Multilevel thresholding satellite image segmentation using chaotic coronavirus optimization algorithm with hybrid fitness function

Abstract

Image segmentation is a critical step in digital image processing applications. One of the most preferred methods for image segmentation is multilevel thresholding, in which a set of threshold values is determined to divide an image into different classes. However, the computational complexity increases when the required thresholds are high. Therefore, this paper introduces a modified Coronavirus Optimization algorithm for image segmentation. In the proposed algorithm, the chaotic map concept is added to the initialization step of the naive algorithm to increase the diversity of solutions. A hybrid of the two commonly used methods, Otsu’s and Kapur’s entropy, is applied to form a new fitness function to determine the optimum threshold values. The proposed algorithm is evaluated using two different datasets, including six benchmarks and six satellite images. Various evaluation metrics are used to measure the quality of the segmented images using the proposed algorithm, such as mean square error, peak signal-to-noise ratio, Structural Similarity Index, Feature Similarity Index, and Normalized Correlation Coefficient. Additionally, the best fitness values are calculated to demonstrate the proposed method's ability to find the optimum solution. The obtained results are compared to eleven powerful and recent metaheuristics and prove the superiority of the proposed algorithm in the image segmentation problem.

1 Introduction

Digital image processing is manipulating digital images through algorithms using digital computers for many purposes, such as image enhancement, image compression, and extracting useful information [1]. Image segmentation is a crucial process in most digital image processing tasks. It isolates the region of interest from the scene [2]. Image segmentation has been successfully applied to several fields, such as image denoising [3], medical image diagnosis [4], and satellite image segmentation [5]. In the literature, several techniques have been proposed for image segmentation. These techniques can be categorized as edge detection-based segmentation [6], clustering-based segmentation [7], and thresholding-based segmentation [8]. Thresholding-based segmentation is considered the most popular technique because of its simplicity and efficiency. In thresholding-based segmentation, the histogram information is extracted from the grayscale image and is used to determine threshold values to separate image pixels into different classes [9]. When one threshold value is needed, it is referred to as bi-level thresholding, in which the image is segmented into only two regions.

Multilevel thresholding is more appropriate in images containing many objects with fine details and complex backgrounds because bi-level thresholding fails to distinguish these objects correctly. After all, it divides the image into only two regions [10]. On the other hand, multilevel thresholding involves using more than one threshold to segment the image into several regions [11]. The thresholding process aims to find the best threshold values that precisely determine the image segments. Otsu [12] and Kapur [13] methods are considered the most popular strategies for determining the optimal thresholds. Otsu's method maximizes the variance between classes, while Kapur's method maximizes the histogram entropy to measure homogeneity between segmented regions.

Over the last few years, Swarm intelligence has been extensively applied to solve multilevel thresholding image segmentation problems [14]. Many algorithms have been proposed for satellite image segmentation, such as a modified version of an artificial bee colony (MABC) proposed by Bhandari et al. [15]. The results reveal that MABC has more computational efficiency and accuracy than the standard ABC. For RGB histogram-based color satellite image segmentation, a multi-strategy Emperor Penguin Optimizer (MSEPO) is proposed by Heming et al. [16]. The results showed that the MSEPO algorithm had superior performance, especially for the high dimensional segmentation of complex satellite images. The proposed hybrid Grasshopper Optimization Algorithm and Differential Evolution (GOA-jDE) has been proposed by Heming et al. [17]. The superiority of the proposed algorithm is illustrated in terms of different metrics such as peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), feature similarity index (FSIM), and standard deviation (STD), convergence performance, and computation time. Many other algorithms for satellite image segmentation have been proposed in [18,19,20,21].

Several algorithms have been proposed in medical images, such as ant colony optimization with Cauchy and greedy levy mutations for COVID X-ray images segmentation [22]. Bandyopadhyay et al. [4] proposed an altruistic Harris Hawks’ optimization algorithm to segment brain MRI images. This algorithm combines the chaotic initialization, the concept of altruism, and a hybrid objective function, where the results show superior searchability and convergence speed performance. Also, Abualigah et al. [23] proposed an evolutionary arithmetic optimization algorithm for COVID-19 CT image segmentation. According to the experimental results, the proposed algorithm produces higher-quality solutions than other comparisons. Other techniques for medical image segmentation are proposed in [24,25,26,27].

