5.  The Technology-Oriented Pathway for Auxiliary Diagnosis in the Digital Health Age: A Self-Adaptive Disease Prediction Model.

Authors:  Zhiyuan Hao; Jie Ma; Wenjing Sun
Journal:  Int J Environ Res Public Health       Date:  2022-09-30       Impact factor: 4.614
  5 in total

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