The non-local means (NLM) two-dimensional histogram (Buades et al., 2005) is an essential method for image denoising. The basic principle of NLM filtering is like that of means filtering. They both achieve denoising by averaging the pixels around the current pixel point, except that a weighting strategy is added to the NLM filtering. NML method can achieve denoising and preserve the details of the image edges. The value of NLM O(p) can be obtained according to Eq. 1.

where p and q are two pixels in image I, respectively. X(p) and X(q) represent the values of p and q, respectively. The Gaussian weighting function ω(p, q) is defined as shown in Eq. 2.

where σ is the standard deviation of the Gaussian weighting function. μ(p), μ(q) denote the local means of p, q pixels, respectively, obtained from Eq. 3.

where x is a pixel in the image I, S(x) is a square filter of size n × n around pixel x.

Suppose an image I(x, y) with gray level L and size M × N, x ∈ [1, M], y∈ [1, N]. For each pixel in I, the corresponding grayscale value f(x, y) and NLM g(x, y) can be calculated. Therefore, each pixel in the original image I(x, y) will be associated with two dimensions: grayscale and NLM. Moreover, we can also obtain the number of h(i, j) pixels with the same grayscale and NLM in (s, t), which is calculated by Eq. 4.

where i, j denote the grayscale value f(x, y) and the nonlocal mean g(x, y) of (x, y) pixels. Both i and j are in the range of 0 to L1. c_i,j is the number of pixels with grayscale value and NLM of (i, j). Two-dimensional histogram of the image is obtained by normalizing Eq. (5) according to h(i, j).

The grayscale and NLM values are used as the horizontal and longitude axes of the two-dimensional histogram. The number of normalized pixels is used as the vertical axis of the two-dimensional histogram, as shown in Figure 1.

Entropy is a physical quantity that measures a certain distribution, and a higher entropy value means that the distribution is more uniform. In order to fully consider the source independence and be able to well extract the feature signal in the picture, Kapur’s entropy is selected to measure the amount of information in the target region and background region. The larger Kapur’s entropy indicates the higher quality of image segmentation. The method of MTIS using Kapur’s entropy can be described as follows: {t₁, t₂ …, t_T} represents the grayscale values of the grayscale image and {s₁, s₂ …, s_T} characterizes the grayscale values of the non-local mean image. The objective function is expressed as calculating the entropy of T + 1 image segmentations and then summing them. The expression for the objective function F of Kapur’s entropy is shown in Eqs. 6, 7.

where H_i denotes the entropy of the i-th image segmentation. {ω₁, ω₂, …, ω_T} are the sum of the grayscale levels in the threshold interval.

In order to increase the convergence speed while improving the accuracy. This study introduced the soft besiege strategy. The soft besiege strategy is inspired by HHO (Heidari et al., 2019b). The soft besiege strategy ensures a more reasonable ratio of global exploration and local exploitation of the algorithm, which is conducive to searching for higher quality solution. At the beginning of the optimization process, the algorithm tends to explore the entire search space more globally, which accelerates convergence and reduces the local optimum risk. Toward the end of the optimization process, the algorithm focuses more on further exploitation around the current optimal solution, improving the convergence accuracy of the algorithm. The pursuit strategy can enhance the local exploitation performance of the algorithm.

The adaptive step size facilitates to coordinate the exploration and exploitation of the proposed method. The exploration phase enables the search agents to conduct a global search with a big step size. In the exploitation phase, the search agent performs the local search with a small step size. E is calculated from Eq. 14.

where FEs is the number of current function evaluation. MaxFEs is the maximum number of function evaluation. r₁ is a randomly generated number between 0 and 1. From Eq. 14, as the number of iterations rises, we know that |E| drops from 2 to 0. The variation of E with the increasing number of iterations is shown in Figure 2.

The soft besiege strategy uses the average position of all ants in the colony, the random ant position, and the food position (global optimal solution) to influence the current ant movement. The soft besiege strategy consists of two phases according to the optimization process.

When the current number of iterations is 1, the update formula for the current individual Xi=(xi1,…xidim),i∈[1,N] is shown in Eq. 15.

where xrandj, xbestj, and xmeanj denote the j-th component of the random individual, optimal individual and average individual in the population, respectively. r₂, r₃, r₄, r₅, and r₆ are randomly generated between 0 and 1. ub_j and lb_j denote the maximum and minimum values of the j-th component, respectively. When r₆ > 0.5, xij is updated according to the position of other individual. When r₆ ≤ 0.5, xij updates its position based on the current best individual ant and the mean value of the population. This stage ensures that the colony can explore the search space more extensively and find the global optimal solution more easily.

