Multilevel segmentation of 2D and volumetric medical images using hybrid Coronavirus Optimization Algorithm

Abstract

Medical image segmentation is a crucial step in Computer-Aided Diagnosis systems, where accurate segmentation is vital for perfect disease diagnoses. This paper proposes a multilevel thresholding technique for 2D and 3D medical image segmentation using Otsu and Kapur's entropy methods as fitness functions to determine the optimum threshold values. The proposed algorithm applies the hybridization concept between the recent Coronavirus Optimization Algorithm (COVIDOA) and Harris Hawks Optimization Algorithm (HHOA) to benefit from both algorithms' strengths and overcome their limitations. The improved performance of the proposed algorithm over COVIDOA and HHOA algorithms is demonstrated by solving 5 test problems from IEEE CEC 2019 benchmark problems. Medical image segmentation is tested using two groups of images, including 2D medical images and volumetric (3D) medical images, to demonstrate its superior performance. The utilized test images are from different modalities such as Magnetic Resonance Imaging (MRI), Computed Tomography (CT), and X-ray images. The proposed algorithm is compared with seven well-known metaheuristic algorithms, where the performance is evaluated using four different metrics, including the best fitness values, Peak Signal to Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Normalized Correlation Coefficient (NCC). The experimental results demonstrate the superior performance of the proposed algorithm in terms of convergence to the global optimum and making a good balance between exploration and exploitation properties. Moreover, the quality of the segmented images using the proposed algorithm at different threshold levels is better than the other methods according to PSNR, SSIM, and NCC values. Additionally, the Wilcoxon rank-sum test is conducted to prove the statistical significance of the proposed algorithm.

1. Introduction

Computer-Aided Diagnoses (CAD) tools play a critical role in healthcare [1]. Medical image segmentation is one of the essential steps for disease diagnoses. It refers to extracting objects of interest in medical images to analyze these objects' behavior, which may indicate the existence of a problem or a disease [2]. In the literature, several techniques have been proposed for image segmentation, such as edge detection-based segmentation [3], clustering-based segmentation [4], and thresholding-based segmentation [5]. Image segmentation based on thresholding is considered the most popular technique because it has simple implementation and high accuracy.

Despite the importance of image segmentation in extracting the objects of interest from medical images, some problems cause errors in the medical image segmentation process, such as image acquisition artifacts and corruption by noise. Various smoothing techniques can reduce error or remove noises, such as developing an algorithm or tuning a filter [6].

Depending on the number of thresholds used to segment the image, thresholding-based segmentation is classified into bi-level and multilevel thresholding [7]. In bi-level thresholding, one threshold is used to segment the image into two regions. All pixels with values more significant than the threshold value are classified as region 1, and the other pixels in the image are classified as region 2. On the other hand, multilevel thresholding involves using more than one threshold to segment the image into several regions. Bi-level thresholding fails to correctly identify images containing many objects with colored and complex backgrounds because it divides the image into only two classes. In such cases, multilevel thresholding is more appropriate [8]. The most critical step in the thresholding process is to find the optimum threshold values that efficiently determine the image segments. Over the last few decades, several strategies have been developed for determining the optimal thresholds; Ostu [9] and Kapur [10] methods are the most popular due to their efficiency and simplicity. Otsu's method maximizes the variance between classes, and Kapur's method maximizes the histogram entropy to measure homogeneity between segmented regions.

Image segmentation can be considered an optimization problem in which the objective is to find the optimum thresholds that precisely determine image classes. Traditional thresholding-based image segmentation techniques suffer from several problems, such as exponentially increasing the computational cost with the increase in thresholding levels which makes these methods suitable only for a small number of thresholding levels. This challenge encouraged the authors to use metaheuristics-based image segmentation algorithms as an alternative to the classical methods. Over the last years, several metaheuristic algorithms have been applied to solve image segmentation problems [11]. For multilevel thresholding, several algorithms have been used, such as the Genetic Algorithm (GA) [12], which is based on the theory of natural evolution; Particle Swarm Optimization (PSO) [13], inspired by the behavior of bird flocks and schooling fish; Artificial Bee Colony (ABC) [14] that simulates the behavior of bees in finding food sources; Harmony Search (HS) [15] inspired in musicians improvising new harmonies while playing; Electromagnetism Optimization (EO) [16] that mimics the attraction-repulsion mechanism among charges; and many others [[17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]].

Several algorithms have been proposed for medical image segmentation, such as Harris Hawks' optimization algorithm, which is used to segment brain MRI images [28,28a]. The results showed that the proposed framework outperforms the state-of-the-art methods on the same dataset; however, it suffers from some limitations. It consumes more time than other metaheuristics. Another algorithm is proposed for COVID X-ray image segmentation using ant colony optimization with Cauchy and greedy levy mutations [29]. The results demonstrate the superior performance of the proposed algorithm in terms of search ability and convergence speed. Also, ABD ELAZIZ et al. [30,30a] proposed an algorithm for COVID-19 CT image segmentation based on an improved marine predators algorithm with fuzzy entropy. The experimental results approved the superiority of the proposed algorithm over the existing methods. The recently proposed metaheuristics that are used for solving multilevel thresholding image segmentation problems include the Arithmetic Optimization Algorithm (AOA) [31], which is inspired by the distribution behavior of the main arithmetic operators in mathematics, Remora Optimization Algorithm (ROA) [32] which mimics the parasitic behavior of remora, Black Window Optimization algorithm (BWO) [33], and Equilibrium Optimization Algorithm (EOA) [34]. These algorithms have proved their good performance; however, they may have limitations, such as getting stuck into local optima [35]. Some researchers employ hybridization as a way to avoid these limitations.

In image segmentation, many hybrid optimization algorithms have been proposed. For example, the Grasshopper Optimization Algorithm (GOA) is combined with the Differential Evolution algorithm (DE) for multilevel segmentation of satellite images [36]. The experimental results indicate that the proposed algorithm outperforms the standard GOA and DE algorithms. Also, a combination of the Gravitational Search Algorithm (GSA) and Genetic Algorithm (GA) is proposed for multilevel thresholding image segmentation using entropy and between-class variance as fitness functions [37]. The experimental results showed that the GSA-GA produced superior or comparative segmentation accuracy in entropy and between-class variance criteria. Many other hybrid algorithms are proposed to solve the image segmentation problem [38,39].

This paper proposes a novel hybrid optimization algorithm for multilevel thresholding 2D and 3D medical image segmentation based on combining the novel Coronavirus Disease Optimization Algorithm (COVIDOA) [40] and the Harris Hawks Optimization Algorithm (HHOA) [35]. COVIDOA is a recent evolutionary optimization algorithm that mimics the replication lifecycle of Coronavirus. COVIDOA has three main phases: Virus Entry, Virus Replication, and Virus mutation. Coronavirus uses frameshifting [41,42] to make new virus copies in the Replication phase. Frameshifting produces many viral proteins combined to form new virus particles as many new particles are created, and many human cells are damaged. In addition, the virus uses mutation techniques to escape from the human immunity system. COVIDOA has been applied to many benchmark test functions and real-world problems and showed superior performance. Its advantages include a good balance between exploration and exploitation and high convergence speed.

HHOA algorithm is a novel metaheuristic that mimics the chasing behavior of Harris hawks. HHOA has been applied to solve many real-world problems such as pressure vessel design problems, 3three-bar truss design problems, and welded beam design problems. HHOA shows good exploitative search ability.

This paper proposes combining COVIDOA and HHOA to find the optimum threshold values using Otsu's and Kapur's entropy as fitness functions. This hybridization helps benefit from both algorithms' advantages and overcome their limitations.

The reasons for using hybrid COVID and HHOA are as follows:

• 1.The No Free Lunch (NFL) theorem demonstrates that no single algorithm performs best for all optimization problems. This theorem encouraged the authors to use a hybrid version of the recent COVIDOA to solve the image segmentation problem.
• 2.The COVIDOA [40] and its binary version, BCOVIDOA [43], outperformed most existing optimization algorithms in solving benchmark and real-world optimization problems.
• 3.The idea of the proposed algorithm is to divide the initial population into two half and assign each half to one of the two powerful metaheuristics (COVIDOA and HHOA). The two metaheuristics then work in parallel to update the two half populations. Then the updated two subpopulations are merged into one full population. These stapes are repeated until the maximum number of iterations is reached and finally output the optimum solution found so far. The idea of the proposed algorithm is very simple and can be easily implemented.
• 4.The proposed approach makes parallel hybridization (which is more suited to parallel computer environments) between two powerful metaheuristics (COVIDOA and HHOA) for solving segmentation problems.
• 5.The proposed algorithm can fix the limitations of the two metaheuristics because each technique operates on only half of the population, not the whole population.
• 6.Traditional thresholding-based image segmentation techniques suffer from several problems, such as exponentially increasing the computational cost, which encouraged the authors to use metaheuristics-based image segmentation algorithms as an alternative to the classical methods. Our proposed algorithm achieved superior performance in medical image segmentation, especially at high thresholding levels in which traditional methods are unsuitable due to the high computational cost.
• 7.The proposed hybrid technique can be easily applied to any other metaheuristics. Still, we preferred combining COVIDOA and HHOA algorithms because of their superior performance in solving various optimization problems.

The proposed algorithm works as follows: the population of solutions is divided into two halves, and then each half is assigned to one of the two algorithms. Each algorithm operates in parallel with its sub-population and generates the updated sub-population. The two generated sub-populations are combined to form one new population in which the optimum solution is found. The validity of the proposed algorithm in solving various optimization problems is proved by solving 6 test problems from IEEE CEC 2019 benchmark problems [44]. In Medical image segmentation, the quality of the segmented medical images using the proposed algorithm is evaluated using different metrics such as MSE, PSNR, SSIM, FSIM, and NCC. The proposed algorithm is compared with seven state-of-the-art metaheuristics such as Harris Hawks Optimization Algorithm (HHOA) [35], Bat Algorithm (BA) [23], Harmony Search Algorithm (HS) [15], Cuckoo Search Algorithm (CS) [19], Sine Cosine Algorithm (SCA) [22], Flower Pollination Algorithm (FPA) [18], and Seagull Optimization Algorithm (SOA) [45]. In addition, the Wilcoxon rand sum test is calculated to prove the statistical significance of the proposed algorithm.

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

• •A novel hybrid COVIDOA-HHOA algorithm is proposed for medical image segmentation.
• •The efficiency of COVIDOA-HHOA is demonstrated by solving six IEEE CEC 2019 problems.
• •The performance of COVIDOA-HHOA is compared with seven well-known metaheuristics.
• •The comparison proved the superior performance of COVIDOA-HHOA against its beers.
• •Two datasets are used for testing, including 2D and 3D medical images.
• •Best fitness, PSNR, SSIM, and NCC metrics evaluate performance.
• •The Wilcoxon rank-sum test is conducted to prove the efficiency of COVIDOA-HHOA.

This paper is organized as follows: Section 2 provides a brief overview of multilevel thresholding techniques such as Otsu's method and Kapur's entropy. Sections 3, 4 give an overview of Coronavirus disease optimization and Harris hawks optimization, respectively. The proposed hybrid algorithm for multilevel thresholding is discussed in Section 5. The medical datasets, parameter setting, performance metrics, and experimental results are discussed in Section 6. Finally, conclusions and future work are given in Section 7.

2. Multilevel thresholding

Image thresholding is converting the color or gray scale image into a binary image by setting a threshold value on the pixel intensity of the image [46]. Where pixels below that threshold value are converted to black and pixels above it are converted to white. Image thresholding can be categorized into two classes: bi-level and multilevel. Bi-level thresholding aims to assign each pixel p of a graey-scale image to one of two regions (R1 and R2) using only one threshold value (th) as follows:

$$\begin{array}{l}p\in R_{1}if0\le p<th,\\ p\in {}_{2}ifth\le p<L-1,\end{array}$$ (1)

where $L$ refers to maximum intensity level.

However, multilevel thresholding segments an image into several distinct regions using more than one threshold value as follows:

$$\begin{array}{l}p\in R_{1}if0\le p<t{h}_1,\\ p\in R_{2}ift{h}_1\le p<t{h}_2,\\ p\in R_{i}ift{h}_i\le p<t{h}_{i+1},\\ p\in R_{k}ift{h}_{k-1}\le p<L-1,\end{array}$$ (2)

where {$t{h}_1,t{h}_2,\dots ,t{h}_{k-1}$} represents a vector of different threshold values.

