Multi-Level Thresholding Image Segmentation Based on Improved Slime Mould Algorithm and Symmetric Cross-Entropy

Multi-level thresholding image segmentation divides an image into multiple regions of interest and is a key step in image processing and image analysis. Aiming toward the problems of the low segmentation accuracy and slow convergence speed of traditional multi-level threshold image segmentation methods, in this paper, we present multi-level thresholding image segmentation based on an improved slime mould algorithm (ISMA) and symmetric cross-entropy for global optimization and image segmentation tasks. First, elite opposition-based learning (EOBL) was used to improve the quality and diversity of the initial population and accelerate the convergence speed. The adaptive probability threshold was used to adjust the selection probability of the slime mould to enhance the ability of the algorithm to jump out of the local optimum. The historical leader strategy, which selects the optimal historical information as the leader for the position update, was found to improve the convergence accuracy. Subsequently, 14 benchmark functions were used to evaluate the performance of ISMA, comparing it with other well-known algorithms in terms of the optimization accuracy, convergence speed, and significant differences. Subsequently, we tested the segmentation quality of the method proposed in this paper on eight grayscale images and compared it with other image segmentation criteria and well-known algorithms. The experimental metrics include the average fitness (mean), standard deviation (std), peak signal to noise ratio (PSNR), structure similarity index (SSIM), and feature similarity index (FSIM), which we utilized to evaluate the quality of the segmentation. The experimental results demonstrated that the improved slime mould algorithm is superior to the other compared algorithms, and multi-level thresholding image segmentation based on the improved slime mould algorithm and symmetric cross-entropy can be effectively applied to the task of multi-level threshold image segmentation.


Introduction
Image segmentation is a key part of image processing [1][2][3], which aims to extract the target region of interest from the image. Image segmentation is widely used in various fields, such as medical image processing [4,5], agricultural image processing [6,7], and remote sensing image analysis [8,9], because of its simplicity and effectiveness. At present, the commonly used image processing methods are threshold segmentation [10], region segmentation [11,12], and clustering segmentation [13,14]. Among them, the threshold segmentation method is a popular research direction in the field of image segmentation. For complex images, multiple thresholds are selected to segment the image into multiple effective targets.
Multi-threshold segmentation identifies a set of threshold values in the image to be segmented according to a certain criterion and segments the image into multiple parts. Common threshold selection methods for multi-threshold segmentation criteria include Otsu's method (by maximizing the between-class variance), Kapur's entropy method (by maximizing the entropy of the classes), fuzzy entropy, minimum cross-entropy, etc. [15][16][17]. However, the multi-threshold segmentation of images expands the search space with the increase in the number of thresholds, the computational complexity increases exponentially, and the computational efficiency is low.
In recent years, meta-heuristic algorithms (MAs) have been widely used for data processing and practical problems such as multi-peaked, high-dimensional, and nonlinear complex computations. In the face of uncertainty or a large range of solution spaces, MAs use their stochastic search capability to obtain the optimal solution without traversing the solution space, which can greatly reduce the computational effort and the optimization search time, with examples such as the seagull optimization algorithm (SOA) [18], grey wolf optimizer (GWO) [19], sparrow search algorithm (SSA) [20], moth-flame optimization (MFO) [21], pelican optimization algorithm [22], etc. Due to the superiority of MAs, researchers have applied the optimization algorithms to multilevel threshold image segmentation tasks [23]. Lang et al. [24] used differential evolution (DE) as a local search technique to improve the situation whereby WOA falls into local optimization in the later iterations and combined WOA-DE with Kapur's entropy to obtain an effective image segmentation method. Yu et al. [25] optimized the grey wolf optimizer by modifying the weights of the first three wolves to make full use of the knowledge of the first three wolves and achieved good results in the image segmentation task. Zhao et al. [26] improved the ant colony optimization (ACO) algorithm by the randomized alternate strategy and chaotic enhancement strategy and performed image segmentation experiments at low and high threshold levels, respectively, and the experimental results were satisfactory. Houssein et