Modified Remora Optimization Algorithm for Global Optimization and Multilevel Thresholding Image Segmentation

Abstract

Image segmentation is a key stage in image processing because it simplifies the representation of the image and facilitates subsequent analysis. The multi-level thresholding image segmentation technique is considered one of the most popular methods because it is efficient and straightforward. Many relative works use meta-heuristic algorithms (MAs) to determine threshold values, but they have issues such as poor convergence accuracy and stagnation into local optimal solutions. Therefore, to alleviate these shortcomings, in this paper, we present a modified remora optimization algorithm (MROA) for global optimization and image segmentation tasks. We used Brownian motion to promote the exploration ability of ROA and provide a greater opportunity to find the optimal solution. Second, lens opposition-based learning is introduced to enhance the ability of search agents to jump out of the local optimal solution. To substantiate the performance of MROA, we first used 23 benchmark functions to evaluate the performance. We compared it with seven well-known algorithms regarding optimization accuracy, convergence speed, and significant difference. Subsequently, we tested the segmentation quality of MORA on eight grayscale images with cross-entropy as the objective function. The experimental metrics include peak signal-to-noise ratio (PSNR), structure similarity (SSIM), and feature similarity (FSIM). A series of experimental results have proved that the MROA has significant advantages among the compared algorithms. Consequently, the proposed MROA is a promising method for global optimization problems and image segmentation.

1. Introduction

Image segmentation technique is a primary step in computer vision and pattern recognition for pre-processing and analyzing images in the fields of remote sensing, medicine, etc. [1,2,3,4,5]. This technique divides an image into several homogeneous regions or segments with similar characteristics according to features, color, texture, and contrast [6,7]. In the literature, there exist four common types of image segmentation techniques, which can be categorized into (1) region-based techniques, (2) graph-based techniques, (3) clustering-based techniques, and (4) thresholding-based techniques [8]. Among them, the thresholding-based method has gained more attention from researchers due to its efficiency and ease of implementation. The thresholding-based technique can be categorized into bi-level and multi-level thresholding [9]. As the name suggests, the bi-level thresholding uses a single threshold value to segment an image into two homogeneous foreground and background areas. By contrast, multi-level thresholding divides an image into more than two regions based on pixel intensities [10,11,12]. Thus, the multi-level thresholding technique has widely applied in various fields by researchers. However, selecting suitable threshold values is still the most challenging problem and requires further research.

To determine optimal threshold values to segment a given image into several regions, there are two approaches commonly used, which are Otsu’s method (by maximizing between-class variance) and Kapur’s entropy (by maximizing the entropy of the classes) [13]. These methods are suitable for determining a single threshold value. However, when extended into multi-level thresholding, the fatal shortcomings are the computational time and complexity. Therefore, various nature-inspired MAs were proposed in the literature to handle these problems and gained excellent performance [14,15,16].

MAs are considered stochastic algorithms that use randomly generated search agents and specific operators to determine optimal solutions in the search space instead of gradient information. These operators are inspired by nature, such as swarm behavior, social behavior, physics, evolutionary rules, etc. [17,18,19,20,21,22,23,24,25]. Over the last years, various MAs have been proposed and applied in complex real-world problems. Such MAs can be categorized into three main categories: (1) swarm intelligence-based methods, (2) nature evolution-based methods, (3) physics-based methods. The first category mainly simulates the swarm behavior of biological entities, such as birds, slime mould, or the grey wolf. Particle Swarm Optimization (PSO) [26] is one of the most popular MAs inspired by the foraging behavior of birds. Grey Wolf Optimizer (GWO) [27] simulates the grey wolf’s hunting behavior and leadership relationship. Slime Mould Algorithm (SMA) [28] imitates the slime mould’s contraction and oscillation mode during foraging. Other popular MAs in the first category include the Whale Optimization Algorithm (WOA) [29], Cuckoo Search Algorithm (CSA) [30], Salp Swarm Algorithm (SSA) [31], Ant Colony Optimization (ACO) [32], Harris Hawks Optimization (HHO) [33], Moth Flame Optimization (MFO) [34], and Aquila Optimizer (AO) [35]. The second category mainly imitates the evolution process in nature. Genetic Algorithm (GA) [36] is the most famous one developed by the Darwinian evolution law. Some other popular algorithms include Evolutionary Strategy (ES) [37], Differential Evolution (DE) [38], Genetic Programming (GP) [39], Bio-geography-Based Optimizer (BBO) [40], and Evolution Strategy (ES) [41]. The physics-based method simulates the physical laws of the universe. One of the most popular algorithms in this category is Simulated Annealing (SA) [42], which mimics the principle of simulated annealing. It starts from a higher initial temperature, then reduces with the decrease in temperature parameters. Established by this principle, other algorithms include Gravity Search Algorithm (GSA) [43], Gradient-based method (GBO) [44], Sine Cosine Algorithm (SCA) [45], Golden Sine Algorithm (Gold-SA) [46], Arithmetic Optimization Algorithm (AOA) [47], Henry Gas Solubility Optimization (HGSO) [48], Atom Search Optimization (ASO) [49], Central Force Optimization (CFO) [50], and Multi-Verse Optimizer (MVO) [51]. Although many nature-inspired MAs were proposed to solve numerical optimization problems, as real-world problems grow in complexity and difficulty, we need to present more useful MAs to handle them.

Jia et al. [52] presented a new nature-inspired meta-heuristic, namely the Remora Optimization Algorithm (ROA). The main inspiration of ROA is the parasitic behavior of remora during foraging in the ocean. ROA integrates different position update rules based on different hosts. As one of the intelligent creatures in the world, remora usually adsorbs on different hosts to achieve movement and foraging, so the location update rules are the same as those of the host. Moreover, remora also attempt to move by themselves, which is used to determine whether to change hosts. The experimental results demonstrated that the ROA is superior to other algorithms in terms of optimization accuracy and convergence speed. Zheng et al. [53] proposed an improved ROA, namely IROA, for solving global optimization problems. In this work, a mechanism called autonomous foraging was proposed. This mechanism allows each remora a small chance to find the food randomly or according to the current food position, which was used to improve the ROA’s optimization accuracy. Then, the IROA’s performance was evaluated on function optimization and five constrained engineering design problems. The experimental results demonstrated that IROA has a superior performance to other selected MAs in terms of optimization accuracy and convergence speed. However, the No-Free-Lunch (NFL) theorem [54] stated that no unique MAs are available to solve any optimization problems, which encourages us to develop more efficient methods and apply them in various fields such as multi-level thresholding problems.

This paper presents a modified Remora Optimization Algorithm called MROA for global optimization and multi-level thresholding image segmentation tasks. The motivations of this work are to alleviate ROA’s weaknesses and enhance its performance in solving image segmentation tasks. The two major improvements are contained in this work. First, Brownian motion is used to enhance the ROA’s exploration ability and provide it with a high level of opportunity to find the optimal or near-optimal solution in the search space. Second, the lens opposition-based learning strategy is used to improve the exploitation ability of ROA and help search agents to jump out the local optimal solution. We first used 23 benchmark functions in the experimental design including unimodal and multimodal types to validate its performance.