Fig. 1 shows the difference between bi-level thresholding and multilevel thresholding of the mandrill baboon image.

Fig. 1.

Bi-level and multilevel thresholding.

The optimal threshold values can be obtained by maximizing a fitness function. Otsu's method and Kapur's entropy are two popular techniques used in thresholding. Each technique proposes a different fitness function that must be maximized to obtain the optimal threshold values. The two techniques are briefly described in the following subsections.

2.1. Otsu's method

Otsu is a thresholding method that selects the optimal threshold by maximizing the variance value between different classes [9]. Assume that we have L intensity levels in a gray scale image, where L=256 and a vector V of k-1 thresholds are used to segment the image into k regions as in equation (2), where V = [th ₁ , th ₂ , …, th _k-1 ]. Then the best threshold is obtained by maximizing the Otsu's fitness function as follows:

$$F_{ostu}\left(\text{V}\right)=\text{max}\left({\sigma }_b^2\left(V\right)\right)$$ (3)

where V=[th ₁ , th ₂ , …, th _k-1 ], and ${\sigma }_b^2$ represents the between-class variance which can be expressed as follows:

$${\mathrm{\sigma }}_{\text{b}}^2=\sum _{k=0}^K{\omega }_k.{({\mu }_k-{\mu }_T)}^2$$ (4)

where ${\omega }_k$ is the cumulative probability for region R _k, ${\mu }_k$ is the average intensity in region R _k and ${\mu }_T$ is the average intensity for the whole image as follows:

$${\omega }_k=\sum _{i\in R_k}{P}_i,{\mu }_k=\sum _{i\in R_k}\frac{i.P_i}{{\omega }_k},{\mu }_k=\sum _{i=0}^{L-1}i.P_i$$ (5)

where $P_i$ is the probability of gray level i, which can be represented as follows:

$$P_i=\frac{f_i}{{\sum }_{i=0}^{L-1}{f}_i}$$ (6)

where fi is the frequency of gray level i.

2.2. Kapur's entropy method

Image entropy represents the compactness and separateness between image classes [10]. The Kapur method is another widely used thresholding method that aims to find the optimal threshold value by maximizing the Kapur's entropy as follows:

$$t{h}^{\ast }=max\left(F_{kapur}\left(th\right)\right)$$ (7)

where

$$F_{kapur}\left(th\right)=A_0+A_1,$$

$$\begin{array}{l}{A}_0=-\sum _{i=0}^{th-1}\frac{P_i}{{\omega }_0}ln\frac{P_i}{{\omega }_0},A_1=-\sum _{i=th}^{L-1}\frac{P_i}{{\omega }_1}ln\frac{P_i}{{\omega }_1},\\ {\omega }_0=\sum _{i=0}^{th-1}{P}_i,{\omega }_1=\sum _{i=th}^{L-1}{P}_i\end{array}$$

where $P_i$ is described in Eq. (6).

For muli-level thresholding, Kapur's method can be defined as follows:

$$F_{kapur}\left(\text{V}\right)=A_0+A_1\dots +A_{k-1}$$ (8)

$$A_0=-\sum _{i=0}^{t{h}_1-1}\frac{P_i}{{\omega }_0}ln\frac{P_i}{{\omega }_0},{\omega }_0=\sum _{i=0}^{t{h}_1-1}{P}_i$$

$$A_1=-\sum _{i=t{h}_1}^{t{h}_2-1}\frac{P_i}{{\omega }_1}ln\frac{P_i}{{\omega }_1},{\omega }_1=\sum _{i=t{h}_1}^{t{h}_2-1}{P}_i$$

$$A_2=-\sum _{i=t{h}_2}^{t{h}_3-1}\frac{P_i}{{\omega }_2}ln\frac{P_i}{{\omega }_2},{\omega }_2=\sum _{i=t{h}_2}^{t{h}_3-1}{P}_i$$

$$A_n=-\sum _{i=t{h}_{k-1}}^{L-1}\frac{P_i}{{\omega }_n}ln\frac{P_i}{{\omega }_n},{\omega }_2=\sum _{i=t{h}_{k-1}}^{L-1}{P}_i$$

The symbol V is the vector of thresholds.

3. Coronavirus Disease Optimization Algorithm

COVIDOA is a recently proposed population-based optimization algorithm that simulates the replication mechanism of Coronavirus when getting inside the human body [40]. The replication process of Coronavirus has four main stages as follows:

• 1.Virus entry and uncoating

When a human is infected with COVID, the Coronavirus particles attach to the human cell via one of its structural proteins, called spike protein [42]. After getting inside the human cell, the virus contents are released.

• 2.Virus replication

The virus's replication technique is called the frameshifting technique [41]. Frameshifting is moving the reading frame of a protein sequence of the virus to another reading frame that leads to the creation of many new viral proteins that are then merged to form new virus particles. There are many types of frameshifting techniques; however, the most popular is +1 frameshifting as follows:

• •+1 frameshifting technique

The elements of the parent virus particle (parent solution) are moved in the right direction by 1 step. As a result of +1 frameshifting, the first element is lost. In the proposed algorithm, the first element is set as a random value in the range [$\text{Lb}$, Ub] as follows:

$$S_k\left(1\right)=rand(Lb,Ub),$$ (9)

$$S_k(2:D)=P(1:D-1),$$ (10)

where $Lb$ and $Ub$ are the lower and upper bounds for the variables in each solution.

• 3.Virus mutation

Coronavirus uses mutation to resist the human immune system [47]. In the proposed algorithm, the mutation is applied to the previously created new virus particle (solution) to produce a new one as follows:

$$Z_i=\{\begin{array}{l}rifrand\left(\mathrm{0,1}\right)<MR\hfill \\ X_{i}otherwise\hfill \end{array}$$ (11)

The symbol X referes to the solution before mutation, Z is the mutated solution, X _i and Z _i are the ith element in the old and new solutions, respectively, i = 1, …, D, and r is a random value in the range [$\text{Lb}$, $\text{Ub}$], MR is the mutation rate.

• 4.New virion release

The newly created virus particle leaves the infected cell targeting new healthy cells. In the proposed algorithm, if the fitness of the new solution is better than the parent solution fitness, the parent solution is replaced by the new one. Otherwise, the parent solution remains.

The pseudocode of the COVID optimization algorithm is shown in Fig. 2 .

Fig. 2.

pseudocode of COVID optimization algorithm.

4. Harris hawks optimization

HHOA [35] is a population-based algorithm inspired by the chasing behavior of Harris hawks to capture prey. The two main phases of HHOA are exploration and exploitation, which are explained in the following subsections.

4.1. Exploration

In this phase, The Harris Hawks update their position based on two strategies with equal chance P. If p < 0.5, the position is updated based on the position of another family member. If p > 0.5, the Harris Hawks perch on random tall trees and wait to find prey. These two strategies are modeled in equation (1).

$$X\left(t+1\right)=\{\begin{array}{l}{X}_{rand}\left(t\right)-r_1\left|X_{rand}\left(t\right)-2r_{1}X\left(t\right)\right|P\ge 0.5\hfill \\ \left(X_{rabbit}\left(t\right)-X_m\left(t\right)\right)-r_3\left(Lb+r_4\left(Ub-Lb\right)\right)P<0.5\hfill \end{array}$$ (12)

where $X\left(t+1\right)$ is the position of a hawk in the next iteration, $X\left(t\right)$ is the position of hawk of the current iteration, $X_{rand}\left(t\right)$ is a randomly selected hawk from the current population, $X_{rabbit}\left(t\right)$ is the position of the intended prey, $r_1$, $r_2$, $r_3$, and $r_4$ are random numbers in the range [0,1]. Lb and Ub are the lower and upper bounds, $X_m\left(t\right)$ is the average position vector of all hawks in the population, which can be calculated as follows:

$$X_m\left(t\right)=\frac{1}{N}\sum _{i=1}^N{X}_i\left(t\right)$$ (13)

where N is the total number of hawks in the population, t refers to the current iteration, $X_i$ represents the position of the ith hawk in the population.

4.2. The transition from exploration to exploitation

While the prey is escaping from the attack, its energy continuously decreases. The energy can be modeled as follows:

$$E=2E_0\left(1-\frac{t}{T}\right)$$ (14)

where E refers to the escaping energy of the prey at each iteration. $E_0$ is the initial energy of the prey, which varies between −1 and 1, and T is the maximum number of iterations. When $\left|E\right|\ge 1$, the algorithm is in the exploration phase, where the hawks explore different regions. In contrast, when $\left|E\right|\le 1$, the algorithm is in the exploitation phase.

4.3. Exploitation phase

The exploitation phase can be modeled based on the prey's ability to escape from being hunted and the chasing approaches of the Hawks. If a random number r is less than 0.5, the rabbit has successfully run away. Otherwise, the rabbit failed to escape. On the other hand, the behavior of Harris Hawks depends on the prey's escaping energy E. If $\left|E\right|\ge 0.5$, the Hawks perform soft besiege; otherwise, they perform hard to besiege. Four stages are considered to simulate this phase, as follows:

4.4. Soft besiege

This stage simulated the case in which the rabbit failed to escape from attack ($r\ge 0.5$), although it had enough energy ($\left|E\right|\ge 0.5$). In this case, the Hawks update their location according to the following equations:

$$X\left(t+1\right)=\Delta X\left(t\right)-E\left|J{X}_{prey}\left(t\right)-X\left(t\right)\right|$$ (15)

$$\Delta X\left(t\right)=X_{prey}\left(t\right)-X\left(t\right)$$

$$J=2(1-r_5)$$

The $\Delta X\left(t\right)$ refers to the difference between the rabbit's position and the hawk's current location at iteration t. The J represents the power of the prey's jump while escaping. Finally, $r_5$ is a random number in the range [0, 1].

4.5. Hard besiege

The Hawks perform hard besiege when $r\ge 0.5$ and $\left|E\right|<0.5$ means that the prey becomes very tired and has not enough energy to escape. This case can be modeled as follows:

$$X\left(t+1\right)=X_{prey}\left(t\right)-E\left|\Delta X\left(t\right)\right|$$ (16)

4.6. Soft besiege with progressive rapid dives

In this case, $r<0.5$ and $\left|E\right|\ge 0.5,$ which means that the prey has enough energy to escape, and the Hawks perform soft besiege and find their next position by the following equation:

$$Y=X_{prey}\left(t\right)-E\left|J{X}_{prey}\left(t\right)-X\left(t\right)\right|$$ (17)

If the position is not improved, team rapid dives based on levy flight will be executed as follows:

$$Z=Y+S\times LF\left(D\right)$$ (18)

where S is a randomly selected vector of dimension D. LF represents the levy flight function and can be calculated as follows:

$$LF\left(D\right)=0.01\times \frac{\mu \times \sigma }{{\left|\vartheta \right|}^{\frac{1}{\beta }}}$$

$$\sigma ={\left(\frac{\mathrm{\Gamma }(1+\mathrm{\beta })\times \sin \left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+\mathrm{\beta }}{2}\right)\times \mathrm{\beta }\times 2^{\left(\frac{\beta -1}{2}\right)}}\right)}^{\frac{1}{\beta }}$$

This stage can be summarized as follows:

$$X\left(t+1\right)=\{\begin{array}{l}YF\left(Y\right)<F\left(X\left(t\right)\right)\hfill \\ ZF\left(z\right)<F\left(X\left(t\right)\right)\hfill \end{array}$$ (19)

4.7. Hard besiege with progressive rapid dives

In this case, $r<0.5$ and $\left|E\right|<0.5,$ which means that the prey does not have enough energy to escape, and a hard besiege is executed by the Hawks as follows:

$$X\left(t+1\right)=\{\begin{array}{l}YF\left(Y\right)<F\left(X\left(t\right)\right)\hfill \\ ZF\left(z\right)<F\left(X\left(t\right)\right)\hfill \end{array}$$ (20)

where

$$Y=X_{prey}\left(t\right)-E\left|J{X}_{prey}\left(t\right)-X_m\left(t\right)\right|$$

$$Z=Y+S\times LF\left(D\right)$$

The $X_m\left(t\right)$ is the mean location defined in equation (13). The pseudocode of the HHOA algorithm is shown in Fig. 3 .