When the number of iterations is greater than one, the main factor affecting the update of individual position is the current optimal individual. In addition, to improve the convergence accuracy and prevent falling into local optimum, random step J and Lévy flight are introduced. The updated formula for J is shown in Eq. 16.

where r₇ means random numbers between 0 and 1.

J and Lévy flight ensures that an individual can search locally at the current position and jump out of the local optimum by the feature of Lévy flight between long and short steps. At the next iteration, the individual X_temp that may replace the current X_i is generated by Eqs. 17, 18.

where xtempj denotes the j-th component of X_temp. r₈ is randomly generated between 0 and 1.

In addition, when r₈ < 0.5, the current individual’s position is further developed based on local information to obtain a better solution. If the fitness value of X_temp is better, the next search step is performed according to Eq. 19. Otherwise, the original position is retained.

where Levy denotes Lévy flight. r₉ denotes a random number between 0 and 1.

As a result, the individual with the best fitness value of X_temp and X_i is retained for the next iteration based on the current individual. The pseudo-code of the soft besiege strategy is shown in Algorithm 1.

The chase strategy mainly updates the current individual’s position based on the guided individual Sg=(sg1,…sgdim) and the global optimal X_best. Since the two best individuals influence the current position, the strategy gradually devotes more computing resources to local location exploitation. The chase strategy is divided into two cases as shown in Eq. 20, by comparing the function value of S_g and X_i.

where r₁₀, r₁₁, r₁₂, r₁₃ denote the random numbers between 0 and 1. The pseudo-code of the chase strategy is shown in Algorithm 2.

This subsection describes the process of implementing EACOR. Step 1: the parameters of the algorithm are defined. Step 2: the archive is initialized and the fitness value is calculated. Step 3: ACOR’s core update formula of is executed. Step 4: the chase strategy is used to further exploit the population position. Step 5: update the population according to the soft besiege strategy. Step 6: remove N bad solutions from k + N individuals in this iteration. Step 7: the optimal individuals are output. The flow chart of EACOR is shown in Figure 3.

The benchmark function experiments in subsection “Experiments on benchmark functions” focused on testing the global optimization performance of EACOR. IEEE CEC2014 test suite consists of unimodal, multimodal, hybrid, and composition functions that effectively validate the exploration and exploitation capabilities of the algorithms. Details of all IEEE CEC2014 benchmark functions can be referred to Liang et al. (2013). In order to ensure the fairness of the algorithm, all common parameters were standardized, as shown in Table 1. In addition, Table 2 shows the average ranking of the key parameters tested on the 30 functions. It was finally determined that ξ was set to 1, k was set to 10, and q was set to 0.5.

Enhanced ant colony optimization for continuous domains introduced the soft besiege strategy and the chase strategy. In order to verify whether the two enhanced strategies can improve EACOR’s optimization capabilities, a strategy comparison experiment was set up in this subsection. The four combinations of the two strategies are shown in Table 3, where ‘1’ means that the strategy was used and ‘0’ means that the strategy was not used.

Table 4 shows the rankings of WSRT and FT and the comparison results. For WSRT, EACOR ranked the best of the four algorithms with an average ranking of 1.93 on the 30 benchmark functions. For FT, EACOR ranked first, and its average ranking was 1.98. In the last column of Table 4, EACOR had six results that were better than ACOR_S, four results that were worse than ACOR_S, and other results that were like ACOR_S. Compared to ACOR without both strategies, EACOR won in 27 functions and failed in only 2 functions.

The convergence curves for the four different combinations of the two techniques are shown in Figure 5 and Supplementary Figures B.1, B.2. From the convergence curves, we can see that EACOR has better convergence accuracy than the other three algorithms. Furthermore, it can be seen at F11, F29, and F30 that there was a rapid downward trend in the curves of EACOR and ACOR_S for evaluation numbers between 250,000 and 300,000. It is due to the process of EACOR escaping from the local optimum solution when dealing with some multi-peaked functions. It shows that the soft besiege strategy introduced by the algorithm improved the ability of the algorithm to avoid falling into the local optimum solution.

According to the above analysis, only the ACOR algorithm with both the soft besiege strategy and the chase strategy can reach the optimal state.

To analyze the impact of the introduced strategy on ACOR, the search process of EACOR and ACOR was analyzed through 1000 iterations. Figure 6 shows the search process of some functions of EACOR on IEEE CEC2014. Figure 6A shows the 3-D search space of the global optimization problems. Figure 6B shows all exploration trajectories in the search space, where the red dot indicates the optimal positions. Figure 6C depicts the algorithm’s search trajectory in the first dimension. EACOR’s average fitness value is seen in Figure 6D.