Furthermore, we selected eight grayscale images as the benchmark images, and cross-entropy is employed as the objective function to evaluate the segmentation quality of search agents. We compared MROA with other well-known algorithms and its original in all the tests such as ROA, RSA, AOA, AO, SSA, SCA, and GWO. The experimental results reveal that MROA significantly improved compared with these others in segmentation precision and convergence speed. Overall, it can be proved that MROA is an effective method for global optimization and multi-level thresholding.

The main contributions of this paper can be summarized as follows:

• A new modified ROA, namely MROA, is first proposed based on BM and LOBL strategies.
• The optimization performance of the MROA is evaluated on 23 benchmark functions.
• MROA is applied for thresholding segmentation using the cross-entropy method.
• Validate segmentation quality of MROA in terms of PSNR, SSIM, FSIM, and statistical test.
• The performance of MROA is compared with seven well-known MAs.

The rest of this paper is organized as follows. Section 2 introduces the related work of multi-level thresholding image segmentation using MAs. Section 3 briefly describes the background knowledge. Section 4 presents the proposed MROA. The experimental results are introduced and analyzed in Section 5 and Section 6. Finally, Section 7 concludes this paper and gives future research directions.

2. Related Works

The MAs-based multi-level thresholding segmentation technique has gained more attention from scholars because of its reliable performance, small computational cost, and ease of implementation. In this field, determining the threshold values is the core of the entire segmentation process in the multi-level thresholding technique. In many application scenarios, many scholars use the exhaustive method to determine the optimal threshold values. However, this method has fatal disadvantages, such as complexity, time-consuming computation, and poor threshold values. Therefore, to alleviate these shortcomings, many works show the efficiency and performance of MAs in obtaining optimal threshold values; the following are a few outstanding research works. In [55], Jia et al. presented an efficient satellite image segmentation approach, called DHHO/M, based on dynamic HHO with a mutation mechanism. This work introduces the dynamic control parameter mechanism and mutation operators to improve HHO performance and avoid falling into local optimal solutions. Kapur’s entropy, Otsu between-class variance, and Tsallis entropy were employed to determine optimal threshold values. In the experiments, eight color satellite images and four oil pollution images validate DHHO/M’s performance. The experimental results demonstrated that the DHHO/M provided competitive performance in fitness evaluation, segmentation quality, and statistical tests compared with eight advanced thresholding techniques.

Ewees et al. [56] proposed a new multi-level thresholding method, using modified artificial ecosystem-based optimization (AEO), namely AEODE. The AEODE integrated differential evolution as a local search operator to overcome AEO’s shortcomings. In this way, the AEODE was used to determine a set of optimal threshold values and fuzzy entropy as the objective function. Then, the AEODE was accessed by different grayscale images at six various threshold values. The experimental results show that AEODE outperforms others, including original AEO and DE, in terms of fitness values, PSNR, SSIM, and FSIM.

In [57], an improved version of MPA with an opposition-based learning strategy called MPA-OBL was proposed to determine optimal threshold values at different thresholds. The Otsu between-class variance and Kapur’s were employed as the objective function in this work. The proposed method was evaluated on CEC 2020 test suite and several benchmark images. Compared with ELSHADE-SPACMA-OBL, CMA-ES-OBL, DE-OBL, HHO-OBL, SCA-OBL, SSA-OBL, and standard MPA, the proposed technique MPA-OBL produced better results in different evaluation measurements.

Su et al. [58] presented a variant ABC, namely CCABC, which introduced vertical search and horizontal search mechanisms to improve ABC’s optimization performance. Furthermore, the proposed method, CCABC, was used to find the appropriate threshold values in COVID-19 X-ray images based on Kapur’s entropy. The proposed method was compared with a set of advanced and classical MAs. The evaluation results show that the CCABC is an excellent method in most quality measurements.

Liu et al. [59] proposed a novel multi-level thresholding segmentation technique based on a modified evolution algorithm, called MDE, and applied it to a breast cancer image segmentation task. In this work, the slime mold foraging behavior is introduced to enhance DE’s optimization accuracy and convergence speed. Furthermore, three image quality metrics, including PSNR, SSIM, and FSIM, are used to evaluate its segmentation results to show its segmentation performance.

Li et al. [60] proposed a partitioned and cooperative quantum-behaved particle swarm optimization namely SCQPSO for processing stomach CT images. In this work, SCQPSO was used to determine the optimal threshold values according to Otsu’s method. Additionally, the inter-class variance of Otsu and its variance was used to show SCQPSO’s segmentation quality.

In the literature, we find that more and more scholars presented variants of the multi-level thresholding image segmentation method using MAs to determine optimal threshold values instead of the traditional exhaustive method. Consequently, we confirm that the meta-heuristic-based multi-level thresholding method is a promising work and interesting theories need further research. However, as real-world problems grow in complexity and difficulty, these methods will inevitably fail to find the optimal threshold values during segmentation. Therefore, image segmentation using MAs has more room for improvement to obtain better segmentation quality.

3. Background Knowledge

3.1. Remora Optimization Algorithm (ROA)

ROA [52] is a well-known bionics-based meta-heuristic algorithm inspired by the parasitic behavior of remora during foraging in the ocean. Unlike other fishes, remora usually attach to other hosts (humpback whales or sailfishes) to complete long and short-distance movement in the ocean. Like other MAs, ROA also has three different phases: initialization, exploration, and exploitation.

3.1.1. Initialization

Like other various meta-heuristic algorithms, ROA initializes the search agents using a random approach in the search space, which is calculated by:

$$X_i=lb+rand\times (ub-lb); i\in \left\{1,2, \dots ,N\right\}$$

(1)

where rand denotes a random variable between [0, 1]. ub and lb indicate the search space’s upper and lower bounds. i represents the number of Remora, and N denotes population size.

3.1.2. Free Travel (Exploration Phase)

ROA achieves long-distance and small global movements, respectively, in the search space based on the free travel method. This method includes two main sub-methods: SFO strategy and experience attack.

• SFO strategy

Sailfish: one of the fastest fish in the ocean. Thus, remora attaches to them to achieve quick long-distance movement, and its position update is the same as its host, which is calculated by:

$$X(t+1)=X_{best}(t)-(rand\times (\frac{X_{best}(t)+X_{rand}(t)}{2})-X_{rand}(t))$$

(2)

where X_best(t) indicates global best position of remora and X_rand(t) represents the random position.

• Experience attack

To determine whether the remora changes the host, it needs to take a small step around it. This activity is considered an experience attack and the formula is expressed by:

$$X_{att}(t+1)=X(t)+randn\times (X(t)-X_{pre}(t))$$

(3)

where X_pre(t) represents the position of the previous generation. randn indicates typically distributed random numbers between 0 to 1.

3.1.3. Eat Thoughtfully (Exploitation Phase)

• WOA strategy

When remora is attached to the humpback whale, its attack strategy is the same as the bubble-net attack strategy in WOA. The mathematical formula can be described as follows:

$$X(t+1)=D\times e^a\times \cos (2\pi \alpha )+X(t)$$

(4)

$$\alpha =rand\times (a-1)+1$$

(5)

$$a=-(1+\frac{t}{T})$$

(6)

$$D=\left|X_{best}(t)-X(t)\right|$$

(7)

where D denotes the distance between the best position and current position. α is a random value between [−1, 1].