In recent years, chaotic maps were incorporated into the swarm intelligence algorithms to increase the diversity of solutions and avoid falling into local optimum [28]. Hongwei et al. [29] proposed a Chaos-enhanced moth-flame optimization (MFO) algorithm for global optimization. The statistical results demonstrate that the appropriate chaotic map (singer map) embedded in the appropriate component of MFO can significantly improve the performance of MFO. [30], two different chaotic maps were incorporated into the original elephant herding optimization algorithm. Test results proved that the proposed chaotic elephant herding optimization algorithm performs better and obtains better results. Aggarwal et al. [31] used the chaotic sequence to initialize the social spider optimization algorithm, enhancing its performance. Many other researchers have embedded the chaotic concept into their native algorithms to enhance their search ability [32,33,34,35,36].

Coronavirus Optimization Algorithm (COVIDOA) is a recent metaheuristic inspired by the replication lifecycle of Coronavirus [37]. COVIDOA has three main phases: Virus Entry, Virus Replication, and Virus mutation. Coronavirus uses frameshifting [38,39,40] 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.

This paper introduces the chaotic map concept into the novel Coronavirus Disease Optimization Algorithm (COVIDOA) to increase the diversity of solutions. The proposed algorithm is applied to solve the multilevel thresholding image segmentation problem of satellite images and a set of benchmark images. The proposed algorithm used a hybrid fitness function to find the optimum threshold values by adding weights to the Otsu and Kapur methods. The results showed that using the hybrid fitness function and adding the chaotic maps yields significantly better results than the other proposed algorithms. The motivation for using modified COVIDOA for satellite image segmentation is as follows: The No Free Lunch (NFL) [41] theorem demonstrates that no single algorithm performs best for all optimization problems; this encouraged us to use a modified version of the recent COVIDOA to solve image segmentation problem.

Additionally, the basic and the binary versions of COVIDOA have performed much better in solving many benchmark and real-world problems [37, 42]; real world it can be assumed that, if the basic version is improved, it can also perform well in solving complex optimization problems such as multilevel thresholding problem. It is observed from the literature work that most of the authors used either the Otsu method or Kapur’s entropy as a fitness function for solving multilevel thresholding problems, which encouraged the authors to use a new hybrid fitness function with a modified COVIDOA to achieve better results in solving the multilevel thresholding image segmentation problem.

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

• 1. The chaotic logistic map is used to initialize COVIDOA to increase the diversity of solutions.

• 2. A new hybrid fitness function is used for finding the optimum thresholds by assigning weights to the Otsu and Kapur methods.

• 3. The superiority of the proposed algorithm is validated by applying it to six satellite and six benchmark images.

• 4. The proposed method for image segmentation results is compared with many state-of-the-art algorithms focusing on the recently proposed metaheuristics.

• 5. Several measures are used to evaluate the performance of the proposed algorithm in solving multilevel thresholding problems, such as best fitness value, MSE, PSNR, SSIM, FSIM, and NCC, and conducting the Wilcoxon rank-sum test to prove the efficiency of the proposed algorithm.

This paper is organized as follows: Sect. 2 provides a brief overview of multilevel thresholding techniques such as Otsu’s method, Kapur’s entropy, and the hybrid of the two objective functions. The proposed Coronavirus disease optimization with chaotic map initialization for multilevel thresholding is discussed in Sect. 3. The datasets, parameter setting, performance metrics, and experimental results are discussed in Sect. 4. Finally, conclusions and future work are given in Sect. 5.

2 Multilevel thresholding

Image thresholding is a simple and effective method for splitting the image into regions to make the image easier to analyze. Setting the threshold value t is based on the pixel intensity of the image, where pixels whose intensity values below t are assigned to region 1, and the other pixels are assigned to region 2 [43]. If only one threshold value is needed, this is known as bi-level thresholding, where the image is divided into two regions.

$$ \begin{aligned} & {\text{pixel}}_{i,j} \in R_{1} \quad {\text{if}}\quad 0 \le {\text{pixel}}_{i,j} < t, \\ & {\text{pixel}}_{i,j} \in R_{2} \quad {\text{if}}\quad t \le {\text{pixel}}_{i,j} < L - 1, \\ \end{aligned} $$

(1)

where ${\text{pixel}}_{i,j}$ refers to the gray level at the (i, j)th pixel, t is the value of the threshold, $R_{1}$ and $R_{2}$ refer to region 1 and region 2, respectively, and $L$ refers to maximum intensity level.