Figure 6B demonstrates that most search agents focus their efforts on a local search around the global optimal solution, whereas only a tiny number of search agents conduct a global search for the optimal solution. This shows that EACOR has not only local but also global search capability. Figure 6C shows that EACOR has a sharp oscillation in the search trajectory curve in the early stage of the search, and then the curve becomes smooth. This change ensures that the algorithm improves both the convergence accuracy and the speed of convergence. Figure 6D shows that EACOR has a variety of search agents during the preliminary iterations. The diversity of search agents, on the other hand, decreases as the number of iterations increases. This confirms the transformation of the algorithm from the exploration phase to the exploitation phase.

To further analyze how the introduction mechanism improves the ACOR search capability. We have conducted balanced experiments on the Exploration phase and Exploitation phase of EACOR and ACOR based on 30 functions of IEEE CEC2014. In addition, experiments on the diversity of the two algorithms were conducted. The experimental result is shown in Figure 7 and Supplementary Figures B.3–B.7. The red and blue lines in Figures 7A,B represent the Exploration and Exploitation phases, respectively. The first and second column images show the balance test results for EACOR and ACOR, respectively. From the first two columns of test results, it can be seen that the exploitation phase of EACOR and ACOR dominates and facilitates the algorithm to explore the known solutions further. By comparing the exploration and exploitation phases of the two methods, it can be seen that EACOR enhances the algorithm’s exploration of the search space and improves the probability of obtaining an optimal solution. A comparison of Figures 7A,B in the same row shows that the search ratio of EACOR in the Exploration phase is significantly higher than that of ACOR. This leads to the situation shown in Figure 7C, where the population diversity of EACOR is significantly larger than that of ACOR at the beginning of the iteration and gradually decreases at the end of the iteration. From the above analysis, we can conclude that EACOR can jump out of the local optimal solution and achieve better convergence accuracy because the global search capability is stronger than that of ACOR.

To verify the effect of the real problem dimension component on the EACOR optimization performance (Zhu B. et al., 2018), this subsection analyzed the experimental results of the EACOR algorithm for 30 benchmark functions in high dimensions (dim was set to 50 and 100).

Table 5 shows the ranking of EACOR and ACOR for WSRT and FT in dealing with high dimensional problems and the comparison results. From the table, EACOR was the best in both the WSRT and FT statistical tests in both 50 and 100 dimensions. Furthermore, EACOR achieved stronger optimization results than ACOR for 26 functions in both dimensions, indicating that EACOR can still show excellent optimization performance when dealing with high-dimensional complex problems.

In addition, Figures 8, 9 show the convergence curve of the algorithm in dealing with high-dimensional problems. Although the algorithm’s convergence accuracy was reduced due to the increase in problem dimensionality, EACOR was less affected than the original algorithm. It is worth noting that the convergence curves for F9, F10, F29, and F30 can be observed; EACOR can still jump out of the local optimal solution at the late stage of the iteration in the high dimension. As a result, EACOR was superior in terms of convergence performance and escape from local optimum solutions.

To enable a more objective evaluation of EACOR’s performance, it was compared to ten well-known competitors in this subsection. These algorithms included ACOR, DE, FA, GWO, WOA, HHO, MFO, SCA, SFS, and TSA.

Table 6 shows the average ranking of the optimization results for the 30 benchmark functions. Among the 11 methods, EACOR ranked first in both WSRT and FT, with an average ranking of 2.37 and 2.77, respectively. The suboptimal method is DE, which has an average ranking of 2.97 and 3.2 for WSRT and FT, respectively. Furthermore, EACOR can beat DE on 18 functions, while DE outperformed EACOR on only 10 functions, and the other two results were considered equal. It is noteworthy that the worst-performing method of both statistical methods was ACOR, which shows that the improvement strategy proposed in this study can adequately improve the algorithm’s performance.

Figure 10 and Supplementary Figures B.12, B.13 show the convergence of the 11 algorithms on 30 benchmark functions. We can see that EACOR is at the bottom of the convergence curve. In F1, F6, F9, F21, F30, and EACOR finds better solutions and converges faster than the other algorithms. In F1, F9, F17, F20, F21, F30, and EACOR still finds better solutions at the end of the iterations, especially in F21, EACOR converges at the beginning of the iteration with accuracy second to that of DE, but EACOR’s advantage of jumping out of the local optimum at the end of the iteration makes the result better than DE’s result.