• Host feeding

This method is a subdivision in the exploitation process. In this phase, remora moving around the host can be thought of as a small step, which can be calculated as follows:

$$X(t+1)=X(t)+A$$

(8)

$$A=B\times (X(t)-C\times X_{best}(t))$$

(9)

$$B=2\times V\times rand-V$$

(10)

$$V=2\times (1-\frac{t}{T})$$

(11)

where A denotes a small step movement of remora, which is related to the volume of host and remora. A remora factor (C) is utilized to limit the position of remora to distinguish between the host and remora.

Figure 1 vividly illustrates the main process of ROA, and the pseudo-code of ROA is presented in Algorithm 1.

Algorithm 1 the pseudo-code of ROA

Inputs: N and T Outputs: best solution Initialize the positions of remora Xi (i = 1, 2, …, N); While (t ≤ T) Check if any search agent goes beyond the search space and amend it; Calculate the fitness of each remora; For each remora indexed by i H(i) = round(rand); If H(i) == 0 then Using Equation (4) to update the position of attached whales; Elseif H(i) == 1 then Using Equation (2) to update the position of attached sailfishes; End if Make a one-step prediction by experience attack (Equation (3)); If f(Xatt) < f(X) then X = Xatt; H(i) = round(rand); Else Using Equation (8) to update the position of search agents; End if End for End While

Figure 1. The different process of ROA.

3.2. Brownian Motion

Brownian motion (BM) is a stochastic process in which step length is drawn from the probability function defined by a normal distribution with zero mean (μ = 0) and unit variance (σ² = 1) [61,62]. The probability density function at point x for BM is calculated via:

$$f_B(x;\mu ,\sigma )=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{{(x-\mu )}^2}{2{\sigma }^2}\right)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{x^2}{2}\right)$$

(12)

where x indicates a point following this motion and the distribution, 2D, and 3D trajectory of BM as shown in Figure 2. As the related figures show, we can see that BM can over areas of the domain with more uniform and controlled steps in the search space. On the other hand, BM’s trajectory can trace and explore distant areas of the neighborhood.

Figure 2. Brownian distribution, 2D Brownian trajectory, and 3D Brownian trajectory.

3.3. Lens Opposition-Based Learning

Lens opposition-based learning (LOBL) [63] is a novel strategy that integrates both opposition-based learning (OBL) [64] strategy and lens imaging principle to enhance the searchability of MAs. The LOBL strategy shows more effective performance during the optimization process than OBL when finding optimal or near-optimal solutions in the search space. The mathematical principle of LOBL is described as follows:

Lens imaging theory illustrates that when the distance from the object to the lens exceeds two times the focal length, an inverted and constricted real image will be formed between 1–2 times the focal length on the other side of the lens. As shown in Figure 3, where O represents the midpoint of the interval [lb, ub] and the y-axis is treated as a convex lens. In addition, an object of height h is located at point x, which is twice the lens’s focal length. Through lens imaging, the coordinates of the image vertices become (x*, h*). The formula is listed as follows:

$$\frac{(lb+ub)/2-x}{x^{*}-(lb+ub)/2}=\frac{h}{h^{*}}$$

(13)

Figure 3. Diagram of LOBL.

Let k = h/h*, Equation (14) can be simplified as follows:

$$x^{*}=\frac{lb+ub}{2}+\frac{lb+ub}{2k}-\frac{x}{k}$$

(14)

In general, the optimization tasks are multi-dimensional, so the Equation (15) can be extended as follows:

$$x_i{}^{*}=\frac{l{b}_i+u{b}_i}{2}+\frac{l{b}_i+u{b}_i}{2k}-\frac{x_i}{k}$$

(15)

where $x_i{}^{*}$ denotes the opposite point of $x_i$ in i-th dimension. Interestingly, if k = 1, Equation (16) is the same as the OBL strategy. Therefore, the LOBL can be considered as a variant of OBL. The difference is that LOBL used an adjusting parameter k to realize dynamic search behavior when solving optimization tasks, which further improves the probability of the algorithm escaping from the local optimal solution.

3.4. Cross Entropy

In 1968, cross-entropy was proposed by Kullback [65]. As an essential concept in Shannon’s information theory, cross-entropy is mainly used to measure the difference in information between two probability distributions [66,67]. Let P = {p₁, p₂, …, p_n} and Q = {q₁, q₂, …, q_n} be two probability distributions defined over the same set of values. The cross-entropy between P and Q can be calculated as follows:

$$D\left(P, Q\right)={\displaystyle \sum _{i=1}^N{p}_i\log \frac{p_i}{q_i}}$$

(16)

The minimum cross-entropy algorithm determines the threshold value by minimizing the cross-entropy between the original image and the thresholded image [68]. The lower the cross-entropy value, the lower the uncertainty and the greater the homogeneity. Let the original image be I, and h(i), i = 1, 2, …, L, is the corresponding histogram with L being the number of grey levels. Then the thresholded image I_th is created by the threshold value th using the following equations:

$$I_{th}=\{\begin{array}{l}\mu (1, th),\hspace{1em}\hspace{1em} if I(x,y)<th\\ \mu (th, L+1), if I(x,y)\ge th\end{array}$$

(17)

$$\mu (a,b)=\frac{{\displaystyle \sum _{i=a}^{b-1}ih(i)}}{{\displaystyle \sum _{i=a}^{b-1}h(i)}}$$

(18)

The thresholded image is generated based on Equation (17), then we can calculate the cross-entropy by rewriting Equation (17) as an objective function (also called fitness), which is listed below:

$$f_{cross}(th)={\displaystyle \sum _{i=1}^{th-1}ih(i)\log \left(\frac{i}{\mu (1, th)}\right)}+{\displaystyle \sum _{i=th}^{L}ih(i)\log \left(\frac{i}{\mu (th, L+1)}\right)}$$

(19)

The above objective function considers a single threshold value to segment a given image, i.e., bi-level thresholding, it also can be extended to a multi-level approach for image segmentation, called multi-level thresholding image segmentation. Thus, the Equation (19) can be expressed as:

$$f_{cross}(th)={\displaystyle \sum _{i=1}^{L}ih(i)\log (i)-{\displaystyle \sum _{i=1}^{nt}{H}_i}}$$

(20)

where th = [th₁, th₂, …, th_nt] indicates an array containing nt different threshold values. H_i is defined as follows:

$$H_1={\displaystyle \sum _{i=1}^{t{h}_1-1}ih(i)\log (\mu (1, t{h}_1))}$$

(21)

$$H_k={\displaystyle \sum _{i=t{h}_{k-1}}^{t{h}_k-1}ih(i)\log (\mu (t{h}_{k-1}, t{h}_k))}, 1<k<nt$$

(22)

$$H_{nt}={\displaystyle \sum _{i=t{h}_{nt}}^{L}ih(i)\log (\mu (t{h}_{nt}, L+1))}$$

(23)

4. The Proposed Algorithm

4.1. The Details of MROA

Although ROA is easy to implement, suitable for solving optimization tasks widely, and provides more reliable results than other Mas, it has insufficient searchability (exploration and exploitation) in solving complex optimization problems; for example, it easily stagnates into optimal local solutions and has a slow convergence speed. Motivated by these considerations, this paper proposes a modified ROA algorithm called MROA for global optimization and multi-level thresholding image segmentation. There are two major strategies for modifying. First, Brownian motion is introduced to improve the exploration ability and provide a greater opportunity to find the optimal or near-optimal solution in the search space. Second, LOBL is used to improve the exploitation ability and accelerate the convergence rate of ROA. Therefore, the exploration phase of ROA is modified using Brownian motion, which is expressed as:

$$X(t+1)=\mathrm{Brownian}\times X_{best}(t)-(rand\times (\frac{X_{best}(t)+X_{rand}(t)}{2})-X_{rand}(t))$$

(24)

where Brownian indicates Brownian motion.