Fig. 3.

Pseudocode of HHOA algorithm.

5. The proposed hybrid algorithm (COVID-HO)

Because of some limitations in each metaheuristic, researchers tend to use a hybridization strategy to improve the performance and overcome these limitations. According to the execution order, metaheuristic hybridization is classified into sequential and parallel [48]. In Sequential hybridization, the output of the first algorithm is used as input to the second. On the other side, parallel hybridization approaches similarly apply the algorithms. Although most of the proposed hybrid metaheuristics are sequential, researchers of these sequential metaheuristics indicate the parallelization of their algorithms as future work because parallel hybrid metaheuristics are more suited to parallel computer environments [49].

Although HHOA and COVID algorithms effectively solve various engineering problems, each one separately can suffer from limitations such as falling into local optimum in HHOA and being time-consuming in COVID. This paper combines the two algorithms to produce a parallel hybrid COVID-HHOA algorithm to minimize the drawbacks of both algorithms and benefit from their advantages. It then solves the thresholding problem using Otsu's method and Kapur's entropy as objective functions.

The proposed hybrid algorithm can be summarized as follows:

• 1.The population of solutions of size nPop is randomly initialized.
• 2.The fitness function is calculated for each solution, and the solution with the best fitness is set as the rabbit location.
• 3.The population is divided into two sub-populations such that solutions from 1 to $\frac{N}{2}$ The initial population is considered the first sub-population, and the remaining solutions are considered the second subpopulation.
• 4.The first sub-population is assigned to the COVID algorithm, and the second subpopulation is assigned to the HHOA algorithm.
• 5.The two algorithms operate in parallel to produce two updated sub-populations.
• 6.The two new sub-populations are combined into one population.
• 7.Steps 2–6 are repeated until the maximum number of iterations is reached.

The flow chart of the proposed COVID-HHOA algorithm for multi-thresholding segmentation for 2D images is shown in Fig. 4 .

Fig. 4.

Flowchart of the proposed COVID- HHOA algorithm.

In the case of 3D medical images, the image is divided into several 2D slices, and each slice is segmented using the proposed algorithm. The segmented slices are concatenated together to form the segmented 3D image, as shown in Fig. 5 .

Fig. 5.

Diagram of multilevel thresholding image segmentation for 3D images using the proposed COVID-HHOA algorithm.

The advantages of medical image segmentation using our proposed approach are as follows:

• 1.Segmentation has a crucial role in medical imaging as it helps to separate the objects of interest from the whole body, simplifying medical decisions.
• 2.The segmentation approach, which gives perfect results for one type of imaging modality, might not even work for another. Our proposed approach achieved very good segmentation results for several medical images from different modalities (CT, MRI, and X-ray).
• 3.The proposed approach makes parallel hybridization (which is more suited to parallel computer environments) between two powerful metaheuristics (COVIDOA and HHOA) for solving segmentation problems.
• 4.It can be applied to any two population-based metaheuristics.
• 5.Its computational time is half the computational time of sequential hybrid metaheuristics because the initial population is divided into two halves, and each algorithm operates on one half in parallel.

To prove the superiority of the proposed Hybrid COVID-HHOA algorithm over the native HHOA and COIVD algorithms, we utilized 6 test functions from IEEE CEC 2019 benchmark problems. These are a group of modern test functions known as “The 100-Digit Challenge” intended to be used in single objective numerical optimization IEEE competitions. The description of these functions in terms of problem dimension, range of possible values, and the global optimum is discussed in Ref. [50].

The convergence curves of the proposed hybrid algorithm, HHOA, and COVID algorithms are presented in Fig. 6 . The figure clearly shows the superiority of the proposed hybrid algorithm as it reaches the optimum fitness values compared to COVID and HHOA algorithms. The pseudocode of the proposed COVID-HHOA algorithm for multilevel thresholding is shown in Fig. 7 .

Fig. 6.

Comparison of convergence curves of hybrid COVID-HHOA, HHOA, and COVID algorithms using CEC 2019 test problems.4.

Fig. 7.

Pseudocode of the proposed COVID_HHOA optimization algorithm for multilevel thresholding.

6. Experimental results and discussion

In this section, we firstly provide a brief description of the medical datasets used for testing. Then, we show the parameter settings for the proposed and state-of-the-art algorithms. After that, the evaluation metrics used for comparing the results are explained in detail. Then, we present the numerical results obtained from running the proposed algorithm and its peers. Finally, we conducted a comparative analysis of the obtained results.

6.1. Datasets

6.1.1. 2D medical images

We used six 2D medical images to prove the efficiency of the proposed algorithm in medical image segmentation. Medical1, Medical2, and Medical4 images are MRI images, Medical3 are X-ray images, and Medical5 and Medical6 are CT images. These images have many variations, such as size, resolution, and modality. The images and their histograms are shown in Table 1 .

Table 1.

Test images and their histograms.

6.1.2. Volumetric (3D) medical images

The advanced medical imaging technologies available today, such as MRI, CT, X-ray, and ultrasound, make it possible to view the detailed structure of human anatomy by acquiring efficient volumetric medical images. These images make it easier for medical experts to examine, detect, and diagnose diseases. Volumetric medical images can be represented by a group of 2D image slices [51], as shown in Fig. 8 . A set of 3D medical images are selected for testing from the open-source dataset available in Ref. [52]. The selected image slices are shown in Fig. 9 .

Fig. 8.

3D (volumetric) medical image.

Fig. 9.

Original volumetric medical images used for testing.

6.2. Parameter setting

The results of multilevel thresholding using the proposed algorithm are compared with those of seven well-known metaheuristic algorithms in different evaluation criteria. These algorithms are: Harris Hawks Optimization Algorithm (HHOA) [35], Bat Algorithm (BA) [23], Harmony Search Algorithm (HSA) [15], Cuckoo Search Algorithm (CSA) [19], Sine Cosine Algorithm (SCA) [22], Flower Pollination Algorithm (FPA) [18], and Seagull Optimization Algorithm (SOA) [45].

The reasons for selecting these algorithms for comparison are as follows:

• ⁃They have proved their superior performance in solving various optimization problems, especially image segmentation.
• ⁃Most of them are recent and published in reputable sources.
• ⁃Their MATLAB implementations are publicly available on the MATLAB website (https://www.mathworks.com/).

All the experiments were run on a laptop with the following specifications: Intel(R) Core(TM) i7-1065G7 processor, RAM of 8.0 GB size, and Windows 10 Ultimate 64-bit operating system. All the algorithms are developed using MATLAB R2016a development environment.

6.3. Performance metrics

The performance of the proposed algorithm is evaluated using several performance metrics, including Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), Normalized Correlation Coefficient (NCC), and best fitness, in addition to the Wilcoxon rank-sum test.

PSNR, SSIM, and NCC are used to measure the quality of the segmented images, while best fitness is measured to prove the ability of the proposed algorithm to find optimum solutions, and the Wilcoxon rank-sum test is utilized to prove the statistical significance of the proposed algorithm as follows:

• a)Best Fitness

The maximum fitness obtained from running the proposed ad state-of-the-art algorithms with Otsu's method and Kapur's entropy functions is measured using equations (3), (7).

• b)Peak signal-to-noise ratio (PSNR)

PSNR is commonly used to quantify the quality of images. It refers to the ratio between the segmented image power and noise power. PSNR for 2D and 3D images is calculated as follows:

$$PSNR=10Lo{g}_{10}\left(\frac{{255}^2}{MSE}\right)$$ (21)

where $MSE$ of a 2D image is calculated as follows:

$$MSE=\frac{1}{M\times N}\sum _{i=1}^M\sum _{j=1}^N{\left[F\left(i,j\right)-f(i,j)\right]}^2$$ (22a)

F(i,j) is the original image, f(i,j) is the segmented image, and $M\times N$ refers to the 2D image size.

For 3D images, the MSE is calculated as follows:

$$MSE=\frac{1}{M\times N\times L}\sum _{i=1}^M\sum _{j=1}^N\sum _{k=1}^L{\left[F\left(i,j,k\right)-f(i,j,k)\right]}^2$$ (22b)

where F(i,j,k) is the original 3D image, f(i,j,k) is the segmented 3D image, and $M\times N\times L$ refers to the size of the 3D image.

• c)Structural similarity index (SSIM)

SSIM is used to quantify the structural similarity between the original and segmented images; the SSIM for 2D and 3D images is calculated as follows:

$$SSIM(F,f)=\frac{(2{\mu }_F{\mu }_f+C_1)(2{\sigma }_{Ff}+C_2)}{({\mu }_F^2{\mu }_f^2+C_1)({\sigma }_F^2{\sigma }_f^2+C_2)}$$ (23)

The F and f refer to the original and segmented images. The ${\mu }_F$ and ${\mu }_f$ are the mean intensity of F and f, while ${\sigma }_F^2$ and ${\sigma }_f^2$ refer to the variance of F and f, respectively. The values of C ₁ = 6.502 and C2 = 58.522 are used.

• d)Normalized correlation coefficient (NCC)

NCC is used to measure the extent to which two images are related. The absolute value of NCC ranges from 0 to 1, where 0 indicates that the two images have no relation and 1 indicates the strongest possible relation. The higher the absolute value of NCC, the stronger the relationship between the two images.

NCC between the original and segmented 2D images F(i,j) and f(i,j) is calculated as follows:

$$NCC=\frac{{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}(F(i,j)\times f(i,j))}{\sqrt{{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}(F(i,j)\times F(i,j))\times {\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}(f(i,j)\times f(i,j))}}$$ (24)

where $M\times N$ is the size of the 2D image.

The NCC between two 3D images F(i,j,k) and f(i,j,k) is calculated as follows:

$$NCC=\frac{{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}{\sum }_{k=0}^{L-1}(F(i,j,k)\times f(i,j,k))}{\sqrt{{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}{\sum }_{k=0}^{L-1}(F(i,j,k)\times F(i,j,k))\times {\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}{\sum }_{k=0}^{L-1}(f(i,j,k)\times f(i,j,k))}}$$

• e)Wilcoxon rank-sum test

The Wilcoxon rank-sum test is a non-parametric statistical test used to measure the statistical difference between two related methods [51]. We conducted the Wilcoxon rank-sum test with a 5% significance level to prove the proposed algorithm's statistical significance compared to the other algorithms.

6.4. Results

This section presents the numerical results of running the proposed algorithm to select the optimum threshold values using Otsu's method and Kapur's Entropy. These results are compared with the state-of-the-art algorithms regarding best fitness, PSNR, SSIM, NCC, and Wilcoxon rank-sum test. This section is divided into two subsections for presenting the results of using the proposed algorithm to segment 2D and 3D medical images as follows:

6.4.1. Experimental results for 2D medical images

Table 2, Table 3 show the segmentation results of the 2D medical test images using the proposed COVID-HHOA algorithm. These images are segmented using Otsu's method and Kaur's Entropy fitness functions at 6, 8, 10, 12, and 14 threshold levels.

Table 2.

Medical Images segmented by the proposed COVID-HHOA algorithm using Otsu's method.

Table 3.

Medical Images segmented by the proposed COVID-HHOA algorithm using Kapur's entropy method.

Table 4, Table 5 show the PSNR values produced by the proposed and state-of-the-art algorithms for all test images using Otsu's method and Kapur's entropy at different threshold levels. It's shown from the tables that the proposed algorithm has the highest PSNR in 59 from 60 cases.

Table 4.