al. [27] proposed an image-thresholding method based on the black widow optimization algorithm to extract the optimal threshold values for the selected images using Otsu's and Kapur's entropy methods, respectively. Chen et al. [28] used a multi-strategy optimized shuffled frog leaping algorithm (SFLA) combined with Kapur's entropy method for the multi-threshold image segmentation of common breast cancers, which outperformed the other competitors in terms of the solution efficiency and time complexity. MAs show a good segmentation performance in the field of multi-threshold image segmentation. The slime mould algorithm (SMA) [29] is a newly proposed meta-heuristic swarm intelligence algorithm with the advantages of a high merit-seeking ability and few parameters. However, in the late iterations of the algorithm, SMA, like other intelligent algorithms, is prone to fall into the local optimal solution. Örnek et al. [30] proposed an enhanced slime mould algorithm (ESMA) with a better ability to avoid local optimization by updating the position of the sticky bacterium using sine and cosine trigonometric functions. Hu et al. [31] proposed a dispersed foraging SMA (DFSMA) with a dispersed foraging strategy, which outperformed the other optimizers in terms of the convergence speed and accuracy.
In this paper, we propose an improved slime mould algorithm, called ISMA, for the multilevel thresholding image segmentation task. First, elite opposition-based learning (EOBL) was used to improve the quality and diversity of the initial population and accelerate the convergence speed. The adaptive probability threshold was used to adjust the selection probability of the slime mould so as to enhance the ability of the algorithm to jump out of the local optimum. The historical leader strategy, which selects the optimal historical information as the leader for the position update, was found to improve the convergence accuracy. Moreover, the optimization ability and solution accuracy of ISMA were verified through single-peak and multi-peak benchmark test functions. ISMA was combined with symmetric cross-entropy multi-threshold segmentation to solve the problems of the complicated calculation and low computational efficiency of multi-threshold image segmentation and realize multi-threshold image segmentation. We selected eight grayscale images as the reference images and performed a comparison of the different segmentation criteria and different MAs. The experimental results demonstrated that multi-level thresholding image segmentation based on ISMA and symmetric cross-entropy outperforms the other methods in terms of the PSNR, SSIM, and FSIM and showed significant improvements in the convergence speed and segmentation accuracy.
The main contributions of this paper can be summarized as follows: We optimized SMA through multiple strategies and propose an improved SMA, called ISMA; We evaluated the performance of ISMA through single-peak and multi-peak benchmark test functions; We combined ISMA with symmetric cross-entropy for threshold segmentation and compared it with Otsu's and Kapur's entropy methods and minimum cross-entropy; We compared the performance of ISMA with other MAs and evaluated the image segmentation quality through PSNR, SSIM, and FSIM.
The rest of the paper is organized as follows. Section 2 introduces the slime mould algorithm. Section 3 introduces the improvement strategy for the slime mould algorithm, elite opposition-based learning, the adaptive probability threshold, and the historical leader. Section 4 discusses ISMA performance tests. Section 5 introduces the threshold segmentation technology combining ISMA and symmetric cross-entropy. Section 6 describes image segmentation tests and analyzes the experimental results. Finally, Section 7 summarizes this paper and provides the future research directions.

Slime Mould Algorithm
The slime mould algorithm (SMA) establishes a mathematical model based on the foraging behavior of physarum polycephalum, which adjusts its position by oscillating reactions to search for the optimal food position.
The slime mould approach the food according to the odor in the air, and some individuals separate in order to search for higher-quality food in other domains after identifying the food source, as shown in the following equation: Smell Index = sort(S) where i ∈ 1, 2, . . . , n, S(i) denotes the fitness value of the slime mould; DF denotes the optimal fitness value; the condition is the rank first-half fitness of the search agent of S(i); r denotes the random value in [0, 1]; max_t denotes the maximum iteration; bF and wF denote the best fitness and the worst fitness currently obtained, respectively; and SmellIndex denotes the slime mould individuals sorted by fitness.
vc in the slime mould algorithm are used to simulate the oscillatory response of the slime mould so that the slime mould can approach in order to grasp quality food faster; → vb oscillates randomly between [−a, a], gradually approaching zero; and → vc oscillates between [−1, 1] and eventually tends toward zero.

