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Multilevel thresholding image segmentation plays a crucial role in various image processing applications. However, achieving optimal segmentation results often poses challenges due to the intricate nature of images. In this study, a novel metaheuristic search algorithm named Weighted Chimp Optimization Algorithm with Fitness–Distance Balance (WChOA-FDB) is developed. The algorithm integrates the concept of Fitness–Distance Balance (FDB) to ensure balanced exploration and exploitation of the solution space, thus enhancing convergence speed and solution quality. Moreover, WChOA-FDB incorporates weighted Chimp Optimization Algorithm techniques to further improve its performance in handling multilevel thresholding challenges. Experimental studies were conducted to test and verify the developed method. The algorithm’s performance was evaluated using 10 benchmark functions (IEEE_CEC_2020) of different types and complexity levels. The search performance of the algorithm was analyzed using the Friedman and Wilcoxon statistical test methods. According to the analysis results, the WChOA-FDB variants consistently outperform the base algorithm across all tested dimensions, with Friedman score improvements ranging from 17.3% (Case-6) to 25.2% (Case-4), indicating that the FDB methodology provides significant optimization enhancement regardless of problem complexity. Additionally, experimental evaluations conducted on color image segmentation tasks demonstrate the effectiveness of the proposed algorithm in achieving accurate and efficient segmentation results. The WChOA-FDB method demonstrates significant improvements in Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Feature Similarity Index (FSIM) metrics with average enhancements of 0.121348 dB, 0.012688, and 0.003676, respectively, across different threshold levels (m = 2 to 12), objective functions, and termination criteria.

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1. Introduction

Image segmentation is a fundamental process in computer vision and image processing that aims to partition an image into distinct and meaningful regions or objects. This division enables a computer to understand and analyze the visual content within an image by identifying and categorizing different entities. Essentially, image segmentation breaks down an image into its constituent parts, allowing for more detailed analysis and interpretation. The significance of image segmentation lies in its ability to provide a more detailed understanding of visual data. By identifying individual objects or regions within an image, segmentation facilitates various applications, including object recognition, scene understanding, and image editing. In an image recognition system, segmentation is an important stage that helps to extract the object of interest from an image. In this context, segmentation is extensively used in medical imaging. For example, in tumor/lesion detection, segmentation helps to isolate tumors/lesions from surrounding healthy tissue, facilitating accurate diagnosis and tailored treatment planning [1,2]. In autonomous driving, image segmentation is critical for scene understanding and safe navigation [3]. Also, segmentation is essential for monitoring crop health in agriculture and managing resources more effectively [4]. By segmenting fields into different categories based on vegetation index or soil moisture levels, farmers can optimize irrigation, fertilization, and harvesting strategies.

Despite its importance, image segmentation poses several challenges. Variability in illumination, complex object boundaries, and the presence of noise make the segmentation process more complex. Additionally, scalability to handle diverse types of images and adaptability to different contexts are ongoing challenges in the field. Researchers are exploring innovative approaches and algorithms to address these issues and enhance the precision and efficiency of image segmentation. For example, symmetry is an important cue for machine perception that involves high-level knowledge of image components, so it was used to improve region-based image segmentation performance [5]. For clustering-based image segmentation, the membership values of points to different clusters were computed based on point symmetry-based distance [6]. By adding symmetric masks in several layers, deep convolutional symmetric neural networks were designed for image segmentation [7].

In the context of multilevel image analysis, segmentation becomes even more critical. Multilevel segmentation involves the identification and categorization of objects at multiple levels of granularity within an image. This is essential for applications such as medical image analysis, satellite imagery interpretation, and industrial inspection. Achieving accurate multilevel segmentation often requires sophisticated algorithms that can handle the complexities associated with diverse image datasets. Accurate image segmentation plays a pivotal role in various domains, contributing significantly to the advancement of computer vision and image processing. The precision and reliability of segmentation techniques have far-reaching implications across different applications.

Image thresholding techniques are widely used in image segmentation due to their simplicity and effectiveness. These methods utilize predefined threshold values to segment specific objects, features, or information within an image. There are two methods for thresholding-based image segmentation [8]. Bilevel thresholding is the process of partitioning an image into two classes using a predetermined threshold value: the gray levels below the threshold value are categorized into one class, while the gray levels greater than the threshold value are categorized into another class. In multilevel thresholding, the image is subdivided into multiple classes using a set of predefined threshold values. That is, these thresholds are used to identify discrete clusters within the image. For color images, segmentation could be achieved via the utilization of a separate threshold value for each color channel (red, green, and blue). The employment of multilevel thresholding for each channel tends to yield improved outcomes in terms of image segmentation accuracy. However, this approach significantly increases computational complexity. Due to this computational complexity, metaheuristic optimization algorithms are used to solve the multilevel thresholding image segmentation problem.

Several approaches have been proposed to determine the optimum threshold values from image histograms. The most frequently used methods are Kapur’s entropy [9] and Otsu’s criterion (between class variance) [10]. Among the thresholding algorithms, Kapur’s entropy outperforms other functions [11]. Kapur’s entropy determines the threshold values that maximize the histogram entropies of segmented classes using the probability distribution of gray levels. Otsu’s method; on the other hand, selects the threshold values that maximize the variance among the gray-level classes. In color image segmentation, threshold values must be determined for all color channels (red, green, and blue) separately.

Thresholding approaches successfully and effectively solve bilevel segmentation problems, but the calculation complexity and computational cost increase exponentially for multilevel segmentation problems. Therefore, in recent years, researchers have proposed the use of metaheuristic search (MHS) algorithms to reduce computational complexity and increase segmentation performance [12,13,14]. Metaheuristic optimization algorithms can perform efficient search in a solution space without having difficulty with local optimums. However, multilevel thresholding image segmentation becomes a complex operation due to the multimodality of the histogram. Additionally, each image histogram and the number of threshold levels are considered as separate problems. Because it is difficult to find an algorithm suitable for many optimization problems, several metaheuristic algorithms have been developed in the literature.

This study introduces a novel approach to multilevel thresholding image segmentation: Weighted Chimp Optimization Algorithm based on Fitness–Distance Balance (WChOA-FDB). Drawing inspiration from the intelligent group hunting behavior of chimps, this metaheuristic optimization algorithm directly overcomes the challenges. WChOA-FDB operates by delicately balancing fitness and distance within the search space, enabling it to navigate effectively and avoid traps of local optima. Furthermore, through the incorporation of a novel weighting scheme, WChOA-FDB further enhances its ability to find optimal solutions. The performance of the proposed algorithm was evaluated using 10 benchmark functions (IEEE CEC 2020) of different types and complexity levels. The search performance of the algorithm was analyzed using the Friedman and Wilcoxon statistical test methods.

This research also investigates the complexities of multilevel segmentation and meticulously details the theoretical foundations, implementation specifics, and evaluation of WChOA-FDB on benchmark datasets. In this paper, we applied WChOA-FDB to perform Multilevel Thresholding Image Segmentation. To evaluate WChOA-FDB’s performance, we employed Kapur’s entropy and Otsu’s between-class variance methods. The performance of WChOA-FDB has been compared with other well-known metaheuristic algorithms, namely Artificial Bee Colony (ABC) [15], Cuckoo Search (CS) [16], Grey Wolf Optimizer (GWO) [17], Harris Hawks Optimization (HHO) [18], Moth Flame Optimization (MFO) [19], Sine Cosine Algorithm (SCA) [20] and Salp Swarm Algorithm (SSA) [21].

Various problems are encountered when studies in the field of multilevel thresholding image segmentation with MHS algorithms are examined. In multilevel image segmentation studies, the number of iterations is generally used as the termination criterion, and these iteration numbers vary. For example, one study can use 100 iterations while the other 300 iterations as termination criteria. So, it is unknown whether the method is still successful for more iterations. In summary, the termination criterion has a significant effect on the results of the optimization process. The value of this parameter can change the quality of the solutions and the algorithms that find the best solutions. Therefore, in this study, the performance of the proposed algorithm was evaluated for two different termination criteria. To compare the algorithms fairly, the maximum number of objective function evaluations, maxFEs = 500 × m (m: number of threshold levels), and maxFEs = 1000 × m were used as termination criteria. Another problem in multilevel thresholding image segmentation literature is the number of threshold levels used in the segmentation of images. In most of the studies, a low number of threshold levels (2, 3, 4, 5) were used. Therefore, it is not known how the algorithm will perform when a medium or high number of threshold levels is used. In this study, segmentation of 10 images with m = 2, 4, 6, 8, 10, 12 levels were performed with the proposed and competitor algorithms.

The results showcase the algorithm’s higher performance compared to existing methods, demonstrating its potential in multilevel image segmentation across a wide range of applications. By harnessing the power of WChOA-FDB, we pave the way for a future where machines can extract deeper insights and unlock greater understanding from the vast realm of visual information. The main contributions of this study are given below.

Methodological Innovation: This research presents the first application of the Weighted Chimp Optimization Algorithm (WChOA) to multilevel image segmentation, enhanced through integration with the Fitness–Distance Balance (FDB) method to develop WChOA-FDB, a novel hybrid metaheuristic optimization approach.

Comprehensive Experimental Framework: The proposed WChOA-FDB algorithm was systematically evaluated through rigorous experimentation encompassing ten IEEE CEC 2020 benchmark functions of varying complexity, multilevel segmentation scenarios ranging from low to high threshold levels (m = 2, 4, 6, 8, 10, 12), comparative analysis against seven established metaheuristic algorithms, and assessment under different computational constraints (500 × m and 1000 × m maximum function evaluations).

Empirical Validation and Performance Assessment: The efficacy of the proposed approach was thoroughly validated through multi-dimensional evaluation criteria, incorporating both optimization performance metrics (fitness function values) and established image quality assessment measures (Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Feature Similarity Index (FSIM)), demonstrating superior performance across diverse segmentation complexity levels.

The rest of the paper is organized as follows. In Section 2, a comprehensive literature review on metaheuristic-based multilevel thresholding image segmentation is presented. The multilevel thresholding problem is clarified, and the proposed method is explained in Section 3. The experimental studies and findings are given in Section 4, and the results are discussed in Section 5.

2. Related Work

Image segmentation, particularly multilevel thresholding, has attracted significant attention in recent years due to its wide range of applications in various fields such as medical imaging, remote sensing, and object recognition. Numerous metaheuristic algorithms have been employed to address the challenges associated with multilevel thresholding, aiming to improve segmentation accuracy and computational efficiency. In this section, we review some of the prominent works in the literature that have explored the application of metaheuristic algorithms in multilevel thresholding image segmentation.

Multilevel image thresholding studies based on meta-heuristic optimization can be divided into two groups. The studies in the first group include the application of an existing MSA to the solution of the problem and its comparison with the competing algorithms. The studies in the second group aim to develop a new algorithm by modifying an existing MSA or hybrid MSA and thus increase the image segmentation performance. However, each algorithm has its limitations. Therefore, researchers try to increase the performance and reduce the complexity by modifying the method by changing the search process strategy in the algorithm or by developing hybrid methods by using the algorithms together.

Su et al. [22] conducted a study on the application of the Artificial Bee Colony algorithm (ABC) and its improved version (CCABC) for the segmentation of COVID-19 X-ray images. They introduced an enhanced ABC algorithm, CCABC, which incorporates horizontal and vertical search mechanisms to improve the optimization performance. Additionally, they proposed a multilevel thresholding image segmentation method based on CCABC to enhance the effectiveness of the segmentation process. The study compared CCABC with 15 algorithms using 30 benchmark functions to demonstrate its performance, showing that CCABC outperforms ABC and other similar algorithms, indicating its ability to find optimal solutions and improve image segmentation accuracy. Liu et al. [23] employed a chimp-inspired remora optimization algorithm to determine optimal threshold values, thereby reducing computational costs and enhancing segmentation accuracy. They used the cross-entropy method for image segmentation. Sharma et al. [24] introduced a modified firefly algorithm with Kapur’s, Tsalli’s, and Fuzzy entropy methods and between-class variance for optimal thresholding. The authors also employed an opposition-based learning method for initializing the candidate solution population and incorporated Levy flight and local search techniques to determine optimal threshold values for multilevel image segmentation. H. Houssein et al. [25] introduced a new optimization algorithm called Improved Gray Wolf Optimization (IGJO) for tumor identification in medical images. The robustness and accuracy of IGJO were tested on different engineering and real-world problems with unknown search spaces. Gharehchopogh et al. [26] introduced an improved African Vultures Optimization Algorithm (AVOA) for multilevel thresholding image segmentation. They utilized Quantum Rotation Gate (QRG) and Association Strategy (AS) mechanisms to enhance the performance of AVOA by increasing diversity and optimizing local trap escapes. Kumar et al. [27] combined the Cuckoo Search Algorithm (CSA) with Recursive Minimum Cross Entropy (R-MCE) to obtain the best threshold values for crop image segmentation. The proposed technique was tested on 20 crop images and demonstrated improved accuracy and segmented image quality compared to other algorithms. Agrawal et al. [28] proposed a Dominant Color Component and Adaptive Whale Optimization Algorithm (DCC-AWOA) method for multilevel thresholding color image segmentation. They compared their proposed technique with existing approaches. They utilized entropic and non-entropic objective functions, such as Kapur’s entropy and edge magnitude-based thresholding. Tan et al. [29] proposed an Improved Cuckoo Search algorithm (ICS) for multilevel color image thresholding based on modified fuzzy entropy. They addressed the time-consuming nature of traditional exhaustive search methods for optimal multilevel thresholds in color images. The ICS incorporates two modifications to enhance the performance of the standard Cuckoo Search Algorithm: an adaptive control parameter mechanism and a hybrid search strategy. The study compared the ICS with six other optimization algorithms and demonstrated its high performance in terms of objective function value, convergence speed, and parametric statistical tests.

Houssein et al. [30] integrated the Improved Heap-Based Optimizer (IHBO) with opposition-based learning to improve the convergence rate and local search efficiency for multilevel thresholding image segmentation. The study evaluates the IHBO-based method on the CEC’2020 test suite and compares it against seven well-known metaheuristic algorithms, demonstrating its effectiveness in solving complex multilevel thresholding problems. The authors provide a comprehensive analysis of the IHBO algorithm’s performance using the Otsu method as the objective function, presenting graphical and statistical results for benchmark images. Wang et al. [31] proposed an improved Golden Jackal Optimization (HGJO) for multilevel thresholding image segmentation. They conducted experiments using the CEC2022 test suite to measure the performance of their algorithm, comparing it with other existing algorithms. The results showed that the proposed HGJO algorithm outperformed the other algorithms in terms of overall performance, demonstrating its effectiveness in reducing image complexity while preserving original features. Abualigah et al. [32] proposed a hybrid Marine Predators Algorithm (MPA) with the Salp Swarm Algorithm (SSA) (MPASSA) to determine the optimal multilevel threshold image segmentation. They employed various benchmark images to validate the proposed algorithm’s performance. The results showed that the proposed MPASSA outperformed other well-known optimization algorithms published in the literature. Eisham et al. [8] proposed a hybrid algorithm named the Spotted Hyena-based Chimp Optimization Algorithm (SSC), which combines the sine-cosine function and the attacking strategy of Spotted Hyena Optimizer (SHO) for solving engineering problems. This algorithm was designed to address the challenges of sluggish convergence speed and entrapment in local optima in high-dimensional problem-solving. Swain et al. [33] introduced a novel objective function named Differential Exponential Entropy (DEE) and a new method called the Equilibrium-Cuckoo Search Optimizer (ECSO) for multilevel threshold selection in color satellite images. They compared the performance of their proposed technique with traditional methods such as Otsu’s between-class variance and Kapur’s entropy. The results showed that the proposed technique outperformed the traditional methods in terms of image segmentation quality. Thomas et al. [34] proposed a novel segmentation model, Harris Hawks Optimizer (HHO) for MR brain images, particularly for Alzheimer’s disease. The model integrates multilevel thresholding with an optimization algorithm to select optimal threshold values for segmentation. They tested the model on MR brain images of patients with Alzheimer’s disease and found that they achieved satisfactory results in terms of performance metrics. Additionally, they discussed various segmentation approaches and machine learning models for Alzheimer’s disease detection, emphasizing the significance of accurate segmentation in healthcare applications.