PSNR results of Otsu's method for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	28.0241	28.0015	27.8603	27.9327	25.9398	27.6060	27.4521	28.1034
	8	29.5159	30.5702	29.8950	30.5846	29.4844	28.8106	30.0270	30.6115
	10	31.0962	32.0855	31.4062	31.1762	30.3947	29.7557	31.2010	32.3208
	12	32.2296	33.6432	31.1533	31.7412	31.2790	30.6256	33.1004	33.8829
	14	32.5824	34.6792	33.6707	33.0175	31.8011	32.6048	34.0983	35.1157
Medical2	6	23.8976	24.3012	23.9940	24.2262	22.7669	22.6919	22.7570	24.3882
	8	26.4672	26.5729	26.0080	26.5961	25.0116	24.2180	26.1634	26.6181
	10	27.0786	27.9677	27.0960	27.4578	25.8210	25.6833	27.5117	28.1228
	12	28.8093	29.4139	28.9450	29.3669	28.4395	27.3844	28.9853	29.6817
	14	29.2411	30.7983	29.7738	29.3305	28.4481	28.4797	29.8231	30.9323
Medical3	6	24.0112	24.5539	24.5079	24.5771	23.6808	22.7121	23.8876	24.5965
	8	25.9225	26.0874	25.5731	25.9562	24.9593	24.0463	25.5063	26.1987
	10	27.7211	28.0173	26.7435	27.8348	26.5461	26.1655	27.4362	28.1225
	12	28.4870	29.2997	29.4472	28.8344	27.0262	27.2857	27.4541	29.6674
	14	29.9596	30.7411	29.8566	30.1048	28.9854	29.0777	30.5004	31.1412
Medical4	6	26.2450	26.5800	25.7530	26.0731	24.7475	25.8652	25.6488	26.6784
	8	26.5107	28.3621	27.5579	27.7016	26.5912	27.6891	28.0003	28.4004
	10	29.2529	30.3418	29.8796	28.3021	28.2514	28.7927	28.6717	30.3635
	12	30.1704	31.3994	30.3462	30.4476	28.8059	29.2508	29.5124	31.6963
	14	31.0825	32.3168	31.8551	31.5092	30.4896	29.6222	32.2716	32.8176
Medical5	6	24.7579	25.7082	25.6246	25.7676	23.1043	25.1616	24.7941	25.9592
	8	27.7621	27.7409	27.9747	28.0629	27.3604	26.3591	27.0363	28.2565
	10	28.3140	29.4341	28.8750	29.4337	28.1992	28.5822	28.2220	29.7908
	12	29.3404	30.7092	29.7841	31.0181	29.8806	29.6569	29.5604	31.8339
	14	31.1842	32.3781	32.2351	32.6095	31.9152	31.9355	30.4210	33.0516
Medical6	6	24.4866	24.8421	24.5311	24.9414	24.3269	24.4083	23.3405	25.1480
	8	25.9630	27.2037	25.9113	26.8320	24.4700	25.8269	26.4225	27.6991
	10	27.2134	28.9124	28.6518	29.0542	26.9230	27.7604	28.4883	29.7148
	12	29.0199	30.6417	29.9051	30.1581	29.1440	28.4976	29.7973	31.1191
	14	29.7714	31.6065	31.2646	31.3577	30.5977	30.6166	31.0194	32.1560

Table 5.

PSNR results of Kapur's entropy for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	25.2524	25.4433	26.3512	24.5522	25.2684	25.2804	24.7321	25.4509
	8	26.4326	26.9740	27.3109	27.0291	27.2530	26.2667	26.7912	27.5253
	10	28.0860	28.5479	27.7927	27.2030	28.0816	28.5909	28.6896	28.8268
	12	30.3107	30.4151	29.7459	29.3509	28.9154	29.4261	29.5701	30.5083
	14	31.4768	31.0771	30.1544	30.5344	30.5967	31.2580	31.0262	31.9442
Medical2	6	21.5328	21.0665	19.9647	21.1612	20.7791	20.3524	20.3740	21.5760
	8	22.7437	22.9694	22.0828	23.6234	23.4112	22.8048	23.3296	23.4328
	10	25.1282	24.7891	24.9807	23.7998	25.1116	24.8474	24.0673	25.6612
	12	26.9097	26.2715	26.4183	25.8614	26.0482	25.5683	27.0147	27.6624
	14	27.4481	28.6800	28.5732	26.2504	26.6410	27.8178	28.7076	29.1074
Medical3	6	21.4500	21.5384	21.4728	21.5716	20.7611	21.1768	20.9559	21.5922
	8	23.6874	24.0349	23.3270	24.1519	24.1593	23.7441	23.2662	24.1918
	10	25.9861	26.4415	24.6536	26.7867	24.5258	26.1975	25.3053	26.7991
	12	26.8566	28.0038	25.9627	27.1257	25.8697	26.9955	28.3338	28.5621
	14	26.8949	29.1298	27.4718	29.0164	27.2153	27.4595	28.8884	29.8025
Medical4	6	21.8727	22.1638	21.4247	21.0408	21.8694	21.5044	22.0440	22.3226
	8	24.8756	24.4229	23.8503	23.4907	23.8887	24.3239	23.9722	24.9068
	10	25.8773	25.6581	24.4352	25.2046	25.9492	25.9190	24.7383	26.3789
	12	28.4434	27.5950	25.5445	25.5175	27.1250	27.4470	28.1482	28.4808
	14	28.9051	28.8699	28.2158	28.0796	28.3505	28.8813	28.7985	29.3621
Medical5	6	22.4242	22.5562	21.5969	22.9651	20.6772	22.5832	21.5968	22.9713
	8	25.3659	24.8873	24.6034	25.6448	24.4785	25.3410	24.9054	26.2635
	10	27.6808	27.4961	25.9944	27.6617	28.0819	27.3302	26.8027	28.3719
	12	28.2827	28.4007	27.4091	28.7241	28.3098	28.2413	28.7864	29.7731
	14	30.7075	31.1078	30.0344	30.8091	29.4999	28.8087	30.1128	32.0764
Medical6	6	23.2536	23.7728	20.8852	24.8318	21.1701	23.6189	22.5869	23.7954
	8	24.5919	25.7041	22.5375	25.7105	23.7390	25.3574	25.4927	25.7812
	10	26.3945	26.3455	25.6227	26.6879	24.1615	25.8539	26.5609	27.2621
	12	26.7619	27.7460	26.9430	27.7070	27.1467	27.0896	27.2924	28.0202
	14	27.6779	28.3430	28.1736	28.2404	27.7577	27.9679	27.8562	28.6281

The SSIM results of Otsu's method and Kapur's entropy are presented in Table 6, Table 7 . The closer the SSIM values to 1, the better the quality of the segmented images. The tables show that the proposed algorithm has superior results in 27 from 30 cases in the case of Otsu's method; however, it has the best results in 21 from 30 cases in the case of Kapur's entropy.

Table 6.

SSIM results of Otsu's method for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.5815	0.5817	0.5796	0.5808	0.5466	0.5733	0.5945	0.5827
	8	0.6115	0.6237	0.6153	0.6222	0.6024	0.6988	0.7371	0.6256
	10	0.7519	0.8630	0.7505	0.6388	0.8533	0.7315	0.8871	0.9053
	12	0.7743	0.9055	0.7562	0.7539	0.9054	0.7549	0.9051	0.9190
	14	0.7841	0.9169	0.8659	0.9063	0.9120	0.7652	0.9147	0.9532
Medical2	6	0.5941	0.6004	0.5943	0.5998	0.5902	0.5892	0.5861	0.6004
	8	0.6709	0.6735	0.6688	0.6647	0.6527	0.6739	0.6698	0.6816
	10	0.7340	0.7339	0.7203	0.7299	0.6956	0.7192	0.7336	0.7342
	12	0.7607	0.7722	0.7427	0.7508	0.7195	0.7513	0.7757	0.7735
	14	0.7848	0.8087	0.7980	0.7923	0.7708	0.7729	0.8280	0.8151
Medical3	6	0.8261	0.8259	0.7993	0.7977	0.8051	0.8299	0.8399	0.8768
	8	0.6902	0.6957	0.6951	0.6961	0.6827	0.6859	0.6817	0.6961
	10	0.7996	0.8219	0.8125	0.8197	0.7072	0.7128	0.7797	0.8257
	12	0.8185	0.8550	0.8340	0.8528	0.8352	0.7662	0.8356	0.8562
	14	0.8497	0.8811	0.8738	0.8719	0.8374	0.8307	0.8571	0.8827
	6	0.8688	0.9032	0.8820	0.8910	0.8688	0.8533	0.8984	0.9038
Medical4	8	0.6929	0.6939	0.6762	0.6854	0.6508	0.6775	0.6835	0.6946
	10	0.7199	0.8297	0.7206	0.7223	0.7474	0.7570	0.7935	0.8412
	12	0.8475	0.8793	0.8022	0.7453	0.7929	0.7857	0.7987	0.8848
	14	0.8498	0.8854	0.8211	0.7895	0.8438	0.7948	0.8302	0.9090
	6	0.8748	0.8891	0.8728	0.8721	0.8835	0.8017	0.9020	0.9236
Medical5	8	0.8840	0.9021	0.9022	0.9020	0.8587	0.8741	0.8841	0.9029
	10	0.9355	0.9361	0.9367	0.9367	0.9201	0.9237	0.9279	0.9373
	12	0.9477	0.9541	0.9452	0.9547	0.9256	0.9327	0.9546	0.9551
	14	0.9484	0.9649	0.9596	0.9580	0.9468	0.9529	0.9624	0.9655
	6	0.9552	0.9726	0.9679	0.9692	0.9607	0.9573	0.9711	0.9739
Medical6	8	0.7027	0.8478	0.8118	0.8183	0.8683	0.6648	0.7568	0.8672
	10	0.7217	0.8651	0.8401	0.8784	0.8715	0.6813	0.8186	0.8859
	12	0.7654	0.8799	0.8577	0.8900	0.8730	0.8025	0.8215	0.9037
	14	0.8159	0.8948	0.8660	0.8987	0.8996	0.8749	0.8490	0.9239
	K	0.8440	0.9013	0.9153	0.9146	0.9407	0.9246	0.9061	0.9451

Table 7.

SSIM results of Kapur's entropy for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.5585	0.5633	0.5626	0.5126	0.5394	0.5548	0.5601	0.5641
	8	0.5823	0.5806	0.5638	0.5801	0.5785	0.5732	0.5779	0.5855
	10	0.5989	0.5995	0.5909	0.5816	0.5871	0.5956	0.5973	0.6048
	12	0.6166	0.6217	0.6054	0.6027	0.6008	0.6082	0.6253	0.6244
	14	0.6392	0.6342	0.6336	0.6354	0.6169	0.6332	0.6345	0.6447
Medical2	6	0.5761	0.5558	0.5814	0.6086	0.6108	0.5853	0.5688	0.5788
	8	0.6565	0.6564	0.6581	0.6488	0.6691	0.6681	0.6533	0.6720
	10	0.7200	0.7021	0.6883	0.6527	0.7017	0.6964	0.7025	0.7270
	12	0.7445	0.7422	0.7418	0.7230	0.7038	0.7456	0.7599	0.7649
	14	0.7577	0.7848	0.7718	0.7506	0.7567	0.7740	0.7807	0.7907
Medical3	6	0.6471	0.6464	0.6453	0.6352	0.6387	0.6397	0.6513	0.6482
	8	0.6888	0.6949	0.6910	0.6918	0.6952	0.6841	0.6950	0.6985
	10	0.7388	0.7480	0.7097	0.7437	0.7085	0.7379	0.7385	0.7477
	12	0.7523	0.7687	0.7408	0.7579	0.7409	0.7634	0.7748	0.7789
	14	0.7899	0.7924	0.7499	0.7897	0.7800	0.7671	0.8067	0.8424
Medical4	6	0.5623	0.5478	0.5994	0.5531	0.5854	0.5529	0.5346	0.5566
	8	0.6601	0.6193	0.6186	0.6129	0.6284	0.6478	0.6628	0.6614
	10	0.6743	0.6872	0.6274	0.6773	0.6901	0.6729	0.6712	0.7033
	12	0.7491	0.7311	0.6919	0.7003	0.7278	0.7297	0.7454	0.7537
	14	0.7764	0.7603	0.7676	0.7382	0.7320	0.7659	0.7666	0.7777
Medical5	6	0.8266	0.8357	0.8094	0.8598	0.8313	0.8300	0.8326	0.8385
	8	0.8763	0.8703	0.8681	0.8831	0.8771	0.8879	0.8814	0.8913
	10	0.9187	0.9128	0.8942	0.9206	0.9227	0.9109	0.9057	0.9233
	12	0.9262	0.9267	0.9235	0.9321	0.9322	0.9146	0.9305	0.9376
	14	0.9453	0.9543	0.9481	0.9518	0.9387	0.9296	0.9461	0.9613
Medical6	6	0.6015	0.6064	0.5905	0.6248	0.6047	0.6142	0.6019	0.6101
	8	0.6299	0.6296	0.5931	0.6347	0.6115	0.6281	0.6309	0.6325
	10	0.6421	0.6379	0.6299	0.6445	0.6189	0.6410	0.6459	0.6485
	12	0.6513	0.6561	0.6566	0.6603	0.6398	0.6502	0.6531	0.6627
	14	0.6655	0.6663	0.6592	0.6642	0.6625	0.6542	0.6625	0.6717

In addition to PSNR and SSIM results, the NCC results are presented to prove the high quality of the segmented images produced by the proposed algorithm. The NCC values produced by all algorithms for all 2D images using Otsu's method and Kapur's entropy are shown in Table 8, Table 9 . It can be seen that the proposed algorithm exceeds all state-of-the-art algorithms in terms of NCC values in all cases except one case in Otsu's method results and 3 cases in Kapur's entropy results.