On the other hand, multilevel thresholding partitions the image into several distinct regions using more than one threshold value as follows:

$$ \begin{aligned} & {\text{pixel}}_{i,j} \in R_{1} \quad {\text{if}}\quad 0 \le {\text{pixel}}_{i,j} < t_{1} , \\ & {\text{pixel}}_{i,j} \in R_{2} \quad {\text{if}}\quad t_{1} \le {\text{pixel}}_{i,j} < t_{2} , \\ & {\text{pixel}}_{i,j} \in R_{j} \quad {\text{if}}\quad t_{j} \le {\text{pixel}}_{i,j} < t_{j + 1} , \\ & {\text{pixel}}_{i,j} \in R_{k} \quad {\text{if}}\quad t_{k} \le {\text{pixel}}_{i,j} < L - 1, \\ \end{aligned} $$

(2)

where $\left\{ {t_{1} ,t_{2} , \ldots , t_{k} } \right\}$ represents a vector of different threshold values.

The result of applying bi-level versus multilevel thresholding on the Lena image is shown in Fig. 1.

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 [12]. Assume that we have L intensity levels in a grayscale image, where L = 256 and a vector V of k − 1 thresholds are used to segment the image into K regions as in Eq. (2), where V = [th₁, th₂, …, th_{k − 1}]. Then the best threshold is obtained by maximizing the Otsu’s fitness function as follows:

$$ F_{{{\text{ostu}}}} (V) = \max \left( {\sigma_{b}^{2} (V)} \right) $$

(3)

where $\sigma_{b}^{2}$ represents the between-class variance which can be expressed as follows:

$$ \sigma_{b}^{2} = \mathop \sum \limits_{k = 0}^{K} \omega_{k} \cdot \left( {\mu_{k} - \mu_{T} } \right)^{2} $$

(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:

$$ \omega_{k} = \mathop \sum \limits_{{i \in R_{k} }} P_{i} ,\quad \mu_{k} = \mathop \sum \limits_{{i \in R_{k} }} \frac{{i \cdot P_{i} }}{{\omega_{k} }},\quad \mu_{k} = \mathop \sum \limits_{i = 0}^{L - 1} i \cdot P_{i} $$

(5)

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

$$ P_{i} = \frac{{f_{i} }}{{\mathop \sum \nolimits_{i = 0}^{L - 1} f_{i} }} $$

(6)

where f_i is the frequency of gray level i.

2.2 Kapur’s entropy method

Image entropy represents the compactness and separateness between image classes [13]. 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:

$$ {\text{th}}^{*} = \max (F_{{{\text{kapur}}}} ({\text{th}})) $$

(7)

where

$$ \begin{aligned} & F_{{{\text{kapur}}}} ({\text{th}}) = A_{0} + A_{1} , \\ & A_{0} = - \mathop \sum \limits_{i = 0}^{{{\text{th}} - 1}} \frac{{P_{i} }}{{\omega_{0} }}\ln \frac{{P_{i} }}{{\omega_{0} }}, \\ & A_{1} = - \mathop \sum \limits_{{i = {\text{th}}}}^{L - 1} \frac{{P_{i} }}{{\omega_{1} }}\ln \frac{{P_{i} }}{{\omega_{1} }}, \\ & \omega_{0} = \mathop \sum \limits_{i = 0}^{{{\text{th}} - 1}} P_{i} ,\quad \omega_{1} = \mathop \sum \limits_{{i = {\text{th}}}}^{L - 1} P_{i} , \\ \end{aligned} $$

where ${P}_{i}$ is described in Eq. (6).