Renugambal et al. [35] proposed a hybrid Sine-Cosine Crow Search Algorithm (SCCSA) to improve multilevel image segmentation efficiency. They evaluated the hybrid algorithm on 12 standard image sets and compared its performance with other state-of-the-art algorithms such as SCA, CSA, and ABC. The experimental results showed that the proposed SCCSA consistently outperformed the other algorithms in terms of various metrics. Additionally, they conducted a Wilcoxon test to detect significant differences between the proposed algorithm and the other algorithms, and the findings indicated that the SCCSA succeeded in outperforming the other well-known algorithms. Akay et al. [36] conducted a study on nature-inspired optimization algorithms for multilevel thresholding image segmentation. They introduced modifications to the Teaching–Learning-Based Optimization Algorithm (TLBO) to improve its performance, including the use of Levy flight equations and adding an indicator of the fertile area to the framework strategy. A modified version of the Teaching–Learning-Based Artificial Bee Colony (MTLABC) was proposed in this work. The Friedman test and the Wilcoxon signed-rank test were used to determine whether there was a significant difference between the proposed algorithm and other optimization algorithms. Akan et al. [37] proposed a multilevel image thresholding approach using the Battle Royal Optimization (BRO) algorithm. The authors compare the performance of the BRO algorithm with four other optimization algorithms (AMO, PSO, BFO, and GA) on ten benchmark images. The results show that the BRO algorithm is stable and provides satisfactory results, ranking first in terms of the standard deviation (SD) metric. Zhang et al. [38] proposed a hybrid Golden Jackal Optimization with a Sine Cosine Algorithm (SCGJO) for tackling multilevel thresholding image segmentation problems. They compared the SCGJO with other evolutionary algorithms and demonstrated its high segmentation quality and overall optimal value. Additionally, they conducted Wilcoxon’s rank-sum test to confirm the effectiveness and disparity between SCGJO and other algorithms. Kang et al. [39] proposed a multilevel thresholding image segmentation algorithm based on the Mumford–Shah (MSh) model. They used the segmentation method to solve the multitarget segmentation problem and improve the processing accuracy of the algorithm. The algorithm utilizes the MSh model to determine the optimum thresholds and convert the problem into a multi-objective function optimization problem. The results showed that the proposed algorithm had the best objective function value and performed well in terms of segmentation quality and convergence time. Wang et al. [40] introduced the Mixed-Strategy-Based Improved Whale Optimization Algorithm (MSWOA) for multilevel thresholding image segmentation. This innovative algorithm combines the principles of the Whale Optimization Algorithm (WOA) with mixed strategies to enhance the quality and efficiency of image segmentation, with a specific focus on color image segmentation applications. The MSWOA method incorporates several key features to optimize the segmentation process, including k-point initialization, a non-linear convergence factor, and an adaptive weight coefficient.

3. Materials and Methods

3.1. Multilevel Thresholding Problem

Image thresholding problem involves determining the optimal threshold values to effectively segment the image into distinct classes. In other words, the goal is to find a set of thresholds that divides the image into regions, each representing a distinct intensity level or class. If m threshold values are used in the segmentation process, the image will be divided into m + 1 regions. When $m$ equals 1, the thresholding problem is referred to as bilevel thresholding. In bilevel thresholding, the image is partitioned into two regions: foreground and background, based on a single threshold value. Conversely, when $m$ is greater than one, the problem transforms into multilevel thresholding. Multilevel thresholding extends the concept by segmenting the image into multiple intensity levels or classes using multiple threshold values. This enables a finer segmentation of images with complex intensity distributions or multiple regions of interest. Multilevel thresholding is an optimization problem trying to find the best set of threshold values, represented as $[t_{1}, t_{2}, \dots, t_{m}]$ , to split an image into $m + 1$ regions. Each region corresponds to a specific range of pixel intensities. The intensity $f (x, y)$ at the location $(x, y)$ in image I is segmented into $m + 1$ sub-images as follows:

(1) $\begin{matrix} I_{0} = \{f (x, y) \in I| 0 \leq f (x, y) \leq t_{1} - 1\} \\ I_{1} = \{f (x, y) \in I| t_{1} \leq f (x, y) \leq t_{2} - 1\} \\ ⋮ \\ I_{i} = \{f (x, y) \in I| t_{i} \leq f (x, y) \leq t_{i + 1} - 1\} \\ ⋮ \\ I_{m} = {f (x, y) \in I | t_{m} \leq f (x, y) \leq 255}, \end{matrix}$

Equation (1), which is typically used for gray-level images, can also be extended to apply to the red, green, and blue channels of RGB color images. In this case, each channel of the RGB image is considered as a separate individual image.

Various methods have been proposed to determine the threshold values that will make a good distinction between the objects and the background in the image. Kapur’s entropy [9] and Otsu’s method [10] are the most frequently used non-parametric thresholding functions. These methods are briefly described in the next sub-sections.

3.1.1. Kapur’s Entropy-Based Thresholding

The Kapur function [9], also known as the Maximum Entropy Thresholding, aims to find the threshold that maximizes the entropy or uncertainty between the foreground and background regions. By maximizing entropy, the Kapur function effectively captures the overall information content in the image and selects the threshold that maximizes the discriminative power between classes. Kapur’s entropy is calculated based on the probability distributions of the gray levels in the image histogram. Assuming $[{t h}_{1}, {t h}_{2}, \dots . {t h}_{n}]$ are the threshold values to be used to classify the image, the objective function is defined as,

(2) $H ({t h}_{1}, {t h}_{2}, \dots . {t h}_{n}) = H_{1} + H_{2} + \dots + H_{n}$

where

(3) $\begin{matrix} H_{1} = - \sum_{j = 0}^{{t h}_{1} - 1} \frac{p_{j}}{ω_{0}} l n \frac{p_{j}}{ω_{0}}, ω_{0} = \sum_{j = 0}^{{t h}_{1} - 1} p_{j} \\ H_{2} = - \sum_{j = {t h}_{1}}^{{t h}_{2} - 1} \frac{p_{j}}{ω_{1}} l n \frac{p_{j}}{ω_{1}}, ω_{1} = \sum_{j = {t h}_{1}}^{{t h}_{2} - 1} p_{j} \\ ⋮ \\ H_{n} = - \sum_{j = {t h}_{n}}^{L - 1} \frac{p_{j}}{ω_{n}} l n \frac{p_{j}}{ω_{n}}, ω_{n} = \sum_{j = {t h}_{n}}^{L - 1} p_{j} \end{matrix}$

In Equations (3) and (4), $H_{1}, H_{2} \dots H_{n}$ are the class entropies and, $ω_{1}, ω_{2} \dots ω_{n}$ are the class probabilities. The optimal threshold values are obtained by maximizing the objective function in Equation (3). The complexity of solving this problem increases exponentially depending on the number of thresholds. For this reason, determining the threshold values that will maximize the objective function given in Equation (4) can be considered as an n-dimensional optimization problem and expressed as follows:

(4) $t^{*} = argmax (\sum_{i = 1}^{n} H_{i})$

3.1.2. Between-Class Variance-Based Thresholding (Otsu’s Method)

Otsu’s method [10] is a non-parametric segmentation method based on minimizing the intra-class variance while maximizing the between-class variance. It calculates the optimal threshold by maximizing the ratio of between-class variance to within-class variance. The method defines the between-class variance as the sum of sigma functions calculated as:

(5) $f (t) = σ_{0} + σ_{1} + \dots + σ_{n}$

where

(6) $σ_{0} = ω_{0} {(μ_{0} - μ_{T})}^{2}, σ_{1} = ω_{1} {(μ_{1} - μ_{T})}^{2}, \dots ., σ_{n} = ω_{n} {(μ_{n} - μ_{T})}^{2}$

In Equation (6), $μ_{T}$ is the average brightness value of the input image, $μ_{i}$ is the average of the i-th class and is calculated using Equation (8).

(7) $μ_{0} = \sum_{j = 0}^{{t h}_{1} - 1} \frac{i p_{i}}{ω_{0}}, μ_{1} = \sum_{j = {t h}_{1}}^{{t h}_{2} - 1} \frac{i p_{i}}{ω_{0}}, \dots, μ_{n} = \sum_{j = {t h}_{n}}^{L - 1} \frac{i p_{i}}{ω_{0}}$

Here, $[{t h}_{1}, {t h}_{2}, \dots . {t h}_{n}]$ are the threshold values. The optimal threshold values are obtained by maximizing the objective function in Equation (8) given below:

(8) $t^{*} = argmax (\sum_{i = 1}^{n} σ_{i})$

3.2. Proposed Method

Thresholding image segmentation is a fundamental technique used in image processing to partition an image into foreground and background regions based on intensity levels. The principle behind thresholding is straightforward and is defined as follows: pixels with intensities above a certain threshold value are assigned to one class (often considered as foreground), while pixels with intensities below the threshold belong to the other class (typically background). However, traditional thresholding methods face challenges when dealing with images affected by varying illumination conditions, noise, or complex background structures. In such cases, simple thresholding techniques may lead to sub-optimal segmentation results, with regions of interest being incorrectly classified or merged with background noise. Also, the computational cost of traditional thresholding methods significantly increases for multilevel thresholding problems. Metaheuristic algorithms offer a promising approach to address the limitations of traditional thresholding methods. These algorithms are heuristic search techniques inspired by natural phenomena or social behavior. Unlike exact optimization methods, metaheuristic algorithms do not guarantee finding the global optimum but are capable of efficiently exploring large solution spaces and finding near-optimal solutions. Each type of metaheuristic algorithm has its own set of rules for generating new solutions and deciding which solutions to use [33]. For example, in a genetic algorithm, new solutions are created by combining parts of existing solutions, mimicking the process of natural selection and evolution. Because of these differences in how solutions are generated and updated, each metaheuristic algorithm may produce different results when applied to the same problem. This diversity can be beneficial because it allows researchers and practitioners to explore various approaches and find the most suitable one for their specific problem. In this section, we will present a thorough introduction to a novel metaheuristic algorithm called the WChOA-FDB. We will explore its principles, mechanisms, and characteristics, providing a comprehensive understanding of how it operates and its applications in optimization tasks. This section first introduces the ChOA and WChOA algorithms. Then, the proposed algorithm, a mixed-strategy-based Fitness–Distance Balance and WChOA optimization algorithm (WChOA-FDB) is explained in detail.

3.2.1. WChOA Algorithm

Khishe et al. [41] proposed the Weighted Chimp Optimization Algorithm (WChOA), which is a modified version of ChOA [42], developed in 2021. ChOA is a standard algorithm inspired by the hunting mechanism of chimps in nature. Compared with other nature-inspired methods, ChOA requires the adjustment of only a few operators and can be easily implemented. There are four kinds of chimps—driver, barrier, chaser, and attacker—in a chimp colony. Each role is assigned the responsibility of guiding and coordinating the other chimpanzees toward the prey, aiming for an optimal solution. Achieving the best solution to the optimization problem requires maintaining a balanced equilibrium between exploring and exploiting the search space.

In conventional ChOA, only the initial four solutions—driver, chaser, attacker, and barrier—are employed to update the positions of the remaining chimps. In essence, the remaining chimps are drawn toward these four best solutions. Despite the inherent ability of attackers to forecast prey movement, it is not guaranteed that the solution provided by attackers is always the best. This uncertainty arises because chimps may deviate from their assigned tasks during the hunting process or persist with the same duties throughout. Consequently, updating the positions of other chimps based solely on attackers may lead them into local optima, hindering the exploration of new areas within the search space, given the significant concentration of their solution space around attacker-derived solutions. Similar concerns apply to the other primary solutions (driver, chaser, and barrier).

To address this challenge, the WChOA method was developed. WChOA addresses the limitations of the ChOA by introducing a position-weighted equation based on weights to improve exploration and exploitation. Equations (9)–(11) are employed to update the positions of the remaining chimps. Essentially, this forces the other chimps to adjust their positions based on those of the driver, chaser, attacker, and barrier. Recognizing the limitations mentioned previously opens avenues for novel approaches to update the positions of other chimps. The proposed solution involves a corresponding weighting method based on the Euclidean distance of the step size, as follows:

(9) ${\vec{D}}_{A t t a c k e r} = |{\vec{C}}_{1} {\vec{X}}_{A t t a c k e r} - {\vec{M}}_{1} \vec{X}| {\vec{D}}_{B a r r i e r} = |{\vec{C}}_{2} {\vec{X}}_{B a r r i e r} - {\vec{M}}_{2} \vec{X}|$

(10) $\begin{matrix} {\vec{D}}_{C h a s e r} = |{\vec{C}}_{3} {\vec{X}}_{C h a s e r} - {\vec{M}}_{3} \vec{X}| {\vec{D}}_{D r i v e r} = |{\vec{C}}_{4} {\vec{X}}_{D r i v e r} - {\vec{M}}_{4} \vec{X}| \\ {\vec{X}}_{1} = {\vec{X}}_{A t t a c k e r} - {\vec{A}}_{1} ({\vec{D}}_{A}), {\vec{X}}_{2} = {\vec{X}}_{B a r r i e r} - {\vec{A}}_{2} ({\vec{D}}_{B}) \\ {\vec{X}}_{3} = {\vec{X}}_{C h a s e r} - {\vec{A}}_{3} ({\vec{D}}_{C}), {\vec{X}}_{4} = {\vec{X}}_{D r i v e r} - {\vec{A}}_{4} ({\vec{D}}_{D}) \end{matrix}$

${\vec{X}}_{A t t a c k e r}$ , ${\vec{X}}_{B a r r i e r}$ , ${\vec{X}}_{C h a s e r}$ and ${\vec{X}}_{D r i v e r}$ vectors indicate the current positions of the attacker, barrier, chaser, and driver, respectively. $\vec{X}$ vector is the current position of other chimps. Also, $\vec{C}$ , $\vec{M}$ , and $\vec{A}$ vectors contribute greatly to ChOA. These vectors are calculated by Equations (11)–(13):

(11) $\vec{A} = 2 . f . {\vec{r}}_{1} - f$

(12) $\vec{C} = 2 . {\vec{r}}_{2}$

(13) $\vec{M} = Chaotic_value$

where

{\vec{r}}_{1}

and

{\vec{r}}_{2}

are random vectors that can vary in the range [0, 1], and f represents a control value that decreases non-linearly from 2.5 to 0. The chaotic vector

\vec{M}

is used to model the sexual motivation in ChOA and is calculated based on a chaotic map [42]. This chaotic behavior helps chimps alleviate the problems of entrapment in local optima and slow convergence rate in solving high-dimensional problems. Six chaotic maps (Chebyshev map, Circle map, Gauss/mouse map, Iterative map, Logistic map, and Piecewise map) have been used in the study. These chaotic maps are deterministic processes that also have random behavior.

(14) $w_{1} = \frac{|{\vec{X}}_{1}|}{|{\vec{X}}_{1}| + |{\vec{X}}_{2}| + |{\vec{X}}_{3}| + |{\vec{X}}_{4}|}$

(15) $w_{2} = \frac{|{\vec{X}}_{2}|}{|{\vec{X}}_{1}| + |{\vec{X}}_{2}| + |{\vec{X}}_{3}| + |{\vec{X}}_{4}|}$

(16) $w_{3} = \frac{|{\vec{X}}_{3}|}{|{\vec{X}}_{1}| + |{\vec{X}}_{2}| + |{\vec{X}}_{3}| + |{\vec{X}}_{4}|}$

(17) $w_{4} = \frac{|{\vec{X}}_{4}|}{|{\vec{X}}_{1}| + |{\vec{X}}_{2}| + |{\vec{X}}_{3}| + |{\vec{X}}_{4}|}$

where

w_{1}

w_{2}

w_{3}

, and

w_{4}

are the learning rates of other chimps from the attacker, barrier, chaser, and driver, respectively. Also, |.| indicates the Euclidean distance. In ChOA the relationship between positions was calculated as follows:

(18) $\vec{X} (t + 1) = \frac{{\vec{X}}_{1} + {\vec{X}}_{2} + {\vec{X}}_{3} + {\vec{X}}_{4}}{4}$

However, in WChOA the position-weighted relationship is calculated using Equation (19):

(19) $\vec{X} (t + 1) = \frac{1}{w_{1} + w_{2} + w_{3} + w_{4}} \times \frac{w_{1} {\vec{X}}_{1} + w_{2} {\vec{X}}_{2} + w_{3} {\vec{X}}_{3} + w_{4} {\vec{X}}_{4}}{4}$

In WChOA, the position-weighted relationship Equation (19) can be utilized instead of Equation (18) in the standard ChOA. The main difference between Equation (19) and the traditional position-weighted relationship Equation (18) is the application of the corresponding learning rate. As mentioned previously, since there is a possibility that some chimps do not have any sexual motivation in the process of hunting, a probability of 50% can be considered to indicate whether the position-weighted strategy of chimps will be normal (Equation (19)) or not (chaotic model). Thus, the following relationship is applied:

(20) $\vec{X_{C h i m p}} (t + 1) = \{\begin{matrix} Equation (19) i f μ < 0.5 \\ Chaotic - Value i f μ \geq 0.5 \end{matrix}$

where t denotes the current iteration and

μ

denotes the probability value in range [0,1] which indicates the position-weighted strategy of chimps.

It is noteworthy that the learning rates in the position-weighted relationship change dynamically. This means that these parameters are not constant during every iteration of WChOA. It enhances the convergence speed and avoidance of local optima where attackers, barriers, chasers, and drivers are less likely to be knowledgeable about the position of the prey.

3.2.2. Fitness–Distance Balance Selection

The Fitness–Distance Balance (FDB) selection method [43] is a fundamental approach employed in population-based and evolutionary-based MHS algorithms. It serves as a mechanism to guide the selection process by evaluating solution candidates based on both their fitness values and distances from the optimal solution. By calculating FDB scores for each candidate using normalized fitness and distance values, the method effectively strikes a balance between selecting high-quality solutions and maintaining diversity within the population. This balance ensures that the algorithm explores various regions of the solution space, thereby enhancing its ability to locate multiple optima. Ultimately, the FDB selection method contributes to the efficiency and effectiveness of MHS algorithms by enabling them to efficiently navigate complex search landscapes. This selection method achieves a balance between fitness and distance by considering candidates with high scores as suitable alternatives for the population to converge towards [44]. In the initial stage of FDB, the distance of the population from the best individual is calculated following Equation (21).