Table 8.

NCC results of Otsu's method for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	4	0.9727	0.9734	0.9722	0.9734	0.9730	0.9723	0.9855	0.9734
	6	0.9862	0.9862	0.9858	0.9860	0.9866	0.9851	0.9867	0.9868
	8	0.9921	0.9934	0.9917	0.9930	0.9893	0.9857	0.9919	0.9937
	10	0.9945	0.9948	0.9927	0.9946	0.9920	0.9914	0.9937	0.9954
	12	0.9952	0.9964	0.9946	0.9956	0.9942	0.9939	0.9961	0.9968
	14	0.9961	0.9972	0.9960	0.9963	0.9949	0.9952	0.9968	0.9975
Medical2	6	0.9894	0.9900	0.9890	0.9894	0.9854	0.9844	0.9854	0.9905
	8	0.9933	0.9942	0.9931	0.9941	0.9900	0.9899	0.9936	0.9944
	10	0.9949	0.9958	0.9948	0.9950	0.9926	0.9927	0.9950	0.9961
	12	0.9965	0.9970	0.9966	0.9967	0.9956	0.9942	0.9966	0.9972
	14	0.9966	0.9977	0.9968	0.9970	0.9954	0.9957	0.9972	0.9979
Medical3	6	0.9900	0.9918	0.9914	0.9918	0.9913	0.9892	0.9918	0.9920
	8	0.9945	0.9945	0.9938	0.9941	0.9929	0.9916	0.9940	0.9947
	10	0.9957	0.9961	0.9940	0.9959	0.9946	0.9950	0.9961	0.9966
	12	0.9965	0.9970	0.9971	0.9963	0.9954	0.9956	0.9964	0.9977
	14	0.9974	0.9980	0.9974	0.9977	0.9970	0.9969	0.9979	0.9983
Medical4	6	0.9871	0.9893	0.9863	0.9877	0.9838	0.9866	0.9870	0.9893
	8	0.9889	0.9921	0.9914	0.9920	0.9889	0.9911	0.9918	0.9925
	10	0.9938	0.9949	0.9944	0.9924	0.9924	0.9933	0.9930	0.9951
	12	0.9943	0.9963	0.9956	0.9952	0.9934	0.9938	0.9944	0.9965
	14	0.9960	0.9970	0.9965	0.9966	0.9949	0.9948	0.9970	0.9973
Medical5	6	0.9918	0.9934	0.9929	0.9934	0.9916	0.9912	0.9922	0.9938
	8	0.9956	0.9960	0.9959	0.9957	0.9934	0.9930	0.9958	0.9964
	10	0.9963	0.9970	0.9965	0.9966	0.9957	0.9961	0.9957	0.9974
	12	0.9965	0.9979	0.9971	0.9978	0.9975	0.9967	0.9974	0.9983
	14	0.9984	0.9985	0.9983	0.9983	0.9982	0.9982	0.9976	0.9989
Medical6	6	0.9900	0.9911	0.9887	0.9917	0.9849	0.9909	0.9831	0.9919
	8	0.9933	0.9939	0.9904	0.9934	0.9894	0.9928	0.9932	0.9961
	10	0.9943	0.9959	0.9959	0.9958	0.9923	0.9943	0.9962	0.9973
	12	0.9965	0.9976	0.9966	0.9968	0.9964	0.9962	0.9972	0.9980
	14	0.9977	0.9980	0.9979	0.9975	0.9972	0.9976	0.9979	0.9984

Table 9.

NCC results of Kapur's entropy for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.9759	0.9796	0.9800	0.9750	0.9750	0.9781	0.9755	0.9801
	8	0.9855	0.9841	0.9818	0.9850	0.9842	0.9846	0.9847	0.9864
	10	0.9896	0.9894	0.9873	0.9872	0.9886	0.9895	0.9893	0.9905
	12	0.9931	0.9935	0.9909	0.9890	0.9919	0.9913	0.9930	0.9938
	14	0.9947	0.9946	0.9923	0.9912	0.9931	0.9941	0.9947	0.9956
Medical2	6	0.9767	0.9724	0.9751	0.9756	0.9765	0.9747	0.9728	0.9774
	8	0.9865	0.9872	0.9832	0.9798	0.9871	0.9868	0.9874	0.9885
	10	0.9918	0.9912	0.9912	0.9847	0.9908	0.9899	0.9894	0.9929
	12	0.9944	0.9946	0.9934	0.9854	0.9913	0.9921	0.9950	0.9959
	14	0.9942	0.9966	0.9959	0.9935	0.9937	0.9942	0.9966	0.9970
Medical3	6	0.9863	0.9866	0.9861	0.9866	0.9843	0.9863	0.9861	0.9867
	8	0.9922	0.9929	0.9909	0.9948	0.9920	0.9914	0.9923	0.9929
	10	0.9955	0.9956	0.9912	0.9959	0.9925	0.9945	0.9947	0.9959
	12	0.9959	0.9969	0.9941	0.9969	0.9935	0.9948	0.9969	0.9972
	14	0.9964	0.9976	0.9961	0.9974	0.9953	0.9962	0.9974	0.9978
Medical4	6	0.9709	0.9691	0.9779	0.9693	0.9755	0.9679	0.9663	0.9719
	8	0.9843	0.9821	0.9789	0.9728	0.9824	0.9822	0.9842	0.9855
	10	0.9877	0.9877	0.9805	0.9845	0.9864	0.9870	0.9856	0.9895
	12	0.9932	0.9924	0.9858	0.9892	0.9909	0.9913	0.9934	0.9939
	14	0.9942	0.9942	0.9926	0.9924	0.9916	0.9926	0.9942	0.9950
Medical5	6	0.9870	0.9857	0.9840	0.9916	0.9844	0.9843	0.9839	0.9866
	8	0.9934	0.9920	0.9907	0.9950	0.9931	0.9925	0.9942	0.9946
	10	0.9954	0.9954	0.9929	0.9960	0.9953	0.9953	0.9953	0.9964
	12	0.9962	0.9962	0.9962	0.9969	0.9963	0.9962	0.9970	0.9975
	14	0.9978	0.9980	0.9962	0.9980	0.9972	0.9972	0.9976	0.9983
Medical6	6	0.9871	0.9882	0.9765	0.9913	0.9828	0.9879	0.9850	0.9884
	8	0.9916	0.9920	0.9789	0.9920	0.9852	0.9898	0.9911	0.9923
	10	0.9932	0.9929	0.9893	0.9933	0.9861	0.9911	0.9932	0.9942
	12	0.9937	0.9946	0.9935	0.9946	0.9922	0.9934	0.9941	0.9950
	14	0.9945	0.9952	0.9945	0.9950	0.9936	0.9944	0.9947	0.9955

The relationship between the number of thresholds and PSNR, SSIM, and NCC for Otsu's method and Kapur's entropy is shown in Fig. 10 to Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15 , respectively. Each one is split into two figures (a and b) for more clarity.

Fig. 10.

(a): PSNR results of Otsu's method for all algorithms at each threshold level for Medical1, Medical2, and Medical3 images.

(b): PSNR results of Otsu's method for all algorithms at each threshold level for Medical4, Medical5, and Medical6 images.

Fig. 11.

(a): PSNR results of Kapur's entropy for all algorithms at each threshold level for Medical1. Medical2 and Medical3 images.

(b): PSNR results of Kapur's entropy for all algorithms at each threshold level for Medical4, Medical5, and Medical6 images.

Fig. 12.

(a): SSIM results of Otsu's method for all algorithms at each threshold level for Medical1, Medical2, and Medical3 images.

(b): SSIM results of Otsu's method for all algorithms at each threshold level for Medical4, Medical5, and Medical6 images.

Fig. 13.

(a): SSIM results of Kapur's entropy for all algorithms at each threshold level for Medical1, Medical2, and Medical3 images.

(b): SSIM results of Kapur's entropy for all algorithms at each threshold level for Medical4, Medical5, and Medical6 images.

Fig. 14.

(a): NCC results from Otsu's method for all algorithms at each threshold level for Medical1, Medical2, and Medical3 images.

Fig. 14 (b): NCC results from Otsu's method for all algorithms at each threshold level for Medical4, Medical5, and Medical6 images.

Fig. 15.

(a): NCC results in Kapur's entropy for all algorithms at each threshold level for Medical1, Medical2, and Medical3 images.

(b): NCC results in Kapur's entropy for all algorithms at each threshold level for Medical4, Medical5, and Medical6 images.

According to the fitness function, we compared the proposed algorithm to the other algorithms in terms of the best fitness value obtained from running each algorithm 30 times with Otsu's method and Kapur's entropy. Higher fitness function values indicate higher quality of the solutions produced by the algorithm. The best fitness values for all algorithms using Otsu's method and Kapur's entropy are shown in Table 10, Table 11 . The relationship between the thresholds and the fitness values is shown in Fig. 16 (a, b) and 17 (a, b) for Otsu's method and Kapur's entropy, respectively. Although all algorithms have relative fitness values, the proposed algorithm slightly exceeds them in almost all cases, indicating its ability to find high-quality solutions.

Table 10.