Elite Opposition-Based Learning
The population initialization of the SMA randomly generates the population positions, which causes the population to have large randomness and uncertainty and affects the final convergence speed and accuracy. Opposition-based learning introduces the reverse solution, which effectively increases the diversity and quality of the algorithm population. However, the reverse solution generated by the reverse learning may render it more difficult to search for the optimal value than the current search space. Elite oppositionbased learning takes advantage of elite individuals carrying more effective information compared with ordinary individuals so as to improve the diversity and population quality of the mucilage population and enhance the global search performance and convergence accuracy of the algorithm. In this paper, we apply EOBL to the initialization of SMA, take advantage of the feature according to which elite individuals contain more valid information with which to construct inverse populations, and select the better individuals from the current population and the inverse population as the initial solution so as to improve the quality and diversity of the initial populations.
Assuming that elite individuals X i,j = (x i,1 , x i,2 , · · · , x i,d )(i = 1, 2, · · · , N; j = 1, 2, · · · , d), the inverse solution X i,j = x i,1 , x i,2 , · · · , x i,d is defined as follows: where the dynamic coefficient K ∈ (0, 1), X i,j ∈ α j , β j , α j = min(X i, j ), β j = max(X i, j ), α j, β j denotes the dynamic boundary. The dynamic boundary overcomes the disadvantage of the difficulty in preserving the search experience at the fixed boundary, so that the elite inverse solution can be located in a narrow search space, which is conducive to the convergence of the algorithm. If the dynamic boundary operation causes X i, j to cross the boundary and become an infeasible solution, it will be reset using the random generation method in the following way:

The Adaptive Probability Threshold
The SMA balances the different movements of the slime mould surrounding the food and grasping food by the adaptive parameter p. During the iteration, when the current individual fitness differs significantly from the optimal fitness, p is 1, and the slime mould individual updates its position using the movement of surrounding the food. However, when the value range of the test function is small, it will probably update the position by the movement method of grasping food, and it will be impossible to choose a predatory strategy that is suitable for the current slime mould population, causing a slow convergence and low accuracy. Therefore, this paper introduces a new adaptive probability threshold to cause the slime mould to select the appropriate predation strategy for the current population, thus improving the convergence speed. The adaptive probability threshold mathematical model is described as shown in Equation (8): where ε is the minimum constant preventing the denominator from being zero.

The Historical Leader
During the search process of SMA, the update of the ith slime mould concentration at the t+1 iteration mainly depends on the best global slime mould concentration at the current iteration number t, resulting in an insufficient global search, rendering it easy to fall into the local extreme value region, and, sometimes, causing the low convergence accuracy of the algorithm. In this paper, the first-and second-best positions of the previous generation and the global optimal position are introduced as leaders in the slime mould position update formula, and the magnitude and direction of the slime mould surrounding the food are controlled according to the optimal historical information and current state, a method which effectively avoids the problem of the basic algorithm's tendency to easily fall into the local extreme value region and improves the algorithm's search accuracy. The ISMA location update formula is as follows: where → X C is the current global optimal position, and → X D and → X E are the first-and secondbest positions of the previous generation, respectively.

Pseudo-Code of ISMA
The pseudo-code of ISMA is shown in Algorithm 1.

ISMA Performance Evaluation Experiments and Analysis
The experiments were conducted on a computer with Intel(R) Core(TM) i7-10870H CPU @ 2.20 GHz, 8 GB of RAM, the Windows10 operating system, and MATLAB 2020a compiler software.