(21) ${}_{i = 1}^{n}\forall P_{i} \neq P_{b e s t}, D_{i} = \sqrt{\sum_{j = 1}^{d} {(x_{j b e s t} - x_{j i})}^{2}}$

In Equation (21), $P$ is the set of solutions, $n$ is the number of vectors in $P$ , and D represents the distance vector, with $x_{b e s t}$ and $x_{i}$ denoting the solution vectors of the best and $i_{t h}$ solutions, respectively. $d$ represents the problem dimension (number of variables). To evaluate the scores of solution candidates, the normalized fitness values ( $n o r m F$ ) and the normalized distance values ( $n o r m D$ ) of these candidates were utilized. To avoid dominance between two parameters, their normalized values are used. Finally, fitness values and distance values vectors were used to calculate the FDB score using Equation (22) [43].

(22) ${}_{i = 1}^{n}\forall P_{i}, {F D B_S c o r e}_{P_{i}} = w \times {n o r m F}_{i} + (1 - w) \times {n o r m D}_{{P b e s t}_{i}}$

The vector ${n o r m D}_{{P b e s t}_{i}}$ is the normalized value of the distance of the $i_{t h}$ solution candidate in P to $P_{b e s t}$ , which has the best fitness value in $P$ . When computing the scores of the vectors, a weight coefficient (w) is employed, which determines the impact of fitness and distance values. Based on these descriptions and the explanations 0 < w < 1, the score of each candidate in the P-population is determined. For more information on the FDB method, please review the study referenced in [43].

3.2.3. Proposed WChOA-FDB Scheme

The search capabilities of metaheuristic search algorithms mainly depend on the design of convergence equations. Convergence equations are used to determine the positions of solution candidates in the search space. The positions of solution candidates change depending on the guides used in the convergence equations. Guides determine the direction and behavior of the population in the search space. The guides selected randomly from the population create a diversity effect, while guides selected using the greedy method create an exploitation effect. The FDB selection method considers the fitness values of the solution candidates and their distance values to the best individual in the population. Of these two values, the fitness value reflects the exploitation effect, while the distance value reflects the diversity effect. In this way, the FDB selection method is quite effective in determining the guides that provide the exploitation–exploration balance. According to the studies in the literature, significant improvements have been achieved in the search capabilities of many MHS algorithms whose guidance mechanisms are designed using the FDB selection method [44,45,46,47]. So, in this study, we applied the FDB selection method to improve the design of the guidance mechanism of the WChOA algorithm. Thus, we ensured that the solution candidates that provide the exploitation–exploration balance in the search process are used as guides in the convergence equations of WChOA.

WChOA-FDB represents an evolution of WChOA. While WChOA demonstrated effectiveness in optimization tasks, the integration of Fitness–Distance Balance (FDB) introduces a novel dimension to further enhance its capabilities. The novel aspect of WChOA-FDB lies in the incorporation of FDB principles. FDB is a mechanism designed to balance exploration and exploitation dynamically, ensuring that the algorithm maintains diversity in the solution space while converging toward optimal solutions.

FDB is seamlessly integrated into WChOA to address the subtle challenges of balancing exploration and exploitation. This integration ensures that the algorithm adapts its strategies to explore diverse regions of the solution space while exploiting promising solutions for optimization. FDB introduces a mechanism to measure the fitness of solutions in relation to their distances within the solution space. This measurement guides the algorithm in prioritizing solutions that contribute to both exploration and exploitation goals. The integration of FDB with WChOA adds a layer of dynamic adaptation. As the algorithm progresses, FDB influences the exploration–exploitation trade-off, allowing WChOA-FDB to respond effectively to changes in the optimization landscape.

The WChOA-FDB algorithm works in three main stages: Select, Search, and Update. In the Select phase, the entire search space is explored to find a potential solution. The Search phase focuses on exploiting the selected area, while the Update phase focuses on targeting the best solution. The formulation of the three phases is given in Equations (9)–(19). All these steps together provide the WChOA-FDB. The algorithmic steps and processes of the proposed algorithm are given below:

Initialization with FDB Parameters: WChOA-FDB begins by initializing a population of potential solutions, incorporating parameters specific to FDB.

Evaluation and Fitness Assignment: Each solution is evaluated based on the objective function, and the fitness is assigned. FDB principles contribute to fitness assessment by considering the distances between solutions.

Weighted Solution Evaluation with FDB: The weighted evaluation mechanism, enriched by FDB, assigns weights to solutions based on both their fitness and distances. This weighted approach guides subsequent decision-making processes.

Collaborative Decision-Making Enhanced by FDB: Solutions engage in collaborative decision-making, now with added guidance from FDB. The algorithm prioritizes solutions not only based on their fitness but also considering their contribution to maintaining diversity within the population.

Dynamic Adaptation with FDB: WChOA-FDB dynamically adapts its parameters throughout the optimization process, influenced by FDB. This ensures a continuous and adaptive exploration–exploitation balance.

Iterative Optimization Enhanced by FDB: The algorithm iteratively optimizes solutions, leveraging the integrated FDB principles to achieve a balance between exploration and exploitation. Iterations continue until the termination criterion is met.

Termination and Solution Retrieval: WChOA-FDB concludes when the termination criterion is satisfied. The best solutions, considering both fitness and distances, are retrieved as the final optimized outcomes.

In this study, multiple WChOA-FDB variants (Case-1 through Case-72) were created by applying the FDB method to different components of the original WChOA algorithm’s equations. Then approximately 10% of the most successful cases (8 cases) were selected and their details are given below. The FDB-based WChOA variants (Case-1, Case-2, Case-3, Case-4, Case-5, Case-6, Case-7, and Case-8) were created to improve the exploration and balanced search ability of the WChOA algorithm. The equations where the FDB method is applied and the mathematical models of the WChOA-FDB variants are given in Table 1. Case-1 replaces the ${\vec{X}}_{D r i v e r}$ component in the ${\vec{D}}_{D r i v e r}$ equation with the FDB method ( ${\vec{X}}_{F D B}$ ). This modifies how the distance for the driver chimp is calculated. Case-2 replaces the $\vec{X}$ component in the ${\vec{D}}_{D r i v e r}$ equation with the FDB method ( ${\vec{X}}_{F D B}$ ). This changes how the current position of other chimps affects the driver-distance calculation. Case-3 applies FDB to both the ${\vec{D}}_{A t t a c k e r}$ and ${\vec{D}}_{D r i v e r}$ equations simultaneously. It replaces both ${\vec{X}}_{A t t a c k e r}$ and ${\vec{X}}_{D r i v e r}$ with ${\vec{X}}_{F D B}$ . This modifies how both the attacker and driver guide other chimps. Case-4 applies FDB to the attacker and driver equations, but differently from Case-3. It replaces ${\vec{X}}_{A t t a c k e r}$ in the attacker equation and $\vec{X}$ in the driver equation with ${\vec{X}}_{F D B}$ . This creates a different balance between exploration and exploitation. Case-5 uses FDB to modify the current position component $\vec{X}$ in the attacker equation and the ${\vec{X}}_{D r i v e r}$ component in the driver equation. Case-6 uses FDB to replace the current position component $\vec{X}$ in both the attacker and driver equations. This creates a consistent modification of how current chimp positions influence both equations. Case-7 applies FDB to the barrier and driver roles by replacing ${\vec{X}}_{B a r r i e r}$ and ${\vec{X}}_{D r i v e r}$ in the barrier and driver equations, respectively. And finally, Case-8 applies FDB to modify the current position component $\vec{X}$ in the barrier equation and the ${\vec{X}}_{D r i v e r}$ component in the driver equation. The general framework of the proposed WChOA-FDB algorithm and the equations where the FDB selection method was applied are explained step by step in Algorithm 1. Also, the flowchart of the WChOA-FDB algorithm is given in Figure 1.

Algorithm 1. Pseudo-code of proposed WChOA-FDB algorithm

1. Input: Chimp population (p), Number of iteration (Max_iter),

Problem dimension (number of variables) (d)

2. Output: X_Attacker

3. Begin:

4. Initialize the constants f, m, a and c

5. Calculate the initial position (population) for each chimp and evaluate fitness

6. Randomly divide the chimps into separate groups

7. for i = 1:p do

8. Evaluate the fitness of each chimp

9. end for

10. xattacker = the best search agent

11. xbarrier = the second best search agent

12. xchaser = the third best search agent

13. xdriver = the fourth best search agent

14. //Meta-heuristic search process//

15. while (l < Max_iter) //Number of iteration is not achieved

16. for i = 1: p do

17. for each chimp:

18. Extract the chimp’s group

19. Use its group strategy to update f, m and c

20. Use f, m _and c to calculate a then d

21. end for

22. //Implementation of FDB selection method//

23. for i = 1:p do

24. Calculate Euclidean distance for each chimp using Equation (21)

25. Calculate FDB score for each chimp using Equation (22)

26. end for

27. Create D and S as vectors using Equation (21) and Equation (22), respectively

28. Determine x_FDB based on FDB philosophy by using Equation (9)

29. Calculate d_Driver using Equation (23)//Case-1//

30. Calculate d_Driver using Equation (24)//Case-2//

31. Calculate d_Attacker and d_Driver using Equation (25)//Case-3//

32. Calculate d_Attacker and d_Driver using Equation (26)//Case-4//

33. Calculate d_Attacker and d_Driver using Equation (27)//Case-5//

34. Calculate d_Attacker and d_Driver using Equation (28)//Case-6//

35. Calculate d_Barrier and d_Driver using Equation (29)//Case-7//

36. Calculate d_Barrier and d_Driver using Equation (30)//Case-8//

37. Calculate d_Chaser using Equation (9)

38. end for

39. for i = 1: p do

40. if (φ < 0.5)

41. if (|a| < 1)

42. Update the position of the driver by Equation (19)//Case-1//

43. Update the position of the driver by Equation (19)//Case-2//

44. Update the position of the attacker and driver by Equation (19)//Case-3//

45. Update the position of the attacker and driver by Equation (19)//Case-4//

46. Update the position of the attacker and driver by Equation (19)//Case-5//

47. Update the position of the attacker and driver by Equation (19)//Case-6//

48. Update the position of the barrier and driver by Equation (19)//Case-7//

49. Update the position of the barrier and driver by Equation (19)//Case-8//

50. Update the position of the chaser by Equation (19)

51. else if (|a| > 1)

52. Select a random search agent

53. end if

54. else if (φ > 0.5)

55. Update the position of the present search agent by Equation (20)

56. end if

57. end for

58. Update the constants f, m, a and c

59. Update xattacker, xbarrier, xchaser and xdriver

60. end while

61. return xattacker

The most significant theoretical contribution of WChOA-FDB is the dynamic improvement of the exploration and exploitation balance, which is one of the most critical challenges in metaheuristic algorithms. The FDB approach considers both the fitness values of candidate solutions and their distribution in the solution space. It enables candidate solutions to be evaluated not only based on their fitness values but also according to their distances from each other. This helps maintain diversity during the search process while also supporting convergence toward good solutions. Modifying the behaviors of different guide chimps in various variants (Cases 1–8) enables more effective exploration of different regions of the solution space.

The complexity of WChOA relies on the number of chimps (n), the number of iterations (maxFEs), and the sorting method for each iteration [41]. In each iteration, the first four best solutions were found and assigned as driver, chaser, attacker, and barrier. Quick sort employed by this method means that, in the best and worst cases, complexity is $O (n \times \log n)$ and $O (n^{2})$ , respectively. The computational complexity of WChOA is defined as follows:

(31) $\begin{matrix} O (W C h O A) = O (m a x F e s \times (O (q u i c k s o r t) + O (p o s i t i o n u p d a t e) \\ = (m a x F e s \times (n^{2} + n \times d) = O (n^{2}) \end{matrix}$

For the proposed algorithm, WChOA-FDB, the Euclidean distances and FDB scores for each chimp are calculated using Equations (21) and (22). So, the computational complexity of WChOA-FDB is defined as:

(32) $\begin{matrix} O (W C h O A - F D B) = O (m a x F e s \times (O (q u i c k s o r t) + O (F D B c a l c u l a t i o n) + \\ O (p o s i t i o n u p d a t e) \\ = (m a x F e s \times (n^{2} + 2 \times n + n \times d) = O {(n}^{2}) \end{matrix}$

4. Experimental Results

4.1. Experimental Settings

A brief description of the experimental working environment and settings is given below.

The IEEE CEC 2020 suite [48], which has four different types of benchmark problems, was used to investigate the performance of the proposed algorithm.

The optimization process termination criterion is determined by the maximum number of objective function evaluations, denoted as maxFEs. This criterion is set as maxFEs = 1000 × D, where D represents the problem dimension.

The experiments were conducted independently for four varying problem dimensions (D = 10, D = 30, D = 50, and D = 100) to analyze and evaluate the performance of the proposed algorithm in low, medium, and high-dimensional optimization scenarios.

The experiments were repeated 25 times for the optimization of each problem.

Statistical test methods were employed to analyze the data obtained from experimental studies.

Pairwise comparisons were conducted using the Wilcoxon test, while multiple comparisons were performed using the Friedman test.

Experiments were conducted utilizing MATLAB R2018b on a Core i7 1065G7 1.30 GHz CPU with 12 GB of RAM and running Windows 11.

4.2. Testing and Analyzing the Performance of the Proposed WChOA-FDB Algorithm

The experimental results of the WChOA-FDB are provided, wherein several cases of the proposed WChOA-FDB algorithm are compared with the base algorithm. The performance of the developed WChOA-FDB algorithm is tested on IEEE CEC 2020 [48] benchmark problems. Ten different problems taken from CEC 2020 presented in Table 2 were analyzed in dimensions of 10, 30, 50, and 100 using the Friedman and Wilcoxon pairwise comparison tests.

Table 2 presents the function types, search range, and the problem sizes of the test functions used in the study. These functions are designed to test the exploration, exploitation, and balanced search capability of meta-heuristic search algorithms. Unimodal and multimodal test functions aim to determine the exploitation and exploration capabilities of an algorithm, respectively. Hybrid and composition test functions are more complex. Hybrid functions are used to investigate the algorithm’s ability to balance between exploration and exploitation. Composition functions are the same as real-world optimization problems because they have many local minima.

Statistical Analyses on CEC’2020 Test Suite

The Friedman test is a non-parametric statistical test used to detect differences in treatments across multiple test attempts. This test is particularly useful in situations where multiple measurements are taken from different methods, algorithms, or conditions on the same subjects. Our statistical analyses included 10 test functions, 25 runs, and 9 algorithms. Table 3 displays the Friedman test results of the experimental data. The results indicated that the WChOA-FDB variants demonstrated better search performance compared to the base WChOA algorithm in all experiments. According to the mean rank index, the Case-4 variant was identified as the most successful among the WChOA-FDB variants. The dimensional scalability analysis reveals that Case-4 achieves the best overall performance with a mean rank of 4.690, demonstrating good optimization capability across varying problem dimensions, particularly excelling in high-dimensional scenarios (D = 100) where it attains the lowest mean rank of 4.492. Case-3 emerges as the second-best performer (mean rank = 4.708) with notably strong performance in higher dimensions (D = 50 and D = 100), suggesting enhanced scalability characteristics compared to other variants. Case-2 maintains consistent performance across all dimensional levels with minimal variance (4.678–4.804), indicating robust stability regardless of problem complexity, while Case-8 shows dimension-dependent behavior with exceptional performance at moderate dimensions (4.478 at D = 30) but degraded performance at higher scales. The analysis demonstrates a clear performance hierarchy where Cases 1–4 significantly outperform Cases 5–8, with the original WChOA exhibiting the poorest scalability (mean rank = 6.272), particularly deteriorating at higher dimensions (6.709 at D = 100). Case 5 and Case 6 consistently rank among the worst performers across all dimensions, suggesting that their specific algorithmic modifications may introduce computational overhead without corresponding optimization benefits. The dimensional analysis indicates that problem complexity significantly influences algorithmic ranking, with high-dimensional scenarios (D ≥ 50) favoring Case 3 and Case 4, while lower dimensions show more balanced performance distribution among the top-performing variants.

The Wilcoxon method is a non-parametric statistical test used to compare two related samples to assess whether their population mean ranks differ. It serves as a robust alternative to the paired t-test, especially when the data does not meet the normality assumptions. This makes it particularly useful for evaluating the performance of various algorithms or their variants across multiple data items. In the context of MHS algorithms, the Wilcoxon signed-rank test can be used to determine the best algorithm by comparing their performance metrics pairwise. Table 4 presents the results of the Wilcoxon pairwise comparison of the base algorithm and its FDB variants. In Table 4, the sum of the items in every cell represents the total number of test functions, which is 10 in this study. For example, “Case-1 vs. WChOA” represents the pairwise comparison between a WChOA-FDB variant and the base algorithm WChOA. The pairwise (+/=/−) in each cell indicates the test problem number in which the associated WChOA-FDB variant showed better, similar, or worse performance, respectively, when compared to the original WChOA algorithm. For example, the pairwise (7_3_0) in the last column (D = 100) of Case-4 means that the Case-4 variant performed better than the base algorithm WChOA in seven problems, showed similar performance in three problems, and performed worse in zero problems. In Table 4, the information about the performance of all WChOA-FDB variants for 10, 30, 50, and 100 dimensions was presented. In D = 10 dimensions, Case-1, Case-2, and Case-5 achieved the best score (5-5-0) in all pairwise comparisons. For D = 30 dimensions, Case-1 and Case-2 obtained the best score as (6-4-0). Also, Case-3, Case-4, Case-7, and Case-8 performed better than the base algorithm with a score of (5-5-0). In D = 50 dimensions, Case-8 achieved the best score (7-3-0) among all cases. Besides Case-4, Case-6 outperformed the base algorithm with score (5-5-0), while Case-1, Case-3, and Case-7 performed better with score of (6-4-0). Finally, we can conclude that the experimental results for all WChOA-FDB variants in D = 100 dimensions performed better than the original algorithm.