Best fitness values of Ostu's method for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	1.5041e+03	1.5049e+03	1.5041e+03	1.5042e+03	1.4955e+03	1.5016e+03	1.5022e+03	1.5050e+03
	8	1.5101e+03	1.5140e+03	1.5128e+03	1.5135e+03	1.5107e+03	1.5092e+03	1.5130e+03	1.5142e+03
	10	1.5164e+03	1.5191e+03	1.5175e+03	1.5163e+03	1.5161e+03	1.5122e+03	1.5175e+03	1.5195e+03
	12	1.5209e+03	1.5223e+03	1.5207e+03	1.5183e+03	1.5166e+03	1.5159e+03	1.5216e+03	1.5227e+03
	14	1.5210e+03	1.5239e+03	1.5232e+03	1.5209e+03	1.5169e+03	1.5218e+03	1.5235e+03	1.5246e+03
Medical2	6	3.9554e+03	3.9577e+03	3.9566e+03	3.9577e+03	3.9395e+03	3.9312e+03	3.9459e+03	3.9580e+03
	8	3.9808e+03	3.9821e+03	3.9811e+03	3.9817e+03	3.9721e+03	3.9654e+03	3.9805e+03	3.9829e+03
	10	3.9855e+03	3.9930e+03	3.9901e+03	3.9922e+03	3.9752e+03	3.9754e+03	3.9900e+03	3.9940e+03
	12	3.9980e+03	4.0008e+03	3.9988e+03	3.9992e+03	3.9916e+03	3.9913e+03	3.9995e+03	4.0016e+03
	14	4.0010e+03	4.0061e+03	4.0028e+03	4.0020e+03	3.9941e+03	3.9980e+03	4.0032e+03	4.0067e+03
Medical3	6	6.1387e+03	6.1128e+03	6.1383e+03	6.1128e+03	6.1219e+03	6.1240e+03	6.1267e+03	6.1402e+03
	8	6.1603e+03	6.1352e+03	6.1594e+03	6.1337e+03	6.1430e+03	6.1436e+03	6.1539e+03	6.1633e+03
	10	6.1744e+03	6.1482e+03	6.1717e+03	6.1470e+03	6.1652e+03	6.1627e+03	6.1721e+03	6.1757e+03
	12	6.1780e+03	6.1549e+03	6.1804e+03	6.1497e+03	6.1650e+03	6.1711e+03	6.1719e+03	6.1841e+03
	14	6.1809e+03	6.1590e+03	6.1854e+03	6.1564e+03	6.1801e+03	6.1792e+03	6.1863e+03	6.1882e+03
Medical4	6	2.2081e+03	2.2089e+03	2.2077e+03	2.2079e+03	2.1824e+03	2.2009e+03	2.1957e+03	2.2088e+03
	8	2.2108e+03	2.2201e+03	2.2181e+03	2.2174e+03	2.2065e+03	2.2144e+03	2.2193e+03	2.2200e+03
	10	2.2242e+03	2.2280e+03	2.2258e+03	2.2239e+03	2.2181e+03	2.2226e+03	2.2202e+03	2.2285e+03
	12	2.2277e+03	2.2331e+03	2.2292e+03	2.2289e+03	2.2227e+03	2.2231e+03	2.2276e+03	2.2333e+03
	14	2.2322e+03	2.2358e+03	2.2332e+03	2.2328e+03	2.2328e+03	2.2280e+03	2.2353e+03	2.2367e+03
Medical5	6	5.3898e+03	4.8687e+03	5.3931e+03	4.8687e+03	5.3681e+03	5.3812e+03	5.3847e+03	5.3936e+03
	8	5.4061e+03	4.8825e+03	5.4065e+03	4.8824e+03	5.4006e+03	5.4007e+03	5.4038e+03	5.4077e+03
	10	5.4108e+03	4.8893e+03	5.4122e+03	4.8893e+03	5.4034e+03	5.4066e+03	5.4125e+03	5.4145e+03
	12	5.4122e+03	4.8929e+03	5.4161e+03	4.8915e+03	5.4131e+03	5.4115e+03	5.4160e+03	5.4184e+03
	14	5.4160e+03	4.8955e+03	5.4192e+03	4.8941e+03	5.4164e+03	5.4167e+03	5.4186e+03	5.4208e+03
Medical6	6	4.3948e+03	4.2056e+03	4.3961e+03	4.2055e+03	4.3687e+03	4.3865e+03	4.3865e+03	4.3978e+03
	8	4.4059e+03	4.2211e+03	4.4117e+03	4.2207e+03	4.4002e+03	4.4012e+03	4.4113e+03	4.4139e+03
	10	4.4157e+03	4.2287e+03	4.4193e+03	4.2282e+03	4.4129e+03	4.4129e+03	4.4199e+03	4.4209e+03
	12	4.4230e+03	4.2326e+03	4.4230e+03	4.2318e+03	4.4188e+03	4.4153e+03	4.4231e+03	4.4257e+03
	14	4.4246e+03	4.2343e+03	4.4259e+03	4.2340e+03	4.4215e+03	4.4219e+03	4.4265e+03	4.4278e+03

Table 11.

Best fitness values of Kapur's entropy for all algorithms.

Image	K	Algorithms
		SOA [45]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	29.1150	29.1440	28.4842	28.5451	28.4039	28.4511	28.9466	29.1497
	8	36.4624	36.6540	35.1770	35.7679	35.6282	36.1923	36.0033	36.7475
	10	42.8551	43.2945	42.1935	42.2456	41.7597	41.3143	42.9482	44.0402
	12	48.0268	49.6176	46.7989	46.3681	45.9796	46.5262	47.9183	50.1440
	14	53.1311	54.8167	52.5195	52.4872	50.0722	51.4593	53.2681	55.7106
Medical2	6	30.0560	30.0624	29.2634	29.3301	29.1013	29.5610	29.7242	30.1228
	8	37.4817	37.6122	36.3408	36.4637	36.4182	37.0426	37.3252	37.5760
	10	44.1656	44.5172	41.9516	43.6734	42.7747	42.7167	43.6806	44.7035
	12	50.0224	50.2347	49.0840	48.5610	47.7153	49.4137	50.2842	50.9972
	14	55.3181	56.5864	53.2115	53.1273	52.3780	52.7230	55.0156	56.9942
Medical3	6	29.4506	29.4480	28.6691	23.1739	28.4564	29.4011	29.2656	29.4703
	8	37.2324	37.3536	35.0245	28.3095	35.1488	36.4795	36.0121	37.3482
	10	43.5538	44.2176	42.3845	32.7091	41.7464	42.8749	42.8232	44.3337
	12	49.1426	50.3112	47.7115	36.5965	47.5394	48.0504	49.5033	50.4694
	14	52.4016	56.0881	52.5454	40.3468	52.2408	52.9522	54.6020	56.1094
Medical4	6	29.7038	29.8214	29.0196	29.4504	29.3854	29.2795	29.4105	29.8370
	8	36.5791	36.8520	35.7648	35.8712	35.9158	36.3379	36.2300	36.9593
	10	42.8729	43.6696	40.4616	41.3581	41.8134	42.2490	42.7692	43.7918
	12	47.4509	49.5428	46.0389	47.2014	46.8216	47.7198	48.9814	49.5914
	14	54.1334	55.1970	51.3232	52.4321	51.3109	52.4020	54.7969	55.4478
Medical5	6	27.5392	27.6409	26.9876	21.7552	26.9903	27.1946	27.1707	27.6422
	8	35.0168	35.1453	33.7456	27.3895	34.5749	34.5868	33.8298	35.2417
	10	41.1285	42.0444	40.8043	31.4742	39.7802	40.7294	41.1127	42.1571
	12	47.0732	48.4408	46.2316	35.6166	45.5961	46.2532	48.1818	48.5939
	14	52.6506	54.3979	50.5047	39.2780	50.1560	50.3272	53.5915	54.5965
Medical6	6	28.9177	29.0004	27.9225	22.9562	34.5683	28.6647	28.2243	29.0004
	8	36.0479	36.9304	34.5297	27.9091	36.3630	35.8314	36.3014	36.9257
	10	43.0778	43.5536	40.2129	32.3120	40.6346	42.0560	42.1871	43.6132
	12	48.8374	49.6670	47.4302	36.4025	45.0181	46.1182	48.5252	49.8300
	14	52.3678	55.3890	51.7384	39.7488	51.7612	50.7992	52.9835	54.4504

Fig. 16.

(a): The best fitness values of Otsu's method for all algorithms for Medical1, Medical2, and Medical3 images.

(b): The best fitness values of Otsu's method for all algorithms for Medical4, Medical5, and Medical6 images.

Fig. 17.

(a): The best fitness values of Kapur's method for all algorithms for Medical1, Medical2, and Medical3 images.

(b): The best fitness values of Kapur's method for all algorithms for Medical4, Medical5, and Medical6 images.

6.4.2. Experimental results for 3D medical images

In this section, the performance of the proposed algorithm in 3D medical image segmentation is assessed according to the same evaluation metrics mentioned above. The proposed algorithm and seven well-known metaheuristic techniques are applied to 6 different volumetric medical images to determine the optimal threshold values to segment the slices of these images. We utilized the same parameter settings and threshold levels as in the case of 2D medical images.

The segmented image slices from applying the proposed algorithm for Otsu's method and Kapur's entropy at different threshold levels are shown in Table 14, Table 15 , respectively. The high quality of the segmented image slices is evident from their visual appearance.

Table 14.

Slices of 3D medical images segmented by the proposed COVID-HHOA algorithm using Otsu's method.

Table 15.

Slices of 3D medical images segmented by the proposed COVID-HHOA algorithm using Kapur's entropy.

The results of PSNR, SSIM, and NCC of the proposed algorithm against its peers for Otsu's method and Kapur's entropy are given in Table 16 to Table 17, Table 18, Table 19, Table 20, Table 21 . The values in these tables, highlighted in bold, indicate the best results.

Table 16.

PSNR results of 3D medical image segmentation using Otsu's method for all algorithms.

Image	K	Algorithms
		SOA [41]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
3D_imag1	6	30.1961	30.4401	30.6070	29.5351	29.8018	30.1225	29.8267	30.6455
	8	32.0807	32.4235	32.5736	31.5420	31.0633	31.4139	31.3611	32.7306
	10	33.7221	34.2130	34.1211	33.3435	32.6111	32.6553	32.3934	34.5134
	12	34.3873	35.2726	35.1746	34.6316	34.4152	34.4275	34.2041	35.6317
	14	35.6631	36.3439	36.3487	36.4534	35.2466	34.2358	34.5301	37.3705
3D_imag2	6	28.4792	28.2612	28.5816	28.1514	28.2390	28.1231	27.4920	28.5700
	8	29.9857	30.3715	30.4281	30.2278	29.2314	29.5801	29.6223	30.8414
	10	31.8611	32.0457	32.0356	32.1523	30.8432	30.8721	30.1978	32.5086
	12	32.9469	33.3460	33.1822	33.1939	31.9144	31.2241	31.4527	33.7971
	14	33.2306	34.3034	34.3424	34.4636	33.2121	32.7286	32.0812	34.9959
3D_imag3	6	23.8765	24.5766	24.7164	24.7615	23.6154	23.5595	22.5702	25.3023
	8	26.1167	26.7796	27.0332	27.1784	26.7589	27.2304	24.7527	27.6537
	10	27.9192	28.8867	28.3241	29.1432	27.2341	27.9804	26.4567	29.6654
	12	28.9860	30.3745	30.0778	31.0939	28.5432	28.4431	27.4352	31.4758
	14	29.2986	31.6990	31.2865	32.0764	29.4811	29.8310	28.4833	32.7302
3D_imag4	6	26.9810	26.9321	27.1231	26.6620	26.8743	26.3212	25.7158	27.3305
	8	29.2953	28.8902	29.5442	28.2524	28.2630	28.1423	27.5351	29.5858
	10	31.0143	30.5272	31.0023	31.0154	30.1232	29.5087	29.2950	31.4076
	12	32.1998	32.0870	32.4472	32.1555	31.1611	30.9122	30.1142	32.8436
	14	32.5924	33.2061	33.3724	33.2594	31.2641	32.3986	31.1415	33.8989
3D_imag5	6	27.4326	28.4923	28.3731	28.1460	27.4155	28.2378	27.7748	28.4717
	8	29.7636	30.2675	29.4632	30.1205	28.2261	29.4793	28.4571	30.5245
	10	31.3146	31.8957	31.8638	32.1162	30.7573	30.4451	30.4593	32.7594
	12	32.2885	33.4235	32.5232	32.7948	31.9792	31.9715	31.6864	33.7197
	14	32.7714	34.2666	33.6360	34.1648	33.7886	31.9899	32.0876	34.8721
3D_image6	6	26.3354	26.4546	26.2072	26.4751	26.0888	26.1373	25.4470	26.7845
	8	28.5190	28.1292	28.0865	28.3212	27.9585	27.1920	27.7022	28.4532
	10	29.4571	30.2222	29.8765	30.2512	29.3425	28.9353	28.234	30.5218
	12	30.4919	31.4218	31.0183	31.7098	30.2065	30.2410	29.9403	31.7267
	14	31.7326	32.2109	32.0645	32.5123	31.2352	31.1132	30.9337	32.6480
Average		30.3646	30.8583	30.8478	30.8634	29.8618	29.8178	29.2882	31.4667

Table 17.

PSNR results of 3D medical image segmentation using Kapur's method for all algorithms.