For multilevel thresholding, Kapur’s method can be defined as follows:

$$ \begin{aligned} & F_{{{\text{kapur}}}} \left( V \right) = A_{0} + A_{1} + \cdots + A_{k - 1} \\ & A_{0} = - \mathop \sum \limits_{i = 0}^{{{\text{th}}_{1} - 1}} \frac{{P_{i} }}{{\omega_{0} }}\ln \frac{{P_{i} }}{{\omega_{0} }},\quad \omega_{0} = \mathop \sum \limits_{i = 0}^{{{\text{th}}_{1} - 1}} P_{i} \\ & A_{1} = - \mathop \sum \limits_{{i = {\text{th}}_{1} }}^{{{\text{th}}_{2} - 1}} \frac{{P_{i} }}{{\omega_{1} }}\ln \frac{{P_{i} }}{{\omega_{1} }},\quad \omega_{1} = \mathop \sum \limits_{{i = {\text{th}}_{1} }}^{{{\text{th}}_{2} - 1}} P_{i} \\ & A_{2} = - \mathop \sum \limits_{{i = {\text{th}}_{2} }}^{{{\text{th}}_{3} - 1}} \frac{{P_{i} }}{{\omega_{2} }}\ln \frac{{P_{i} }}{{\omega_{2} }},\quad \omega_{2} = \mathop \sum \limits_{{i = {\text{th}}_{2} }}^{{{\text{th}}_{3} - 1}} P_{i} \\ & A_{k - 1} = - \mathop \sum \limits_{{i = {\text{th}}_{k - 1} }}^{L - 1} \frac{{P_{i} }}{{\omega_{k - 1} }}\ln \frac{{P_{i} }}{{\omega_{k - 1} }},\quad \omega_{2} = \mathop \sum \limits_{{i = {\text{th}}_{k - 1} }}^{L - 1} P_{i} \\ \end{aligned} $$

(8)

The vector V refers to thresholds to be determined.

2.3 Hybrid fitness function

A hybrid fitness function calculates COVID solutions' fitness in image segmentation problems. This hybrid function is formulated by assigning weights to Otsu and Kapur functions in Eq. 9.

$$ F_{{{\text{hybrid}}}} = aF_{{{\text{Otsu}}}} + bF_{{{\text{Kapur}}}} $$

(9)

where a and b $\in$ [0, 1] are weights associated with the two fitness functions and a + b = 1. The proposed hybrid fitness function optimizes Otsu and Kapur methods simultaneously and performs more efficiently.

3 Coronavirus disease optimization algorithm

COVIDOA is a recent evolutionary optimization algorithm inspired by the replication mechanism of Coronavirus when getting inside the human body [37]. The replication process of Coronavirus has four main stages as follows, see Fig. 2:

• 1. Virus entry and uncoating

  When a human is infected with COVID, the Coronavirus particles attach to the human cell via spike protein which is one of its structural proteins [39]. After getting inside the human cell, the virus contents are released.

• 2. Virus replication

  The virus tries to make more copies to hijack other human healthy cells. The virus's replication technique is called the frameshifting technique [38, 39]. 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. The frameshifting technique is presented in Fig. 3. As shown in the figure, in the replication process, the virus's mRNA (messenger Ribonucleic Acid) is translated into viral proteins by reading tri-nucleotides (e.g., ACU). Each tri-nucleotide is translated into single amino acid. Thus, shifting (backward or forward) the reading frame of the nucleotides sequence by any number (not divisible by 3) will create different sequences that will be translated into different viral proteins. According to this technique, the virus can create millions of new particles than will damage millions of human cells. There are many types of frameshifting techniques; however, the most popular is +1 frameshifting as follows [40]:

  • +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 a random value in the range [Lb, Ub] as follows:

  $$ S_{k} \left( 1 \right) = {\text{rand}}\left( {{\text{Lb}},{\text{Ub}}} \right), $$

  (10)
  $$ S_{k} \left( {2:D} \right) = P\left( {1:D - 1} \right), $$

  (11)

  where P refers to the parent solution, $S_{k}$ is the kth generated viral protein, D is the problem dimension, and Lb and Ub are the lower and upper bounds for the variables in each solution.