4.3. Image Segmentation Performance of the Proposed WChOA-FDB Method

In this section, the multilevel image segmentation performance of the proposed WChOA-FDB algorithm is analyzed. Experimental studies were carried out on 10 color images taken from the Berkeley image segmentation database [49]. The test images and their histograms are presented in Figure 2. The images with complex histograms were chosen to investigate the solution of complex problems. When the histograms were examined, it can be seen that the test images have various brightness and color distributions.

To evaluate WChOA-FDB’s multilevel thresholding image segmentation performance, we employed Kapur’s entropy and Otsu’s between-class variance methods. The performance of WChOA-FDB has been compared with other well-known metaheuristic algorithms, namely ABC, CS, GWO, HHO, MFO, SCA, and SSA. Experimental studies were carried out using 10 color images, 2 thresholding functions (Kapur’s entropy and Otsu’s method), 16 MHS algorithms (WChOA, 8 WChOA variants, and 7 well-known competitor algorithms), 6 different threshold levels (m = 2, 4, 6, 8, 10, 12), and two different search process termination criterion (maxFEs = 500 × m and maxFEs = 1000 × m, m is the number of threshold levels, maxFEs is the search process termination criterion). As a result, a total of 11,520 = 10 image × 3 channel (RGB) × 2 thresholding functions × 16 MHS algorithms × 6 threshold levels × 2 search process termination criteria (500 × m, 1000 × m) experiments were conducted to evaluate multilevel thresholding image segmentation performance of the proposed WChOA-FDB method. The algorithms execute 25 runs for each segmentation task. The original settings of the algorithms were used in the experiments. These settings are presented in Table 5.

4.3.1. Statistical Analysis on Test Images

In this sub-section, the analysis results of the segmentation process using Kapur’s entropy and Otsu’s method as objective functions were given. The fitness values of 16 MHS algorithms for 10 test images, 6 different threshold levels, and R-G-B channels were analyzed. The Friedman test was used for statistical comparisons. Analysis results obtained using Otsu’s method and Kapur’s entropy as objective functions for 2 different maxFEs values were presented in Table 6 and Table 7, respectively. The performance analysis reveals distinct algorithmic behavior patterns across varying experimental conditions, with Case-8 demonstrating the most robust performance by achieving first rank in 7 out of 24 scenarios, particularly excelling at higher threshold levels (m ≥ 8), regardless of fitness function or computational cost. Case-2 emerges as the second most successful variant with 7 first-place rankings, showing particular strength in lower threshold scenarios (m = 2.4) and maintaining competitive performance across both Otsu and Kapur fitness functions. The data indicates a clear threshold-dependent performance hierarchy where simpler segmentation tasks (m = 2.4) favor Case-2’s approach, while complex multilevel scenarios (m ≥ 8) consistently benefit from Case-8’s enhanced optimization mechanism. Computational cost allocation significantly influences algorithmic ranking, as increased function evaluations (1000 × m vs. 500 × m) enable previously underperforming cases like Case-1 and Case-4 to achieve better results, suggesting that these variants require extended exploration phases to reach optimal solutions. The fitness function selection also demonstrates notable impact on case performance, with Kapur’s entropy optimization exhibiting more diverse case rankings compared to Otsu’s method, indicating that the entropy-based objective landscape provides different optimization challenges that favor varied algorithmic strategies across different cases.

In multilevel image segmentation, the increase in the number of threshold levels increases the complexity of the problem. The Friedman rankings in Table 6 and Table 7 indicate that the increase in the complexity of the search space has made distinctive differences between the top-ranking algorithms and their competitors. The change in the threshold level value changes the boundaries and geometric structure of the search space. The success of different algorithms at different threshold levels is related to the algorithms’ exploitation and exploration capabilities. However, the results show that the WChOA-FDB method proposed in the study is more successful than the base algorithm and other competitive algorithms for two thresholding functions, in all threshold levels and maxFEs values, except m = 2 and maxFEs = 1000 × m for Otsu’s thresholding function. This means that the proposed method exhibits a more balanced search capability compared to its competitors for all segmentation problems with different complexities used in the experiments. In addition, the WChOA algorithm, which was used for the first time in the field of multilevel image segmentation, was generally the second most successful algorithm for both objective functions, according to the Friedman scores.

4.3.2. Performance Metrics

Fitness function, PSNR (Peak Signal-to-Noise ratio), SSIM (Structural Similarity Index Measure), and FSIM (Feature Similarity Index) criteria were used as evaluation metrics in the study for image segmentation. Otsu’s method and Kapur’s entropy functions were used as fitness functions. Algorithms were trying to maximize these fitness functions. Therefore, the highest fitness value shows the success of that algorithm. PSNR, SSIM, and FSIM criteria were used to evaluate the quality of segmented images.

PSNR is defined as the ratio between the maximum possible power of signal and noise introduced. It is used to measure the reconstruction quality of an image. Mean Square Error (MSE) measures the error at each pixel location and generates the mean value which is further utilized to generate the PSNR of the image. MSE is defined as the difference between the intensity of the original image and the segmented image. A higher value of PSNR denotes good segmentation. PSNR value was calculated with Equation (33). In the equation, x is the original image, y is the segmented image, and (m, n) is the number of rows and columns of images.

(33) $P S N R = {10 l o g}_{10} \frac{255^{2}}{\frac{1}{m n} \sum_{i = 0}^{m - 1} \sum_{j = 0}^{n - 1} {‖ x (i, j) - y (i, j) ‖}^{2}}$

SSIM is a perception-based model that considers image degradation as a perceived change in structural information. SSIM is generally used to find out the inter-dependencies between original and segmented images. The SSIM is calculated as in Equation (34):

(34) $S S I M (x, y) = \frac{(2 µ_{x} µ_{y} + C_{1}) (2 σ_{x y} + C_{2})}{(µ_{x}^{2} + µ_{y}^{2} + C_{1}) (σ_{x}^{2} + σ_{y}^{2} + C_{2})}$

In the equation, μ_x and μ_y represent mean value, σ_x and σ_y show standard deviation, σ_xy refers to the correlation between x and y images, respectively. C₁ and C₂ are constants chosen to avoid division by zero error. It can be concluded that higher quality thresholding is made at large values of the SSIM.

FSIM is the metric that is used to estimate the structural similarity of the original and segmented image. It is defined as:

(35) $F S I M = \frac{\sum_{x \in Ω} S_{L} (x) . P C_{m} (x)}{\sum_{x \in Ω} P C_{m} (x)}$

where Ω represents the entire image and

S_{L} (x)

indicates the similarity between the segmented images obtained through multilevel thresholding task and the input image. The FSIM parameter of an RGB color image is calculated as the mean of the FSIM parameter of each color channel.

4.3.3. Performance Evaluation Using the Fitness Function

During the optimization process, Otsu’s method and Kapur’s entropy are used as the objective functions in the study. Algorithms were trying to maximize these fitness functions. Therefore, the highest fitness value shows the success of that algorithm. The average fitness function values of each algorithm using Otsu’s method for maxFEs = 500 × m and maxFEs = 1000 × m are presented in Table 8 and Table 9, respectively. It is seen that the WChOA and the proposed WChOA-FDB variants are the most outstanding in almost all experiments. In Table 8 and Table 9, WChOA and the proposed WChOA-FDB variants give higher fitness function values in 53 out of 60 experiments for maxFEs = 500 × m and in 49 out of 60 experiments for maxFEs = 1000 × m. Of these successful experiments, WChOA-FDB algorithm gave better fitness value results in 39 experiments for maxFEs = 500 × m and in 35 experiments for maxFEs = 1000 × m.

The average fitness function values of each algorithm using Kapur’s entropy for maxFEs = 500 × m and maxFEs = 1000 × m are presented in Table 10 and Table 11, respectively. It is seen that the WChOA and the proposed WChOA-FDB variants give better results in almost all experiments. In Table 10 and Table 11, WChOA and the proposed WChOA-FDB variants give higher fitness function values in 46 out of 60 experiments for maxFEs = 500 × m and 48 out of 60 experiments for maxFEs = 1000 × m. Of these successful experiments, the proposed WChOA-FDB algorithm gave better fitness value results in 38 and 37 experiments for maxFEs = 500 × m and maxFEs = 1000 × m, respectively.

This experimental analysis demonstrates the superior performance of the proposed WChOA-FDB algorithm in multilevel image segmentation optimization. The results show that WChOA and WChOA-FDB variants consistently outperform competing algorithms across both objective functions (Otsu’s method and Kapur’s entropy) and maximum number of fitness evaluations. Specifically, the WChOA family achieves success rates of 88% (53/60) and 82% (49/60) for Otsu’s method at maxFEs = 500 × m and 1000 × m, respectively, and 77% (46/60) and 80% (48/60) for Kapur’s entropy under the same conditions. Notably, within these successful experiments, the proposed WChOA-FDB algorithm demonstrates improved optimization capability, achieving the highest fitness values in approximately 73% of successful cases for Otsu’s method and 80% for Kapur’s entropy across both maxFEs values. These findings indicate that WChOA-FDB exhibits enhanced search balance and convergence characteristics, making it particularly effective for complex multilevel thresholding problems with varying difficulty levels, thereby establishing its strong practical applicability in color image segmentation tasks.

4.3.4. Performance Evaluation Using PSNR, SSIM and FSIM

In this section, the PSNR, SSIM, and FSIM metrics were used to evaluate the multilevel image segmentation performances of the proposed and competitor algorithms. The average PSNR, SSIM, and FSIM results obtained by each algorithm for each number of threshold levels, thresholding functions, and termination criteria on 10 images are shown graphically.

The average PSNR values of the MHS algorithms for Otsu’s method and Kapur’s entropy for both termination criteria (maxFEs = 500 × m and maxFEs = 1000 × m) and a number of threshold levels are presented in Figure 3. The figure shows the graphs of the average PSNR values of the Otsu and Kapur functions on 10 test images for the maxFEs = 500 × m and maxFEs = 1000 × m for each number of threshold levels. As can be seen from the figure, the WChOA algorithm, which was first applied to the image segmentation problem and its proposed variants, gives higher PSNR results at all number of threshold levels compared to the competing algorithms. By using the PSNR metric, the image distortion degree can be compared. This is also used to evaluate the image quality. The smaller image distortions result in a higher PSNR value. According to the results, the proposed method shows higher performance in image segmentation when compared to other algorithms.

When the PSNR values, including 16 algorithms, 10 images, and 6 threshold levels, 2 termination criteria for Otsu’s method are examined, the WChOA and WChOA variants give higher PSNR values in 49 of 60 experiments. For maxFEs = 1000 × m the number of higher PSNR values is 48. When Kapur’s entropy is used as an objective function, the number of experiments in which the PSNR values are higher than other algorithms for the proposed method are 45 for maxFEs = 500 × m and 44 for maxFEs = 1000 × m. These results establish the proposed method’s consistent success in producing higher quality segmented images, with an overall success rate of approximately 78% across both objective functions and maxFEs values, confirming its robust performance in practical image segmentation applications.

The SSIM is an important metric that is used to evaluate the similarity of the original image and the segmented image visually. It is based on contrast, brightness, and structural information of the images. If the segmented image and the original image are more similar, the SSIM value is higher. Also, the value of SSIM increases as the number of thresholds increases, which means that the segmented images are more similar to the original images and the interested objects can be extracted more accurately.

The average SSIM values of the MHS algorithms for Otsu’s method and Kapur’s entropy for both termination criteria (maxFEs = 500 × m and maxFEs = 1000 × m) and the number of threshold levels are presented in Figure 4. The figure shows the graphs of the average SSIM values of the Otsu and Kapur functions on 10 test images for the maxFEs = 500 × m and maxFEs = 1000 × m for each number of threshold levels. As can be seen from the figure, the WChOA algorithm and the proposed WChOA-FDB variants give higher SSIM results at almost all threshold levels.

The experimental results indicate that the SSIM of the proposed method has the highest values for the majority of experiments and outperforms all the other algorithms. When maxFEs = 500 × m, the number of experiments in which WChOA and the proposed WChOA-FDB method achieved higher SSIM values were 55 and 43 (out of 60 experiments) for Otsu’s method and Kapur’s entropy, respectively. The number of successful experiments for Otsu’s method and Kapur’s entropy are 56 and 49 when maxFEs = 1000 × m. The consistently high success rates across both objective functions and maxFEs values, with an average of 84.6% better performance, demonstrate the proposed algorithm’s remarkable capability in preserving structural information during image segmentation, indicating excellent correlation between optimized thresholds and perceptual image quality.

FSIM is based on the fact that the human visual system understands an image mainly according to its low-level features. Specifically, phase congruency, which is a dimensionless measure of the significance of a local structure, is used as the primary feature in FSIM. Considering that PC is contrast invariant while the contrast information does affect the human visual system’s perception of image quality, the image gradient magnitude is employed as the secondary feature in FSIM. Therefore, the FSIM is an important metric that is used to evaluate the feature similarity of the original image and the segmented image. If the segmented image and the original image are more similar, the FSIM value is higher. Also, the value of FSIM increases as the number of thresholds increases.

The average FSIM values of the MHS algorithms for Otsu’s method and Kapur’s entropy for both termination criteria (maxFEs = 500 × m and maxFEs = 1000 × m) and the number of threshold levels are presented in Figure 5. The figure shows the graphs of the average FSIM values of the Otsu and Kapur functions on 10 test images for the maxFEs = 500 × m and maxFEs = 1000 × m for each number of threshold levels. As can be seen from the figure, the WChOA algorithm and the proposed WChOA-FDB variants also give higher FSIM results at almost all number of threshold levels.

The experimental results indicate that the FSIM index of the proposed method has the highest values for the majority of experiments and outperforms all the other algorithms. When maxFEs = 500 × m, the number of experiments in which WChOA and the proposed WChOA-FDB method achieved higher FSIM values were 54 and 48 (out of 60 experiments) for Otsu’s method and Kapur’s entropy, respectively. The number of successful experiments for Otsu’s method and Kapur’s entropy are 51 and 48 when maxFEs = 1000 × m. The high success rates across both objective functions, with an overall average of 83.8% higher performance, confirm the proposed algorithm’s remarkable capability in preserving critical image features during segmentation.

In the study, a huge number of segmented images, 3840 in total, were obtained for 10 images, 6 threshold levels, 16 algorithms, 2 thresholding functions, and 2 maxFEs values. The top-ranking algorithms according to the Friedman scores given in Section 4.3.1. are presented in Table 12. This table shows the success of the proposed method in image segmentation at different threshold levels, thresholding functions, and termination criteria. Images segmented with the algorithms in Table 12 are given in Figure 6, Figure 7, Figure 8 and Figure 9.

Based on the computational performance analysis, Table 13 presents the execution times in seconds for the base WChOA algorithm and its eight FDB variants across different problem dimensions (m = 2, 6, 12) with maxFEs = 500 × m. The computational performance analysis reveals that all FDB variants demonstrate comparable efficiency to the base WChOA algorithm, with execution times exhibiting predictable scaling characteristics as problem dimensionality increases from m = 2 to m = 12. The algorithms show sub-exponential computational complexity, with approximately 4-fold and 2.8-fold increases in execution time for m = 6 and m = 12, respectively, indicating favorable scalability properties. Among the variants, Case-6 consistently demonstrates the most efficient performance across all tested dimensions (0.723 s, 2.986 s, 8.817 s), while Case-4 and Case-5 exhibit marginally higher computational overhead.

The tight clustering of execution times within approximately 3.5–6.3% variance across all variants suggests that the FDB enhancements introduce negligible computational penalty while potentially offering distinct algorithmic advantages, thereby positioning computational efficiency as a secondary consideration in variant selection compared to optimization effectiveness and convergence characteristics.

5. Discussion and Conclusions

In this study, a novel MHS algorithm named WChOA-FDB is developed. The algorithm’s performance was evaluated using 10 benchmark functions (IEEE CEC 2020) of different types and complexity levels. The search performance of the algorithm was analyzed using the Friedman and Wilcoxon statistical test methods. The results indicated that the WChOA-FDB variants demonstrated higher search performance when compared to the base WChOA algorithm in all experiments on benchmark functions. In the study, many FDB variants were produced, the eight most successful cases were selected, and their results were given. Then, the performance of these algorithms in the image segmentation problem was evaluated.