In order to test the optimization ability and solution accuracy of ISMA, 14 benchmark functions [32] were selected to test the algorithm's performance. The information of the benchmark test functions is shown in Table 1 which are divided into single-peak functions (F1-F7) and multi-peak functions (F8-F14). D, UM, and MM denote the function dimension, Entropy 2023, 25, 178 6 of 29 single-peak function, and multi-peak function, respectively. The function visualization is shown in Figure 1. These functions are defined in Table 1. It can be seen that the singlepeak function has only one global optimal solution, which can verify the search ability of the algorithm. The multi-peak function has many local optimal solutions and only one global optimal solution, which can verify the escape ability of the local optimal solution of the algorithm. The experiments were conducted on a computer with Intel(R) Core(TM) i7-10870H CPU @ 2.20 GHz, 8 GB of RAM, the Windows10 operating system, and MATLAB 2020a compiler software.
In order to test the optimization ability and solution accuracy of ISMA, 14 benchmark functions [32] were selected to test the algorithm's performance. The information of the benchmark test functions is shown in Table 1 which are divided into single-peak functions (F1-F7) and multi-peak functions (F8-F14). D, UM, and MM denote the function dimension, single-peak function, and multi-peak function, respectively. The function visualization is shown in Figure 1. These functions are defined in Table 1. It can be seen that the single-peak function has only one global optimal solution, which can verify the search ability of the algorithm. The multi-peak function has many local optimal solutions and only one global optimal solution, which can verify the escape ability of the local optimal solution of the algorithm.   To verify the performance of the proposed ISMA, it was compared with seven other algorithms, including SMA, SOA, MFO, POA, GWO, ESMA, and DFSMA. Table 2 illustrates the parameter settings of each algorithm. For all the algorithms in the comparison, the population size n = 50 and the maximum number of iterations max_t = 500. Due to the fact that the search strategy of the MAs is random, it was run 30 times independently on all the benchmark functions, respectively, and the evaluation criteria were the average fitness (mean), standard deviation (std), and computation time. The experimental results are provided in Tables 3-5, where the best values are marked in bold. Table 2. Parameter settings for each algorithm.

Algorithm
Parameters  F14, ISMA achieves the theoretical optimal value with the smallest std value, ESMA and DFSMA achieve the mean values close to the std value, and SMA is slightly larger than ESMA and DFSMA. For F10, F12, and F13, none of the algorithms achieve the optimal solution stably, but ISMA achieves a better performance. The analysis of the experimental results shows that ISMA outperforms the other comparison algorithms in relation to the 14 benchmark functions tested. Table 5. Computation time(s) statistical results of the algorithms based on 14 benchmark functions.
The average CPU times of the different algorithms in the 14 benchmark functions are shown in Table 5. As can be seen from the table, ISMA takes a relatively longer time to compute; however, ISMA can still outperform some algorithms with less time spent, such as ESMA and DFSMA. In general, ISMA still has a great advantage over the other algorithms.
To reflect the dynamic convergence characteristics of ISMA, the convergence curves of seven optimization algorithms under 14 benchmark functions are shown in the Figure 2. For F1, F2, F3, F4, F8, F9, F10, and F11, ISMA is obviously superior to the other algorithms in terms of the convergence speed and optimization accuracy, and the search performance in the early iteration and the exploitation performance at the end of the iteration are also superior to those of the other algorithms. This shows that EOBL causes ISMA to ensure the exploitation ability and the search ability without losing the population diversity and search stability. For F5, F6, F7, F8, F12, F13, and F14, with the increase in the iterations, various algorithms stalled to different degrees and fell into local optimum. However, due to the introduction of the adaptive threshold, ISMA could effectively jump out of local optimum and obtain a better search accuracy.