The means and standard deviations of 30 independent runs of the experiments indicate that the algorithm can obtain excellent and stable optimization results. WSRT and FT demonstrate the statistical significance of the experimental results. Overall, EACOR is more competitive than some advanced and original algorithms.

To further demonstrate the superiority of the EACOR algorithm in terms of performance, EACOR was compared with ten other excellent improved algorithms. These peers were CDLOBA, CBA, HGWO, ASCA_PSO, SCADE, m_SCA, IGWO, OBLGWO, ACWOA, and BMWOA.

Table 7 shows the ranking of EACOR with these ten peers for both statistical methods. EACOR performed best in WSRT and FT, with an average ranking of 2.17 and 2.62, respectively. In addition, EACOR can beat the second-ranked IGWO on 23 functions and be disadvantaged on only 4 functions. The various comparisons in Table 7 show that EACOR has better optimization performance and can handle different optimization problems better.

Figure 11 and Supplementary Figures B.14, B.15 show the convergence of EACOR with ten advanced peer methods on IEEE CEC2014, and it can be seen that EACOR can achieve better optimization results in most functions compared to other methods. In addition, by looking at F3, F16, and F29, we can see that the convergence curves of EACOR are still significantly skewed in the late iterations, which indicates that EACOR has an excellent ability to jump out of the local optimal solution.

Based on the above analysis, we can conclude that EACOR still has significant advantages not only compared with the original algorithm but also compared with the improved algorithm in terms of convergence accuracy, convergence speed, and prevention of premature convergence.

In order to judge the quality of the segmented image, three methods, PSNR, SSIM, and FSIM, were used to evaluate the segmentation results. Table 8 describes the evaluation methods PSNR, SSIM, and FSIM.

After obtaining the above three criteria, the evaluation data were statistically analyzed using the mean, standard deviation, WSRT, and FT. The final evaluation data of the image segmentation effect were obtained.

Since melanoma can be easily confused with a pigmented nevus, there are some mistakes in the process of pathological image detection. To segment melanoma effectively, we use the MTIS technique to segment melanoma pathological images. To see more clearly the value of EACOR in melanoma image segmentation, 9 pathological images of melanoma were segmented at 5 thresholds by EACOR with 9 similar algorithms, namely ACOR, CS, GWO, HHO, SCA, ACWOA, IGWO, m_SCA, and SCADE, according to the evaluation method in subsection “Validation criteria for image quality.”

Figure 12 shows the jet colormap and gray images obtained after segmentation of nine melanoma images by each method. It is easy to observe that the image segmentation obtained by SCA and IGWO was inferior to the other methods. However, EACOR, ACOR, and CS were visually difficult to distinguish the segmentation quality. Therefore, in subsequent experiments, the performance of the methods was compared more visually through the three evaluation methods.

Supplementary Tables A.1–A.3 compare FSIM, PSNR, and SSIM at different thresholds. As can be seen from the table, EACOR ranked first for all thresholds. Furthermore, the difference between EACOR and ACOR for segmentation of 9 images was small at thresholds 4, 8, and 12. However, at 16 and 20 thresholds, most images segmented using EACOR were significantly better than those obtained with ACOR. Figures 13–15 show the three image evaluation metric scores at each threshold. The mean values of FSIM, PSNR, and SSIM were the highest for EACOR, indicating that the EACOR-based image segmentation method can achieve high-quality segmentation of melanoma images. And by comparing the thresholding results of the five levels of thresholding an increase in the threshold level between the experimentally set threshold levels is beneficial in improving the segmentation results. To further verify the significance of the obtained results, the experimental results were further analyzed by the Friedman test. Figures 16–18 show the FT results of the three evaluation criteria. The three FT results show that EACOR was the best compared to the other algorithms with the same conditions. The combination of FSIM, PSNR, and SSIM shows that the melanoma images obtained by this method retain more useful image features and have less image distortion.

As the proposed image segmentation framework used Kapur’s entropy as the objective function of the segmentation threshold. Therefore, the larger Kapur’s entropy value indicates that the maximum amount of information is retained between the background and the target, which is more conducive to improving image segmentation quality. Supplementary Table A.4 shows that the maximum of Kapur’s entropy is obtained for different thresholds, and EACOR can still obtain the optimal Kapur’s entropy value in most cases. Figure 19 shows the convergence curves for each algorithm for nine images at 20 levels of thresholding. Based on the above analysis of Kapur’s entropy, the most reasonable solution was obtained by EACOR, followed by ACOR. Finally, Supplementary Figures B.17–B.25 show the optimal set of thresholds for the 9 images at 8 levels of thresholding.