The pseudo-code of MROA is presented in Algorithm 2, and Figure 4 illustrates the flowchart of MROA.

Algorithm 2 the pseudo-code of MROA

Inputs: N and T Outputs: best solution Initialize the positions of remora Xi (i = 1,2, …, N); While (t ≤ T)   Check if any search agent goes beyond the search space and amend it;   Apply LOBL strategy to generate lens-opposite solution by Equation (15);   Evaluate candidate solution and select the best position by greedy strategy.   For each remora indexed by i     If H(i) == 0 then       Using Equation (4) to update the position of attached whales;     Elseif H(i) == 1 then       Using Equation (24) to update the position of attached sailfishes;     End if     If f(Xatt) < f(X) then       X = Xatt;       H(i) = round(rand);     Else       Using Equation (8) to update the position of search agents;     End if   End for End While

Figure 4. Flowchart of the proposed MROA.

4.2. The Proposed MROA for Solving Multi-Level Thresholding Image Segmentation Task

In this subsection, the MROA is employed to solve the multi-level thresholding image segmentation task and cross-entropy is used to determine optimal threshold values. We present the different steps of this method as below:

Step 1: Read image I and convert it to grayscale I_g, calculate the histogram h of I_g.

Step 2: Initialize the parameters of MROA: N, T, C, k, and random array H.

Step 3: Initialize the location of search agents with N population size and nTh dimensions.

Step 4: If any search agent goes beyond the search space and amend it.

Step 5: Evaluate the cross-entropy with Equation (20) for each search agent.

Step 6: Apply LOBL strategy to generate lens opposite solution and select the best one between the original and its opposite solution according to the fitness value.

Step 7: Select the appropriate host to update the location of the search agent according to the value of the H.

Step 8: The t index is increased in 1, if the stop criteria (t ≥ T) are satisfied then output the best threshold values, otherwise jump to Step 5.

Step 9: Generate the segmented image Seg with the best threshold values obtained by MROA.

4.3. Computation Complexity of MROA

In the initialization phase, MROA produces the search agents randomly in the search space, so the computational complexity of this phase is O(N × D), where N denotes the number of population and D denotes the dimension size. Afterward, MROA evaluates each individual’s fitness during the whole iteration with the complexity O(T × N × D), where T indicates the number of iterations. Finally, for position updating, the complexity is summarized as O(T × N × D). In summary, the total computational complexity of MROA is O(T × N × D), which is the same as the original.

5. Experimental Results for Global Optimization

This section evaluates the optimization performance of the proposed MROA using 23 benchmark functions. First, the definitions of the 23 benchmark functions are introduced. Second, the experimental setup and comparison group including other well-known MAs are described in detail. Finally, the experimental results are analyzed and discussed.

5.1. Definition of 23 Benchmark Functions

To evaluate the searchability of the proposed MROA, 23 benchmark functions are introduced in this paper [15]. These functions are categorized into: (1) unimodal functions (F1–F7), (2) multimodal functions (F8–F14), and (3) fixed-dimension multimodal functions. These functions are defined in Table 1, where D, UM, and MM indicate the dimension, unimodal, and multimodal functions, respectively. Figure 5 shows the visualization of these functions. We can see that the unimodal functions have only one global optimal solution, which is available to evaluate the exploitation ability of MAs. By contrast, multimodal and fixed-dimension multimodal functions have many local optimal and only one global optimal solution, which is available to evaluate the exploration and ability to escape from the local optimal solution of MAs.

Table 1. Definition of 23 benchmark functions.

No	Name	D	Range	Type	fmin
F1	Sphere	30	[−100, 100]	UM	0
F2	Schwefel 2.22	30	[−10, 10]	UM	0
F3	Schwefel 1.2	30	[−100, 100]	UM	0
F4	Schwefel 2.21	30	[−100, 100]	UM	0
F5	Rosenbrock	30	[−30, 30]	UM	0
F6	Step	30	[−100, 100]	UM	0
F7	Quartic	30	[−1.28, 1.28]	UM	0
F8	Schwefel	30	[−500, 500]	MM	−12,569.487
F9	Rastrigin	30	[−5.12, 5.12]	MM	0
F10	Ackley	30	[−32, 32]	MM	0
F11	Griewank	30	[−600, 600]	MM	0
F12	Penalized	30	[−50, 50]	MM	0
F13	Penalized 2	30	[−50, 50]	MM	0
F14	Foxholes	2	[−65.536, 65.536]	MM	0.998004
F15	Kowalik	4	[−5, −5]	MM	0.0003075
F16	Six-hump camel-back	2	[−5, −5]	MM	−1.0316285
F17	Branin	2	[−5, −5]	MM	0.398
F18	Goldstein-Price	2	[−2, 2]	MM	3
F19	Hartman 3	3	[−1, 2]	MM	−3.862782
F20	Hartman 6	6	[0, 1]	MM	−3.32236
F21	Shekel 5	4	[0, 10]	MM	−10.1532
F22	Shekel 7	4	[0, 10]	MM	−10.4029
F23	Shekel 10	4	[0, 10]	MM	−10.5364

Figure 5. View of 23 benchmark functions.

5.2. Experimental Setup

As stated above, the 23 benchmark functions are utilized to evaluate the optimization performance of the proposed work. To show the representativeness of experimental results, the proposed algorithm MROA is compared with the original ROA [52] and other well-known MAs, including RSA [69], AOA [47], AO [35], SSA [31], SCA [45], and GWO [27]. For fair evaluation, we set the number of maximum iteration T = 500, population size N = 30, and dimension size D = 30, respectively. Moreover, all the tests are run 30 times independently. The best results are highlighted in bold. Table 2 represents the parameter setting of each algorithm and the details of these algorithms are listed as follows:

• ROA [52]: Simulates remora’s parasitic behavior on hosts, such as whales and swordfish, to finish the predation behavior.
• RSA [69]: mimics the encircling and hunting mechanism of a crocodile swarm during foraging.
• AOA [47]: simulates the behavior of arithmetic operators in nature, which include multiplication (×), division (÷), addition (+), subtraction (−).
• AO [35]: inspired by the different hunting behavior of Aquila hawks in nature.
• SSA [31]: simulates the swarming behavior of salps during moving and foraging in the ocean.
• SCA [45]: simulates the common mathematical models, sine and cosine functions.
• GWO [27]: mimics the hunting strategy and leadership hierarchy of gray wolves.

Table 2. Parameter settings for each algorithm.

Algorithm	Parameters
MROA	c = 0.1; k = 10,000
ROA [52]	c = 0.1
RSA [69]	α = 0.1; β = 0.1
AOA [47]	α = 5; μ = 0.5;
AO [35]	U = 0.00565; c = 10; ω = 0.005; α = 0.1; δ = 0.1;
SSA [31]	c1 = [1, 0]; c2 ∈ [0, 1]; c3 ∈ [0, 1]
SCA [45]	A = 2
GWO [27]	a = [2, 0]

Table 2. Parameter settings for each algorithm.