Image	K	Algorithms
		SOA [41]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	28.9391	28.7182	26.7359	27.8832	27.8761	28.5393	28.2466	29.0592
	8	30.789	31.7338	27.3459	30.2013	28.3848	29.2610	31.9910	32.5919
	10	33.4462	34.5674	28.8927	32.7649	30.2011	31.6723	33.9310	34.3256
	12	34.8542	35.6831	30.5443	33.9639	31.5210	32.4643	35.436	36.2500
	14	36.1017	37.0353	32.4340	36.6387	33.1574	33.7446	36.6467	37.4015
Medical2	6	27.8760	28.1027	24.0359	25.3058	26.2936	26.8300	27.1412	28.8769
	8	29.6887	30.2295	25.5423	28.4327	27.5561	28.4240	29.0898	30.9285
	10	30.1092	31.8796	27.3453	31.9067	29.3428	30.9737	31.9503	32.6432
	12	33.2782	34.2882	28.5334	33.3440	30.1108	31.6471	32.6461	34.9733
	14	33.7369	35.6173	29.2686	34.7503	30.1138	31.9441	33.5732	35.9389
Medical3	6	25.3868	23.6456	19.1499	22.6928	20.8765	25.0683	23.5777	25.8508
	8	27.6444	25.2165	21.3553	25.7434	22.1535	26.3433	25.8927	27.3553
	10	28.8815	26.7871	24.6891	27.9029	25.1566	27.8761	28.0051	28.3343
	12	30.3123	30.4122	25.2345	30.6865	26.4503	28.4920	29.3453	32.4365
	14	31.2641	31.3217	26.4080	32.5818	28.6095	30.4881	30.5706	33.6475
Medical4	6	27.1636	29.3512	23.3768	22.7454	20.3453	28.3452	27.3451	29.4851
	8	29.6801	30.1578	24.2954	25.1042	22.9910	29.4786	29.1662	30.3341
	10	31.1514	32.6423	25.9872	27.0787	25.2696	30.9397	30.4599	32.6304
	12	31.9376	34.1111	27.6909	30.4325	27.3560	31.6453	31.3344	34.2147
	14	32.8808	35.2234	30.5359	33.1648	28.0357	32.5760	32.6905	35.4816
Medical5	6	27.0996	29.8734	26.0212	23.6755	25.0232	26.4998	26.8532	30.1234
	8	29.5678	31.1420	27.7549	24.6086	26.7692	28.2310	29.0765	31.6352
	10	31.2849	32.1204	29.4533	28.9561	29.6891	29.3244	30.4819	32.5461
	12	32.4544	32.6433	30.0826	30.5643	30.7765	30.6543	32.5921	33.7241
	14	33.8361	34.3116	31.8976	31.3037	31.4373	31.5773	33.4335	34.9109
Medical6	6	25.6432	25.2261	22.3990	24.5574	23.4597	24.6637	25.0497	25.6262
	8	27.1155	28.1435	23.5334	26.7452	24.7891	27.5356	27.3455	28.5462
	10	29.1776	30.1078	25.5249	29.4057	26.9886	29.0901	29.6829	30.4483
	12	30.1092	31.5349	26.3562	30.6543	28.3455	29.6740	30.5332	31.8232
	14	30.5725	32.2195	27.5853	31.9613	29.8339	30.1687	31.8188	32.4613
Average		30.3994	31.0255	26.7121	29.0911	27.2672	29.4724	30.1968	31.7027

Table 18.

SSIM results of 3D medical image segmentation using Otsu's method for all algorithms.

Image	K	Algorithms
		SOA [41]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.9727	0.9740	0.9737	0.9734	0.9686	0.9719	0.9731	0.9741
	8	0.9776	0.9783	0.9778	0.9781	0.9750	0.9759	0.9776	0.9785
	10	0.9795	0.9797	0.9797	0.9800	0.9770	0.9777	0.9790	0.9801
	12	0.9805	0.9811	0.9806	0.9811	0.9790	0.9794	0.9806	0.9812
	14	0.9811	0.9815	0.9814	0.9816	0.9800	0.9798	0.9810	0.9822
Medical2	6	0.9600	0.9599	0.9600	0.9597	0.9546	0.9555	0.9588	0.9602
	8	0.9673	0.9673	0.9664	0.9674	0.9643	0.9627	0.9663	0.9676
	10	0.9707	0.9711	0.9698	0.9701	0.9674	0.9646	0.9703	0.9719
	12	0.9721	0.9733	0.9719	0.9727	0.9693	0.9697	0.9720	0.9736
	14	0.9721	0.9736	0.9734	0.9738	0.9714	0.9711	0.9732	0.9745
Medical3	6	0.9153	0.9171	0.9173	0.9179	0.9022	0.9042	0.8993	0.9200
	8	0.9279	0.9325	0.9323	0.9287	0.9220	0.9265	0.9260	0.9347
	10	0.9373	0.9416	0.9387	0.9354	0.9289	0.9301	0.9378	0.9419
	12	0.9404	0.9448	0.9431	0.9439	0.9387	0.9341	0.9418	0.9460
	14	0.9442	0.9488	0.9453	0.9469	0.9415	0.9425	0.9436	0.9487
Medical5	6	0.8381	0.8402	0.8400	0.8388	0.8322	0.8312	0.8365	0.8410
	8	0.8460	0.8513	0.8458	0.8484	0.8345	0.84001	0.8455	0.8527
	10	0.8501	0.8567	0.8497	0.8512	0.8388	0.8439	0.8512	0.8573
	12	0.8557	0.8603	0.8537	0.8550	0.8471	0.8511	0.8559	0.8606
	14	0.8560	0.8613	0.8562	0.8568	0.8474	0.8542	0.8601	0.8624
Medical6	6	0.9612	0.9642	0.9646	0.9660	0.9577	0.9545	0.9646	0.9659
	8	0.9701	0.9717	0.9678	0.9701	0.9621	0.9678	0.9698	0.9722
	10	0.9734	0.9742	0.9730	0.9738	0.9707	0.9700	0.9734	0.9751
	12	0.9741	0.9762	0.9740	0.9751	0.9725	0.9716	0.9744	0.9767
	14	0.9751	0.9766	0.9756	0.9767	0.9753	0.9735	0.9747	0.9774
Medical7	6	0.9394	0.9417	0.9402	0.9414	0.9334	0.9378	0.9385	0.9410
	8	0.9531	0.9501	0.9529	0.9531	0.9481	0.9412	0.9507	0.9534
	10	0.9567	0.9603	0.9598	0.9587	0.9500	0.9554	0.9602	0.9617
	12	0.9623	0.9635	0.9621	0.9630	0.9536	0.9601	0.9629	0.9638
	14	0.9644	0.9664	0.9638	0.9658	0.9611	0.9631	0.9655	0.9666
Average		0.9424	0.9446	0.9430	0.9437	0.9374	0.9387	0.9421	0.9454

Table 19.

SSIM results of 3D medical image segmentation using Kapur's method for all algorithms.

Image	K	Algorithms
		SOA [41]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.9649	0.9632	0.9552	0.9680	0.9601	0.9624	0.9635	0.9650
	8	0.9621	0.9738	0.9621	0.9701	0.9668	0.9700	0.9734	0.9755
	10	0.9743	0.9765	0.9673	0.9732	0.9723	0.9733	0.9767	0.9788
	12	0.9792	0.9800	0.9720	0.9768	0.9756	0.9775	0.9785	0.9809
	14	0.9807	0.9813	0.9750	0.9807	0.9779	0.9795	0.9810	0.9818
Medical2	6	0.9588	0.9578	0.9244	0.9529	0.9500	0.9448	0.9513	0.9600
	8	0.9635	0.9631	0.9477	0.9638	0.9549	0.9588	0.9650	0.9664
	10	0.9697	0.9688	0.9545	0.9686	0.9600	0.9671	0.9694	0.9701
	12	0.9713	0.9729	0.9622	0.9721	0.9640	0.9702	0.9710	0.9735
	14	0.9725	0.9742	0.9648	0.9734	0.9681	0.9688	0.9735	0.9745
Medical3	6	0.9044	0.9012	0.8415	0.8968	0.8648	0.9026	0.8894	0.9055
	8	0.9123	0.9187	0.8829	0.9106	0.8835	0.9107	0.9059	0.9240
	10	0.9298	0.9276	0.9074	0.9220	0.9038	0.9168	0.9266	0.9323
	12	0.9321	0.9332	0.9100	0.9324	0.9235	0.9215	0.9322	0.9424
	14	0.9386	0.9373	0.9166	0.9421	0.9326	0.9335	0.9404	0.9453
Medical4	6	0.8301	0.8327	0.8008	0.8314	0.8109	0.8420	0.8450	0.8470
	8	0.8470	0.8378	0.8165	0.8373	0.8149	0.8433	0.8480	0.8535
	10	0.8531	0.8409	0.8268	0.8463	0.8227	0.8485	0.8499	0.8557
	12	0.8540	0.8421	0.8342	0.8525	0.8402	0.8531	0.8541	0.8566
	14	0.8558	0.8456	0.8452	0.8565	0.8416	0.8552	0.8599	0.8644
Medical5	6	0.9583	0.9610	0.9504	0.9371	0.9485	0.9542	0.9543	0.9620
	8	0.9675	0.9653	0.9598	0.9441	0.9592	0.9624	0.9661	0.9688
	10	0.9719	0.9743	0.9637	0.9648	0.9652	0.9659	0.9734	0.9750
	12	0.9732	0.9751	0.9688	0.9682	0.9695	0.9702	0.9747	0.9758
	14	0.9750	0.9769	0.9664	0.9707	0.9719	0.9720	0.9760	0.9772
Medical6	6	0.9213	0.9204	0.8700	0.9160	0.9131	0.9191	0.9211	0.9244
	8	0.9420	0.9412	0.9145	0.9415	0.9327	0.9265	0.9419	0.9430
	10	0.9569	0.9565	0.9277	0.9589	0.9476	0.9519	0.9554	0.9579
	12	0.9590	0.9604	0.9412	0.9591	0.9516	0.9645	0.9608	0.9612
	14	0.9618	0.9648	0.9479	0.9632	0.9576	0.9577	0.9665	0.9652
Average		0.9380	0.9374	0.9192	0.9350	0.9268	0.9348	0.9381	0.9421

Table 20.

The NCC results of 3D medical image segmentation using Otsu's method for all algorithms.

Image	K	Algorithms
		SOA [41]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.9943	0.9947	0.9946	0.9947	0.9898	0.9964	0.9968	0.9950
	8	0.9964	0.9968	0.9967	0.9968	0.9944	0.9955	0.9962	0.9970
	10	0.9975	0.9978	0.9977	0.9977	0.9958	0.9967	0.9969	0.9982
	12	0.9980	0.9981	0.9980	0.9982	0.9972	0.9972	0.9978	0.9986
	14	0.9983	0.9986	0.9983	0.9981	0.9975	0.9977	0.9978	0.9990
Medical2	6	0.9955	0.9954	0.9955	0.9959	0.9933	0.9939	0.9952	0.9958
	8	0.9971	0.9973	0.9970	0.9972	0.9953	0.9962	0.9969	0.9975
	10	0.9978	0.9980	0.9979	0.9980	0.9966	0.9966	0.9972	0.9983
	12	0.9981	0.9985	0.9982	0.9984	0.9973	0.9972	0.9980	0.9987
	14	0.9983	0.9986	0.9985	0.9986	0.9980	0.9979	0.9981	0.9989
Medical3	6	0.9959	0.9965	0.9964	0.9965	0.9943	0.9955	0.9947	0.9968
	8	0.9975	0.9980	0.9979	0.9980	0.9967	0.9972	0.9972	0.9984
	10	0.9984	0.9986	0.9981	0.9986	0.9975	0.9978	0.9978	0.9990
	12	0.9986	0.9990	0.9989	0.9990	0.9981	0.9981	0.9980	0.9992
	14	0.9987	0.9992	0.9989	0.9991	0.9986	0.9987	0.9985	0.9994
Medical4	6	0.9971	0.9975	0.9974	0.9978	0.9970	0.9970	0.9975	0.9977
	8	0.9983	0.9987	0.9984	0.9985	0.9974	0.9978	0.9984	0.9986
	10	0.9988	0.9991	0.9990	0.9990	0.9980	0.9985	0.9991	0.9994
	12	0.9990	0.9993	0.9992	0.9993	0.9986	0.9989	0.9993	0.9996
	14	0.9992	0.9994	0.9993	0.9994	0.9988	0.9990	0.9994	0.9998
Medical5	6	0.9934	0.9941	0.9940	0.9940	0.9912	0.9931	0.9937	0.9944
	8	0.9954	0.9961	0.9950	0.9960	0.9941	0.9944	0.9948	0.9964
	10	0.9969	0.9972	0.9968	0.9973	0.9961	0.9959	0.9964	0.9978
	12	0.9970	0.9974	0.9971	0.9975	0.9970	0.9967	0.9968	0.9980
	14	0.9978	0.9982	0.9976	0.9981	0.9975	0.9968	0.9971	0.9984
Medical6	6	0.9926	0.9930	0.9930	0.9932	0.9899	0.9920	0.9924	0.9934
	8	0.9955	0.9959	0.9954	0.9955	0.9938	0.9945	0.9950	0.9959
	10	0.9963	0.9966	0.9961	0.9965	0.9943	0.9956	0.9960	0.9969
	12	0.9968	0.9970	0.9968	0.9970	0.9955	0.9961	0.9967	0.9974
	14	0.9970	0.9973	0.9972	0.9972	0.9961	0.9966	0.9972	0.9978
Average		0.9970	0.9973	0.9971	0.9973	0.9958	0.9964	0.9968	0.9977

Table 21.