In summary, whether single-peak or multi-peak functions are applied, ISMA shows a better overall search performance and a better solution accuracy and stability than the seven representative comparison algorithms, with a superior solution performance. It was shown that ISMA can explore the search space sufficiently and efficiently and ensure the global search capability and local exploration capability. ISMA solves the problem of the susceptibility of the SMA algorithm to fall into the local extreme value region, with an unstable optimization performance and low precision, when solving complex functions. In summary, whether single-peak or multi-peak functions are applied, ISMA shows a better overall search performance and a better solution accuracy and stability than the seven representative comparison algorithms, with a superior solution performance. It was shown that ISMA can explore the search space sufficiently and efficiently and ensure the global search capability and local exploration capability. ISMA solves the problem of the susceptibility of the SMA algorithm to fall into the local extreme value region, with an unstable optimization performance and low precision, when solving complex functions.

Symmetric Cross-Entropy Threshold Segmentation
In 1968, Kullback proposed cross-entropy for the measurement of the difference in information between two probability distributions [33]. Let P = {p 1 , p 2 , . . . , p n } and Q = {q 1 , q 2 , . . . , q n } be two probability distributions defined based on the same set of values. The cross-entropy between P and Q can be calculated as follows: Multi-level threshold segmentation identifies a set of thresholds in the image to be segmented according to a certain criterion and segments the image into multiple parts. The minimum cross-entropy algorithm determines the threshold value by minimizing the cross-entropy between the original image and the threshold image [34].
In this paper, we use symmetric cross-entropy to determine the threshold values. Symmetric cross-entropy takes into account both gray-level and neighborhood average gray-level information and provides better results for the segmentation of real images [35]. Let the original image be I and h(i), i = 1, 2, . . . , L be the corresponding histogram, with L being the number of grey levels. Assuming that t thresholds need to be selected, the object function of symmetric cross-entropy can be defined as: where: Above, H 0 , H 1 , . . . , H n denote the entropies of distinct classes. In order to obtain the optimal threshold values, the fitness function in Equation (12) is minimized:

Multi-Level Threshold Segmentation Based on ISMA and Symmetric Cross-Entropy
In order to improve the accuracy and computational speed of the multi-threshold segmentation technique, multi-level threshold segmentation based on ISMA and symmetric cross-entropy is proposed. The method determines the optimal threshold value by minimizing the objective function given in Equation (12). The steps are as follows: (a) Read the image to be segmented (grayscale image). (b) Find the grayscale histogram of the image. (c) Initialize the parameters of ISMA, the size of the population of slime mould (n), the maximum number of iterations (max_t), the initial values of the upper bound (LB) and lower bound (UB), and the number of desired partition thresholds (d). (d) Find the optimal fitness value using symmetric cross-entropy as the ISMA objective function. (e) If ISMA reaches the maximum number of iterations max_t, the optimization is completed, and the slime mould location information regarding the best fitness is returned, which is the best segmentation threshold. Otherwise, skip to step (d). (f) Perform grayscale image segmentation with the best threshold and obtain the segmented image.

Threshold Segmentation Experiment for the Segmentation Criteria
To verify the effectiveness of symmetric cross-entropy threshold segmentation, Lena, Cameraman, Butterfly, Lake, Barbara, Columbia, Milkdrop, and Man, the classic threshold segmentation images, were selected as the test images to test the segmentation effect of this paper's algorithm, and the four segmentation criteria based on Otsu, Kapur's entropy, minimum cross-entropy, and symmetric cross-entropy were compared. Here, the eight benchmark images are grayscale. These images and their histograms are presented in Figure 3. The experimental parameters of the algorithm are set as follows: the population size n = 50, the maximum number of iterations max_t = 100, the upper and lower bounds of the individuals are taken as [0, 255], and the dimension (d) is taken as 2, 3, 4, and 5, corresponding to 2, 3, 4, and 5 thresholding, respectively. Figure 3 shows the grayscale histograms of the eight selected images, and it can be seen that they have different histogram distributions and can represent different types of complex, multi-target images.