• 3. Virus mutation

  Coronavirus uses the mutation technique to resist the human immune system [40]. In the proposed algorithm, the mutation is applied to the previously created new virus particle (solution) to produce a new one as follows:

  $$ Z_{i} = \left\{ {\begin{array}{*{20}l} r \hfill & {{\text{if rand}}\left( {0,1} \right) < {\text{MR}}} \hfill \\ {X_{i} } \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right. $$

  (12)

  where X is the solution before mutation, Z is the mutated solution, X_i and Z_i are the i^th element in the old and new solutions, respectively, i =1, …, D, and r is a random value in the range [Lb, 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 algorithm is as follows:

Fig. 2

Coronavirus replication lifecycle

Fig. 3

Frameshifting technique

4 COVIDOA with a chaotic map

In COVIDOA, each virus particle represents a solution in the population. The dimension of each solution is equal to the number of threshold values needed for segmentation plus 1. The first population solution is initialized randomly, where each element in the solution vector is assigned a value within the range of pixel intensities of the grayscale image. For the remaining solutions in the population, the initialization is done using chaotic maps to generate a uniformly distributed initial population [44, 45]. We used eight chaotic maps to enhance the quality of the initial population.

In the chaotic initialization, given the solution vector ${S}_{j.}$ The solution vector ${S}_{j+1}$ can be driven by the following formula:

• 1. Sine Chaotic map:

  $${S}_{j+1}=\frac{q}{4}\mathrm{sin}\left(\pi {S}_{j}\right), q=4$$

  (13)

• 2. Singer Chaotic Map:

  $${S}_{j+1}=\beta \left(7.86{S}_{j}-23.31{{S}_{j}}^{2}+28.75{{S}_{j}}^{3}-13.302875{{S}_{j}}^{4}\right), \beta =1.07$$

  (14)

• 3. Sinusoidal Chaotic Map:

  $${S}_{j+1}=u{{S}_{j}}^{2}\sin(\pi {S}_{j}), u=2.3$$

  (15)

• 4. Chebyshev Chaotic Map:

  $${S}_{j+1}=\cos(\hbox{arccos}{S}_{j})$$

  (16)

• 5. Tent Chaotic Map:

  $${S}_{j+1}=\left\{\begin{array}{ll}\frac{{S}_{j}}{0.7}&\quad {S}_{j}<0.7\\ \frac{10}{3}\left(1-{S}_{j}\right)&\quad {S}_{j}\ge 0.7\end{array}\right.$$

  (17)

• 6. Logistic Chaotic Map:

  $${S}_{j+1}=u{S}_{j}\left(1-{S}_{j}\right),u=4$$

  (18)

• 7. Iterative Chaotic Map:

  $${S}_{j+1}=sin\frac{u\pi }{{S}_{j}},u=0.7$$

  (19)

• 8. Gauss/Mouse Chaotic Map:

  $${S}_{j+1}={e}^{-\alpha {{S}_{j}}^{2}}+\beta,\quad \alpha =4.90,\quad \beta =-0.58$$

  (20)

Chaotic initialization is a modern technique used to ensure that the solutions of the initial population are uniformly distributed, which helps avoid the problem of getting stuck into local minima or maxima [46]. As discussed in the results section, we found that the Logistic chaotic map is the one that gives the best results.

5 Results and discussion

In this section, we firstly provide a brief description of the 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. Finally, we present the numerical results of running the proposed algorithm and its peers.

5.1 Datasets

Six satellite images are selected from “NASA Visible Earth” [47] to prove the efficiency of the proposed algorithm in image segmentation. In addition to six benchmark images. These images have many variations, such as size and resolution. The test images and their histograms are shown in Table 1.

Table 1 Test images and their histograms

5.2 Parameter setting

The results of multilevel thresholding using the proposed algorithm are compared with eleven well-known metaheuristic algorithms. In comparison, we focused on the recently proposed algorithms to prove the superiority of the proposed algorithm. These algorithms are: Harris Hawks Optimization algorithm (HHO) [48], Reptile Search Algorithm (RSA) [49], Seagull Optimization algorithm (SOA) [50], Black Widow Optimization Algorithm (BWOA) [51], Marine Predators Algorithm (MPA) [52], Aquila optimizer (AO) [53], Slime Mold Algorithm (SMA) [54], Arithmetic Optimization Algorithm (AOA) [55], Jellyfish Optimization algorithm (JOA) [56], Moth–flame optimization algorithm (MFO) [57], Sine Cosine Algorithm (SCA) [58]. The reasons for selecting these algorithms for comparison are as follows:

• They have proved their superior performance in 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/).