Algorithm	Parameters
MROA	c = 0.1; k = 10,000
ROA [52]	c = 0.1
RSA [69]	α = 0.1; β = 0.1
AOA [47]	α = 5; μ = 0.5;
AO [35]	U = 0.00565; c = 10; ω = 0.005; α = 0.1; δ = 0.1;
SSA [31]	c1 = [1, 0]; c2 ∈ [0, 1]; c3 ∈ [0, 1]
SCA [45]	A = 2
GWO [27]	a = [2, 0]

5.3. Statistical Results on 23 Benchmark Functions

This subsection compared MROA with seven basic algorithms on 23 benchmark functions in terms of the mean value (Mean) and standard deviation (Std). The experimental results are given in Table 3. By observing this table, we can see that MROA has the smallest Mean and Std values among the most functions. Specifically, for F1–F4, MROA obtains the theoretical optimal solution, where ROA only finds the optimal solution, which shows the LOBL strategy effectively enhances the exploitation ability of MROA, and the lens opposite solution increases the diversity of the population, helping the search agents to jump out of the local optimal solution. For MM functions, MROA also provides more competitive results than seven well-known algorithms; the Brownian motion strategy improves the ROA’s exploration capability and provides search agents with a greater opportunity to find the optimal or near-optimal solution in the search space. For F8, although MROA only finds the near-optimal solution, its convergence accuracy is also better than others. For F9, F10, and F11, MROA obtains the theoretically optimal solution. For F12, F13, and F4, the accuracy of MROA is weaker than AO and SSA, respectively. For F15–F23, MROA also provides a higher convergence accuracy than others. Moreover, Figure 6 shows the MROA and competitor algorithms’ ranking in UM and MM functions. We can see from the results that the proposed MROA outperforms in most benchmark functions.

Table 3. Statistical results of algorithms on 23 benchmark functions.

Function		MROA	ROA	RSA	AOA	AO	SSA	SCA	GWO
F1	Mean	0.00 × 10+00	8.88 × 10−314	0.00 × 10+00	5.22 × 10−06	6.58 × 10−102	2.03 × 10−07	1.76 × 10+01	1.23 × 10−27
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	2.58 × 10−06	3.59 × 10−101	2.30 × 10−07	5.31 × 10+01	2.21 × 10−27
F2	Mean	0.00 × 10+00	1.01 × 10−168	0.00 × 10+00	1.86 × 10−03	1.72 × 10−59	2.02 × 10+00	2.95 × 10−02	1.19 × 10−16
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	1.77 × 10−03	9.40 × 10−59	1.56 × 10+00	5.07 × 10−02	8.16 × 10−17
F3	Mean	0.00 × 10+00	1.15 × 10−268	0.00 × 10+00	1.10 × 10−03	3.11 × 10−101	1.67 × 10+03	1.01 × 10+04	1.60 × 10−04
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	8.16 × 10−04	1.64 × 10−100	8.80 × 10+02	6.49 × 10+03	8.16 × 10−04
F4	Mean	0.00 × 10+00	4.59 × 10−164	0.00 × 10+00	1.86 × 10−02	2.13 × 10−53	1.07 × 10+01	3.79 × 10+01	6.77 × 10−07
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	1.30 × 10−02	9.90 × 10−53	4.18 × 10+00	1.10 × 10+01	6.21 × 10−07
F5	Mean	2.74 × 10+01	2.63 × 10+01	2.13 × 10+01	2.81 × 10+01	7.19 × 10−03	3.22 × 10+02	6.92 × 10+04	2.70 × 10+01
	Std	6.30 × 10−01	4.96 × 10+00	1.30 × 10+01	2.22 × 10−01	1.18 × 10−02	6.84 × 10+02	1.81 × 10+05	6.69 × 10−01
F6	Mean	5.72 × 10−01	1.05 × 10−01	7.35 × 10+00	3.10 × 10+00	1.23 × 10−04	2.33 × 10−07	1.96 × 10+01	7.63 × 10−01
	Std	2.48 × 10−01	7.73 × 10−02	2.90 × 10−01	2.37 × 10−01	1.84 × 10−04	3.20 × 10−07	2.77 × 10+01	3.27 × 10−01
F7	Mean	5.95 × 10−05	1.64 × 10−04	1.02 × 10−04	7.00 × 10−05	1.12 × 10−04	1.68 × 10−01	9.58 × 10−02	1.80 × 10−03
	Std	6.00 × 10−05	1.88 × 10−04	1.16 × 10−04	7.00 × 10−05	8.14 × 10−05	6.99 × 10−02	8.56 × 10−02	9.21 × 10−04
F8	Mean	−1.24 × 10+04	−1.23 × 10+04	−5.34 × 10+03	−5.50 × 10+03	−8.43 × 10+03	−7.10 × 10+03	−3.80 × 10+03	−6.09 × 10+03
	Std	2.60 × 10+02	4.96 × 10+02	3.88 × 10+02	4.21 × 10+02	3.55 × 10+03	8.77 × 10+02	2.56 × 10+02	8.92 × 10+02
F9	Mean	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	1.60 × 10−06	0.00 × 10+00	5.90 × 10+01	2.98 × 10+01	3.34 × 10+00
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	1.51 × 10−06	0.00 × 10+00	2.59 × 10+01	2.55 × 10+01	4.09 × 10+00
F10	Mean	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	4.71 × 10−04	8.88 × 10−16	2.49 × 10+00	1.42 × 10+01	1.07 × 10−13
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	1.40 × 10−04	0.00 × 10+00	9.35 × 10−01	8.86 × 10+00	1.60 × 10−14
F11	Mean	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	3.13 × 10−03	0.00 × 10+00	1.97 × 10−02	1.17 × 10+00	5.43 × 10−03
	Std	0.00 × 10+00	0.00 × 10+00	0.00 × 10+00	1.19 × 10−02	0.00 × 10+00	1.53 × 10−02	7.83 × 10−01	8.53 × 10−03
F12	Mean	4.41 × 10−02	1.11 × 10−02	1.55 × 10+00	7.46 × 10−01	2.73 × 10−06	6.64 × 10+00	4.05 × 10+04	3.83 × 10−02
	Std	2.34 × 10−02	1.09 × 10−02	3.31 × 10−01	3.10 × 10−02	3.94 × 10−06	3.35 × 10+00	1.54 × 10+05	1.52 × 10−02
F13	Mean	2.44 × 10+00	2.13 × 10−01	3.85 × 10−01	2.96 × 10+00	6.09 × 10−05	1.81 × 10+01	3.82 × 10+05	6.87 × 10−01
	Std	1.07 × 10+00	1.29 × 10−01	9.99 × 10−01	1.99 × 10−02	1.03 × 10−04	1.52 × 10+01	1.23 × 10+06	1.90 × 10−01
F14	Mean	9.57 × 10+00	3.22 × 10+00	4.54 × 10+00	1.07 × 10+01	2.82 × 10+00	1.10 × 10+00	1.66 × 10+00	5.50 × 10+00
	Std	4.85 × 10+00	3.89 × 10+00	3.51 × 10+00	3.15 × 10+00	3.97 × 10+00	3.03 × 10−01	9.48 × 10−01	4.28 × 10+00
F15	Mean	4.32 × 10−04	4.33 × 10−04	2.85 × 10−03	1.29 × 10−02	4.57 × 10−04	2.81 × 10−03	1.22 × 10−03	3.75 × 10−03
	Std	1.59 × 10−04	1.96 × 10−04	1.63 × 10−03	2.88 × 10−02	9.57 × 10−05	5.96 × 10−03	3.42 × 10−04	7.56 × 10−03
F16	Mean	−1.03 × 10+00	−1.03 × 10+00	−1.03 × 10+00	−1.03 × 10+00	−1.03 × 10+00	−1.03 × 10+00	−1.03 × 10+00	−1.03 × 10+00
	Std	2.30 × 10−08	3.47 × 10−08	1.97 × 10−03	2.22 × 10−11	3.91 × 10−04	1.81 × 10−14	4.59 × 10−05	2.85 × 10−08
F17	Mean	3.98 × 10−01	3.98 × 10−01	4.17 × 10−01	4.03 × 10−01	3.98 × 10−01	3.98 × 10−01	3.99 × 10−01	3.98 × 10−01
	Std	7.33 × 10−06	5.30 × 10−06	1.59 × 10−02	2.50 × 10−02	1.88 × 10−04	1.12 × 10−14	1.15 × 10−03	9.65 × 10−07
F18	Mean	3.00 × 10+00	3.00 × 10+00	6.68 × 10+00	3.06 × 10+01	3.04 × 10+00	3.00 × 10+00	3.00 × 10+00	3.00 × 10+00
	Std	1.23 × 10−04	7.58 × 10−05	9.55 × 10+00	3.49 × 10+01	4.04 × 10−02	2.32 × 10−13	1.92 × 10−04	3.62 × 10−05
F19	Mean	−3.86 × 10+00	−3.86 × 10+00	−3.76 × 10+00	−3.77 × 10+00	−3.86 × 10+00	−3.86 × 10+00	−3.85 × 10+00	−3.86 × 10+00
	Std	1.27 × 10−04	2.73 × 10−03	1.65 × 10−01	5.23 × 10−01	7.21 × 10−03	2.50 × 10−12	8.81 × 10−03	3.81 × 10−03
F20	Mean	−3.30 × 10+00	−3.24 × 10+00	−2.71 × 10+00	−3.27 × 10+00	−3.16 × 10+00	−3.23 × 10+00	−2.89 × 10+00	−3.27 × 10+00
	Std	4.51 × 10−02	8.27 × 10−02	2.93 × 10−01	5.93 × 10−02	1.09 × 10−01	5.83 × 10−02	3.16 × 10−01	7.81 × 10−02
F21	Mean	−1.02 × 10+01	−1.01 × 10+01	−5.06 × 10+00	−7.71 × 10+00	−1.01 × 10+01	−7.23 × 10+00	−2.53 × 10+00	−8.98 × 10+00
	Std	3.00 × 10−03	1.45 × 10−02	3.61 × 10−07	2.69 × 10+00	1.02 × 10−02	3.48 × 10+00	1.81 × 10+00	2.44 × 10+00
F22	Mean	−1.04 × 10+01	−1.04 × 10+01	−5.09 × 10+00	−8.26 × 10+00	−1.04 × 10+01	−8.76 × 10+00	−3.26 × 10+00	−1.04 × 10+01
	Std	2.02 × 10−03	2.45 × 10−02	7.16 × 10−07	2.92 × 10+00	1.45 × 10−02	3.05 × 10+00	1.85 × 10+00	2.40 × 10−03
F23	Mean	−1.05 × 10+01	−1.05 × 10+01	−5.13 × 10+00	−7.35 × 10+00	−1.05 × 10+01	−8.85 × 10+00	−3.65 × 10+00	−1.01 × 10+01
	Std	2.19 × 10−03	1.97 × 10−02	1.44 × 10−06	3.38 × 10+00	2.93 × 10−02	2.90 × 10+00	1.86 × 10+00	1.85 × 10+00