To verify the effectiveness of symmetric cross-entropy threshold segmentation, Lena Cameraman, Butterfly, Lake, Barbara, Columbia, Milkdrop, and Man, the classic thresh old segmentation images, were selected as the test images to test the segmentation effec of this paper's algorithm, and the four segmentation criteria based on Otsu, Kapur's en tropy, minimum cross-entropy, and symmetric cross-entropy were compared. Here, th eight benchmark images are grayscale. These images and their histograms are presented in Figure 3. The experimental parameters of the algorithm are set as follows: the popula tion size n = 50, the maximum number of iterations max_t = 100, the upper and lowe bounds of the individuals are taken as [0, 255], and the dimension (d) is taken as 2, 3, 4 and 5, corresponding to 2, 3, 4, and 5 thresholding, respectively. Figure 3 shows the gray scale histograms of the eight selected images, and it can be seen that they have differen histogram distributions and can represent different types of complex, multi-target images In order to objectively evaluate the stability of the segmentation algorithm and th effect of multi-threshold image segmentation, each image was run 30 times inde pendently, and the peak signal to noise ratio (PSNR), structural similarity (SSIM), and feature similarity (FSIM) were selected as the evaluation criteria. PSNR was used to eval uate the image degradation, according to which the larger the value is, the smaller th image degradation and the better the image segmentation effect will be. SSIM evaluate the similarity between images based on the image brightness, contrast, and structure in formation. The SSIM value range is [0, 1], and the larger the value is, the more similar th In order to objectively evaluate the stability of the segmentation algorithm and the effect of multi-threshold image segmentation, each image was run 30 times independently, and the peak signal to noise ratio (PSNR), structural similarity (SSIM), and feature similarity (FSIM) were selected as the evaluation criteria. PSNR was used to evaluate the image degradation, according to which the larger the value is, the smaller the image degradation and the better the image segmentation effect will be. SSIM evaluates the similarity between images based on the image brightness, contrast, and structure information. The SSIM value range is [0, 1], and the larger the value is, the more similar the image after threshold segmentation will be to the original image, and the better the image segmentation effect will be. FSIM uses image gradient features and phase consistency features for image quality evaluation, and the larger the value is, the better the image segmentation quality will be.
PSNR is computed by the following equation: where x and y denote the original and segmented images, respectively. M and N are the sizes of the images. SSIM is computed by the following equation: where µ x and µ y indicate the mean intensities of the original and segmented images, respectively. σ x and σ y are the standard deviations of original and segmented images. C 1 and C 2 are two constants equal to 0.065. FSIM is computed by the following equation: where Ω indicates the entire domain of the image. C 3 and C 4 are constants which are equal to 0.85 and 160, respectively. G indicates the gradient magnitude of an image, and PC denotes the phase congruence. ISMA was combined with Otsu, minimum cross-entropy, Kapur's entropy, and symmetric cross-entropy, respectively, and the experimental results are shown in Table 6. It can be seen from the results in Table 6 that a significant difference in the image quality is obtained according to the different image segmentation criteria. The quality of the images is gradually enhanced with the increase of the number of thresholds in the segmentation results, and the PSNR, SSIM, and FSIM values are gradually increased, and the image segmentation performance is gradually enhanced. Among the values, the PSNR, SSIM, and FSIM obtained from the images segmented by Kapur's entropy thresholding at the low threshold (d = 2, 3) are the lowest, and the PSNR, SSIM, and FSIM values obtained from the images segmented by symmetric cross-entropy, Otsu, and minimum cross-entropy thresholding have little difference, which proves the feasibility of symmetric cross-entropy image segmentation. At the high threshold (d = 4, 5), the symmetric cross-entropy method outperforms Otsu, minimum cross-entropy, and Kapur's entropy segmentation methods in terms of the PSNR, SSIM, and FSIM, proving that the results of the multi-threshold image segmentation based on symmetric cross-entropy have less distortion and retain the feature information of the original image in a more complete manner. Table 7 denotes the best threshold values obtained according to the different segmentation criteria. It can be seen from the results in Table 7 that the optimal threshold obtained by symmetric cross-entropy differs less from that obtained by minimum cross-entropy and more from those obtained by Otsu and Kapur's entropy at the low thresholds (2, 3). At the high thresholds (4, 5), the optimal thresholds obtained by the different image segmentation criteria are significantly different, resulting in different image qualities.   