Experiments conducted on color image segmentation tasks demonstrate the effectiveness of WChOA-FDB in achieving accurate and efficient segmentation results. The results of image segmentation experiments show that the WChOA-FDB method proposed in the study is more successful than the base algorithm and other competitive algorithms for two thresholding functions, in all threshold levels and maxFEs values. This means that the proposed method exhibits a more balanced search capability compared to its competitors for all segmentation problems with different complexities. This shows the proposed algorithm’s considerably strong practicability in color image segmentation. Also, according to the PSNR, SSIM, and FSIM results, the proposed method demonstrated enhanced performance in image segmentation when compared to other algorithms. It has been observed that among the proposed FDB variants, Case-1, Case-2, Case-4, Case-7, and Case-8 are more successful in the segmentation problem. The synergy between WChOA-FDB and WChOA enhances the algorithm’s robustness, adaptability, and overall performance. The algorithm’s convergence speed, solution quality, and resilience in diverse and dynamic optimization landscapes are notably improved with the integration of FDB.

The comprehensive analysis of both performance results (Table 3, Table 6 and Table 7) reveals a complex algorithmic behavior pattern where Case-8’s dominance in multilevel image segmentation tasks (achieving first rank in 7 out of 24 scenarios) contrasts sharply with its moderate performance in benchmark function optimization (4th place with mean rank 4.752), indicating that algorithm effectiveness is highly task-dependent and cannot be generalized across different optimization domains. Case-4 demonstrates the most robust scalability characteristics by achieving the best overall performance in dimensional analysis (mean rank 4.690) and showing particular strength in high-dimensional problems (rank 4.492 at D = 100), yet it only achieves top performance in two specific image segmentation scenarios, primarily at higher threshold levels with increased computational cost (m = 10,12 with 1000 × m evaluations). Case-2 exhibits remarkable consistency across both evaluation frameworks, maintaining competitive performance in image segmentation tasks (7 first-place rankings) while achieving third place in dimensional scalability (mean rank 4.722), suggesting it represents the most balanced and reliable algorithmic variant for diverse optimization challenges. The stark performance degradation of the original WChOA (worst performer in dimensional analysis with mean rank 6.272, and only one first-place ranking in segmentation tasks) validates the necessity of the proposed hybrid modifications, while Cases 5–8 generally underperform in benchmark optimization despite Case-8’s segmentation success, highlighting the critical importance of algorithm-problem matching in metaheuristic optimization design.

It is observed that the methods proposed in image segmentation studies are generally tested on a small number of images. This is a general limitation of these studies. A larger and more diverse set of test images would provide stronger evidence of the algorithm’s effectiveness. In this study, 10 test images are used for the image segmentation experiments. In future studies, it is aimed to increase the number of images and evaluate the method performance on images with different characteristics such as medical and satellite images. In this study, the performance of the proposed method was compared with seven well-known MHS methods. It aims to compare the proposed method with different MHS algorithms in future studies. Also, parameter optimization for the proposed algorithm can be performed.

Author Contributions

Conceptualization, A.G.Y.; methodology, A.G.Y.; software, S.A.; validation, A.G.Y. and S.A.; writing—original draft preparation, A.G.Y. and S.A.; writing—review and editing, A.G.Y.; supervision, A.G.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original data presented in the study are openly available in https://doi.org/10.5281/zenodo.15590550 (accessed on 3 July 2025).

Conflicts of Interest

The authors have declared no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Flowchart of the proposed WChOA-FDB algorithm.

Figure 2 Test images (a) and their histograms (b) (The image names are from left to right and top to bottom: 12003.jpg, 108073.jpg, 124084.jpg, 232038.jpg, 323016.jpg, 106024.jpg, 291000.jpg, 12084.jpg, 61060.jpg, 65010.jpg).

Figure 3 The average PSNR values of the MHS algorithms for Otsu’s method and Kapur’s entropy for both termination criteria (maxFEs = 500 × m and maxFEs = 1000 × m) and number of threshold levels (m). (Otsu 500: Otsu’s method for maxFEs = 500 × m, Otsu 1000: Otsu’s method for maxFEs = 1000 × m, Kapur 500: Kapur’s entropy for maxFEs = 500 × m, Kapur 1000: Kapur’s entropy for maxFEs = 1000 × m).

Figure 4 The average SSIM values of the MHS algorithms for Otsu’s method and Kapur’s entropy for both termination criteria (maxFEs = 500 × m and maxFEs = 1000 × m) and number of threshold levels (m). (Otsu 500: Otsu’s method for maxFEs = 500 × m, Otsu 1000: Otsu’s method for maxFEs = 1000 × m, Kapur 500: Kapur’s entropy for maxFEs = 500 × m, Kapur 1000: Kapur’s entropy for maxFEs = 1000 × m).

Figure 5 The average FSIM values of the MHS algorithms for Otsu’s method and Kapur’s entropy for both termination criteria (maxFEs = 500 × m and maxFEs = 1000 × m) and number of threshold levels (m). (Otsu 500: Otsu’s method for maxFEs = 500 × m, Otsu 1000: Otsu’s method for maxFEs = 1000 × m, Kapur 500: Kapur’s entropy for maxFEs = 500 × m, Kapur 1000: Kapur’s entropy for maxFEs = 1000 × m).

Figure 6 Segmentation results of the best algorithms for Otsu’s method and maxFEs = 500 × m.

Figure 7 Segmentation results of the best algorithms for Otsu’s method and maxFEs = 1000 × m.

Figure 8 Segmentation results of the best algorithms for Kapur’s method and maxFEs = 500 × m.

Figure 9 Segmentation results of the best algorithms for Kapur’s method and maxFEs = 1000 × m.

Table 1

Mathematical model of the WChOA-FDB variants (proposed).

Algorithm	Explanation	Mathematical Model
Case-1	In this case, the proposed development process was achieved by replacing the ${\vec{X}}_{D r i v e r}$ component in driver equation with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{F D B} - {\vec{M}}_{4} \vec{X}\|$	(23)
Case-2	In this case, the proposed development process was achieved by replacing the $\vec{X}$ component in driver equation with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{D r i v e r} - {\vec{M}}_{4} {\vec{X}}_{F D B}\|$	(24)
Case-3	In this case, the proposed development process was achieved by replacing the ${\vec{X}}_{A t t a c k e r}$ and ${\vec{X}}_{D r i v e r}$ components in attacker and driver equations, respectively, with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{A t t a c k e r} = \|{\vec{C}}_{1} {\vec{X}}_{F D B} - {\vec{M}}_{1} \vec{X}\|$ ${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{F D B} - {\vec{M}}_{4} \vec{X}\|$	(25)
Case-4	In this case, the proposed development process was achieved by replacing the ${\vec{X}}_{A t t a c k e r}$ and $\vec{X}$ components in attacker and driver equations, respectively, with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{A t t a c k e r} = \|{\vec{C}}_{1} {\vec{X}}_{F D B} - {\vec{M}}_{1} \vec{X}\|$ ${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{D r i v e r} - {\vec{M}}_{4} {\vec{X}}_{F D B}\|$	(26)
Case-5	In this case, the proposed development process was achieved by replacing the $\vec{X}$ and ${\vec{X}}_{D r i v e r}$ components in attacker and driver equations, respectively, with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{A t t a c k e r} = \|{\vec{C}}_{1} {\vec{X}}_{A t t a c k e r} - {\vec{M}}_{1} {\vec{X}}_{F D B}\|$ ${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{F D B} - {\vec{M}}_{4} \vec{X}\|$	(27)
Case-6	In this case, the proposed development process was achieved by replacing the $\vec{X}$ component in attacker and driver equations, respectively, with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{A t t a c k e r} = \|{\vec{C}}_{1} {\vec{X}}_{A t t a c k e r} - {\vec{M}}_{1} {\vec{X}}_{F D B}\|$ ${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{D r i v e r} - {\vec{M}}_{4} {\vec{X}}_{F D B}\|$	(28)
Case-7	In this case, the proposed development process was achieved by replacing the ${\vec{X}}_{B a r r i e r}$ and ${\vec{X}}_{D r i v e r}$ components in barrier and driver equations, respectively, with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{B a r r i e r} = \|{\vec{C}}_{2} {\vec{X}}_{F D B} - {\vec{M}}_{2} \vec{X}\|$ ${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{F D B} - {\vec{M}}_{4} \vec{X}\|$	(29)
Case-8	In this case, the proposed development process was achieved by replacing the $\vec{X}$ and ${\vec{X}}_{D r i v e r}$ components in barrier and driver equations, respectively, with FDB method ( ${\vec{X}}_{F D B}$ ).	${\vec{D}}_{B a r r i e r} = \|{\vec{C}}_{2} {\vec{X}}_{B a r r i e r} - {\vec{M}}_{2} {\vec{X}}_{F D B}\|$ ${\vec{D}}_{D r i v e r} = \|{\vec{C}}_{4} {\vec{X}}_{F D B} - {\vec{M}}_{4} \vec{X}\|$	(30)

Table 2

Summary of the CEC’2020 test suite.

No.	Function Type	Function Description	${F_{i}}^{} = F_{i} (x^{})$
F1	Unimodal	Shifted and Rotated Bent Cigar Function	100
F2	Multimodal Functions	Shifted and Rotated Schwefel’s Function	1100
F3	Multimodal Functions	Shifted and Rotated Lunacek bi-Rastrigin Function	700
F4	Multimodal Functions	Expanded Rosenbrock’s plus Griewangk’s Function	1900
F5	Hybrid Functions	Hybrid Function 1 (N = 3))	1700
F6	Hybrid Functions	Hybrid Function 2 (N = 4)	1600
F7	Hybrid Functions	Hybrid Function 3 (N = 5)	2100
F8	Composition Functions	Composition Function 1 (N = 3)	2200
F9	Composition Functions	Composition Function 2 (N = 4)	2400
F10	Composition Functions	Composition Function 3 (N = 5)	2500
Search range: ${[- 100, 100]}^{D}$ , D: number of dimensions. (10, 30, 50, 100)

Table 3

Friedman test rankings for WChOA and WChOA-FDB variants.

Algorithm	Dimension = 10	Dimension = 30	Dimension = 50	Dimension = 100	Mean Rank
Case-1	4.681	4.938	4.785	4.802	4.801 5th
Case-2	4.678	4.683	4.804	4.723	4.722 3rd
Case-3	4.826	4.795	4.623	4.590	4.708 2nd
Case-4	4.864	4.709	4.697	4.492	4.690 1st
Case-5	4.921	5.190	5.119	4.933	5.040 7th
Case-6	5.316	5.183	5.185	5.057	5.185 8th
Case-7	4.897	4.842	4.697	4.852	4.822 6th
Case-8	4.916	4.478	4.778	4.838	4.752 4th
WChOA	5.897	6.178	6.307	6.709	6.272 9th

Table 4

Wilcoxon pairwise comparison results between WChOA-FDB variants and WChOA.

vs. WChOA+/=/−	Dimension10	Dimension 30	Dimension 50	Dimension100
Case-1	5_5_0	6_4_0	6_4_0	5_5_0
Case-2	5_5_0	6_4_0	4_6_0	7_3_0
Case-3	4_6_0	5_5_0	6_4_0	7_3_0
Case-4	4_6_0	5_5_0	5_5_0	7_3_0
Case-5	5_5_0	4_6_0	4_6_0	6_4_0
Case-6	4_5_1	4_6_0	5_5_0	6_4_0
Case-7	4_6_0	5_5_0	6_4_0	5_5_0
Case-8	4_6_0	5_5_0	7_3_0	5_5_0

Table 5

Algorithm settings.

Algorithm	Settings
WChOA	search agents = 30, ….
ABC	colony size = 50, SN = colony size/2, limit = D × SN
CS	number of nests = 25, probability = 0.25, beta = 1.5
GWO	number of search agents = 30
HHO	N = 30, Bfid = 1, nD = 10, Jr = 0.25
MFO	number of moths = 30
SCA	number of search agents = 30, a = 2
SSA	salp population size = 30

Table 6

Friedman rankings of algorithms at different threshold levels for maxFEs = 500 × m and maxFEs = 1000 × m (Otsu’s method).

3 × *25 Run (RGB), maxFEs* = 500** × m,
m = 2	m = 4	m = 6	m = 8	m = 10	m = 12
Case-2 (6.98)	Case-2 (7.92)	Case-7 (8.03)	Case-8 (8.07)	Case-7 (8.07)	Case-8 (8.22)
WChOA (7.60)	WChOA (8.21)	WCHOA (8.19)	WCHOA (8.34)	WCHOA (8.21)	WCHOA (8.33)
HHO (9.50)	GWO (8.93)	GWO (8.89)	SCA (8.40)	MFO (8.68)	HHO (8.56)
SSA (9.63)	SSA (9.01)	MFO (8.98)	MFO (8.83)	SCA (8.98)	SCA (8.72)
SCA (9.73)	MFO (9.21)	SSA (9.37)	GWO (9.13)	SSA (8.99)	SSA (8.79)
MFO (9.84)	HHO (9.29)	HHO (9.41)	SSA (9.23)	HHO (9.13)	MFO (9.226)
GWO 9.85)	SCA (9.85)	SCA (9.42)	HHO (9.34)	GWO (9.44)	GWO (9.23)
ABC (10.33)	CS (9.98)	ABC (9.73)	CS (9.99)	ABC (9.78)	CS (9.53)
CS (10.79)	ABC (10.23)	CS (9.92)	ABC (10.26)	CS (10.05)	ABC (10.35)
3 × *25 Run (RGB), maxFEs* = 1000** × m,
m = 2	m = 4	m = 6	m = 8	m = 10	m = 12
WCHOA (7.05)	Case-1 (7.61)	Case-2 (7.89)	Case-8 (8.18)	Case-1 (8.27)	Case-8 (7.84)
Case-1 (7.31)	WCHOA (7.72)	WCHOA (8.12)	WCHOA (8.53)	WCHOA (8.33)	WCHOA (8.27)
SSA (9.61)	MFO (9.22)	GWO (9.03)	MFO (8.85)	SSA (8.91)	MFO (8.51)
SCA (9.89)	GWO (9.30)	MFO (9.11)	GWO (9.00)	HHO (9.069)	SSA (8.73)
MFO (10.00)	SCA 9.32)	HHO (9.12)	SCA (9.19)	GWO (9.074)	GWO (9.06)
HHO (10.09)	HHO (9.62)	SCA (9.38)	HHO (9.38)	SCA (9.13)	HHO (9.24)
GWO (10.14)	SSA (9.77)	SSA (9.79)	CS (9.54)	MFO (9.30)	SCA (9.35)
CS (10.67)	ABC (10.15)	CS (9.94)	SSA (9.58)	ABC (9.46)	CS (10.22)
ABC (10.80)	CS (10.36)	ABC (10.13)	ABC (10.10)	CS (9.89)	ABC (10.23)

Table 7

Friedman rankings of algorithms at different threshold levels for maxFEs = 500 × m and maxFEs = 1000 × m (Kapur’s entropy).

3 × *25 Run (RGB), maxFEs* = 500** × m,
m = 2	m = 4	m = 6	m = 8	m = 10	m = 12
Case-8 (7.80)	Case-2 (8.44)	Case-7 (8.33)	Case-8 (7.98)	Case-2 (8.47)	Case-1 (8.59)
WCHOA (8.23)	WCHOA (8.77)	WCHOA (8.61)	SSA (8.28)	WCHOA (8.59)	GWO (8.62)
MFO (9.05)	GWO (8.80)	HHO (8.66)	MFO (8.67)	SSA (8.69)	HHO (8.82)
GWO (9.22)	MFO (8.83)	SSA (8.86)	GWO (8.73)	MFO (8.83)	WCHOA (8.826)
SSA (9.33)	SSA (9.07)	GWO (9.11)	HHO (8.79)	GWO (8.88)	SCA (8.83)
SCA (9.63)	HHO (9.15)	SCA (9.11)	WCHOA (8.95)	SCA (8.96)	SSA (9.07)
HHO (9.68)	SCA (9.22)	MFO (9.30)	SCA (9.00)	HHO (9.24)	MFO (9.09)
CS (10.23)	ABC (9.38)	ABC (9.38)	CS (9.65)	CS (9.57)	ABC (9.27)
ABC (10.34)	CS (9.66)	CS (9.68)	ABC (9.70)	ABC (9.59)	CS (9.72)
3 × *25 Run (RGB), maxFEs* = 1000** × m,
m = 2	m = 4	m = 6	m = 8	m = 10	m = 12
Case-2 (7.65)	Case-7 (8.16)	Case-8 (8.48)	Case-2 (8.38)	Case-4 (8.44)	Case-4 (8.18)
WCHOA (7.93)	WCHOA (8.22)	WCHOA (8.60)	GWO (8.67)	SCA (8.79)	HHO (8.53)
HHO (9.16)	GWO (8.82)	SCA (8.70)	MFO (8.72)	MFO (8.93)	MFO (8.61)
SSA (9.41)	MFO (8.91)	GWO (8.82)	SCA (8.73)	HHO (9.02)	GWO (8.71)
SCA (9.48)	HHO (8.99)	SSA (8.84)	WCHOA (8.74)	WCHOA (9.14)	WCHOA (8.86)
GWO (9.83)	SSA (9.07)	HHO (8.96)	HHO (8.78)	GWO (9.15)	SSA (8.98)
MFO (9.95)	ABC (9.41)	MFO (8.97)	SSA (9.17)	SSA (9.22)	SCA (9.33)
ABC (10.44)	CS (9.51)	ABC (9.38)	ABC (9.31)	ABC (9.28)	CS (9.49)
CS (10.48)	SCA (9.57)	CS 9.99)	CS (9.41)	CS (9.61)	ABC (9.59)

Table 8

The average fitness values of Otsu’s method in comparison with other algorithms (maxFEs = 500 × m).

m	I	ABC	CS	GWO	HHO	MFO	SCA	SSA	WChOA	Case-1	Case-2	Case-3	Case-4	Case-5	Case-6	Case-7	Case-8
2	1	2737.074	2725.243	2743.743	2739.341	2737.367	2735.023	2742.085	2752.066	2748.824	758.227	2742.976	2746.675	2741.095	2733.923	2743.837	2744.737
	2	1546.111	1542.659	1543.179	1550.087	1549.543	1542.648	1550.242	1558.874	1562.89	1560.921	1557.5	1556.029	1555.361	1545.926	1557.683	1558.725
	3	2045.823	2048.159	2052.438	2055.056	2056.045	2058.083	2056.986	2063.669	2069.752	2060.543	2064.757	2063.925	2063.697	2062.156	2060.129	2064.917
	4	3751.211	3750.514	3746.45	3755.181	3745.171	3753.348	3750.586	3768.945	3767.619	3769.786	3772.374	3767.559	3767.873	3765.901	3764.98	3769.406
	5	6543.776	6539.734	6556.53	6553.129	6548.085	6555.473	6550.013	6572.956	6572.608	6573.75	6572.767	6565.286	6563.162	6564.433	6576.648	6566.533
	6	3410.207	3432.674	3433.317	3432.827	3422.742	3432.178	3426.886	3432.669	3437.171	3438.64	3422.988	3429.121	3427.679	3425.944	3429.048	3430.714
	7	2001.345	1992.559	2006.84	2006.68	2006.063	2002.733	2008.456	2014.906	2011.195	2009.464	2015.995	2010.7	2010.275	2009.741	2005.913	2016.538
	8	1332.685	1325.152	1328.754	1339.794	1332.781	1337.124	1338.239	1346.524	1349.764	1346.32	1349.134	1348.85	1341.322	1345.048	1346.207	1349.081
	9	3078.593	3086.274	3090.238	3097.18	3090.702	3088.602	3096.844	3108.937	3099.905	3107.093	3096.787	3099.8	3099.324	3099.298	3095.716	3097.638
	10	1525.829	1500.797	1526.837	1531.625	1508.113	1522.823	1522.144	1532.401	1532.89	1529.045	1523.838	1524.909	1533.122	1526.415	1530.328	1529.312
4	1	2971.224	2983.725	2978.576	2979.185	2982.445	2979.029	2979.672	2980.495	2985.098	2983.454	2978.855	2984.145	2983.573	2982.657	2983.677	2985.035
	2	1685.928	1685.786	1690.94	1690.277	1690.988	1684.644	1691.281	1691.914	1692.159	1694.731	1693.811	1690.774	1695.597	1688.21	1690.137	1691.956
	3	2237.984	2233.492	2242.192	2236.82	2242.401	2238.648	2242.114	2246.91	2251.695	2247.267	2243.703	2246.085	2245.638	2245.366	2242.789	2247.555
	4	3907.647	3904.117	3915.465	3916.674	3905.069	3911.686	3916.22	3916.376	3916.036	3920.918	3919.852	3918.136	3920.576	3913.992	3920.143	3919.481
	5	6850.046	6856.642	6856.163	6861.359	6856.216	6858.982	6857.505	6865.578	6864.933	6866.469	6862.937	6863.268	6860.461	6854.138	6860.196	6863.986
	6	3626.963	3621.681	3635.886	3629.191	3635.181	3634.584	3630.871	3639.573	3631.285	3632.235	3639.309	3630.197	3634.974	3633.786	3636.253	3640.083
	7	2215.266	2220.331	2228.699	2224.114	2233.559	2220.861	2220.534	2226.323	2229.603	2222.014	2222.431	2225.457	2223.379	2229.035	2230.55	2221.228
	8	1458.24	1460.839	1466.011	1461.55	1462.692	1461.551	1465.346	1466.807	1462.093	1463.385	1462.209	1464.689	1467.757	1458.666	1468.058	1461.075
	9	3326.803	3329.522	3340.704	3340.955	3336.021	3336.222	3341.613	3342.281	3338.818	3350.156	3340.224	3340.15	3335.643	3334.665	3344.094	3341.944
	10	1731.037	1725.522	1732.67	1728.978	1733.961	1735.643	1739.335	1735.071	1740.234	1731.957	1731.666	1737.891	1736.247	1739.74	1742.006	1746.334
6	1	3055.786	3062.076	3062.298	3059.377	3060.01	3057.887	3059.611	3067.026	3062.175	3065.018	3060.077	3061.563	3064.167	3064.047	3062.819	3064.734
	2	1739.541	1736.057	1743.046	1740.114	1740.816	1740.505	1738.122	1740.503	1741.137	1739.936	1742.39	1743.033	1736.743	1738.72	1741.42	1740.668
	3	2300.056	2297.708	2301.462	2302.265	2304.24	2302.317	2301.682	2302.744	2305.403	2307.668	2305.133	2306.693	2300.929	2308.062	2307.642	2305.348
	4	3968.216	3966.064	3970.711	3969.996	3965.829	3967.566	3967.846	3973.705	3969.814	3968.732	3970.132	3970.526	3969.221	3968.834	3971.435	3967.14
	5	6944.81	6946.51	6949.056	6948.293	6953.987	6950.394	6951.48	6952.411	6953.982	6951.883	6950.96	6948.6	6952.184	6948.545	6956.471	6952.526
	6	3703.885	3707.077	3707.834	3702.849	3706.919	3709.819	3706.175	3709.408	3707.619	3709.916	3710.537	3707.324	3708.627	3710.73	3709.275	3708.092
	7	2303.496	2297.185	2304.373	2304.355	2303.958	2303.739	2301.594	2304.913	2303.75	2302.027	2305.436	2303.754	2306.8	2305.37	2298.706	2307.224
	8	1505.397	1507.466	1512.165	1512.44	1509.612	1511.83	1509.776	1513.262	1511.194	1508.625	1512.027	1510.219	1511.149	1511.152	1511.32	1511.749
	9	3419.025	3421.206	3424.437	3422.984	3426.666	3422.367	3422.771	3427.851	3429.736	3427.156	3424.266	3425.246	3427.52	3424.029	3427.858	3428.033
	10	1792.242	1791.979	1796.248	1797.419	1794.984	1792.366	1796.761	1795.846	1794.564	1800.367	1797.849	1794.258	1795.875	1797.854	1795.282	1794.056
8	1	3095.035	3094.788	3100.935	3100.012	3103.595	3099.656	3100.506	3098.546	3104.124	3100.637	3098.626	3099.479	3101.677	3101.74	3104.208	3100.35
	2	1763.523	1763.879	1764.596	1764.643	1767.469	1768.115	1764.392	1768.653	1768.017	1767.328	1765.518	1765.958	1766.287	1767.216	1765.996	1768.691
	3	2330.243	2332.263	2334.553	2333.814	2333.914	2335.018	2334.367	2335.825	2332.99	2335.402	2335.729	2334.178	2333.14	2335.624	2334.013	2334.405
	4	3993.417	3994.206	3993.94	3994.909	3993.251	3997.008	3996.942	3999.81	3998.207	3999.083	3997.443	3997.645	3995.869	3994.276	3996.613	3998.257
	5	6992.46	6994.43	6995.409	6994.536	6995.865	6997.45	6995.604	6993.495	6997.185	6993.848	6994.644	6996.019	6993.927	6996.418	6997.189	6997.663
	6	3742.932	3738.304	3742.176	3745.083	3744.734	3747.146	3742.216	3742.347	3742.419	3739.728	3744.674	3744.946	3742	3747.931	3745.343	3743.693
	7	2341.433	2337.658	2341.707	2340.95	2342.599	2343.816	2343.457	2341.77	2341.306	2345.089	2340.648	2341.969	2341.934	2339.032	2341.941	2343.293
	8	1532.37	1533.489	1533.219	1532.896	1535.076	1534.259	1536.2	1536.904	1535.819	1536.228	1536.754	1536.637	1533.963	1533.872	1534.513	1535.599
	9	3462.031	3461.852	3464.573	3466.318	3461.814	3462.475	3464.669	3461.062	3465.976	3463.881	3465.466	3463.109	3465.688	3463	3464.724	3463.39
	10	1823.205	1822.047	1825.318	1822.844	1823.57	1826.354	1823.671	1823.238	1824.776	1823.024	1824.77	1824.682	1823.968	1822.874	1824.185	1823.903
10	1	3121.272	3120.716	3121.541	3122.446	3121.88	3123.461	3121.707	3122.548	3123.24	3124.93	3122.275	3123.788	3123.813	3122.623	3125.303	3124.233
	2	1780.889	1779.301	1780.672	1781.553	1781.717	1779.962	1780.486	1782.688	1780.872	1781.24	1781.569	1781.722	1779.26	1782.714	1781.475	1782.744
	3	2349.223	2347.22	2349.941	2349.661	2350.596	2351.171	2349.778	2352.851	2350.929	2352.054	2350.112	2351.156	2351.022	2351.174	2352.244	2351.718
	4	4010.342	4010.753	4012.625	4011.558	4012.929	4012.716	4013.307	4013.326	4012.187	4011.648	4014.059	4013.29	4010.777	4012.077	4013.891	4013.239
	5	7016.677	7017.247	7018.761	7019.45	7020.418	7018.466	7019.57	7020.04	7018.951	7019.892	7019.9	7020.678	7020.031	7018.165	7020.202	7018.861
	6	3764.082	3767.565	3766.572	3765.698	3764.905	3764.871	3768.077	3764.142	3765.857	3764.868	3766.76	3766.391	3765.913	3764.814	3766.928	3765.589
	7	2363.542	2361.786	2366.986	2365.115	2365.012	2365.636	2365.319	2367.344	2366.109	2364.133	2365.797	2366.46	2362.534	2368.323	2364.999	2365.382
	8	1549.772	1546.985	1550.604	1550.507	1550.854	1551.823	1548.698	1552.82	1551.89	1549.949	1550.952	1549.48	1551.613	1549.012	1551.665	1550.676
	9	3482.569	3485.255	3485.294	3486.925	3486.103	3485.139	3485.89	3487.332	3485.683	3485.421	3484.755	3487.011	3486.388	3484.586	3486.593	3485.943
	10	1841.121	1839.201	1841.381	1841.749	1840.399	1840.823	1839.551	1840.796	1840.817	1840.762	1839.598	1838.908	1839.984	1839.928	1839.385	1842.901
12	1	3136.007	3137.837	3137.857	3136.9	3138.668	3138.144	3137.766	3139.396	3137.933	3137.861	3136.487	3137.712	3137.715	3137.082	3138.769	3138.984
	2	1788.918	1789.947	1789.865	1791.464	1790.576	1790.157	1790.615	1791.131	1791.84	1791.068	1789.69	1791.593	1790.929	1791.854	1791.904	1791.554
	3	2358.881	2360.723	2360.519	2361.172	2361.73	2361.348	2362.522	2363.326	2362.441	2363.241	2362.423	2361.651	2361.495	2361.488	2362.595	2361.647
	4	4021.542	4021.101	4023.276	4023.616	4021.452	4022.395	4023.176	4023.792	4022.839	4021.872	4022.023	4023.028	4023.627	4022.798	4024.05	4024.98
	5	7031.876	7033.484	7033.18	7035.353	7032.047	7035.917	7034.202	7032.981	7034.485	7033.429	7033.632	7032.504	7033.894	7034.956	7033.989	7033.885
	6	3778.532	3779.473	3779.162	3779.449	3779.565	3779.341	3778.395	3780.279	3778.271	3778.942	3778.744	3779.472	3778.63	3777.83	3780.363	3779.821
	7	2378.585	2378.113	2379.155	2379.194	2381.583	2381.161	2379.665	2379.437	2379.498	2381.203	2380.351	2378.184	2379.641	2380.059	2381.388	2379.248
	8	1557.726	1559.154	1560.626	1560.37	1559.783	1560.107	1561.226	1561.398	1558.858	1560.875	1560.791	1560.69	1560.379	1561.393	1561.181	1559.775
	9	3499.918	3500.526	3500.897	3501.652	3501.318	3499.199	3500.579	3500.413	3499.358	3500.897	3501.214	3500.387	3501.753	3499.855	3501.595	3500.624
	10	1848.916	1850.436	1851.596	1851.921	1851.159	1851.458	1852.321	1852.37	1851.271	1850.858	1851.481	1852.153	1851.102	1850.839	1850.91	1849.552

Table 9

The average fitness values of Otsu’s method in comparison with other algorithms (maxFEs = 1000 × m).