Figure 6. The radar graphs of algorithms on 23 benchmark functions.

Furthermore, owing to the MAs being stochastic algorithms, in this paper, we consider the Wilcoxon rank-sum test to supplement our statistical analysis and evaluate the statistical significance of the results [70]. The Wilcoxon rank-sum test is a non-parametric test used to compare the results of each pair of algorithms. The test is based on two hypotheses: null and alternative. The null hypothesis indicates that there is no difference between the two samples. The alternative hypothesis indicates that there is a significant difference between the two samples. We set the significance level as 5%; according to this criterion, if the p-value is higher than 0.05, which indicates the null hypothesis is true, the significant difference is not obvious. If the p-value is less than 0.05, which indicates there is a significant difference, the null hypothesis is rejected. Furthermore, NaN denotes that there is no difference between the two samples. Table 4 shows the p-values obtained using the Wilcoxon rank-sum test, which is calculated by the fitness values between MROA and each of the algorithms included, presenting six different pairs, MROA vs. ROA, MROA vs. RSA, MROA vs. AOA, MROA vs. AO, MROA vs. SSA, MROA vs. SCA, and MROA vs. GWO. This table shows that the results of MROA are statistically significant for most benchmark functions, which indicates that the performance of MROA is not random. Consequently, based on the above analysis, we conclude that the MROA outperforms other algorithms for most benchmark functions in terms of convergence accuracy and statistical significance.

Table 4. Statistical results of algorithms on 23 benchmark functions using Wilcoxon rank-sum test.

Function	MROA vs.
	ROA	RSA	AOA	AO	SSA	SCA	GWO
F1	4.19 × 10−02	NaN	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12
F2	1.21 × 10−12	NaN	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12
F3	1.21 × 10−12	NaN	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12
F4	1.21 × 10−12	NaN	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12	1.21 × 10−12
F5	2.46 × 10−01	2.50 × 10−02	1.70 × 10−08	3.02 × 10−11	5.57 × 10−10	3.02 × 10−11	3.18 × 10−01
F6	3.20 × 10−09	1.79 × 10−11	4.08 × 10−11	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11	8.77 × 10−02
F7	2.13 × 10−05	3.92 × 10−02	6.41 × 10−01	8.50 × 10−02	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11
F8	9.06 × 10−08	3.01 × 10−11	3.02 × 10−11	9.33 × 10−02	8.99 × 10−11	3.02 × 10−11	3.02 × 10−11
F9	NaN	NaN	1.21 × 10−12	NaN	1.21 × 10−12	1.21 × 10−12	1.18 × 10−12
F10	NaN	NaN	1.21 × 10−12	NaN	1.21 × 10−12	1.21 × 10−12	1.02 × 10−12
F11	NaN	NaN	1.21 × 10−12	NaN	1.21 × 10−12	1.21 × 10−12	1.10 × 10−02
F12	3.16 × 10−10	1.10 × 10−11	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11	1.30 × 10−03
F13	3.16 × 10−10	6.71 × 10−05	9.51 × 10−06	3.02 × 10−11	2.75 × 10−03	3.02 × 10−11	1.03 × 10−06
F14	3.37 × 10−06	3.34 × 10−04	3.91 × 10−01	8.07 × 10−07	2.18 × 10−10	3.20 × 10−07	4.56 × 10−05
F15	3.18 × 10−01	5.49 × 10−11	1.22 × 10−01	6.36 × 10−05	6.70 × 10−11	1.86 × 10−09	5.69 × 10−01
F16	1.77 × 10−03	5.57 × 10−10	3.02 × 10−11	5.57 × 10−10	3.01 × 10−11	5.57 × 10−10	1.08 × 10−02
F17	3.71 × 10−01	6.07 × 10−11	3.02 × 10−11	1.17 × 10−09	2.86 × 10−11	3.34 × 10−11	2.90 × 10−01
F18	5.08 × 10−03	1.09 × 10−01	3.99 × 10−04	3.02 × 10−11	3.02 × 10−11	7.98 × 10−02	7.24 × 10−02
F19	9.06 × 10−08	3.02 × 10−11	1.86 × 10−03	6.07 × 10−11	3.02 × 10−11	3.02 × 10−11	2.25 × 10−04
F20	4.01 × 10−02	3.69 × 10−11	6.77 × 10−05	3.50 × 10−03	3.26 × 10−01	3.47 × 10−10	2.05 × 10−03
F21	3.01 × 10−07	3.02 × 10−11	3.37 × 10−04	7.30 × 10−04	1.86 × 10−01	3.02 × 10−11	5.26 × 10−04
F22	1.17 × 10−09	3.02 × 10−11	9.53 × 10−07	1.47 × 10−07	3.99 × 10−04	3.02 × 10−11	6.05 × 10−07
F23	8.48 × 10−09	3.02 × 10−11	1.27 × 10−02	5.19 × 10−07	9.51 × 10−06	3.02 × 10−11	1.75 × 10−05