The parameters of all algorithms are set as mentioned in their original papers. In all algorithms, the population size is 50, and the maximum number of iterations to 100. All algorithms were run 20 times, and the best-obtained results are reported in the results section.

5.3 Performance metrics

The performance of the proposed algorithm is evaluated using several performance metrics, including Mean Square Error (MSE), peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), Feature Similarity Index (FSIM), 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 is obtained from running the proposed ad state-of-the-art algorithms with the proposed hybrid fitness function equations (9). By trial and error approach, we found that the proposed algorithm yields better results at a = 0.5 and b = 0.5.

• (b) Mean Square Error (MSE)

  MSE is commonly used to estimate the error between the original and segmented images. It can be calculated as follows:

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

  (21)

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

• (c) 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

  $$\hbox{PSNR}=10 \hbox{Log}_{10}\left(\frac{{255}^{2}}{MSE}\right)$$

  (22)

• (d) Structural similarity index (SSIM)

  SSIM is used to quantify the structural similarity between the original and segmented images as follows:

  $$\hbox{SSIM}(F,f)=\frac{(2{\mu }_{F}{\mu }_{f}+{C}_{1})(2{\sigma }_{Ff}+{C}_{2})}{({\mu }_{F}^{2}{\mu }_{f}^{2}+{C}_{1})({\sigma }_{F}^{2}{\sigma }_{f}^{2}+{C}_{2})}$$

  (23)

  where F and f are the original and segmented images. ${\mu }_{F}$ and ${\mu }_{f}$ are the mean intensity of F and f, respectively. ${\sigma }_{F}^{2}$ and ${\sigma }_{f}^{2}$ are the variance of F and f, respectively. C₁ = 6.502 and C₂ = 58.522.

• (e) Feature similarity index (FSIM)

  FSIM is used to measure the similarity in the structure of the two images as follows:

  $$\hbox{FSIM}\left(F,f\right)=\frac{{\sum }_{x\in\Omega }{S}_{L}\left(x\right)\cdot\hbox{PC}_{m}(x)}{{\sum }_{x\in\Omega }\hbox{PC}_{m}(x)}$$

  (24)

  where ${\mathrm{S}}_{\mathrm{L}}\left(\mathrm{x}\right)$ refers to the similarity between the two images, PC is the phase congruence, and $\Omega$ refers to the spatial domain of the image. The maximum value of the FSIM that corresponds to complete similarity is 1.

• (f) 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 images F(i, j) and f(i, j) is calculated as follows:

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

  (25)

• g) Wilcoxon rank-sum test

  The Wilcoxon rank-sum test is a nonparametric statistical test used to measure the statistical difference between two related methods [59]. 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.

5.4 Experimental results

This section presents the numerical results of running the proposed algorithm to select the optimum threshold values using the proposed hybrid fitness function with chaotic initialization. These results are compared with the state-of-the-art algorithms in best fitness, MSE, PSNR, SSIM, FSIM, NCC, and Wilcoxon rank-sum test. The experiments have been performed using 6, 10, 14, 18, 22, and 26 thresholds.

Firstly, a comparison between the results of the various chaotic maps is conducted to demonstrate that the logistic map gives the best results among the others, as shown in Table 2, where k represents the number of threshold values. The results in the table are calculated by taking the average value for each criterion for all the images in the two mentioned datasets. It is obvious from the table that using chaotic maps increases the diversity of the solutions and yields better results.

Table 2 The results obtained from using different chaotic maps in the initialization phase of COVIDOA

The higher PSNR, SSIM, FSIM, NCC, and fitness values and lower MSE values resulting from the chaotic logistic map demonstrate its robustness. Hence, the chaotic logistic map is utilized while performing further experiments.

Table 3 proves that the hybrid fitness function is more robust than using the Otsu or Kapur methods separately. It is clear from the table that the quality of the segmented images using COVIDOA with the hybrid fitness function is higher than Otsu and Kapur methods according to MSE, PSNR, SSIM, FSIM, and NCC values.