In order to intuitively understand the effect of multi-level threshold image segmentation, the image segmentation results of the four images based on the four segmentation criteria are shown in Figures 4-11, respectively. From the segmented images, it can be seen that with the increase in the number of thresholds, the details of the image segmentation results are clearer, the information is more complete, and the segmentation quality is higher. When Kapur's entropy is used as the ISMA objective function for image threshold segmentation, some information is lost in the segmentation results, and details such as the character outline are blurred, resulting in a poor segmentation effect. When symmetric cross-entropy, minimum cross-entropy, and Otsu are used as the ISMA objective function, respectively, the facial contour and background information of the person can become clearly segmented. When the thresholds are 2, 3, and 4, the segmentation effects of the three objective functions are similar, and there is almost no difference. When the threshold is 5, the image segmentation results of minimum cross-entropy and Otsu lose part of the image information and appear distorted, while the segmentation results of symmetric cross-entropy still obtain a clearer image and can provide a more complete target region, which proves the superiority of symmetric cross-entropy as the objective function.
In summary, the image quality obtained by image segmentation using symmetric cross entropy as the objective function of ISMA is better than those of the other segmentation criteria, as this method can obtain clearer images and retain more original image information.

Threshold Segmentation Experiment of MAs
To verify the performance of ISMA in multi-threshold image segmentation scenarios, we designed experiments of comparison between ISMA and GWO, SOA, SMA, MFO, POA, ESMA, and DFSMA. All the algorithms were run independently 30 times, and PSNR, SSIM, and FSIM were selected as the evaluation metrics. The best values are marked in bold. Tables 8-10 show the PSNR, SSIM, and FSIM obtained for all the images through the algorithm, respectively. From the comparison results, we can see that the image segmentation quality becomes better as the thresholds increase, and PSNR, SSIM, and FSIM are all proportional to the number of thresholds.       As can be seen from Table 8, the PSNR obtained by image segmentation at different thresholds of ISMA achieved optimal values for all eight images, which were better than those of the other comparison algorithms. When the threshold was low (d = 2, 3), there was little difference in the PSNR values obtained by image segmentation using the other comparison algorithms. At high thresholding (d = 4, 5), SOA and GWO obtained poor PSNR values in most image segmentations, and ESMA and DFSMA obtained lower PSNR values than SMA in most image segmentations.
As can be seen in Table 9, the SSIM values obtained by ISMA achieved optimal values in the segmentation of all eight images, outperforming the other compared algorithms. At low thresholds (d = 2, 3), only ISMA obtained the optimal SSIM value for Milkdrop and Man. Each algorithm obtained the optimal SSIM value for the rest of the images, and only SOA still obtained the poor SSIM value for Lena and Barbara. When d = 4, SMA, POA, ESMA, and DFSMA obtained the optimal SSIM values for Cameraman, Butterfly, Barbara, and Man, respectively.
As can be seen in Table 10, the optimal FSIM values were obtained by ISMA in all eight image segmentation tests, which were better than those of the other comparison algorithms. When d = 2, 3, the other algorithms did not differ much in the case of Lena, Cameraman, Lake, and Barbara, but the FSIM values obtained for the remaining images were lower than those obtained by ISMA. When d = 4, all the algorithms except for SOA obtained the optimal FSIM values for Cameraman. When d = 5, the optimal FSIM values were obtained for Lena by all the algorithms except for SOA and POA. To test the stability of ISMA in the image segmentation task, 30 independent runs were performed on the images in order to obtain the optimal fitness values, and the mean and variance of the optimal fitness values were selected as the evaluation indices. Tables 11 and 12 shows the mean and std of fitness obtained by the algorithms for all the images, respectively.  It can be seen from the table that the image segmentation result of SOA was unstable in the 30 independent operations, and the values of the mean and std were the largest. The mean and std obtained by GWO were slightly better than those obtained by SOA, and the image segmentation was still not stable. When d = 2, 3, 4, SMA, POA, MFO, ESMA, DFSMA, and SMA obtained the same mean values for most of the images, but ISMA obtained a lower std value and was able to complete the image segmentation task stably. When d = 5, only ISMA obtained the optimal mean and std.