5.4. Boxplot Behavior Analysis

To further validate the excellent performance of the proposed MROA, Figure 7 presents the boxplot behavior of MROA and other MAs for some functions, including UM and MM types. Since boxplots indicate the data distribution, it is available to describe the consistency between data. MROA is very narrow compared with other algorithms and maintains the lowest values for most functions. It is observed that ROA outperforms the other algorithms in terms of data distribution, but only on F6, F8, and F15; the performance is mediocre for data distribution.

Figure 7. Boxplot behavior of algorithms on some functions.

5.5. Convergence Behavior Analysis

This subsection analyzed the convergence behavior of the proposed MROA with compared algorithms. The convergence curve of these algorithms on 23 benchmark functions is shown in Figure 8. From this figure, we can conclude that MROA outperforms other well-known MAs in terms of convergence speed and optimization accuracy. For unimodal functions F1, F2, and F4, MROA achieves the fastest convergence speed and obtains the theoretically optimal solutions. However, for F6, the performance of MROA is not excellent and it may be trapped into the local optimal. For F7, although MROA cannot obtain the theoretical optimal solution, it also finds the most near-optimal solution and shows the fastest convergence speed than others. For MM functions, MROA also shows excellent optimization performance than other popular methods. For F8, MROA has a significant convergence capability that exceeds two ROA and SSA methods. For F9 and F11, MROA finds the theoretical optimal solution and has a faster convergence speed than others. We concluded that MROA has an excellent convergence ability in UM and MM functions.

Figure 8. Convergence behavior of algorithms on some functions.

6. Experimental Results for Multi-Level Thresholding Image Segmentation

This section used MROA to minimize the cross-entropy for solving multi-level thresholding image segmentation tasks. First, we selected eight images from the Berkeley segmentation dataset [71] as the benchmark images and the experimental setup described in detail. Second, three image measurements, including PSNR, SSIM, and FSIM are introduced to measure the segmentation performance (quality, consistency, and accuracy) of MROA and other algorithms. Finally, the experimental results are discussed and analyzed.

6.1. Berkeley Segmentation Dataset

The proposed MROA with other algorithms is evaluated on eight images from the Berkeley segmentation dataset [71]. In this work, eight benchmark images are grayscale. These images and their histogram are presented in Figure 9. The purpose of selecting these images is only to test the algorithms’ segmentation performance and quality.

Figure 9. Berkeley image dataset and its histogram.

6.2. Experimental Setup

The proposed MROA is compared with seven well-known algorithms such as ROA [52], RSA [69], AOA [47], AO [35], SSA [31], SCA [45], and GWO [27]. The described parameters and settings of these algorithms are described in Section 4.2. For fair evaluation, the number of maximum iteration T = 500, the population size N = 30, and all the algorithms were run 30 times independently. The best values are highlighted in bold. All the images are segmented using different threshold values nTh = [4, 8, 12, 16].

6.3. Segmentation Quality Evaluation Measurements

6.3.1. Peak Signal-to-Noise Ratio (PSNR)

PSNR is used to calculate the ratio between the maximum possible signal power and the distorted noise power that affects the quality of its representation. The signal is regarded as the original data, and the noise is the error caused by segmentation. The PSNR is the most commonly used quality assessment technique to compare an original image and its segmented images using root mean square error (RMSE) per pixel. A higher PSNR indicates more similarity between the images, which is reflected in a better segmentation process [71,72]. It can be described as follows:

$$PSNR=20lo{g}_{10}\frac{255}{RMSE}$$

(25)

$$RMSE=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^M{\displaystyle {\sum }_{j=1}^N({(I_g(i,j)-Seg(i,j))}^2)}}}{M\times N}}$$

(26)

where I_g and Seg denote the original and segmented image, respectively. M and N are the sizes of the image.

6.3.2. Structural Similarity (SSIM)

The SSIM is used to measure the degree of distortion of the images, and it can also measure the similarity from three perspectives, including luminance, contrast, and structural content between original and segmented images. Unlike mean-square error (MSE) and PSNR, SSIM measures absolute error, which is more in line with the intuitive feeling of the human eye [72,73]. SSIM is computed by the following equation:

$$SSIM(I_g,Seg)=\frac{(2{\mu }_I{}_{{}_g}{\mu }_{Seg}+c_1)(2{\sigma }_{I_g,Seg}+c_2)}{({\mu }_{I_g}{}^2+{\mu }_{Seg}{}^2+c_1)({\sigma }_{I_g}{}^2+{\sigma }_{Seg}{}^2+c_2)}$$

(27)

where μ_I and μ_Seg indicate the mean intensities of the original and its segmented image, respectively. σ_I and σ_Seg are standard deviations of original and its segmented images. σ_{I, Seg} is the covariance between original and segmented images. c₁ and c₂ are two constants equal to 0.065.

6.3.3. Feature Similarity (FSIM)

FSIM is a variant of SSIM that considers low-level features such as the gradient magnitude and phase congruency to measure the similarity of two images [74,75]. It can be described as follows:

$$FSIM=\frac{{\displaystyle {\sum }_{\omega \in \mathrm{\Omega }}{S}_L(\omega )P{C}_m(\omega )}}{{\displaystyle {\sum }_{\omega \in \mathrm{\Omega }}P{C}_m(\omega )}}$$

(28)

$$S_L(\omega )=S_{PC}(\omega )S_G(\omega )$$

(29)

$$S_{PC}(\omega )=\frac{2P{C}_1(\omega )P{C}_2(\omega )+T_1}{P{C}_1{}^2(\omega )+P{C}_2{}^2(\omega )+T_1}$$

(30)

$$S_G(\omega )=\frac{2G_1(\omega )G_2(\omega )+T_2}{G_1{}^2(\omega )+G_2{}^2(\omega )+T_2}$$

(31)

where Ω indicates the entire domain of the image. T₁ and T₂ are constants which equal to 0.85 and 160, respectively. G indicates the gradient magnitude of an image, and PC denotes the phase congruence.