m	I	ABC	CS	GWO	HHO	MFO	SCA	SSA	WChOA	Case-1	Case-2	Case-3	Case-4	Case-5	Case-6	Case-7	Case-8
2	1	2728.046	2736.835	2735.819	2738.899	2732.252	2743.27	2748.984	2750.304	2751.112	2753.223	2751.062	2744.249	2744.117	2748.843	2751.934	2748.661
	2	1541.264	1543.551	1553.007	1542.008	1546.289	1544.833	1547.862	1566.518	1560	1564.981	1558.983	1562.899	1558.124	1554.514	1559.679	1562.125
	3	2045.685	2039.071	2055.435	2049.221	2054.039	2053.009	2052.089	2069.108	2071.535	2065.321	2063.702	2070.788	2062.614	2067.042	2064.544	2064.917
	4	3748.363	3747.132	3756.105	3763.88	3756.725	3758.588	3761.66	3777.123	3769.923	3765.023	3771.937	3766.127	3764.552	3764.588	3767.747	3772.434
	5	6538.647	6541.102	6548.044	6554.843	6551.463	6560.329	6562.061	6577.023	6574.494	6572.791	6561.217	6567.042	6566.947	6568.381	6573.875	6575.547
	6	3418.88	3428.078	3425.07	3424.068	3423.448	3437.035	3431.321	3433.769	3434.632	3425.304	3442.465	3430.343	3434.823	3434.983	3434.992	3433.11
	7	1998.971	1993.185	2001.735	1995.327	2015.309	2011.661	2004.819	2018.947	2014.765	2010.078	2019.249	2009.592	2013.035	2004.64	2015.367	2023.932
	8	1333.447	1338.047	1342.042	1342.825	1340.98	1338.311	1336.657	1345.354	1346.992	1350.59	1342.185	1344.259	1348.665	1336.141	1350.515	1351.343
	9	3076.526	3074.3	3092.04	3093.225	3082.186	3082.65	3080.201	3108.386	3102.983	3102.363	3100.289	3108.197	3104.625	3094.063	3098.907	3103.495
	10	1512.576	1520.529	1522.59	1531.779	1528.389	1529.76	1528.472	1527.622	1529.597	1530.715	1528.477	1534.044	1531.255	1541.085	1528.88	1529.634
4	1	2968.877	2974.673	2976.953	2982.646	2983.781	2983.182	2974.886	2986.18	2993.527	2983.145	2985.545	2979.449	2985.497	2990.744	2984.501	2989.566
	2	1685.734	1684.697	1691.253	1688.241	1688.154	1685.925	1688.085	1692.083	1692.318	1693.647	1688.929	1690.528	1689.897	1694.252	1690.332	1692.058
	3	2236.763	2233.671	2239.522	2237.048	2239.533	2243.753	2241.44	2247.71	2251.86	2244.449	2246.807	2243.106	2241.359	2241.43	2239.921	2249.022
	4	3911.946	3906.841	3913.986	3912.154	3913.167	3914.514	3910.033	3922.988	3918.909	3918.289	3915.913	3922.249	3918.58	3916.518	3921.51	3921.867
	5	6853.341	6853.016	6857.619	6859.257	6860.216	6856.958	6851.583	6870.819	6870.554	6869.962	6867.785	6865.297	6859.28	6859.328	6860.541	6864.823
	6	3622.164	3628.469	3632.114	3629.386	3631.182	3636.282	3632.784	3634.409	3639.17	3632.699	3635.723	3626.903	3632.451	3630.29	3628.696	3629.595
	7	2223.086	2220.79	2221.21	2218.413	2230.996	2229.773	2225.54	2231.388	2226.721	2232.232	2225.85	2227.68	2228.391	2224.434	2225.59	2228.059
	8	1460.392	1453.18	1462.304	1456.405	1464.13	1460.032	1465.229	1465.837	1463.322	1465.049	1463.708	1462.694	1462.984	1464.793	1467.923	1470.102
	9	3334.496	3335.009	3336.26	3342.41	3336.526	3332.321	3332.07	3346.039	3345.475	3343.608	3338.967	3337.257	3339.679	3340.703	3342.745	3343.647
	10	1731.641	1732.62	1739.704	1738.475	1734.856	1734.896	1738.816	1736.581	1737.571	1739.48	1739.4	1735.278	1740.574	1734.675	1733.197	1740.018
6	1	3054.89	3057.875	3065.102	3060.355	3065.534	3058.264	3058.435	3064.305	3062.155	3060.906	3065.791	3065.027	3063.015	3059.596	3061.147	3062.069
	2	1736.295	1740.601	1741.567	1742.609	1741.826	1738.128	1736.916	1741.491	1742.811	1743.101	1742.887	1739.93	1740.656	1740.016	1740.365	1741.52
	3	2297.228	2298.316	2299.478	2300.546	2303.911	2303.289	2302.267	2307.601	2306.669	2309.762	2304.063	2305.343	2305.631	2304.557	2306.897	2304.655
	4	3966.832	3966.346	3968.78	3968.852	3965.158	3968.234	3967.571	3971.748	3971.597	3973.789	3970.312	3971.821	3971.719	3970.195	3971.47	3973.35
	5	6948.413	6946.323	6953.707	6951.962	6949.013	6952.838	6951.862	6958.564	6952.067	6955.2	6956.26	6955.716	6947.41	6954.764	6955.349	6955.144
	6	3704.839	3703.534	3707.762	3706.617	3708.863	3710.317	3706.395	3707.58	3708.317	3706.599	3707.531	3704.282	3708.052	3706.703	3704.996	3708.614
	7	2299.587	2302.118	2304.125	2302.648	2302.855	2300.734	2303.164	2306.07	2307.361	2304.45	2305.295	2306.147	2303.055	2301.439	2302.757	2301.506
	8	1507.623	1505.338	1511.008	1509.325	1513.302	1508.243	1509.08	1513.86	1509.119	1511.701	1508.891	1510.77	1507.269	1508.652	1512.774	1511.701
	9	3418.785	3419.915	3419.67	3423.655	3421.695	3422.716	3421.299	3428.2	3423.16	3423.516	3425.031	3420.209	3425.477	3423.809	3422.04	3426.303
	10	1791.085	1795.063	1797.006	1794.699	1797.044	1795.819	1795.145	1796.587	1795.566	1798.723	1798.105	1799.672	1792.64	1796.022	1796.409	1792.647
8	1	3095.424	3098.226	3096.092	3097.604	3102.578	3099.541	3099.191	3099.212	3101.358	3101.778	3100.179	3100.502	3102.252	3099.098	3101.87	3100.961
	2	1763.605	1764.15	1765.118	1763.981	1764.757	1766.239	1764.589	1766.657	1766.214	1766.362	1767.323	1766.702	1767.332	1767.677	1766.73	1767.469
	3	2331.683	2332.329	2336.407	2333.429	2333.353	2332.681	2330.897	2334.925	2335.595	2335.206	2334.794	2335.313	2331.837	2333.505	2335.347	2335.138
	4	3991.394	3993.546	3996.526	3994.596	3995.894	3995.08	3993.947	3998.584	3996.09	3995.683	3994.939	3997.878	3995.456	3996.091	3997.743	3998.184
	5	6994.574	6992.489	6996.982	6995.766	6996.97	6996.545	6995.52	6996.596	6998.921	6995.723	6995.579	6996.429	6994.3	6996.624	6996.008	6996.598
	6	3740.619	3743.201	3744.652	3747.414	3742.679	3745.083	3742.676	3745.304	3746.505	3743.033	3742.877	3744.669	3741.516	3742.274	3745.246	3744.463
	7	2338.008	2339.88	2341.908	2341.752	2340.785	2343.072	2344.421	2343.893	2344.684	2344.975	2344.928	2343.454	2339.125	2339.513	2343.181	2342.491
	8	1532.206	1534.454	1535.469	1534.594	1536.051	1536.161	1533.073	1534.429	1534.321	1534.774	1534.544	1535.912	1535.287	1535.801	1535.95	1535.511
	9	3460.832	3461.665	3461.833	3464.059	3465.007	3463.435	3460.846	3464.962	3464.063	3464.794	3464.007	3464.592	3463.611	3459.917	3463.195	3462.235
	10	1821.58	1823.373	1825.424	1823.769	1823.732	1823.699	1823.89	1826.243	1824.103	1824.848	1822.675	1822.179	1823.944	1824.598	1821.639	1825.766
10	1	3121.566	3121.436	3122.419	3123.941	3122.662	3123.408	3122.736	3122.229	3125.366	3124.473	3122.204	3123.704	3125.035	3123.791	3122.537	3124.241
	2	1779.86	1778.508	1780.32	1781.454	1780.51	1780.512	1781.793	1781.902	1781.122	1782.374	1781.871	1780.933	1781.002	1780.946	1780.107	1781.327
	3	2348.292	2350.154	2350.028	2348.608	2350.075	2349.618	2349.901	2352.94	2351.65	2352.243	2351.126	2351.07	2352.178	2351.473	2351.447	2351.905
	4	4011.765	4010.708	4012.009	4012.243	4012.024	4012.82	4011.999	4013.106	4013.321	4013.07	4013.452	4013.395	4013.246	4011.017	4013.645	4012.578
	5	7018.332	7016.6	7018.802	7020.395	7019.052	7018.817	7019.594	7019.27	7018.496	7018.313	7017.208	7019.206	7017.496	7019.534	7018.535	7019.683
	6	3765.221	3765.084	3766.547	3764.597	3764.329	3765.063	3766.101	3764.879	3766.703	3767.271	3766.753	3766.111	3763.796	3764.821	3767.254	3767.566
	7	2362.372	2364.721	2367.784	2365.264	2367.074	2365.771	2363.436	2366.816	2364.538	2366.595	2365.153	2365.298	2364.94	2364.768	2366.19	2365.625
	8	1547.35	1548.857	1550.523	1549.75	1550.339	1548.436	1550.902	1548.917	1550.333	1551.165	1549.411	1550.627	1548.317	1550.911	1550.535	1550.003
	9	3484.584	3486.2	3487.555	3486.881	3486.499	3486.478	3488.178	3486.453	3485.418	3487.519	3487.666	3485.411	3485.499	3486.559	3486.85	3486.332
	10	1840.401	1841.179	1839.402	1839.981	1840.279	1841.144	1840.132	1840.866	1842.475	1840.578	1839.741	1840.352	1840.648	1840.58	1839.715	1838.935
12	1	3134.44	3136.6	3136.688	3138.253	3139.368	3137.47	3137.295	3137.719	3137.197	3138.016	3137.149	3136.706	3136.78	3137.356	3137.76	3138.711
	2	1791.053	1790.06	1792.763	1791.236	1792.344	1790.594	1791.359	1791.675	1791.034	1790.569	1792.46	1792.275	1792.191	1790.135	1790.75	1791.757
	3	2359.776	2360.097	2361.101	2359.697	2361.586	2361.25	2362.324	2362.188	2362.023	2363.447	2360.705	2362.297	2360.749	2362.513	2362.617	2362.435
	4	4021.808	4020.994	4022.981	4022.752	4023.066	4021.606	4022.534	4024.395	4022.699	4023.96	4021.734	4021.895	4023.016	4022.624	4021.904	4023.955
	5	7030.88	7030.263	7032.821	7033.733	7033.572	7034.036	7033.335	7034.657	7033.916	7034.09	7035.272	7035.66	7034.446	7034.704	7034.734	7035.379
	6	3778.642	3777.926	3780.491	3780.75	3779.242	3780.082	3779.822	3779.739	3778.284	3777.89	3779.966	3780.146	3778.824	3780.96	3778.195	3779.976
	7	2379.771	2377.931	2379.529	2379.473	2378.306	2380.184	2380.402	2379.799	2379.072	2379.993	2378.707	2378.749	2381.309	2380.605	2380.009	2378.113
	8	1560.064	1560.418	1560.195	1560.076	1560.009	1559.923	1558.566	1560.584	1560.862	1560.471	1559.653	1560.866	1560.978	1560.251	1560.321	1559.517
	9	3499.543	3500.39	3502.058	3499.72	3501.449	3501.565	3500.338	3502.946	3500.596	3500.458	3499.782	3500.154	3500.313	3500.551	3500.373	3501.486
	10	1850.506	1851.965	1851.88	1851.197	1851.601	1850.497	1851.715	1851.901	1850.073	1851.44	1852.009	1852.028	1851.357	1850.521	1851.529	1850.593

Table 10

The average fitness values of Kapur’ entropy in comparison with other algorithms (maxFEs = 500 × m).

m	I	ABC	CS	GWO	HHO	MFO	SCA	SSA	WChOA	Case-1	Case-2	Case-3	Case-4	Case-5	Case-6	Case-7	Case-8
2	1	18.1368	18.1495	18.1504	18.1494	18.1632	18.1736	18.1471	18.2009	18.1946	18.2044	18.1758	18.1809	18.2032	18.1845	18.1916	18.2008
	2	18.0276	18.0032	18.0657	18.0593	18.0543	18.0177	18.047	18.0856	18.1052	18.0902	18.078	18.0693	18.0614	18.0614	18.0838	18.0919
	3	17.8067	17.8181	17.8302	17.8242	17.8262	17.8136	17.8316	17.8491	17.8478	17.8388	17.8436	17.8549	17.8541	17.8398	17.8438	17.8435
	4	17.3608	17.345	17.3875	17.3788	17.3903	17.3786	17.3965	17.4064	17.4134	17.4035	17.3928	17.4041	17.3985	17.3698	17.4153	17.4212
	5	17.3849	17.3863	17.406	17.4029	17.4068	17.4298	17.3984	17.4043	17.4041	17.4353	17.4279	17.4156	17.4117	17.4338	17.4406	17.4414
	6	17.8873	17.8691	17.8656	17.8723	17.8879	17.8913	17.8743	17.9317	17.928	17.9144	17.9088	17.9001	17.9154	17.9083	17.9171	17.9069
	7	18.1286	18.1174	18.1355	18.1574	18.15	18.1378	18.1427	18.1656	18.1567	18.1552	18.1458	18.162	18.1512	18.1626	18.1518	18.1656
	8	17.5944	17.6346	17.635	17.6231	17.626	17.6348	17.6321	17.6589	17.6475	17.6441	17.64	17.6536	17.646	17.6345	17.6525	17.6411
	9	18.3431	18.3341	18.3436	18.3457	18.339	18.3386	18.3415	18.3703	18.3682	18.3587	18.3698	18.367	18.3621	18.363	18.3683	18.3682
	10	18.0356	18.0645	18.0733	18.055	18.0808	18.0447	18.0507	18.1101	18.1507	18.0802	18.0885	18.0821	18.1104	18.0959	18.0825	18.0971
4	1	26.4539	26.4284	26.4731	26.4156	26.4775	26.4549	26.4864	26.5186	26.5047	26.5007	26.5128	26.5246	26.4933	26.4718	26.5715	26.5016
	2	26.3788	26.3047	26.3303	26.4531	26.4889	26.4297	26.4394	26.3991	26.4708	26.4446	26.5001	26.4214	26.4541	26.4777	26.4939	26.4452
	3	26.4427	26.4858	26.5249	26.4666	26.4531	26.4824	26.4223	26.5849	26.5481	26.5693	26.5682	26.5274	26.4975	26.5018	26.4828	26.5274
	4	25.6216	25.6072	25.6957	25.6062	25.7166	25.6618	25.6951	25.6017	25.6222	25.6223	25.6714	25.6784	25.6324	25.6215	25.652	25.6894
	5	25.6124	25.5358	25.6154	25.5895	25.5836	25.5274	25.5314	25.6095	25.6523	25.6135	25.5681	25.578	25.5379	25.5764	25.5522	25.5412
	6	26.1631	26.2231	26.3128	26.2759	26.2423	26.2248	26.2327	26.4739	26.3593	26.3372	26.3865	26.3561	26.2638	26.2528	26.3736	26.3258
	7	26.3218	26.3635	26.4778	26.3858	26.3946	26.4247	26.4358	26.5133	26.5152	26.4276	26.4399	26.4671	26.5647	26.4607	26.4813	26.5471
	8	25.9399	25.9303	26.0078	26.0783	26.063	26	26.0022	26.0315	26.0062	25.993	26.0578	26.0148	26.0233	26.0069	26.0273	26.0043
	9	26.7513	26.879	26.8558	26.8437	26.8399	26.8632	26.8239	26.8969	26.9082	26.9186	26.8429	26.8481	26.8754	26.8951	26.9046	26.875
	10	26.2302	26.2562	26.2434	26.2701	26.2801	26.3174	26.3167	26.3262	26.3428	26.301	26.2635	26.3042	26.3619	26.262	26.4016	26.324
6	1	33.4881	33.4789	33.4362	33.5008	33.455	33.5565	33.5867	33.6363	33.5198	33.608	33.5764	33.6229	33.5652	33.4892	33.5142	33.5267
	2	33.3792	33.3621	33.4635	33.564	33.4128	33.4581	33.4538	33.6097	33.4946	33.4979	33.5459	33.4593	33.5911	33.5293	33.6642	33.5527
	3	33.6629	33.5176	33.6249	33.7471	33.7072	33.5948	33.6511	33.7211	33.6852	33.6837	33.5923	33.623	33.6931	33.6465	33.7448	33.6681
	4	32.3543	32.4357	32.3928	32.4529	32.4432	32.5597	32.4003	32.3804	32.4711	32.5401	32.3967	32.515	32.5093	32.4202	32.4526	32.4715
	5	32.3666	32.2859	32.5567	32.5066	32.3881	32.1976	32.4537	32.3448	32.4425	32.3195	32.4851	32.3525	32.4157	32.4185	32.5036	32.3567
	6	33.3052	33.2489	33.4477	33.3065	33.409	33.4168	33.472	33.4858	33.4425	33.5066	33.425	33.5106	33.4404	33.2458	33.3576	33.3715
	7	33.3317	33.3949	33.4652	33.3841	33.435	33.4475	33.4994	33.4899	33.4827	33.4509	33.5129	33.4304	33.4929	33.4655	33.4772	33.5274
	8	32.8515	32.8558	33.074	32.9753	33.1121	32.9505	33.0525	33.0334	32.9492	32.9474	33.0794	32.9334	32.9713	32.951	32.9555	32.9727
	9	33.741	33.9225	33.9594	33.917	33.8178	33.9487	33.9579	34.0233	33.8619	33.9986	33.9097	33.9555	33.9271	33.9303	33.9552	33.8695
	10	33.2457	33.2254	33.3001	33.1889	33.3985	33.2375	33.3563	33.3365	33.3026	33.3389	33.2596	33.2962	33.3404	33.2061	33.3431	33.3367
8	1	39.6542	39.6046	39.7289	39.6203	39.7214	39.8244	39.7647	39.7776	39.8255	39.7105	39.69	39.7053	39.4951	39.508	39.6522	39.8859
	2	39.485	39.6532	39.6704	39.5308	39.7228	39.583	39.6784	39.5275	39.6197	39.4752	39.437	39.6264	39.6404	39.798	39.7203	39.6696
	3	39.7851	39.6724	39.9963	39.9955	39.9304	39.9017	40.0142	39.8485	40.0934	39.8218	39.945	39.9171	39.8935	39.8617	39.9746	39.9929
	4	38.1632	38.2725	38.4891	38.5041	38.5406	38.2242	38.4625	38.4729	38.3803	38.4304	38.394	38.4037	38.3038	38.4777	38.433	38.6501
	5	38.1944	38.1337	38.2023	38.3373	38.1977	38.3534	38.4304	38.2755	38.3406	38.3951	38.3723	38.3594	38.3531	38.3149	38.3107	38.4203
	6	39.3847	39.4799	39.6881	39.6054	39.4553	39.5818	39.611	39.6223	39.4673	39.4256	39.6146	39.7396	39.6074	39.4367	39.469	39.6697
	7	39.2328	39.51	39.6203	39.6535	39.5779	39.5167	39.5965	39.7636	39.6175	39.8066	39.6526	39.6249	39.6013	39.6118	39.8343	39.6979
	8	38.8108	38.9866	39.0569	39.1751	39.0468	39.1035	39.0333	39.113	38.9369	39.1589	38.9883	39.0233	39.1143	39.0571	39.1615	39.174
	9	39.9452	40.0001	40.2522	40.1259	40.0237	40.1405	40.1069	40.0891	40.0831	40.0731	40.1034	40.1618	40.0762	39.946	40.0216	40.1562
	10	39.2797	39.3825	39.5154	39.4897	39.3565	39.3714	39.3999	39.5358	39.3702	39.4511	39.3866	39.5534	39.6257	39.4653	39.6519	39.4061
10	1	44.8822	44.9997	45.1163	45.1197	45.1888	44.8987	45.1579	45.2963	45.0898	45.3264	45.0398	44.973	45.2482	45.1778	45.1222	45.1331
	2	44.8819	45.0948	45.129	45.179	45.2404	45.0329	45.1829	45.2981	45.2153	45.1134	45.1707	45.1432	45.0196	45.208	45.187	45.2354
	3	45.3413	45.18	45.3556	45.2069	45.2663	45.395	45.268	45.4197	45.3401	45.2721	45.2157	45.4078	45.3667	45.3441	45.5054	45.3096
	4	43.6398	43.4466	43.6701	43.6728	43.627	43.6456	43.7502	43.6682	43.8048	43.8289	43.6656	43.7246	43.6448	43.7593	43.7299	43.6969
	5	43.4534	43.5257	43.5974	43.4677	43.6007	43.687	43.6364	43.5042	43.5713	43.7285	43.6775	43.456	43.687	43.666	43.4754	43.5855
	6	45.0327	44.7496	45.1609	44.9878	44.9379	45.2286	45.1243	45.1544	45.1039	45.0628	44.9768	45.2049	44.8993	44.9117	45.0701	45.093
	7	45.0507	45.0746	45.153	45.1249	45.1072	45.2943	44.9489	45.0178	45.4016	45.2151	45.2035	45.1389	44.9937	45.1838	45.0679	45.0423
	8	44.3647	44.1358	44.4024	44.5367	44.4388	44.51	44.4058	44.3472	44.3247	44.4524	44.2478	44.3878	44.4963	44.7413	44.3989	44.3034
	9	45.3902	45.5319	45.7924	45.602	45.7649	45.5799	45.5285	45.5358	45.5386	45.5355	45.526	45.5584	45.6132	45.6222	45.6588	45.4983
	10	44.6622	44.616	44.81	44.9342	44.6821	45.0056	45.054	44.985	45.0997	44.8781	45.0954	45.0189	44.9863	45.0284	44.9154	44.9266
12	1	50.109	49.9629	50.0388	50.125	50.0391	50.1448	50.0414	49.9765	50.0937	50.0517	50.1259	49.9864	50.0199	50.197	49.889	50.0596
	2	49.9664	49.7223	49.9106	50.0551	49.8345	50.1219	49.934	50.2139	50.1742	50.1666	50.0424	50.3589	49.9034	50.1255	50.04	50.1106
	3	49.9935	50.118	50.417	50.166	50.2849	50.1546	50.279	50.225	50.2676	50.3137	50.1273	50.2022	50.0886	50.3255	50.2872	50.3428
	4	48.5401	48.1495	48.6916	48.5116	48.5148	48.5933	48.4214	48.5913	48.7052	48.6342	48.5408	48.5321	48.5956	48.5771	48.4713	48.6433
	5	48.3381	48.4181	48.5006	48.4079	48.4935	48.2765	48.4177	48.3548	48.3507	48.3383	48.5297	48.315	48.3769	48.2988	48.2412	48.32
	6	49.6864	49.9849	49.7779	50.0558	49.7857	49.9004	49.856	50.1134	49.9286	49.9738	50.0922	50.1288	49.9031	49.8098	49.9603	49.8359
	7	49.9754	49.716	50.0195	49.939	49.8953	49.8407	49.9006	50.0693	49.8907	49.89	49.9685	49.7241	49.9627	49.9248	49.7795	50.1329
	8	49.2573	48.9175	49.1518	49.3701	49.2196	49.0886	49.1584	49.2261	49.2652	49.1936	49.2525	49.218	49.1627	49.0611	49.4244	49.0695
	9	50.2301	50.2229	50.3961	50.4003	50.2643	50.4754	50.5842	50.5423	50.54	50.5797	50.3907	50.6312	50.528	50.6036	50.5787	50.7503
	10	49.6697	49.8004	50.0617	49.7402	49.8911	49.9707	49.9199	49.7979	49.6791	49.7734	49.8626	49.8722	49.7396	49.8031	49.6657	49.705