The average CPU times of the different algorithms, considering all cases, are provided in Table 13. As can be seen from the table, MFO and SOA each achieved the lowest computation time for most of the images. SMA also achieved the optimal computation time for a small number of images. ISMA performed second to SMA and better than the other residual algorithms. POA performed poorly in terms of the image segmentation time. ISMA improved the image segmentation accuracy while maintaining the runtime. To better reflect the convergence of the five algorithms, the five-threshold segmentation convergence curves of the eight images were plotted, as shown in Figure 12. From the figure, it can be seen that it was easy for SOA to fall into the local optimum during image segmentation, and the obtained adaptation value was poor. GWO performed slightly better than SOA for the eight images. For the eight images, all the algorithms except SOA eventually converged to the optimal fitness value. However, ISMA was the first to converge and the fastest to converge, followed by POA. This was made possible by the adaptive probability threshold used by ISMA, which allows the sticklebacks to select a predation strategy suitable for the current population, thus increasing the convergence speed of the algorithm. slightly better than SOA for the eight images. For the eight images, all the algorithms except SOA eventually converged to the optimal fitness value. However, ISMA was the first to converge and the fastest to converge, followed by POA. This was made possible by the adaptive probability threshold used by ISMA, which allows the sticklebacks to select a predation strategy suitable for the current population, thus increasing the convergence speed of the algorithm.  In summary, ISMA can converge to the optimal solution stably, and there are some improvements in the convergence speed and segmentation accuracy compared with the other seven algorithms, and it can obtain high-quality segmented images. Therefore, this paper proposed that multi-level thresholding image segmentation based on the improved slime mould algorithm and symmetric cross-entropy can be effectively applied to image multi-threshold segmentation tasks, with an excellent performance.

Conclusions
In this paper, we introduced an improved slime mould algorithm, ISMA, for multithreshold image segmentation tasks. The slime mould algorithm can easily fall into the local optimum, as in the case of other intelligent algorithms, and cannot solve complex real-world problems. In this work, EOBL improved the quality and diversity of the initial population to accelerate the convergence speed. The adaptive probability threshold adjusted the selection probability of the slime mould to enhance the ability of the algorithm to jump out of the local optimum, and the historical leader strategy selected the optimal historical information as the leader for the position update so as to improve the convergence accuracy.
We evaluated the optimization performance of ISMA using 14 benchmark test functions. The experimental results showed that ISMA has a better overall capability in terms of the optimization accuracy and convergence speed compared with the original SMA, as well as the other well-known MAs. Subsequently, ISMA was applied to solve the multithreshold image segmentation task, and symmetric cross-entropy was used as the objective function to obtain the optimal threshold value. Experimental evaluation metrics such as the PSNR, SSIM, and FSIM were used to test the quality of the segmented images. The experimental results demonstrated that: (1) the image segmentation quality is better than that obtained by Otsu, Kapur's entropy, and minimum cross-entropy when symmetric cross-entropy is taken as the objective function; and (2) ISMA achieves clearer image segmentation results compared with the other MAs. Finally, we conclude that multi-threshold image segmentation based on ISMA and symmetric cross-entropy outperforms the other selected MAs in terms of the segmentation accuracy and can better preserve the edge details of the images.
Although ISMA has achieved excellent results in benchmark function testing and image segmentation, it still has some shortcomings when solving image tasks. Future work will focus on reducing the computation time without degrading the performance of ISMA and applying multi-threshold image segmentation based on ISMA and symmetric cross-entropy to real medical images and remote sensing image testing in order to further demonstrate its performance.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.