6.4. Experimental Results Analysis of Benchmark Images

We compared MROA with other well-known algorithms for solving multi-level thresholding image segmentation tasks in this subsection. Cross-entropy is used as the objective function. Table A1 (in Appendix A) denotes the best threshold values obtained by algorithms at different threshold values. Table A2 contains the segmented images obtained from the proposed MROA at nTh = [4, 8, 12, 16]. Table A3, Table A4, Table A5 and Table A6 show the mean and standard deviation values for fitness, PSNR, SSIM, and FSIM obtained by algorithms, respectively. The higher mean values obtained indicate more optimization accuracy and efficiency of an algorithm.

On the other hand, the smaller the standard deviation values, the more stable the algorithm. Table A7 shows the Wilcoxon rank-sum results for fitness, the significance level is the same as Section 4.3. Moreover, Figure 10, Figure 11, Figure 12 and Figure 13 show the summary average results obtained by algorithms in terms of fitness, PSNR, SSIM, and FSIM, respectively. Figure 14 summarized the best cases in terms of fitness, PSNR, SSIM, and FSIM, respectively.

Figure 10. Average fitness values obtained by the algorithms over all images.

Figure 11. Average PSNR values obtained by the algorithms over all images.

Figure 12. Average SSIM values obtained by the algorithms over all images.

Figure 13. Average FSIM values obtained by the algorithms over all images.

Figure 14. The number of best cases based on the fitness, PSNR, SSIM, FSIM obtained by algorithms over all images.

It can be seen from Table A1 that in most cases, when the threshold level is 4 and 8, similar values are obtained by algorithms, especially for ROA, AO, SSA, and GWO. When the threshold level is 12 and 16, the difference is significant. According to Figure 10 and Table A3, the proposed work outperforms others in terms of average fitness and summarized. We conduct that MROA achieves the smallest fitness value in most cases at threshold values 8, 12, and 16.

Table A4 and Figure 11 show the mean PSNR values and a summary obtained by algorithms with different threshold values, respectively. It is available for evaluating the quality between the segmented image and the original. From this figure and table, we can see that the MROA has a significant advantage against ROA, AO, SSA, and GWO in most cases. The RSA, AOA, and SCA do not show excellent PSNR values. In the case of ROA, compared with other algorithms, as shown in Table A4, the PSNR values obtained for image 1, image 6, and image 8 present higher results at all levels. While segmenting image 2, image 3, image 4, and image 7, MROA outperforms others at three different levels.

Table A5 and Figure 12 indicate the mean SSIM values and a summary obtained by algorithms at different threshold values, respectively. SSIM measures the similarity between two images. It can be seen that MROA outperforms others in terms of PSNR values. Specifically, for image 4, image 5, and image 8, MROA produces higher results at all levels. For image 1, image 2, image 6, and image 7, MROA outperforms others at three levels.

Table A6 and Figure 13 present the mean FSIM values and a summary obtained by algorithms at different threshold values, respectively. FSIM is a variant of SSIM used to evaluate the feature similarity of two images. We notice that MROA obtains more competitive results than others from these results. MROA produces better results at all threshold levels for image 1, image 2, image 5, and image 8, respectively. While for image 4, image 6, and image 7, MROA outperforms others at three levels. For image 3, MROA only provides high accuracy for image 3 at two levels.

Figure 14 summarizes the number of best cases obtained by algorithms in terms of fitness, PSNR, SSIM, and FSIM. According to this figure, MROA is significantly improved upon its original. AO and GWO are ranked second and third, respectively.

Table A7 shows the p-value obtained by algorithms using the Wilcoxon rank-sum test with a 5% significance level. According to this table, we can see that MROA produces statistically significant results compared with other algorithms. Especially when compared with RSA, AOA, and SCA, MROA has a significant difference between them, which indicates MROA has been improved considerably.

Table A8 shows the CPU time of each algorithm under different thresholds. As it is possible to observe, in most cases a shorter time is obtained using RSA, AOA, and SCA, respectively. However, as shown in Table A3, Table A4, Table A5 and Table A6, RSA, AOA, and SCA may not produce threshold values for segmenting given images well. For the proposed MROA, although MROA increases the CPU time over its original when segmenting images, considering the NFL theorem, it is acceptable to increase some CPU time cost to obtain a better solution.

Overall, the proposed method, MROA, can obtain higher fitness, PSNR, SSIM, and FSIM for most images at different levels, showing that the MROA-based multi-level thresholding segmentation technique can obtain better evaluation results, which further indicates that it can better perform the segmentation of complex grayscale images. It is significantly improved upon the basic version of ROA. Second, it also shows significantly superior performance in terms of fitness, PSNR, SSIM, and FSIM compared with other selected well-known algorithms. It is an excellent approach for solving multi-level thresholding image segmentation, as the BM and LOBL strategies significantly enhance the performance of ROA in determining the optimal threshold values, exhibiting excellent searchability and jumping out of the local optimal solution when solving multi-level thresholding segmentation tasks, and we will apply it to handle other datasets such as medical images, remote sensing images, etc.

7. Conclusions and Future Works

This paper introduces a modified remora optimization algorithm, namely MROA, for global optimization and multi-level thresholding image segmentation tasks. The original ROA has some fatal issues such as premature convergence and easy stagnation into local optimal solutions, which means it cannot solve complex real-world problems. In this work, Brownian motion improves the ROA’s exploration ability and provides a higher opportunity to find the optimal or near-optimal solution in the search space. Second, the Lens opposition-based learning enhances the exploitation capability and promotes search agents to jump out the local optimal solution.

We first evaluate the optimization performance of the proposed MROA using 23 benchmark functions. The experimental results demonstrate that MROA has broader searchability and a greater opportunity to obtain high-quality solutions than the original ROA. In addition, based on the analysis and comparison of MROA with other well-known MAs, it can be seen that MROA also has a better overall ability than its peers in optimization accuracy, convergence speed, and significant difference. Therefore, we consider that MROA is an excellent meta-heuristic for function optimization.

Subsequently, the MROA is applied to solve multi-level thresholding image segmentation tasks and obtain optimal threshold values by cross-entropy. The performance of the proposed work is also compared with its peers. Experimental evaluation metrics including PSNR, SSIM, FSIM are used to test the segmented image quality. Furthermore, a statistical test is used to evaluate the significant difference between the two methods. Experimental results proved the following: (1) the MROA significantly improves the original ROA’s performance for the image thresholding segmentation task. (2) MROA produces excellent results in terms of PSNR, SSIM, and FSIM compared with the other compared MAs. (3) Wilcoxon rank-sum results indicate that MROA produces a significant performance compared with other selected MAs. Finally, we conclude that the proposed MROA can generate high-quality segmented images, outperforms other selected MAs in terms of segmentation accuracy, and is more stable and promising.

While the proposed MROA has significant improvements over the original ROA and shows excellent results compared with other selected MAs in terms of benchmark function optimization and image segmentation, the CPU time also increases when solving the image segmentation task. In the future, we will: (1) Reduce the CPU time without degrading MROA’s performance; we can design a new effective strategy or hybrid the proposed method with other MAs. (2) Extend the image dataset to include images such as medical images and remote sensing images and increase threshold values to further prove the performance of MROA. (3) Since MROA is a high-performance optimizer, MROA must be applied in other fields such as feature selection, engineering design problems, and maximum power point tracking.