Table 3 Comparison between the performance of Otsu, Kapur, and hybrid fitness function

All algorithms have been applied to solve multilevel thresholding problems for both the standard and satellite images to show the effectiveness of the proposed algorithm against other proposed methods. The results for the six benchmark images are shown in Tables 4, 5, 6, 7, 8 and 9 for fitness, MSE, PSNR, SSIM, FSIM, and NCC, respectively. In contrast, the results for the six satellite images are shown in Tables 10, 11, 12, 13, 14 and 15. The values in these tables, highlighted in bold, indicate the best results.

Table 4 The fitness results of benchmark image segmentation using hybrid fitness function for all algorithms

Table 5 The MSE results of benchmark image segmentation using hybrid fitness function for all algorithms

Table 6 The PSNR results of benchmark image segmentation using hybrid fitness function for all algorithms

Table 7 The SSIM results of benchmark image segmentation using hybrid fitness function for all algorithms

Table 8 The FSIM results of benchmark image segmentation using hybrid fitness function for all algorithms

Table 9 The NCC results of standard image segmentation using hybrid fitness function for all algorithms

Table 10 The fitness results of satellite image segmentation using hybrid fitness function for all algorithms

Table 11 The MSE results of satellite image segmentation using hybrid fitness function for all algorithms

Table 12 The PSNR results of satellite image segmentation using hybrid fitness function for all algorithms

Table 13 The SSIM results of satellite image segmentation using hybrid fitness function for all algorithms

Table 14 The FSIM results of satellite image segmentation using hybrid fitness function for all algorithms

Table 15 The NCC results of satellite image segmentation using hybrid fitness function for all algorithms

These experiments proved the ability of the proposed algorithm to find the threshold values that most fit segmentation. In terms of the best fitness, it is noticed from Tables 4 and 10 that the proposed algorithm achieved the optimum fitness in 24 from 36 cases for the benchmark images and in 28 from 36 cases for the satellite images. The proposed algorithm produced fitness values very close to the optimum in the remaining cases. The HHO algorithm ranks second after COVID, where it achieved the highest fitness in 8 from 36 cases.

The MSE values in Tables 5 and 11 illustrate that the proposed algorithm has the minimum MSE values in 29 from 36 cases for the benchmark images and 27 for the satellite images. MPA, HHO, and MFO produce results close to the proposed algorithm; however, the proposed algorithm outperforms them significantly. The PSNR is evaluated to measure the power of the segmented image against noise. The PSNR values produced by running all algorithms at different threshold values are shown in Tables 6 and 12 for the benchmark and satellite images.

Regarding PSNR, the proposed algorithm outperforms the other algorithms in 28 from 36 cases for the benchmark images and 30 from 36 cases for the satellite images. Also, the SSIM and FSIM metrics are measured to evaluate the similarity between the original and segmented images. The SSIM results of all algorithms are shown in Tables 7 and 13 for the two datasets. The proposed algorithm is superior in 26 from 36 cases for the benchmark images and 28 from 38 for the satellite images.

According to FSIM, the proposed algorithm is superior in 30 and 29 of 36 cases for the benchmark and satellite images, respectively, as shown in Tables 8 and 14. However, MPA, SMA, and HHO algorithms perform close to the proposed algorithm. The proposed algorithm outperforms them in most the cases.

Finally, the NCC is evaluated to measure the correlation between the original and segmented images. According to the NCC results shown in Tables 9 and 15, it’s clear that the proposed algorithm is superior in 29 and 31 of 36 cases for both datasets. The previous results show that the proposed algorithm's SSIM, FSIM, and NCC values are close to 1, the best possible value. Thus, the proposed algorithm finds the optimum threshold values for image segmentation.

The proposed algorithm is compared to its peers in terms of the total average values for fitness, MSE, PSNR, SSIM, FSIM, and NCC, and the results are shown in Fig. 4. In terms of fitness, the proposed algorithm exceeds all other algorithms, averaging 1519.4 for the standard dataset and 1839.5 for the satellite dataset. However, HHO performs similarly; the proposed algorithm is slightly superior.

Fig. 4

The average fitness, MSE, PSNR, SSIM, FSIM, and NCC results of segmentation of a standard images and b satellite images for all algorithms

As shown in Fig. 4, the proposed algorithm has the minimum total average MSE for both datasets. It is obvious from the figure that there is a clear gap between the average MSE results produced by the proposed algorithm and those produced by the other algorithms. The bar charts for all the six metrics demonstrate that the proposed algorithm is superior. The highest PSNR, SSIM, FSIM, and NCC values achieved by the proposed algorithm demonstrate the high quality of the segmented images produced by the proposed algorithm.

The segmented images produced by the proposed algorithm at different thresholds are shown in Figs. 5, 6, 7 and 8. The high quality of the segmented images is clear from their visual appearance.

Fig. 5

Results for using the proposed algorithm for segmentation of Image2 at different threshold levels. a Original image, b histogram of the original image, c–e, i–k 6-level to 26-level thresholding based segmented images, and f–h, l–n 6-level to 26-level corresponding histograms

Fig. 6

Results for using the proposed algorithm for segmentation of Image4 at different threshold levels. a Original image, b histogram of the original image, c–e, i–k 6-level to 26-level thresholding based segmented images, and f–h, l–n 6-level to 26-level corresponding histograms

Fig. 7

Results for using the proposed algorithm to segment Sat_img1 at different threshold levels. a Original image, b histogram of the original image, c–e, i–k 6-level to 26-level thresholding based segmented images, and f–h, l–n 6-level to 26-level corresponding histograms

Fig. 8

Results for using the proposed algorithm to segment Sat_img3 at different threshold levels. a Original image, b histogram of the original image, c–e, i–k 6-level to 26-level thresholding based segmented images, and f–h, l–n 6-level to 26-level corresponding histograms

Additionally, some convergence curves are displayed in Fig. 9 to show the proposed algorithm's convergence ability. The proposed algorithm has a high convergence rate compared with the other algorithms as it rapidly reaches the highest fitness value.

Fig. 9

Comparison of convergence curves of all algorithms for segmentation of Sat_img3 with number of thresholds: a 6, b 10, c 14, d 18, e 22 and f 26

Due to the random process in optimization algorithms, the results differ at each run. The results of 5 separate runs of the proposed algorithm for segmentation of Image1 and Sat_img1 are shown in Table 16, and the best results are highlighted in bold. However, the results of each run are not the same; they are very close, which ensures the stability of the proposed algorithm.

Table 16 The results of 5 different runs of the proposed algorithm for segmentation Image1 and Sat_img1

In addition to the previously mentioned evaluation criteria, the Wilcoxon rank-sum test is utilized to prove the statistical significance of the proposed algorithm. This test compares two methods based on the null hypothesis, which assumes no significant difference between the two methods. The P values produced by the Wilcoxon rank-sum test must be ≤ 0.05 to be good evidence against the null hypothesis.

The P values produced by comparing the proposed algorithm with all other algorithms are shown in Tables 17 and 18. All the P values shown in the table are ≤ 0.05, which proves the alternative hypothesis that assumes a significant difference between the two methods. The overall results prove the efficiency of the proposed algorithm in image segmentation.

Table 17 The P values computed by Wilcoxon's rank-sum test for segmentation of benchmark images

Table 18 The P values computed by Wilcoxon's rank-sum test for segmentation of satellite images

6 Conclusions and future work

Satellite image segmentation aims to get a map composed of a few categories (buildings, roads, tracks, trees, crops and water, etc.) from a multispectral satellite image in many applications such as geoscience studies, astronomy, and geographical information systems. This paper proposes an improved Coronavirus Disease Optimization algorithm for solving satellite image's multi-level thresholding segmentation problem. The concept of chaotic initialization is embedded into the proposed algorithm to improve the searchability of the initial population and to void the problem of getting stuck into local minima or maxima. Additionally, a hybrid fitness function is utilized to measure the fitness of solutions instead of the classic Otsu and Kapur methods. Two separate datasets are segmented using the proposed algorithm, and several evaluation criteria have been utilized to measure the performance. The experimental results proved that the proposed algorithm with chaotic initialization and the hybrid fitness function results in image segmentation with better performance than other metaheuristics. Future work will apply the proposed algorithm to image segmentation of color images.