Table 11

The average fitness values of Kapur’ entropy in comparison with other algorithms (maxFEs = 1000 × m).

m	I	ABC	CS	GWO	HHO	MFO	SCA	SSA	WChOA	Case-1	Case-2	Case-3	Case-4	Case-5	Case-6	Case-7	Case-8
2	1	18.14767	18.13094	18.1706	18.1763	18.15902	18.15733	18.15842	18.19908	18.18993	18.21243	18.19995	18.1898	18.21218	18.19041	18.21445	18.18947
	2	18.03548	18.04131	18.06446	18.06117	18.0455	18.05398	18.03799	18.10483	18.07465	18.09165	18.05807	18.07813	18.07329	18.05629	18.10201	18.07103
	3	17.80331	17.80336	17.81195	17.82112	17.82046	17.83214	17.82238	17.85262	17.86083	17.8524	17.85095	17.84297	17.85441	17.85617	17.8447	17.85286
	4	17.35137	17.3603	17.36821	17.39731	17.37465	17.37207	17.38428	17.41553	17.40805	17.41799	17.40426	17.41843	17.40648	17.39298	17.41902	17.40711
	5	17.38128	17.37724	17.41204	17.41878	17.40728	17.41961	17.44221	17.42032	17.4561	17.43498	17.43644	17.43847	17.42513	17.44203	17.44836	17.4398
	6	17.85486	17.88621	17.87982	17.85654	17.88892	17.88606	17.87789	17.93224	17.92762	17.92564	17.9212	17.91601	17.92705	17.90224	17.93199	17.9277
	7	18.12772	18.1302	18.14183	18.13664	18.1557	18.12931	18.15508	18.15076	18.15666	18.17117	18.14625	18.15145	18.16522	18.1595	18.16667	18.1595
	8	17.63711	17.61779	17.63876	17.63061	17.62358	17.63015	17.62539	17.64223	17.65881	17.66365	17.63959	17.65032	17.63577	17.64002	17.6584	17.66009
	9	18.31646	18.33367	18.3554	18.34978	18.3481	18.3326	18.36228	18.3818	18.37625	18.37861	18.35392	18.35187	18.36521	18.36164	18.38287	18.36859
	10	17.97393	18.07125	18.08454	18.09579	18.06028	18.06278	18.06706	18.14419	18.11791	18.13396	18.0909	18.10422	18.09185	18.11055	18.13765	18.11907
4	1	26.45411	26.49577	26.46763	26.51827	26.52109	26.37835	26.44376	26.54081	26.46818	26.50042	26.47643	26.48671	26.49013	26.47642	26.49001	26.48164
	2	26.41446	26.38574	26.44305	26.47022	26.45463	26.38107	26.43198	26.42908	26.44045	26.44121	26.5088	26.46203	26.53786	26.43117	26.50607	26.4463
	3	26.46942	26.51004	26.52149	26.50308	26.49363	26.54053	26.53705	26.60181	26.56951	26.56363	26.5526	26.56726	26.5393	26.50209	26.64571	26.55301
	4	25.56042	25.57624	25.68935	25.65565	25.65457	25.59623	25.60746	25.69804	25.611	25.65392	25.71042	25.62291	25.59629	25.57642	25.679	25.68349
	5	25.55369	25.52038	25.61886	25.5496	25.58135	25.58361	25.62801	25.65459	25.61591	25.65066	25.59056	25.63827	25.54597	25.57407	25.59334	25.66629
	6	26.21546	26.33326	26.20094	26.33933	26.21618	26.21344	26.22824	26.45373	26.37172	26.37147	26.36239	26.33055	26.33268	26.31199	26.44998	26.396
	7	26.36208	26.34238	26.38897	26.48381	26.47322	26.46747	26.49504	26.57432	26.50681	26.50331	26.4587	26.45247	26.47756	26.45941	26.57866	26.57071
	8	25.93116	25.93602	26.00725	25.97441	25.97139	25.98878	25.9372	26.07068	26.00795	26.03467	26.04199	25.9663	26.02882	26.01842	26.01283	26.11938
	9	26.77231	26.69385	26.80852	26.8244	26.82993	26.79953	26.84699	26.93628	26.85897	26.85395	26.90196	26.85769	26.81279	26.86008	26.88192	26.81662
	10	26.25558	26.27341	26.25688	26.34134	26.34536	26.37695	26.35838	26.35346	26.33527	26.36767	26.35638	26.2694	26.27992	26.31037	26.38804	26.37315
6	1	33.4515	33.25011	33.55813	33.51956	33.50482	33.55953	33.59693	33.57825	33.54553	33.54547	33.53485	33.55454	33.60941	33.5472	33.52901	33.60797
	2	33.40765	33.45009	33.42256	33.52358	33.5153	33.46041	33.46082	33.44747	33.51116	33.46381	33.54713	33.43534	33.49627	33.37366	33.54357	33.51226
	3	33.58536	33.58532	33.65183	33.65621	33.72072	33.7373	33.66937	33.71241	33.66462	33.74637	33.66762	33.72286	33.69372	33.77121	33.75552	33.70939
	4	32.49754	32.34782	32.46224	32.38508	32.34306	32.58224	32.40728	32.53399	32.44133	32.43247	32.46814	32.46897	32.54414	32.40828	32.49744	32.53452
	5	32.36725	32.26664	32.44736	32.34947	32.387	32.32351	32.45205	32.47504	32.44833	32.3924	32.35943	32.43733	32.33557	32.43614	32.29951	32.40067
	6	33.2055	33.25917	33.37502	33.53466	33.37561	33.29754	33.3959	33.46313	33.45195	33.34519	33.38993	33.47251	33.39113	33.3158	33.43185	33.34999
	7	33.45087	33.39688	33.44662	33.47276	33.48429	33.40911	33.42005	33.53621	33.52511	33.47694	33.40846	33.40971	33.54687	33.47438	33.47561	33.47385
	8	32.9779	32.92014	32.98793	33.05843	32.96037	32.93879	33.07143	33.11083	33.02706	32.97006	33.01841	33.05868	33.0533	32.99018	32.9083	33.01988
	9	33.81094	33.74961	33.783	33.88444	33.95812	33.96249	33.91559	33.96463	33.95603	33.96877	33.92999	33.9384	34.05527	33.93774	33.94578	33.97326
	10	33.28266	33.21682	33.30882	33.24799	33.43251	33.37281	33.20692	33.34553	33.36816	33.32621	33.2521	33.31283	33.42095	33.36549	33.42146	33.36
8	1	39.61619	39.73444	39.80483	39.66656	39.66644	39.5157	39.63022	39.72881	39.80459	39.81017	39.61505	39.55615	39.59445	39.50606	39.69517	39.81722
	2	39.55134	39.43212	39.60872	39.69203	39.66785	39.70041	39.63695	39.65217	39.71897	39.62055	39.6188	39.67131	39.61876	39.59041	39.65323	39.64281
	3	39.86984	39.8125	39.67683	39.97458	39.87572	39.86595	39.77221	39.94652	39.68267	39.94661	39.96453	39.91404	39.72687	39.91332	39.88874	39.90031
	4	38.17033	38.132	38.44714	38.31689	38.34957	38.55186	38.40441	38.33903	38.3483	38.43328	38.4049	38.50501	38.46182	38.35493	38.43056	38.371
	5	38.23064	38.27734	38.41144	38.30036	38.41316	38.36922	38.18967	38.32183	38.35115	38.36294	38.30981	38.10742	38.37185	38.13765	38.36784	38.29102
	6	39.42494	39.40974	39.35673	39.59573	39.51797	39.59392	39.6625	39.6626	39.496	39.62821	39.57474	39.4446	39.63336	39.49125	39.55237	39.59789
	7	39.50614	39.50019	39.58542	39.51079	39.71337	39.53761	39.62085	39.66342	39.59722	39.5828	39.57987	39.60705	39.72523	39.46582	39.67508	39.69874
	8	38.93266	39.00621	39.09022	39.1185	39.08479	39.10099	38.95904	39.08682	39.17427	38.9643	39.09046	39.0037	39.15757	38.98593	39.07067	39.01538
	9	39.98336	40.01724	40.13166	40.03665	39.93635	40.07195	39.9658	40.06786	40.13363	40.16175	40.07089	40.12013	40.10346	40.01587	39.96663	40.1872
	10	39.39958	39.36145	39.50872	39.37392	39.52514	39.56563	39.51435	39.45128	39.48531	39.35781	39.47265	39.45034	39.40272	39.38493	39.31822	39.46983
10	1	44.92595	44.95815	45.21949	45.1099	45.09598	45.12465	45.0463	45.14486	45.12236	45.02631	45.09462	45.24978	45.19072	45.09997	45.07104	45.11422
	2	45.10123	44.90234	44.93527	45.12066	45.03765	45.26456	45.12176	44.92208	45.27619	44.99618	45.13704	44.95219	45.05646	45.11301	45.12826	44.94391
	3	45.33742	45.26626	45.24897	45.30273	45.37859	45.31484	45.04918	45.278	45.37698	45.40136	45.30858	45.39703	45.35297	45.58491	45.22899	45.33917
	4	43.50313	43.53852	43.763	43.70024	43.58653	43.73774	43.73979	43.7704	43.79688	43.76505	43.79787	43.90749	43.72222	43.97637	43.85429	43.73717
	5	43.63578	43.43302	43.6041	43.53632	43.59172	43.63244	43.64133	43.54618	43.64329	43.52081	43.68556	43.77432	43.69026	43.61376	43.61396	43.77545
	6	44.79116	44.69378	44.95352	45.14394	45.05991	45.10844	44.97361	45.12902	45.04685	45.12224	45.06401	45.15218	45.0628	45.04891	44.9441	44.88314
	7	44.72092	44.98178	45.13074	45.26071	44.96883	44.95387	45.18099	45.07318	45.06851	45.10503	44.99298	44.9644	45.01149	44.92416	45.09605	45.09465
	8	44.23667	44.24473	44.61695	44.54812	44.50249	44.29764	44.50204	44.51436	44.1498	44.54256	44.36598	44.61207	44.38549	44.444	44.30322	44.46693
	9	45.3229	45.57873	45.56259	45.61144	45.45204	45.53985	45.45206	45.48565	45.75407	45.56553	45.62937	45.6216	45.44957	45.49804	45.60375	45.50334
	10	44.92842	44.85422	44.84946	45.05544	44.96633	45.03428	44.84723	44.9047	44.7788	45.07599	44.86051	44.89302	44.86029	44.83001	44.98793	44.97868
12	1	49.78205	49.8361	50.04419	50.21734	49.99091	49.82745	50.01663	50.10595	49.79905	49.89117	49.88565	50.06645	50.23328	50.13346	49.95705	50.0251
	2	49.83734	49.95527	49.98204	49.94922	50.08179	50.00675	50.01161	50.21002	50.16448	50.03838	50.02838	50.14471	49.76841	50.07433	49.97784	49.89936
	3	50.06286	49.94907	50.37016	50.19918	50.39154	50.15233	50.24491	50.15173	50.36946	50.46397	50.20066	50.4642	50.24402	50.28325	50.2416	50.18644
	4	48.46481	48.4162	48.38105	48.59382	48.56587	48.41846	48.29729	48.46708	48.541	48.566	48.43065	48.64118	48.53838	48.45138	48.42021	48.5094
	5	48.29496	48.34059	48.51369	48.48596	48.43813	48.28108	48.47765	48.18402	48.38687	48.33268	48.24718	48.55036	48.48688	48.42496	48.28621	48.4933
	6	49.67198	49.90532	49.8098	49.8688	49.98022	50.14472	49.9891	49.90174	49.97372	49.9395	49.96144	49.8182	50.10642	50.10527	49.81942	50.13004
	7	49.70595	49.89153	49.69365	50.0052	49.87504	49.75096	49.91409	50.03998	49.90816	49.9625	49.86982	49.85514	49.94627	50.00567	50.03323	49.95783
	8	49.03439	48.85467	49.17105	49.20734	49.4654	49.03844	49.08351	49.40737	49.23467	49.35353	49.52265	49.18356	49.19272	49.36735	49.2972	49.24936
	9	50.1249	50.47053	50.51145	50.33997	50.38923	50.70062	50.75394	50.66751	50.392	50.63045	50.47724	50.54064	50.4711	50.44517	50.36818	50.20471
	10	49.67426	49.66839	50.04147	49.81449	49.98884	50.04108	49.90323	49.92433	49.9133	49.75571	49.98039	49.77503	49.8586	49.76656	49.83744	49.85563

Table 12

Top ranking algorithms according to the Friedman scores.

		Threshold Levels (m)
Fitness Function	maxFEs	2	4	6	8	10	12
Otsu’s method	500	Case-2	Case-2	Case-7	Case-8	Case-7	Case-8
Otsu’s method	1000	WChOA	Case-1	Case-2	Case-8	Case-1	Case-8
Kapur’s entropy	500	Case-8	Case-2	Case-7	Case-8	Case-2	Case-1
	1000	Case-2	Case-7	Case-8	Case-2	Case-4	Case-4

Table 13

Computation times (in seconds) of the base algorithm and FDB variants for m = 2, 6, 12 and maxFEs = 500d (1 image/1 run).

Algorithm	m = 2	m = 6	m = 12
WChOA	0.746804	3.091970	8.489719
Case-1	0.728755	2.999997	8.689171
Case-2	0.740006	2.986108	8.759747
Case-3	0.733823	2.988904	8.453628
Case-4	0.769428	3.010361	8.455266
Case-5	0.734451	3.002919	8.770119
Case-6	0.723005	2.986360	8.816571
Case-7	0.727183	3.009741	8.463995
Case-8	0.725269	3.007105	8.587301

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Improved Weighted Chimp Optimization Algorithm Based on Fitness–Distance Balance for Multilevel Thresholding Image Segmentation

Content area

Abstract

Full text

1. Introduction

2. Related Work

3. Materials and Methods

3.1. Multilevel Thresholding Problem

3.1.1. Kapur’s Entropy-Based Thresholding

3.1.2. Between-Class Variance-Based Thresholding (Otsu’s Method)

3.2. Proposed Method

3.2.1. WChOA Algorithm

3.2.2. Fitness–Distance Balance Selection

3.2.3. Proposed WChOA-FDB Scheme

4. Experimental Results

4.1. Experimental Settings

4.2. Testing and Analyzing the Performance of the Proposed WChOA-FDB Algorithm

4.3. Image Segmentation Performance of the Proposed WChOA-FDB Method

4.3.1. Statistical Analysis on Test Images

4.3.2. Performance Metrics

4.3.3. Performance Evaluation Using the Fitness Function

4.3.4. Performance Evaluation Using PSNR, SSIM and FSIM

5. Discussion and Conclusions