The NCC results of 3D medical image segmentation using Kapur's entropy method for all algorithms.

Image	K	Algorithms
		SOA [41]	HHOA [35]	FPA [18]	CS [19]	HS [15]	SCA [22]	BA [23]	COVID-HHOA
Medical1	6	0.9858	0.9853	0.9778	0.9900	0.9821	0.9838	0.9861	0.9867
	8	0.9932	0.9932	0.9851	0.9910	0.9882	0.9887	0.9938	0.9946
	10	0.9950	0.9954	0.9882	0.9921	0.9920	0.9921	0.9951	0.9958
	12	0.9970	0.9975	0.9928	0.9939	0.9944	0.9935	0.9970	0.9978
	14	0.9978	0.9981	0.9931	0.9980	0.9960	0.9952	0.9980	0.9984
Medical2	6	0.9940	0.9921	0.9804	0.9914	0.9921	0.9882	0.9912	0.9948
	8	0.9944	0.9949	0.9820	0.9955	0.9932	0.9920	0.9950	0.9956
	10	0.9960	0.9961	0.9927	0.9975	0.9945	0.9956	0.9974	0.9966
	12	0.9978	0.9980	0.9928	0.9978	0.9964	0.9959	0.9980	0.9983
	14	0.9980	0.9984	0.9929	0.9983	0.9968	0.9962	0.9984	0.9988
Medical3	6	0.9910	0.9939	0.9698	0.9943	0.9926	0.9911	0.9886	0.9943
	8	0.9950	0.9949	0.9910	0.9951	0.9935	0.9934	0.9951	0.9956
	10	0.9965	0.9960	0.9919	0.9965	0.9952	0.9953	0.9968	0.9971
	12	0.9974	0.9975	0.9937	0.9977	0.9973	0.9965	0.9978	0.9985
	14	0.9982	0.9979	0.9961	0.9981	0.9975	0.9980	0.9983	0.9989
Medical4	6	0.9939	0.9969	0.9895	0.9968	0.9950	0.9965	0.9954	0.9971
	8	0.9957	0.9981	0.9952	0.9978	0.9957	0.9973	0.9978	0.9981
	10	0.9980	0.9985	0.9957	0.9982	0.9970	0.9978	0.9980	0.9984
	12	0.9954	0.9988	0.9961	0.9988	0.9978	0.9980	0.9985	0.9989
	14	0.9986	0.9990	0.9970	0.9990	0.9984	0.9984	0.9990	0.9993
Medical5	6	0.9888	0.9923	0.9873	0.9710	0.9879	0.9866	0.9898	0.9926
	8	0.9944	0.9939	0.9905	0.9761	0.9925	0.9925	0.9938	0.9941
	10	0.9960	0.9948	0.9934	0.9943	0.9932	0.9930	0.9950	0.9952
	12	0.9969	0.9968	0.9955	0.9945	0.9953	0.9947	0.9975	0.9974
	14	0.9977	0.9978	0.9962	0.9948	0.9964	0.9956	0.9978	0.9980
Medical6	6	0.9851	0.9845	0.9728	0.9890	0.9822	0.9821	0.9851	0.9861
	8	0.9906	0.9922	0.9876	0.9922	0.9913	0.9932	0.9935	0.9937
	10	0.9943	0.9951	0.9883	0.9956	0.9928	0.9940	0.9958	0.9958
	12	0.9948	0.9960	0.9899	0.9960	0.9944	0.9944	0.9964	0.9965
	14	0.9955	0.9968	0.9907	0.9970	0.9955	0.9947	0.9973	0.9972
Average		0.9947	0.9953	0.9895	0.9939	0.9935	0.9934	0.9952	0.9960

6.5. Discussions

In the medical field, the quality of the images is of great importance because any degradation of the image quality may affect the diagnosis process. The higher the PSNR, SSIM, and NCC values, the higher the quality of the segmented images. As shown in Figs. 10–Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15, all algorithms have similar performance at lower threshold values for both Otsu's method and Kapur's entropy; however, the superiority of the proposed algorithms gets more obvious at large threshold numbers, which proves the efficiency of the proposed algorithm in image segmentation, especially at high threshold levels.

According to PSNR, the proposed algorithm has the best performance in almost all cases, followed by HHOA. SCA and CS algorithms have the lowest PSNR values. The SSIM and NCC results show high similarity between the original and the segmented images produced by the proposed algorithm, which means that there is no high degradation in the medical images, which is very important for taking a medical decision. As in the case of PSNR, the superiority of the proposed algorithm in terms of SSIM and NCC gets clearer at high threshold levels.

The obtained fitness function values show the algorithm's ability to find high-quality solutions to the problem. All algorithms have similar fitness values in almost all cases; however, the proposed algorithm slightly outperforms them.

Despite the very good performance of the HHOA algorithm in most cases, it fails to get high fitness in some cases, such as the case of Medial5 and Medical6 images (which are CT images) for Otsu's method. However, the proposed algorithm achieves superior performance for all cases.

From the visual perception, we can notice the high quality of the segmented images at all threshold levels. These results indicate the ability of the proposed COVID-HHOA algorithm to find the threshold values that most fit for segmentation.

Some algorithms such as BA, CS, and HHOA exceed the proposed algorithm when a low threshold level is used. However, in most cases, especially at high threshold levels, the proposed algorithm has superior performance.

The bar charts in Fig. 18 show all algorithms' average PSNR, SSIM, and NCC results, respectively. The superiority of the proposed algorithm against the other algorithms in terms of PSNR is clearly shown in Fig. 18. In the case of SSIM and NCC, all algorithms have similar performance; however, the proposed algorithm is slightly superior.

Fig. 18.

The average PSNR, SSIM, and NCC results of 3D medical image segmentation using (a) Otsu's method and (b) Kapur's entropy for all algorithms.

For a 3D medical image, each image slice is segmented separately, and then the segmented image slices are concatenated to form a segmented 3D image. The PSNR, SSIM, NCC, and fitness values for a 3D image are obtained by computing the average value of all slices. As in the case of 2D medical images, the proposed algorithm shows superior performance in 3D image segmentation.

In addition to the previously mentioned evaluation criteria, we utilized the Wilcoxon rank-sum test to compare the results of the proposed algorithm with other algorithms. As mentioned in Refs. [[28], [29], [30]], the null hypothesis is defined as: there is no significant difference between a pair of algorithms. The p values obtained from the Wilcoxon rank-sum test are applicable to judge whether or not to reject the null hypothesis. Smaller p values (less than 0.05) indicate that the null hypothesis is rejected. And the two compared algorithms are considered significantly different.

The p values produced by comparing the proposed algorithm with all other algorithms are shown in Table 12, Table 13. All the p values shown in the table are ≤0.05, proving the alternative hypothesis that there is a significant difference between the two methods. The overall results prove the efficiency of the proposed COVID-HHOA algorithm in image segmentation.

Table 12.

P values computed by Wilcoxon's rank-sum test for Ostu's method.

Otsu's method
Image	COVID-HHOA VS SOA	COVID-HHOA VS HHOA	COVID-HHOA VS FPA	COVID-HHOA VS CS	COVID-HHOA VS HS	COVID-HHOA VS SCA	COVID-HHOA VS BA
Medical1	4.4434e-35	3.1021e-05	6.5446e-29	1.0462e-14	2.5439e-35	9.5941e-37	2.7809e-23
Medical2	1.6158e-36	2.6163e-13	9.2430e-35	9.7513e-16	2.0902e-34	2.9425e-34	1.0148e-26
Medial3	9.9291e-34	1.8851e-34	1.6339e-30	8.8083e-15	1.0259e-34	4.2759e-36	3.6013e-32
Medical4	9.8717e-37	2.7708e-10	2.0270e-20	6.5579e-15	6.8035e-34	7.5621e-34	7.0235e-34
Medical5	6.0170e-37	8.3702e-35	1.5449e-24	6.4827e-17	2.0012e-34	5.8710e-34	2.1362e-33
Medical6	1.3883e-36	1.0533e-34	9.2511e-30	6.9292e-15	7.9531e-34	1.1623e-35	1.0836e-32
Average	1.05e-34	2.59e-06	4.61e-19	1.11e-10	3.08e-34	2.42e-33	2.32e-07

Table 13.

P values computed by Wilcoxon's rank-sum test for Kapur's entropy.

Kapur's entropy
Image	COVID-HHOA VS SOA	COVID-HHOA VS HHOA	COVID-HHOA VS FPA	COVID-HHOA VS CS	COVID-HHOA VS HS	COVID-HHOA VS SCA	COVID-HHOA VS BA
Medical1	5.6766e-36	8.0007e-08	3.4285e-39	2.4915e-34	1.5488e-34	2.5412e-36	3.6538e-28
Medical2	2.3507e-36	9.0482e-05	2.6352e-36	5.6352e-35	1.3417e-34	1.4324e-37	3.9573e-24
Medial3	3.1155e-37	2.5995e-06	1.4527e-37	1.5482e-34	2.5436e-34	2.0266e-37	4.6850e-22
Medical4	1.1022e-34	1.7699e-16	1.7217e-37	6.7150e-34	8.6519e-34	4.4321e-36	2.6753e-18
Medical5	1.1275e-34	1.5647e-26	6.3613e-36	4.5266e-35	6.7532e-34	4.5176e-39	3.2124e-10
Medical6	8.9301e-33	6.8546e-05	5.8857e-39	1.5724e-34	2.4597e-34	2.5481e-37	1.4326e-15
Average	7.80e-34	1.35e-05	1.43e-36	3.10e-34	3.75e-34	9.17e-37	2.68e-11

Based on the previously mentioned results, we can say that the proposed COVID-HHOA algorithm outperforms all other algorithms in 2D and 3D medical image segmentation at high threshold levels. However, the proposed approach is not the best at lower threshold values. Higher threshold levels are preferable in image segmentation to precisely locate the complex objects in the image.

7. Conclusions and future work

This paper proposes a hybrid algorithm for solving the multilevel thresholding problem for 2D and 3D medical image segmentation. This algorithm is called COVID-HHOA, which combines two robust metaheuristic algorithms to get better quality solutions. The proposed algorithm uses Otsu's and Kapur's entropy as fitness functions to find the best threshold values. The superiority of the proposed COVID-HHOA algorithm is verified using two groups of 2D medical images and volumetric medical images. The hybridization is implemented by splitting the populations into two smaller populations, and each subpopulation is assigned to one of the two algorithms to be updated in parallel. Different evaluation metrics are utilized to compare the performance of the proposed algorithm with seven well-known metaheuristics. These metrics are PSNR, SSIM, NCC, best fitness, and applying the Wilcoxon rank-sum test to prove the significance and superiority of the proposed algorithm. The overall results reveal the efficiency of the proposed COVID-HHOA in solving the medical image segmentation problem. It is worth mentioning that except the methods used in the paper, some of the most representative computational intelligence algorithms can be used to solve the problem, like monarch butterfly optimization (MBO), earthworm optimization algorithm (EWA), elephant herding optimization (EHO), moth search (MS) algorithm, Slime mould algorithm (SMA), hunger games search (HGS), Runge Kutta optimizer (RUN), colony predation algorithm (CPA), and Harris hawks optimization (HHOA). Future work may include hybridizing the novel COVID algorithm with one of these metaheuristics. Also, we can apply the proposed algorithm in the image segmentation of color images.

Declaration of competing interest

The authors have declared no conflict of interest.

Appendix A. Supplementary data

The following is the Supplementary data to this article: