Remote Sensing Imagery Segmentation: A Hybrid Approach

Abstract

In remote sensing imagery, segmentation techniques fail to encounter multiple regions of interest due to challenges such as dense features, low illumination, uncertainties, and noise. Consequently, exploiting vast and redundant information makes segmentation a difficult task. Existing multilevel thresholding techniques achieve low segmentation accuracy with high temporal difficulty due to the absence of spatial information. To mitigate this issue, this paper presents a new Rényi’s entropy and modified cuckoo search-based robust automatic multi-thresholding algorithm for remote sensing image analysis. In the proposed method, the modified cuckoo search algorithm is combined with Rényi’s entropy thresholding criteria to determine optimal thresholds. In the modified cuckoo search algorithm, the Lévy flight step size was modified to improve the convergence rate. An experimental analysis was conducted to validate the proposed method, both qualitatively and quantitatively against existing metaheuristic-based thresholding methods. To do this, the performance of the proposed method was intensively examined on high-dimensional remote sensing imageries. Moreover, numerical parameter analysis is presented to compare the segmented results against the gray-level co-occurrence matrix, Otsu energy curve, minimum cross entropy, and Rényi’s entropy-based thresholding. Experiments demonstrated that the proposed approach is effective and successful in attaining accurate segmentation with low time complexity.

1. Introduction

Image segmentation comprises the partitioning of an image into homogenous and non-overlapping regions based on the similarity among image features such as color, intensity value, and regional statistics. Generally, it is a pre-processing step in pattern recognition and computer vision problems such as object detection, biomedical imaging, traffic control system, classification, and video surveillance. On the basis of the principle of segmentation, we built a taxonomy of various segmentation techniques that differentiates segmentation techniques based on region, edge, and thresholding.

1.1. Background

Considering the existing segmentation techniques discussed in the literature review by Sezgin and Sankur [1], the histogram-based thresholding segmentation technique holds a prime position in terms of simplicity, accuracy, and robustness. Furthermore, entropy-based approaches for thresholding are extremely popular in research due to their solid theoretical foundation in physics. Thus, entropy-based thresholding techniques have great implications in real-world applications with effective performance [2,3]. Furthermore, Otsu [4] presented a method that chooses the best thresholds based on the maximum inter-class variance of gray-levels. Another subsequent work in this direction was based on the optimization of the Bayes risk factor [5], wherein the entropy was maximized using a histogram to determine the best thresholds. Later on, other entropy-based thresholding techniques based on Tsalli’s method [6], the minimum cross-entropy function [7], and fuzzy clustering [8] were developed and were quite successful in the segmentation of grayscale or multichannel images. Generally, Otsu, and Kapur’s fitness functions are commonly used to identify optimal thresholds because of their high accuracy and robustness. However, most research works have also implemented Rényi’s method [9]. To deal with the constraint of multilevel image segmentation, this method incorporates local information with global information of the histogram. Rényi’s entropy tries to maximize the entropy sum method [3] and the entropic correlation method [10] to produce optimal or near-optimal threshold values.

Further, existing bi-level thresholding techniques have been extended for multilevel thresholding to assist in the segmentation of multichannel images. Because of the multimodality and intrinsic nature of multichannel images, multiple regions or objects are required to be identified. Hence, multilevel thresholding has gained increased popularity when used to attain the expected segmentations. However, multilevel thresholding has a high time complexity, which exponentially grows with the large number of threshold values and leads to an exhaustive search to find the best threshold values, especially in the case of color image segmentation. To overcome this, metaheuristic algorithms have gained significant attention in the efficient search for optimal solutions. In optimization, multi-level thresholding can be formulated as a non-convex complex problem. The objective function of metaheuristic algorithms is a criterion used to determine the optimality of the obtained solutions. Over the last decade, the use of entropy as an objective function has attracted much attention [11,12,13,14,15,16]. These parametric approaches attempt to estimate the parameters of distribution that will best fit the given histogram. This typically leads to a nonlinear optimization problem in which the solution is computationally expensive and time-consuming. The performance of metaheuristic algorithms greatly relies on the control parameters of the considered algorithm. The use of high number of control parameters produces premature convergence. In contrast, with a low number of control parameters, there are more chances of becoming trapped in local optima. A number of metaheuristic algorithms have been applied to multilevel thresholding [17,18], such as the modified artificial bee colony (MABC) [19] algorithm, the cuckoo search algorithm (CS) [20], improved particle swarm optimization (IPSO) [21], the fuzzy adaptive gravitational search algorithm (FAGSA) [22], hybrid Harris Hawks optimization (HHHO) [23], the improved electromagnetism optimization algorithm (IEMO) [24], wind-driven optimization (WDO) [25], the crow search algorithm (CSA) [26], the improved flower pollination algorithm (IFPA) [27], the improved harmony search algorithm (IHSA) [28], and improved emperor penguin optimization (IEPO) [29]. The main feature of these algorithms is their derivative-free behavior to obtain optimal solutions, which enhances the quality of previous solutions on the basis of exploitative and exploratory inclinations.

Despite having complex multimodality features, these algorithms have successfully optimized entropy-based objective functions. In the past years, the CS algorithm, inspired by the egg-laying behavior of cuckoo birds in nature [30], has appeared as a popular metaheuristic algorithm. Moreover, CS has proven to be a better algorithm for solving several complex and practical optimization problems such as feature selection [31], numerical optimization [32], and data clustering [33]. The better performance of CS is due to its fewer parameter settings and robustness. Despite having advantages in solving non-convex optimization problems, CS still needs to be improved in terms of achieving better accuracy and convergence speed. Improvements of the algorithm have been mainly based on the control parameters used to balance exploration and exploitation. In the literature, the CS algorithm has been categorized according to different parameter control strategies, such as hybridized, self-adaptive, and adaptive strategies [34].

1.2. Related Work

Remote sensing imageries represent a wealth of information in geoscience and geographical studies such as marine environment and agriculture, climate survey, the monitoring and mapping of forest resources, military, and metrology. The segmentation of such images has become important to attain better analysis. However, segmenting remote sensing images is a very complex task due to low illumination and dense characteristics [35]. To deal with this, researchers have presented numerous histogram-based multilevel thresholding techniques in the literature [22,23,24,25]. These techniques have performed quite well for higher levels of thresholding but with increased time complexity. A context-sensitive thresholding-based segmentation technique using an energy curve instead of an image histogram has been used to formulate the popular Kapur’s entropy, Tsalli’s entropy, and Otsu methods [36,37,38], and meta-heuristic algorithms have been employed to improve the segmentation process. These methods work with the context-sensitive information of the image. Hence, there is an increase in the quality of the segmentation at the cost of high computational complexities. In [39], a gray-level co-occurrence matrix (GLCM) was integrated with spatial correlation among the image pixels to improve the segmentation efficiency. GLCM can more steadily and consistently run because it has a low computation cost and uses second-order statistics and correlation between intensities. Later, the authors of [40,41] performed histogram-based multichannel remote sensing image segmentation using a novel modified fuzzy entropy function (MFE) and meta-heuristic algorithms. The methods have shown good outcomes compared to other similar approaches in the literature. However, the methods have attained poor performance in terms of image quality—e.g., PNSR, MSE, and FSIM—for the lower level of thresholding, whereas the computational time was satisfactory. In the literature, Rényi’s entropy has been a successful approach and has not been considered much for remote sensing image segmentation. This entropy model has high computation complexity in solving multi-dimensional image segmentation problems. In [42,43], the authors showed the use of two-dimensional Rényi’s entropy for the segmentation of general RGB images. In [44], the authors explored two-dimensional Rényi’s entropy for image compression. In [45], the authors presented multilevel thresholding techniques using a 2D histogram with Tsalli’s and Rényi’s entropy for gray-scale regular test images. The major drawback of these 2D entropy models is that the probabilities of object pixels and background pixels tend to be ignored in the second and third regions of two-dimensional histograms, which results in poor image segmentation.

Although the above-discussed methods have promising outcomes, the development of an automatic algorithm with low computation complexity and high robustness is still an open research area in remote sensing. Remote sensing technology has many advantages (e.g., fast update cycle, fewer interference factors, and saving manpower and material resources), and it is still a challenge to extract boundaries, locate objects, and separate regions in high-resolution satellite images [35]. Furthermore, segmenting remote sensing images is a very complex task due to low illumination and dense characteristics. Consequently, the reliable thresholding of such images could act as a fast indicator in further remote sensing image assessment for various geoscience and geographical research areas. The development of a high-resolution remote sensing image segmentation technique is of utmost significance and can give reliable theoretical support for engineering practices.

1.3. Contribution

To solve the above problems, a hybrid model was adopted with the thresholding technique and optimization algorithm for satellite image segmentation. The authors of this paper propose a Rényi’s entropy and MCS-based robust automatic multi-thresholding algorithm for remote sensing image analysis. In the proposed method, a new hybrid representation is used to allow particles to contain different threshold numbers within a given range defined by the minimum and maximum threshold numbers. In this paper, the modified cuckoo search (MCS) was used to reduce the time-complexity to improve the efficiency and applicability of Rényi’s entropy. Rényi’s entropy was used to produce a perfect threshold value on the basis of the intensities, thus reducing the offset. This entropy model combines the maximum entropy and entropic correlation methods. Furthermore, its integration permits us to deal with the drawbacks of multilevel thresholding, i.e., circumventing failure with sub-optimal values. MCS has a number of benefits such as easy execution, fewer parameters, and low computational cost, and it is also effective for parallel processing. The exploration capability as enhanced by opposition-based learning and an escaping strategy. To justify the performance of the proposed algorithm, the modified firefly algorithm (MFA) [46], modified bacterial foraging optimization (MBFO) [47], modified differential evolution (MDE) [48], modified particle swarm optimization (MPSO) [49], and modified artificial bee colony (MABC) [50] algorithms were compared using multilevel Rényi’s entropy as a fitness function. It is important to note that, as argued by the no-free-lunch (NFL) concept, not all evolutionary computation methods can be employed for all similar looking problems. Hence, it was worth determining whether MFA, MBFO, MDE, MABC, or MCS could offer better multilevel thresholding outcomes for image segmentation. In order to show the better performance of the proposed Rényi’s entropy–MCS (REMCS) technique, other existing thresholding methods such as the EC-based Otsu method, GLCM, and MCE entropy based on the above-mentioned meta-heuristic algorithms were compared. Experiments were performed using multiple natural and remote sensing color images at different segmentation levels. A comparison of the algorithms proved that the proposed method had the best efficiency, accuracy, and robustness for the optimal multilevel thresholding of color remote sensing images.

2. Multilevel Thresholding Functions

Consider an image I of size m × n with L distinct gray-levels. L was considered to be 256 in this paper. Multilevel thresholding determines the multiple thresholds and develops an output image with multiple groups as follows [12]:

$$\begin{gathered} C_1\leftarrow p\mathrm{if}0\le p<t{h}_1 \\ {C}_2\leftarrow p\mathrm{if}t{h}_1\le p<t{h}_2 \\ {C}_i\leftarrow p\mathrm{if}t{h}_i\le p<t{h}_{i+1} \\ {C}_k\leftarrow p\mathrm{if}t{h}_k\le p<L-1 \end{gathered}$$

(1)

where C₁,C₂,…,C_n represents the distinct classes separated by pixel p belonging to image I; th₁, th₂,.., th_k are the different threshold values; and k is the number of classes in which the image is segmented. In this section, different entropy-based objective functions that were used to compute the optimum threshold values are discussed.

2.1. Energy Curve—Otsu Method

Otsu’s method is a non-parametric process for segmentation that computes between-class variance to divide an image into various segments (classes) [4]. Assume k many thresholds represented by vector TH = { th₁, th₂,…….,th_k}. These thresholds partition the original image I into k+1 segments. Let P be the probability distribution. Then, at intensity level g (0 ≤ g ≤ L−1) of image I, the energy function value is calculated. The energy function can be expressed as [11,12]:

$$\begin{gathered} E_g^f=-{\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^N{\displaystyle \sum }_{pq\in N_{ij}^2}{b}_{ij}{b}_{pq}+C \\ f=\{\begin{array}{cc}1,2,3& \mathrm{if}\mathrm{RGB}\mathrm{or}\mathrm{Multispectral} \mathrm{image}\\ 1& \mathrm{if}\mathrm{Gray} \mathrm{level} \mathrm{image}\end{array} \end{gathered}$$

(2)

Each of the elements of the 2D binary matrix, Bg, is indicated by b_ij, Bg = {b_ij, 1 ≤ i ≤ M, 1 ≤ j ≤ N}. If g_ij > g, then b_ij = −1, and if g_ij ≤ g, then b_ij = 1. The pixel locations are indicated as i and j. The additional constant C in Equation (2) confirms that energy is always positive, i.e., E_g > 0. The neighborhood system N for order d at spatial position (i, j): $N_{ij}^d$ = {(i+u, j+v), (u,v) єN_d} shows the spatial correlation between the neighboring pixels of image I. Different configurations can be assumed by the neighborhood system in accordance with the value of d. The authors of this paper considered second-order neighborhood systems—i.e., (u,v)ϵ{(±1, 0), (0, ±1), (1, ±1), (−1, ±1)}—for every pixel in I.

Now, if the total mean intensity of I is ${\mu }_T^c={\displaystyle \sum }_{i=0}^{L-1}i{P}_i^c$, then the image variance is computed as:

$${\sigma }_B^{2^c}={\displaystyle \sum }_{i=0}^{L-1}{\sigma }_i^c={\displaystyle \sum }_{i=0}^{L-1}{w}_i^c{({\mu }_i^c-{\mu }_T^c)}^2$$

(3)

where the probability for every class is $w_i^c$ and mean of every class is ${\mu }_i^c$:

$$w_i^c={\displaystyle \sum }_{i=0}^{t{h}_k-1}{P}_i^c,{\mu }_i^c={\displaystyle \sum }_{i=t{h}_k}^{t{h}_k-1}\frac{i{P}_i^c}{w_i^c},$$

(4)

The optimal threshold values are obtained when the fitness function, f_otsu, is maximized:

$$f_{otsu}\left(t{h}_1^c,t{h}_2^c,\dots ,t{h}_k^c\right)=argmax\left\{{\sigma }_B^{2^c}\left(t{h}_1^c,t{h}_2^c,\dots ,t{h}_k^c\right)\right\}$$

(5)

2.2. Multilevel Minimum Cross Entropy

The cross entropy among the original and the segmented images is minimized in the MCE function to determine the optimal thresholds [7].

2.2.1. Cross Entropy

If F= {f₁, f₂,…., f_N} and G= {g₁, g₂,…, g_N} shows two probability distributions over the same set, then:

$$\begin{gathered} D\left(F,G\right)={\displaystyle \sum }_{i=1}^N{f}_i^c\log \frac{f_i^c}{g_i^c}; \\ c=\{\begin{array}{cc}1,2,3& \mathrm{if}\mathrm{Multispectral}\mathrm{or}\mathrm{RGB}\mathrm{image}\\ 1& \mathrm{if}\mathrm{Gray}\mathrm{scale}\mathrm{image}\end{array} \end{gathered}$$

(6)

Equation (6) indicates cross entropy between F and G, where c = 1 for gray-scale images and c = 3 for an RGB image. Now, the thresholded image I_th can be computed using:

$$I_{th}=\{\frac{{\mu }^c\left(1,th\right)\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)\text{<}\mathit{th}}{{\mu }^c\left(th,L+1\right)\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)\ge th}$$

(7)

where th represents the selected threshold to segment the image into two distinct regions (foreground and background) and ${\mu }^c\left(a,b\right)={\displaystyle \sum }_{i=a}^{b-1}i{h}^c\left(i\right)/{\displaystyle \sum }_{i=a}^{b-1}{h}^c\left(i\right)$. h^c(i) represents the histogram of an input image I. The intensity values are represented with a and b.

$$D\left(th\right)=-{\displaystyle \sum }_{i=1}^{th-1}i{h}^c\left(i\right)\log \left(\frac{i}{{\mu }^c\left(1,t\right)}\right)+{\displaystyle \sum }_{i=th}^{L}i{h}^c\left(i\right)\log \left(\frac{i}{{\mu }^c\left(th,L+1\right)}\right)$$

(8)

The MCE function searches for the optimal threshold by minimizing the cross entropy D (th) in Equation (8) to compute optimal thresholds th*:

$$t{h}^{*}=argmi{n}_{t}D\left(th\right)$$

(9)

The evaluation of all possible threshold values in the range [1, L−1] is considered for the computation of the optimal threshold. For bi-level thresholding, the complexity in locating th* is O(L²), which increases in the case of n level thresholding, i.e., O(Lⁿ⁺¹).

2.2.2. Recursive MCE

To decrease the computational complexity, the objective function uses a recursive programming approach that is represented as:

$$D\left(th\right)=-{\displaystyle \sum }_{i=1}^{L}i{h}^c\left(i\right)\log (i)-{\displaystyle \sum }_{i=1}^{th-1}i{h}^c\left(i\right)\log (\mu \left(1,th\right))-{\displaystyle \sum }_{i=th}^{L}i{h}^c\left(i\right)\log (\mu \left(th,L+1\right))$$

(10)

$$\begin{gathered} \eta \left(th\right)=-{\displaystyle \sum }_{i=1}^{th-1}i{h}^c\left(i\right)\log (\mu \left(1,th\right))-{\displaystyle \sum }_{i=th}^{L}i{h}^c\left(i\right)\log (\mu \left(th,L+1\right)) \\ =-\left({\displaystyle \sum }_{i=1}^{th-1}i{h}^c\left(i\right)\right)\log \left(\frac{{{\displaystyle \sum }}_{i=1}^{th-1}i{h}^c\left(i\right)}{{{\displaystyle \sum }}_{i=1}^{th-1}{h}^c\left(i\right)}\right)-\left({\displaystyle \sum }_{i=th}^{L}i{h}^c\left(i\right)\right)\log \left(\frac{{{\displaystyle \sum }}_{i=th}^{L}i{h}^c\left(i\right)}{{{\displaystyle \sum }}_{i=th}^L{h}^c\left(i\right)}\right) \\ =-m^{c1}\left(1,th\right)\log \left(\frac{m^{c1}\left(1,th\right)}{m^{c0}\left(1,th\right)}\right)-m^{c1}\left(th,L+1\right)\log \left(\frac{m^{c1}\left(th,L+1\right)}{m^{c0}\left(th,L+1\right)}\right) \end{gathered}$$

(11)

Over the partial range of the image histogram, $m^{c0}\left(a,b\right)={\displaystyle \sum }_{i=a}^{b-1}{h}^c\left(i\right)$ shows the zero-moment and $m^{c1}\left(a,b\right)={\displaystyle \sum }_{i=a}^{b-1}i{h}^c\left(i\right)$ shows the first-moment.

To divide the image into more than two classes, multilevel MCE can be used. For image I with L gray levels, k thresholds th₁, th₂…, th_k have to be chosen to partition the original image into k+1 segments. Two dummy thresholds th₀ = 0 and th_k+1 = L were chosen such that th₀ < th₁ < … < th_k < th_k+1 to illustrate the problem. Multilevel MCE with recursive programming can be represented by:

$$f_{MCE}\left(t{h}_1^c,t{h}_2^c,\dots ,t{h}_k^c\right)=m^{c1}\left(t{h}_{i-1},t{h}_i\right)\log \left(\frac{m^{c1}\left(t{h}_{i-1},t{h}_i\right)}{m^{c0}\left(t{h}_{i-1},t{h}_i\right)}\right)$$

(12)

So, the objective criterion in Equation (12) can be minimized to obtain the best threshold values

$$\left[t{h}_1^{*},t{h}_2^{*},\dots ,t{h}_{k-1}^{*}\right]=\arg \min \left\{ f_{MCE}\left(t{h}_1^c,t{h}_2^c,\dots ,t{h}_k^c\right)\right\}$$

(13)

subjected to the following constraints:

$$t{h}_1^{*}<t{h}_2^{*}<\cdot \cdot \cdot \cdot <t{h}_{k-1}^{*}<L-1$$

(14)

2.3. Gray-Level Co-Occurrence Matrix

The relative orientation (φ) between a pair of pixels and the relative distance (d) between those pixels are two parameters used to compute GLCM [51]. The relative pixel coordinates that border the central pixel are (0, d), (−d, d), (−d, 0), (−d, −d), where d is the distance. Consider d = (a, b), where a and b are the integer values; then, d shows a displacement vector that indicates the relative pixel positions for coordinates (x, y) and (x + a, y + b). Let C be an L × L matrix and (i, j) elements of the matrix represent the pixel pair count of image I at relative position d and orientation φ, where the gray level for the first pixel (i) and gray level for the second pixel (j) are in a spatial linear relationship. GLCM is computed by taking the average of all directions [51]:

$$GLCM=\frac{1}{4}\left(\left[C_{d,0^0}\right]+\left[C_{d,{45}^0}\right]+\left[C_{d,{90}^0}\right]+\left[C_{d,{135}^0}\right]\right)$$

(15)

GLCM uses the pixel pair frequency to compute image features. In this paper, the edge magnitude q was considered. Other features, such as correlation, contrast, variance, energy, inverse difference moment, and entropy, can also be computed using GLCM. Information regarding the edge magnitude is obtained by contrast computation to determine threshold values. Let the multiple threshold values be [T₁, T₂,…,T_k−1] in GLCM for multilevel thresholding [52]:

$$T_1=\arg \max \left(\frac{1}{{\eta }_1}{\displaystyle \sum }_{m=0}^{q_1}{\displaystyle \sum }_{n=q_1+1}^{q_2}\frac{m+n}{2}GLCM\left(m,n\right)\right)$$

(16)

$$T_2=\arg \max \left(\frac{1}{{\eta }_2}{\displaystyle \sum }_{m=q_1+1}^{q_2}{\displaystyle \sum }_{n=q_2+1}^{q_3}\frac{m+n}{2}GLCM\left(m,n\right)\right)$$

(17)

$$T_{k-1}=\arg \max \left(\frac{1}{{\eta }_{k-1}}{\displaystyle \sum }_{m=q_{k-2}+1}^{q_{k-1}}{\displaystyle \sum }_{n=q_{k-1}+1}^{L-1}\frac{m+n}{2}GLCM\left(m,n\right)\right)$$

(18)

where:

$${\eta }_1={\displaystyle \sum }_{m=0}^{q_1}{\displaystyle \sum }_{n=q_1+1}^{q_2}GLCM\left(m,n\right)$$

(19)

$${\eta }_2={\displaystyle \sum }_{m=q_1+1}^{q_2}{\displaystyle \sum }_{n=q_2+1}^{q_3}GLCM\left(m,n\right)$$

(20)

$${\eta }_{k-1}={\displaystyle \sum }_{m=q_{k-2}+1}^{q_{k-1}}{\displaystyle \sum }_{n=q_{k-1}+1}^{L-1}GLCM\left(m,n\right)$$

(21)

The threshold values correspond to edge magnitudes shown by q₁, q₂,.., q_k−1 and can be represented as:

[T₁, T₂,…,T_k−1]= arg max {f (q₁,q₂,…,q_k−1)}

(22)

To obtain the optimal thresholds, Equation (22) has to be maximized.

3. Modified Cuckoo Search Algorithm

Metaheuristics have been most generally applied to non-parametric problems and to other combinatorial optimization problems for which a polynomial-time solution exists but is not practical. Since their first appearance, metaheuristics have proven their efficiency in solving complex and intricate nonlinear optimization problems arising in various fields [10]. One of the most popular and used approaches is the CS algorithm. The CS algorithm is based on the cuckoo bird’s parasitic breeding behavior. This meta-heuristic is inspired by the cuckoo’s lifestyle, and it is a population-oriented stochastic global search algorithm [30]. A single egg laid by the cuckoo bird is dumped into a host bird’s nest, which is randomly selected. The host bird fails to find the cuckoo’s eggs if they show high similarity with the egg of the host bird, and then these eggs are carried to next generation. Otherwise, the host bird either abandons the nest or kills the eggs. The suitability of the nest is based on the high surviving rate of eggs.

$$x_i\left(t+1\right)=x_i\left(t\right)+\alpha \oplus Lévy\left(\lambda \right)$$

(23)

where step size α (α > 1) is related to the size of the problem. In Equation (23), x_i (t+1) is created by the Lévy flight of the CS algorithm for cuckoo i [34] and ⊕ represents entry-wise multiplications. The Lévy flight-based random walk carries a very long step length which in turn explores a larger search space. Random step lengths using the Lévy distribution are represented by:

$$Lévy\left(\lambda \right)=t^{-\lambda }\mathrm{where}1<\lambda \le 3$$

(24)

The basic CS algorithm has two complications: premature convergence and high computational complexity. In order to improve the performance of the CS algorithm on the basis of the above analysis, Walton et al. [53] introduced the Modified Cuckoo Search (MCS) algorithm. Lévy flight modeling plays a significant role in the convergence rate control of the CS algorithm. MCS uses a new hybrid representation to take different threshold numbers from a given range, which are defined by the minimum and maximum threshold numbers. In Lévy flight CS, a faster convergence cannot be guaranteed because the search entirely depends on random walks. Consequently, to increase the convergence rate, the Lévy flight step size α must be modified. The value of α is kept either constant or 1 in the CS algorithm [37], whereas in MCS, α is decreased with the increase in the number of generations. Initially, a Lévy flight step size A of 1 is chosen, and at every generation, $\alpha =A/\sqrt{G}$ is used to compute a new Lévy flight step, where G represents the generation number.

Secondly, the exchange of information does not happen between individuals, i.e., independent searches are performed in the CS algorithm. This has been modified in MCS, where information exchange among the eggs has been added to increase the speed of convergence to reach the minimum. Unlike CS, a fraction of the eggs that show the best fitness are placed into a group of best eggs. For every best egg, a second egg within the group is randomly chosen. Then, on the line that connects the above two best eggs, a new egg is generated. The inverse of the golden ratio $\varphi =\left(1+\sqrt{5}\right)/2$ is used to compute the distance along the line over which the new egg is positioned in such a way that it becomes closer to the egg with the best fitness. When both eggs hold the same fitness, the new egg is created at the midpoint. The use of the golden ratio shows much better performance compared to the random fraction used in CS. A local Lévy flight search carried out with a randomly picked nest has a step size of $\alpha =A/G^2$ when the same egg is picked twice. Setting the fraction of nests positioned in the top nest group to 0.25 and the fraction of nests to be abandoned p_a to 0.75 produces superior outcomes over various test functions. The pseudo code of the MCS algorithm is presented as Algorithm 1.

4. Proposed Algorithm

A remote sensing image consists of multiple channels, where each color component carries L number of grey-levels and N number of pixels. The best threshold values are be located in [0, L−1]. Every gray level is linked to the image histogram h(i) that is a plot of the frequency of the occurrence of the ith gray pixel. The proposed algorithm represents a hybrid model formed between two stratified methods by Rényi’s entropy function and MCS. The proposed method randomly searches in the histogram as a candidate; then, the quality is evaluated using Rényi’s objective function. MCS operators are evolved on candidate strings until the optimal solutions are determined.

4.1. Multilevel Rényi’s Entropy

Consider an image I with L gray levels with values in the range 0–255. Let

$$p=\left(p_1,p_2,\dots ,p_n\right)\in {\Delta }_n$$

(25)

${\Delta }_n=\left\{\left(p_1,p_2,\dots ,p_n\right)|p_i\ge 0,\mathrm{i}=1,2,\dots ,\mathrm{n},\mathrm{n}\ge 2,{\displaystyle \sum }_{i=1}^n{p}_i=1\right\}$ , which shows a set of discrete probability distributions p [9] in Equation (25). Rényi entropy is defined as [9]:

$$H_{\alpha }\left[P\right]=\frac{1}{1-\alpha }{\log }_2\left({\displaystyle \sum }_{i=1}^n{p}_i^{\alpha }\right)$$

(26)

for additively independent events. Here, entropy order is a positive integer α. The limiting case of Rényi entropy is when α reaches unity. The a priori Rényi entropy for each distribution [9,42,43] is represented as:

$$\begin{gathered} H_{\alpha }\left[C_1\right]=\frac{1}{1-\alpha }\left[\ln {\displaystyle \sum }_{i=0}^{t_1}{\left(\frac{P\left(i\right)}{P\left(C_1\right)}\right)}^{\alpha }\right], \\ {H}_{\alpha }\left[C_2\right]=\frac{1}{1-\alpha }\left[\ln {\displaystyle \sum }_{i=t_1+1}^{t_2}{\left(\frac{P\left(i\right)}{P\left(C_2\right)}\right)}^{\alpha }\right],\dots , \\ {H}_{\alpha }\left[C_k\right]=\frac{1}{1-\alpha }\left[\ln {\displaystyle \sum }_{i=t_N+1}^{L-1}{\left(\frac{P\left(i\right)}{P\left(C_k\right)}\right)}^{\alpha }\right], \end{gathered}$$

(27)

where

$$\begin{gathered} P\left(C_1\right)={\displaystyle \sum }_{i=0}^{t_1}P\left(i\right), \\ P\left(C_2\right)={\displaystyle \sum }_{i=t_1+1}^{t_2}P\left(i\right),\dots \\ P\left(C_k\right)={\displaystyle \sum }_{i=t_k+1}^{L-1}P\left(i\right) \end{gathered}$$

(28)

The normalized histogram is represented by P(i). The best threshold values (t^*= {t₁, t₂,..,t_N}) can be obtained by the maximization of H_R:

$$H_R=H\left[C_1\left]+H\right[C_2\left]+\dots +H\right[C_k\right]$$

(29)

$$t^{*}=argmax(H\left[C_1\left]+H\right[C_2\left]+\dots +H\right[C_k\right])$$

(30)

The exhaustive search process involved in maximizing the objective function limits the application of the multilevel Rényi’s entropy as the computation complexity becomes O(L^N−1). In the case of color images, Rényi’s entropy is computed for every channel of the color image. This in turn increases the computation complexity. Rényi’s entropy incorporates local information embedded in the weights and global information obtained from the gray-level histogram. Thus, Rényi’s entropy is better than the entropic correlation method or maximum entropy sum method. The objective function assesses the band subsets and provides the degree of their goodness. The performance of the system is influenced by the objective function; therefore, it needs careful determination. Consequently, an appropriate optimization algorithm needs to be selected to escape out of the local optimum and converge to the optimal global solution.

4.2. Steps for Rényi’s Entropy–MCS-Based Multilevel Thresholding

Rényi’s entropy serves as the objective criterion to reduce the complexity issues, and the MCS algorithm is implemented. The objective criterion finds the initial solution quality. At the initial step, random threshold values are generated for every candidate solution. Then, the MCS search generates new candidate solutions by exploiting solutions with objective criteria. The pre-determined rule of the MCS algorithm generates a better segmentation quality by determining the best threshold values by optimizing Rényi’s entropy. MCS avoids easy trap local optimization and causes premature convergence. Moreover, the MCS algorithm requires fewer control parameters than other meta-heuristics optimization processes. Using the operators that mimic the behavior of CS with an improved Lévy flight step size and information-exchange process, the MCS evolves solutions until it finds the optimal one. The obtained best solution is chosen and applied for image segmentation at the end of the iterative process. Below are the steps for the proposed algorithm:

Algorithm 2 Proposed Algorithm

Input: Color test image to be segmented, step size (α), mutation probability value (pa), and scale factor (β), population size, number of iterations (stopping criterion), and threshold levels. Step 1: Determine the optimal thresholds by maximizing the objective criterion following the MCS algorithm steps: Step a: Initialize population and define the control parameters. Step b: Evaluate the fitness for each nest. Step c: Adjust the adaptive control parameters α and pa. Step d: Generate a cuckoo egg (xi) by taking a Lévy flight from random nest. Step e: Abandon some worst nests with probability pa. Step f: Build new nests at new locations via Lévy flights to replace nests lost. Step g: Evaluate fitness of new nests and rank all solutions. Step h: If the stopping criteria is satisfied, return the best solution and finish the algorithm; otherwise, repeat again from step b. Step 2: The best solutions are shown by the nests that have the best quality eggs. The set of optimal threshold values (TR, TG, and TB,) corresponds to the current best solution associated with the maximum fitness function value. The individual segmentation of each color channel leads to the corresponding threshold. The segmented image is then created by concatenating the segmented color channels. Output: A segmented color image.

5. Experimental Results and Comparison of Performances

To evaluate the performance of the proposed hybrid REMCS algorithm for the multilevel thresholding of remote sensing images, experiments were performed. The five 512 × 512 remote sensing color images shown in Figure 1 were considered for the evaluation of the proposed algorithm with different multilevel thresholding algorithms. A comprehensive evaluation of the segmentation results is presented in this section. For the analysis of the segmentation results on the test images, Rényi’s entropy, MCE, EC-Otsu, and GLCM were considered as fitness functions and evaluated using meta-heuristic optimization algorithms: MFA, MBFO, MPSO, MABC, and JADE. The remote sensing images used to evaluate the performance of the algorithms were taken from https://landsat.visibleearth.nasa.gov/ (accessed date: 1 February 2021). Multidimensional colored remote sensing images have an inherent multimodal nature because of their different bands—red (R), green (G), and blue (B). Besides, accurate and sophisticated multilevel thresholding algorithms are required for the detection and identification of the regions of interest in remote sensing images with very dense and complex features. The four thresholding levels of 2-level, 5-level, 8-level, and 12-level were used to test the robustness of the proposed REMCS and other compared algorithms. All the algorithms were implemented using MATLAB R2019b on a personal computer with a 3.4 GHz Intel core-i7 CP and 8 GB of RAM running on a Windows 10 system. The experiments were executed 30 independent times to avoid any stochastic discrepancy because of the optimization algorithm’s random nature. Since the performance of any optimization algorithm depends on the choice of the parameters, the best parametric values adopted for MFA [45], MBFO [46], JADE [47], MPSO [48], and MABC [49] from the respective literature of the algorithm are listed in

Table 1. Parameter values for optimization algorithms.

Parameter Values for Optimization Algorithms
MPSO	Initial value of inertia weight	0.95
	Minimum inertia weight (Wmin)	0.4
	Maximum inertia weight (Wmax)	0.9
	Acceleration coefficients (c1,c2) K consecutive generations	2.0 3.0
	Fraction of max. iterations for which W is linearly varied	0.7
	Value of velocity weight at the end of PSO iterations	0.4
MBFO	Bacterium no. (s)	20
	Reproduction steps no. (Nre)	10
	Chemotactic steps no. (Nc)	10
	Swimming length no. (Ns) Elimination of dispersal events no. (Ned)	10 10
	Height of repellent (hrepellant) Width of repellent (wrepellant)	0.1 10
	Depth of attractant (dattract) Width of attract (wattract)	0.1 0.2
	Elimination and dispersal probability (Ped)	0.9
JADE	Scaling factor (f)	0.5
	Crossover probability	0.2
	Maximum allowed speed or velocity limit	0.3
MFA	Randomization(α)	0.01
	Attractiveness (β0)	1.0
	Light absorption coefficient at the source (γ)	1.0
MABC	Value of Fi(φ)	[0,1]
	Max trial limit	10
	Lower bound Upper bound	1 256
MCS	Scale factor (β)	1.5
	Mutation probability (Pa)	0.25

. The population size was set as 25 and the number of iterations was set as 100 to keep fairness when comparing the performances among MCS and other bio-inspired algorithms.

5.1. Fidelity Parameters for Quantitative Evaluation of the Results

As the segmentation level increases, more classes with distinct characteristics are acquired. These characteristics maintain the local features within the original image. It is impossible to determine the performance of each algorithm with the human eye for the same segmentation level, especially when a complex image with multiple objects is used. Consequently, the segmented image quality requires evaluation using some specific metrics. To carry out a comprehensive assessment of the performance of the algorithms, computation time (in seconds), mean square error (MSE), peak signal-to-noise ratio (PSNR), structural similarity (SSIM), and feature similarity (FSIM) indexes are reported in

Table 2. CPU time using different optimization algorithms with the EC-Otsu and MCE entropy methods.

Images	EC-Otsu						MCE
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	162.251	186.753	146.257	168.441	168.088	140.656	15.245	28.450	5.511	16.144	23.167	4.0367
2	173.145	200.451	167.592	187.262	171.054	165.471	16.480	380.146	8.912	19.082	35.400	7.285
3	172.481	188.415	176.481	186.842	171.287	170.210	16.524	29.810	8.927	18.074	27.019	8.125
4	162.574	185.670	157.426	172.254	168.963	145.011	17.080	28.099	6.933	18.089	27.851	4.364
5	163.275	189.933	155.210	183.352	172.401	150.417	18.662	29.126	9.612	24.171	39.353	3.812

,

Table 3. MSE and PSNR values computed using different optimization algorithms with the EC-Otsu method.

Images	MSE						PSNR
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2564.258	2836.548	2555.8412	2554.012	2000.125	1428.183	18.211	12.781	17.535	14.345	18.254	19.983
2	2299.665	2658.731	2236.704	2234.865	1680.325	1486.474	16.295	13.912	16.768	16.994	19.125	19.789
3	2933.681	2997.227	2153.493	2232.824	1457.955	1076.302	14.572	12.709	16.805	17.011	17.894	18.592
4	1891.080	2706.232	1663.274	1871.285	1391.106	1758.850	17.188	16.959	17.579	18.545	18.101	19.370
5	1641.192	2058.713	1481.607	1608.255	1381.914	1689.140	15.894	17.526	18.967	20.113	19.986	19.822

,

Table 4. Comparison of SSIM and FSIM computed by different algorithms using the EC-Otsu method.

Images	SSIM						FSIM
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	0.7310	0.7268	0.7584	0.7361	0.7791	0.7806	0.7381	0.7465	0.7698	0.7307	0.8857	0.8921
2	0.6845	0.7282	0.7558	0.7402	0.7684	0.7892	0.7684	0.7541	0.7891	0.7654	0.8547	0.8899
3	0.7200	0.6954	0.7963	0.7708	0.8245	0.8541	0.7354	0.6511	0.7587	0.7798	0.8541	0.8951
4	0.7355	0.6733	0.7584	0.7456	0.8208	0.8359	0.7341	0.6531	0.7435	0.7424	0.8650	0.8824
5	0.7411	0.6874	0.7624	0.7542	0.8654	0.8714	0.7822	0.7841	0.7909	0.7932	0.8740	0.8854

,

Table 5. MSE and PSNR using different optimization algorithms with minimum cross entropy.

Images	MSE						PSNR
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	5758.324	4663.072	6513.584	6203.093	2400.645	2300.365	14.935	13.485	12.415	13.425	15.189	17.5621
2	5353.061	4578.005	6766.252	6563.295	2856.142	2285.231	16.712	17.113	14.635	15.956	16.348	17.365
3	5084.854	3250.151	6966.120	6795.172	2685.216	2411.589	13.856	14.842	12.771	15.941	16.645	17.156
4	5030.843	3774.451	6868.256	6455.256	2895.200	2795.253	16.000	16.685	12.124	15.841	17.795	18.525
5	5600.028	3664.432	6789.526	6430.256	2000.263	2850.263	19.476	20.002	17.125	18.845	20.826	21.842

,

Table 6. SSIM and FSIM using different optimization algorithms with minimum cross entropy.

Images	SSIM						FSIM
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	0.8516	0.8601	0.8399	0.8413	0.8713	0.8822	0.8652	0.8718	0.8350	0.8518	0.8902	0.8979
2	0.8415	0.8591	0.8385	0.8426	0.8780	0.8883	0.8725	0.8800	0.8490	0.8552	0.8852	0.8956
3	0.8313	0.8446	0.8145	0.8415	0.8585	0.8756	0.8562	0.8652	0.8421	0.8442	0.8852	0.8952
4	0.8513	0.8613	0.8213	0.8476	0.8713	0.8813	0.8662	0.8761	0.8421	0.8584	0.8821	0.8993
5	0.8553	0.8600	0.8201	0.8423	0.8690	0.8919	0.8695	0.8785	0.8458	0.8517	0.8982	0.9065

,

Table 7. CPU time using different optimization algorithms with GLCM and Rényi’s entropy method.

Images	GLCM						Rényi’s Entropy
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	23.697	17.512	14.214	18.867	18.570	17.902	9.807	16.511	7.933	14.854	7.175	2.109
2	24.210	16.798	15.098	18.867	18.846	17.667	9.812	17.422	7.840	15.798	8.968	4.402
3	24.006	19.425	16.632	19.847	18.739	20.099	9.125	19.198	7.521	14.425	8.227	6.512
4	26.443	16.854	11.098	19.811	18.350	17.016	10.082	20.806	8.251	16.854	9.593	3.872
5	22.749	21.825	20.241	20.869	22.541	21.971	10.962	20.993	7.512	18.825	9.399	6.486

,

Table 8. MSE and PSNR using different optimization algorithms with the GLCM method.

Images	MSE						PSNR
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	974.206	1024.852	1011.564	994.308	869.238	800.634	20.396	19.176	18.176	22.462	23.866	25.533
2	989.812	1077.524	1007.213	994.521	864.609	896.329	20.173	19.114	18.366	22.597	25.439	26.636
3	961.518	1066.218	1116.621	996.212	895.586	822.851	21.587	20.493	18.772	24.492	25.466	25.517
4	985.547	1079.527	1014.021	996.527	941.484	806.524	20.956	19.866	19.215	23.482	25.076	26.256
5	975.347	1090.257	1000.624	992.527	911.209	861.624	20.476	19.893	18.216	16.486	19.287	20.483

,

Table 9. SSIM and FSIM using different optimization algorithms with the GLCM method.

Images	SSIM						FSIM
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	0.8388	0.8705	0.7885	0.9092	0.9550	0.9601	0.8491	0.9215	0.8381	0.9481	0.9597	0.9825
2	0.8474	0.8595	0.8114	0.9125	0.9564	0.9622	0.8462	0.9299	0.8835	0.9599	0.9680	0.9735
3	0.8294	0.9046	0.7912	0.9124	0.9587	0.9734	0.8499	0.9222	0.8534	0.9594	0.9735	0.9775
4	0.8422	0.8616	0.7912	0.9274	0.9554	0.9643	0.8427	0.9222	0.8658	0.9698	0.9792	0.9732
5	0.8712	0.8525	0.7952	0.9321	0.9679	0.9712	0.8469	0.9281	0.8981	0.9699	0.9746	0.9777

,

Table 10. MSE and PSNR using different optimization algorithms with Rényi’s entropy.

Images	MSE						PSNR
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	658.328	563.029	513.586	503.037	300.362	200.656	23.035	23.815	24.815	26.741	26.985	27.952
2	453.068	608.009	566.254	363.252	285.234	256.126	24.812	23.213	24.735	25.056	26.448	27.965
3	684.857	600.157	566.124	495.123	311.585	185.265	22.956	23.942	24.871	25.041	26.745	27.856
4	630.843	864.456	568.256	545.256	395.254	295.200	22.695	23.785	24.224	25.941	26.895	28.825
5	700.028	664.436	589.526	440.256	350.268	250.263	22.476	24.082	24.125	25.845	27.826	28.942

and

Table 11. SSIM and FSIM computed by different algorithms using Rényi’s entropy.

Images	SSIM						FSIM
	MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	0.8615	0.9012	0.9399	0.9422	0.9732	0.9881	0.9325	0.9535	0.9671	0.9615	0.9822	0.9996
2	0.8614	0.9080	0.9094	0.9432	0.9769	0.9883	0.9359	0.9442	0.9552	0.9689	0.9905	0.9965
3	0.8712	0.9052	0.9254	0.9424	0.9794	0.9865	0.9326	0.9416	0.9535	0.9622	0.9900	0.9935
4	0.8712	0.9002	0.9522	0.9465	0.9779	0.9812	0.9236	0.9436	0.9426	0.9622	0.9823	0.9901
5	0.8852	0.9095	0.9212	0.9432	0.9889	0.9939	0.9395	0.9489	0.9468	0.9672	0.9895	0.9986

. The values represent averages computed over four levels (2, 5, 8, and 12) for each of the five different test images shown in Figure 1.

Table 12. Statistical analysis (Wilcoxon rank sum test) of 20 runs for each of the 20 independent samples for the experiments.

Images	Threshold Levels	MCS						Rényi’s Entropy
		Rényi’s vs. MCE		Rényi’s vs. GLCM		Rényi’s vs. EC-Otsu		MCS vs. MFA		MCS vs. MBFO		MCS vs. JADE		MCS vs. MPSO		MCS vs. MABC
		p	h	p	h	p	h	p	h	p	h	p	h	p	h	p	h
1	2	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	5	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	8	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	0.084	0	<0.05	1	<0.05	1
	12	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
2	2	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	5	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	8	<0.05	1	0.079	0	<0.05	1	<0.05	1	<0.05	1	<0.05	1	0.085	0	<0.05	1
	12	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
3	2	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	5	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	8	<0.05	1	<0.05	1	<0.05	1	0.067	0	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	12	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	0.09	0	<0.05	1	<0.05	1
4	2	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	5	<0.05	1	<0.05	1	0.061	0	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	8	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	12	<0.05	1	0.072	0	<0.05	1	<0.05	1	<0.05	1	<0.05	1	0.075	0	<0.05	1
5	2	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	5	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	8	<0.05	1	<0.05	1	<0.05	1	0.069	0	<0.05	1	<0.05	1	<0.05	1	<0.05	1
	12	<0.05	1	0.062	0	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1	<0.05	1

shows the Wilcoxon statistical test used to judge the significance of the proposed algorithm.

5.1.1. Computation Time (in Seconds)

The complexity of any algorithm influences its computation time. The mathematical structure of any algorithm and the objective function used defines the complexity of that algorithm. As a result, computation time becomes an essential factor to determine the efficiency of an algorithm. The time involved to generate the segmented image is directly proportional to the algorithm’s complexity.

5.1.2. PSNR and MSE

PSNR and MSE determine the accuracy of the segmentation algorithm, defined as:

$$MSE=\frac{1}{MN}{\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^N{[I\left(i,j\right)-\tilde{I}\left(i,j\right)]}^2$$

(31)

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

(32)

where M and N represent the image size, I is the input image to be segmented, and Ĩ is the output image at pixel position (i,j) after segmentation at a given thresholding level. A high PSNR and a low MSE are desired to indicate the good performance of an algorithm.

5.1.3. SSIM and FSIM

The global similarity between the input and the segmented output image can be measured by using two parameters: SSIM and FSIM. FSIM indicates how well the features are preserved after the processing of the image. This is significant in the classification systems for remote sensing images. SSIM indicates the visible structures of the test image that are likely to be passed over the segmented image. SSIM is a parameter used to assess the quality of the segmented image and is based on structural information degradation. The SSIM compares the input and segmented output structures using [15,39]:

$$\mathrm{SSIM}\left(x,y\right)=\frac{\left(2{\mu }_x{\mu }_y+U_1\right)\left(2{\sigma }_{xy}+U_2\right)}{\left({\mu }^2{}_x+{\mu }^2{}_y+U_1\right)\left({\sigma }^2{}_x+{\sigma }^2{}_y+U_2\right)}$$

(33)

The mean intensity of image x and y is given by µ_x and µ_y, respectively. The standard deviations of x and y are given by σ_x and σ_y, respectively. The local sample correlation coefficient between x and y is given by σ_xy. The constants—U₁ = U₂ = 0.065—are used to circumvent any instability closer to zero. For multichannel images,

$$\mathrm{SSIM}={\displaystyle \sum }_c\mathrm{SSIM}(x^c,y^c)$$

(34)

where x^c and y^c represent the cth channel of the input image and segmented output image, respectively, where c (i.e., c = 1, 2, 3 in true color RGB images) shows the channel number.

FSIM computes the feature similarity between the input and segmented images [39] as follows:

$$\mathrm{FSIM}=\frac{{{\displaystyle \sum }}_{X\in \Omega }{S}_L\left(X\right)P{U}_m\left(X\right)}{{{\displaystyle \sum }}_{X\in \Omega }P{U}_m\left(X\right)}$$

(35)

where the entire image is indicated by Ω and S_L(x) shows the similarity between the segmented output image and input image. For multichannel images,

$$\mathrm{FSIM}={\displaystyle \sum }_c\mathrm{FSIM}(x^c,y^c)$$

(36)

SSIM and FSIM vary between 1 and 0, where 1 indicates the maximum similarity or a high segmentation quality and 0 represents the minimum similarity or a poor segmentation quality of the output. The fidelity parameters for each channel of the multichannel images are computed separately, and their averages can be taken as the final values.

5.2. Comparison Using the Otsu Energy (EC-Otsu) Method as an Objective Function

The results obtained by using the MFA [45], MBFO [46], JADE [47], MPSO [48], MABC [49], and MCS [29] with the EC-Otsu method as a fitness function are shown in Table 2, Table 3 and Table 4, and Figure 2, Figure 3 and Figure 4 show their graphical representations. Figure 5 shows a visual comparison of the results. Detailed tables indicating the results computed over each of the segmentation levels (2, 5, 8, and 12) are shown in Appendix

Table A1. CPU time using different optimization algorithms with the EC-Otsu and MCE entropy methods.

Images	m	EC-Otsu						Minimum Cross Entropy
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	161.408	181.39	146.256	167.512	163.846	141.291	10.562	22.411	5.213	12.826	16.047	4.227
	5	161.376	188.249	146.428	167.963	168.227	144.871	12.654	25.821	5.084	14.044	18.011	4.183
	8	163.102	188.993	146.658	173.425	169.519	146.931	18.854	28.005	5.367	16.441	24.117	4.634
	12	164.142	190.182	146.989	174.324	170.265	144.415	20.845	32.411	6.589	16.511	32.152	6.863
2	2	168.8	189.068	165.225	175.429	166.229	171.707	12.865	34.54	5.852	13.66	21.15	5.871
	5	170.736	189.539	167.285	178.321	167.539	170.565	15.652	36.469	5.485	15.823	30.472	7.36
	8	172.058	198.587	167.484	183.867	171.284	173.962	19.5	40.34	5.458	19.28	38.4	7.984
	12	172.815	200.958	168.182	188.216	171.689	184.275	20.798	43.809	7.809	19.4	46.676	10.086
3	2	160.322	187.419	175.958	175.427	169.605	174.447	11.658	23.514	7.273	10.007	11.42	5.871
	5	160.567	188.4	176.153	179.541	170	170.541	10.851	26.915	7.206	13.854	20.31	7.36
	8	165.335	188.839	176.282	186.147	171.387	170.147	16.125	28.754	8.972	18.074	27.572	8.1
	12	166.735	189.265	176.862	186.865	171.958	181.254	20.425	30.632	8.982	21.198	35.245	8.086
4	2	160.67	180.437	156.624	166.156	167.369	144.106	12.652	21.981	6.031	10.751	18.089	4.227
	5	161.667	188.213	156.858	171.102	167.475	145.101	17.465	24.351	6.386	13.792	25.768	4.183
	8	162.535	180.157	158.901	174.728	173.297	145.301	17.487	27.9	6.393	18.098	30.702	4.634
	12	163.789	192.658	149.265	175.524	173.689	145.458	20.854	30.098	8.621	19.86	36.016	6.863
5	2	161.037	178.301	150.258	180.265	166.394	180.686	12.285	23.264	8.148	17.748	31.353	2.973
	5	163.174	179.312	151.648	183.795	166.976	182.57	14.285	26.662	9.321	23.371	29.389	2.216
	8	163.135	189.339	155.021	184.543	171.102	183.783	18.865	29.624	9.754	28.097	37.612	4.725
	12	165.893	190.256	156.958	195.425	172.524	194.201	20.825	33.241	10.253	31.399	45.971	5.83

,

Table A2. MSE and PSNR values computed using different optimization algorithms with the EC-Otsu method.

Images	m	MSE						PSNR
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	1954.548	1075.554	2855.01	2465.452	2846.154	2085.551	19.215	11.133	13.524	12.534	17.991	19.592
	5	1745.154	1415.842	2455.026	2945.515	2236.658	2736.442	14.598	12.594	13225	14.557	17.224	19.558
	8	1564.846	1658.013	2655.324	2761.445	2454.954	2918.841	15.484	13.866	15.866	15.135	20.807	20.522
	12	1658.562	1563.325	2123.756	2193.045	2150.151	2860.152	15.549	13.527	15527	15.156	20.596	20.263
2	2	1856.794	1203.774	2324.954	2375.336	2622.155	2341.111	16.902	14.534	17.527	15.465	17.328	19.264
	5	1765.984	1418.015	2014.856	2044.995	2950.481	2163.145	16.527	12.135	14.468	16.855	17.801	19.861
	8	1645.215	1845.351	2850.852	2465.145	2305.848	2198.484	14.523	12.658	15.532	16.987	20.52	19.461
	12	1453.345	1478.759	2956.154	2053.985	2756.442	2495.922	17.228	16.321	15.546	18.669	20.853	20.573
3	2	1567.021	1003.751	2755.256	2006.143	1425.454	1768.461	14.542	10.263	13.322	15.493	15.375	17.523
	5	1215.341	1085.953	2256.181	2883.954	1106.853	2166.222	14.862	13.852	13.158	15.695	16.845	18.527
	8	1065.278	1065.153	1635.754	1395.354	1850.945	2078.896	14.216	13.466	14.554	18.699	19.803	18.658
	12	1984.182	1150.351	1966.784	1445.845	1205.656	2921.145	14.669	13.258	14.187	18.159	19.556	19.661
4	2	1745.068	1352.254	1745.215	1111.654	1965.784	1350.333	17.494	15.863	17.462	17.794	17.866	18.125
	5	1945.042	1895.256	1148.55	1953.784	1748.446	1814.951	16.116	15.225	18.266	18.632	17.551	18.165
	8	1943.986	1912.854	1820.848	1735.955	1425.494	1054	16.637	18.152	19.158	18.594	18.483	20.657
	12	1930.227	1874.856	1938.484	1915.748	1685.205	1345.142	18.505	18.596	19.432	19.158	18.507	20.535
5	2	1909.984	1878.951	1717.446	1256.451	1757.942	1196.365	17.825	13.822	16.511	18.534	17.151	18.546
	5	1654.001	1745.159	1170.985	1749.454	1862.145	1315.256	14.497	14.257	17.264	18.499	19.341	18.189
	8	1500.215	1567.852	1298.448	1965.648	1989.215	1705.142	15.572	17.566	20.558	21.864	21.815	21.296
	12	1500.571	1564.456	1739.552	1460.948	2625.551	1310.896	15.684	17.299	21.535	21.558	21.638	21.258

and

Table A3. Comparison of SSIM and FSIM computed by different algorithms using the EC-Otsu method.

Images	m	SSIM						FSIM
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	0.6114	0.7456	0.7134	0.7261	0.7567	0.7523	0.7123	0.6805	0.7612	0.7315	0.8116	0.8535
	5	0.6713	0.7568	0.7256	0.7345	0.7823	0.7634	0.7234	0.7356	0.7645	0.7302	0.8589	0.8478
	8	0.7301	0.7689	0.7389	0.739	0.7798	0.7789	0.7567	0.7517	0.7689	0.737	0.8875	0.8734
	12	0.7512	0.7702	0.7398	0.7412	0.7825	0.7881	0.7615	0.7597	0.7705	0.7381	0.8812	0.8821
2	2	0.6257	0.7144	0.7298	0.7592	0.7655	0.7645	0.7267	0.6824	0.7587	0.7545	0.8238	0.8016
	5	0.6735	0.7457	0.7345	0.7991	0.7774	0.7765	0.7568	0.7872	0.7789	0.7698	0.8546	0.8654
	8	0.7417	0.7589	0.7456	0.8034	0.7889	0.7834	0.7689	0.8349	0.7867	0.7712	0.8769	0.8829
	12	0.7425	0.759	0.7484	0.8125	0.7952	0.7852	0.7714	0.8365	0.7899	0.7825	0.8789	0.8882
3	2	0.6003	0.7089	0.7592	0.749	0.8034	0.8145	0.7665	0.6357	0.7945	0.7245	0.8055	0.8073
	5	0.6051	0.7246	0.7991	0.7519	0.8245	0.7209	0.7678	0.7297	0.8167	0.7356	0.8356	0.8998
	8	0.7374	0.7238	0.7034	0.7629	0.8456	0.7876	0.7789	0.7502	0.7478	0.7467	0.8504	0.8726
	12	0.7425	0.7245	0.7144	0.7714	0.8526	0.7825	0.7112	0.7525	0.7512	0.7524	0.8584	0.8755
4	2	0.627	0.7078	0.7245	0.7256	0.8256	0.7267	0.7024	0.6539	0.7397	0.7423	0.8243	0.8064
	5	0.673	0.7234	0.7359	0.7378	0.8345	0.7356	0.7124	0.6754	0.7413	0.7534	0.8544	0.8703
	8	0.7486	0.7367	0.7398	0.7456	0.8456	0.7398	0.7345	0.7874	0.7477	0.7612	0.8783	0.8804
	12	0.7512	0.7412	0.7412	0.7475	0.8526	0.7416	0.7412	0.7892	0.7512	0.7648	0.8812	0.8812
5	2	0.6539	0.7821	0.7682	0.768	0.8878	0.7813	0.7867	0.8672	0.7815	0.7845	0.8515	0.8228
	5	0.6754	0.7823	0.7612	0.7688	0.8867	0.7834	0.7867	0.7978	0.7902	0.7898	0.864	0.8838
	8	0.6874	0.7912	0.7642	0.7801	0.8978	0.7912	0.7923	0.7982	0.799	0.7967	0.8892	0.8904
	12	0.6985	0.7925	0.7702	0.7622	0.8995	0.7958	0.7952	0.7956	0.7982	0.7971	0.8899	0.8918

. The analysis of the algorithms is discussed below.

5.2.1. Assessment Based on Computation Time (CPU Time)

Figure 2 shows the graphical analysis of average CPU time computed using the EC-Otsu-based algorithms. Table 2 shows the quantitative results. The MCS algorithm presented the best computation time. In terms of efficiency, MBFO took the largest time due to its complex strategy of searching the optimal solutions. The complexity of this method depends on the mathematical modeling of the objective function, as well as the architecture and search strategy of the optimization algorithms. Therefore, different algorithms lead to different results. For EC-Otsu, the optimization algorithms in terms of increasing time complexity could be arranged as MCS < JADE < MFA < MABC < MPSO < MBFO.

5.2.2. Assessment Based on PSNR, MSE, SSIM, and FSIM

Figure 3 shows a graphical comparison of the average PSNR and MSE computed using EC-Otsu-based algorithms. Table 3 shows a quantitative comparison. According to the results, the average PSNR and MSE values for the MCS algorithm were the best. In other algorithms, JADE obtained somewhat better values, while the results of MABC and MPSO were nearly equal. MBFO obtained the lowest PSNR values. In terms of increasing PSNR and decreasing MSE values, the order of the algorithms was MBFO < MPSO < JADE < MABC < MFA < MCS.

Figure 4 shows a graphical comparison of the average SSIM and FSIM computed using EC-Otsu-based algorithms. Table 4 shows the quantitative results. The maximum SSIM and FSIM were obtained in the case of MCS, followed by JADE, MABC, MPSO, MFA, and MBFO, which indicates the excellent optimization ability of MCS in comparison to the other metaheuristic approaches. The substantial difference between the performances indicates that the segmentation performance obtained by the MBFO, JADE, and MPSO algorithms deteriorated due to the randomness introduced in the selection of the initial population. Moreover, the Lévy flight strategy of MCS had a greater influence on the optimization ability of the algorithm. In terms of increasing SSIM and FSIM values, the order of the algorithms was MBFO < MFA < MPSO < JADE < MABC < MCS.

5.2.3. Visual Analysis of the Results

Figure 5 shows the segmented outputs using the EC-Otsu method. MCS obtained the best segmented output, but MBFO was not able to properly distinguish the pixels among different classes based on their gray levels. The obtained results were not satisfactory. The rest of the algorithms obtained good results at higher thresholding levels. At lower thresholding levels, the segmented output was not very satisfying when using MFA. Generally, as the thresholding level increased, the image quality also improved. From the figure, it can be seen that MCS exhibited excellent optimization performance and searching ability, making it the best choice to solve the segmentation problem.

5.3. Comparison Using MCE Method as an Objective Function

The results obtained using MABC, MPSO, JADE, MFA, MBFO, and MCS using the MCE method as a fitness function are shown in Table 3, Table 5 and Table 6, and Figure 6, Figure 7 and Figure 8 show the average values computed over four different threshold levels (2, 5, 8, and 12). Figure 9 shows a visual comparison of the results. Detailed tables indicating the results computed over each of the segmentation levels (2, 5, 8, and 12) are shown in Table A1,

Table A4. MSE and PSNR computed using different optimization algorithms with minimum cross entropy.

Images	m	MSE						PSNR
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	3054.122	6975.124	4745.428	5845.225	3156.347	2815.622	12.162	12.321	12.415	13.425	15.189	17.5621
	5	5645.548	6515.784	4895.485	5695.452	3256.116	2516.664	13.975	13.485	12.512	13.745	16.412	16.845
	8	5664.123	5758.12	4645.855	3641.155	3354.099	2408.002	14.844	14.658	13.658	14.521	20.798	17.215
	12	5758.324	5663.072	4513.584	3203.093	2400.645	2300.365	14.935	14.715	13.715	14.641	20.685	17.352
2	2	5956.732	5303.425	4654.653	3945.572	3102.276	2651.741	15.296	15.425	15.715	14.554	15.813	16.452
	5	5865.124	5518.455	4684.411	3654.596	3350.828	2103.325	15.265	16.521	16.854	15.548	15.198	17.158
	8	5745.155	5945.128	4710.569	3715.185	3485.629	2198.823	16.315	16.846	17.225	15.879	17.015	17.154
	12	5353.061	5578.005	4766.252	3563.295	2856.142	2285.231	16.712	17.113	17.635	16.956	17.348	17.365
3	2	6667.759	5103.785	5105.378	6416.448	3205.365	1658.244	13.235	13.352	11.213	12.384	14.563	17.9315
	5	5315.765	5185.894	5006.122	4103.394	3026.372	2486.812	13.258	14.248	11.841	13.586	15.215	14.715
	8	6165.563	5165.645	4865.471	5785.439	3140.593	2398.773	13.602	14.654	12.445	13.986	18.398	17.846
	12	6084.854	5250.151	4966.12	3795.172	3685.216	2411.589	13.856	14.842	12.771	13.941	18.645	17.156
4	2	5545.372	5552.577	4985.468	4951.495	2285.601	1620.312	15.384	16.358	15.254	15.487	16.658	17.511
	5	6055.57	5705.566	5798.456	4523.577	2898.492	1024.09	16.501	16.512	16.652	16.126	17.145	17.251
	8	6043.572	5022.563	5790.663	4595.554	3785.498	2784.526	16.726	17.241	17.841	17.385	17.374	18.566
	12	6030.843	5774.451	4868.256	5455.256	3895.2	2795.253	17.595	17.685	18.124	17.841	17.795	18.525
5	2	6009.567	5778.348	4767.345	5686.456	3997.353	2976.182	16.518	17.228	18.105	19.425	16.14	12.635
	5	6754.345	5845.902	4890.567	5589.456	3882.659	2935.967	18.784	17.742	18.452	19.984	18.133	18.971
	8	5600.565	6667.156	4808.123	5445.686	2889.189	2885.16	20.265	20.655	20.845	21.458	21.508	21.682
	12	6600.028	6664.432	4789.526	6430.256	2850.263	2850.263	21.476	21.982	21.125	21.845	21.826	21.842

, and

Table A5. SSIM and FSIM computed using different optimization algorithms with minimum cross entropy.

Images	m	SSIM						FSIM
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	0.8124	0.8457	0.8135	0.8263	0.8617	0.8634	0.8676	0.8732	0.8721	0.8851	0.8802	0.8915
	5	0.8235	0.8569	0.8257	0.8346	0.858	0.8479	0.8632	0.8643	0.8754	0.882	0.8831	0.8965
	8	0.8568	0.869	0.839	0.8391	0.8876	0.8735	0.8689	0.8698	0.8798	0.8807	0.8831	0.8971
	12	0.8616	0.8701	0.8399	0.8413	0.8813	0.8822	0.8652	0.8618	0.875	0.8818	0.8902	0.8979
2	2	0.8268	0.8145	0.8299	0.8593	0.8239	0.8817	0.8655	0.8654	0.8778	0.8854	0.8875	0.8842
	5	0.8569	0.8458	0.8346	0.8992	0.8547	0.8653	0.8647	0.8656	0.8798	0.8889	0.8853	0.8827
	8	0.869	0.859	0.8457	0.8035	0.876	0.882	0.8698	0.8643	0.8876	0.8821	0.8871	0.8894
	12	0.8715	0.8591	0.8485	0.8126	0.878	0.8883	0.8625	0.8625	0.889	0.8852	0.8852	0.8956
3	2	0.8666	0.859	0.8593	0.8491	0.8656	0.8074	0.8643	0.8657	0.8754	0.8854	0.873	0.8975
	5	0.8679	0.8247	0.8992	0.851	0.8357	0.8997	0.8654	0.869	0.8876	0.8865	0.8815	0.8979
	8	0.879	0.8239	0.8035	0.863	0.8505	0.8727	0.8665	0.8667	0.8887	0.8876	0.8947	0.892
	12	0.8113	0.8246	0.8145	0.8715	0.8585	0.8756	0.8662	0.8652	0.8821	0.8842	0.8952	0.8952
4	2	0.8025	0.8079	0.8246	0.8257	0.8244	0.8065	0.8665	0.8676	0.8807	0.8832	0.8898	0.8929
	5	0.8125	0.8235	0.836	0.8379	0.8545	0.8704	0.8654	0.8665	0.8803	0.8843	0.8831	0.8945
	8	0.8346	0.8368	0.8399	0.8457	0.8784	0.8805	0.8665	0.8689	0.8868	0.8821	0.8878	0.8947
	12	0.8413	0.8413	0.8413	0.8476	0.8813	0.8813	0.8662	0.8661	0.8821	0.8884	0.8921	0.8993
5	2	0.8868	0.8322	0.8683	0.8681	0.8516	0.8229	0.9878	0.8631	0.8893	0.8854	0.8951	0.8927
	5	0.8868	0.8824	0.8613	0.8689	0.8641	0.8839	0.9867	0.8643	0.8745	0.8889	0.892	0.8987
	8	0.8924	0.8913	0.8645	0.8802	0.8893	0.8901	0.9978	0.8621	0.8847	0.8876	0.899	0.8928
	12	0.8953	0.8926	0.8701	0.8623	0.889	0.8919	0.9995	0.8685	0.8858	0.8817	0.8982	0.8965

. The analysis of the algorithms is discussed below.

5.3.1. Assessment Based on Computation Time (in Seconds)

Multilevel MCE was maximized using different optimization algorithms to obtain a segmented image. Figure 6 shows the graphical comparison of the average CPU time computed using MCE-based algorithms. Table 3 shows the quantitative results. According to the obtained results, the MCS algorithm was the fastest due to the use of few tuning parameters. JADE also obtained faster results than the MABC and MPSO, which were trapped into local minima. Other algorithms also performed well in the case of MCE. For MCE, the optimization algorithms in terms of increasing time complexity could be arranged as MCS < JADE < MFA < MPSO < MABC < MBFO.

5.3.2. Assessment Based on PSNR, MSE, SSIM, and FSIM

Figure 7 shows a graphical comparison of the average MSE and PSNR computed using the MCE-based algorithms. Table 5 shows the quantitative values. The best results were obtained using the MCS algorithm, followed by ABC and DE. The PSNR and MSE values computed using MBFO were nearly the same as those obtained using MFA, whereas the JADE outputs followed those of the MCS. The high PSNR indicated the better segmentation quality. In terms of increasing PSNR and decreasing MSE values, the order of the algorithms was JADE < MPSO < MFA < MBFO < MABC < MCS.

Figure 8 shows a graphical comparison of the average SSIM and FSIM computed using MCE-based algorithms. Table 6 shows the quantitative results. Both SSIM and FSIM are the essential parameters in the analysis of the segmentation quality of any algorithm. In the case of MCE, SSIM and FSIM were at maximum when using MCS. The other algorithms showed fair results. JADE obtained good results. The performance of MBFO and MFA was good at lower threshold levels. In terms of increasing SSIM and FSIM values, the order of the algorithms was JADE < MPSO < MFA < MBFO < MABC < MCS.

5.3.3. Visual Analysis of the Results

From Figure 9, it can be seen that the MBFO and MFA failed to be efficient in accurately finding the threshold values. This led to poor segmentation in some of the cases. The average computed values show that MCS resulted in good outputs. Furthermore, the searching ability of the CS algorithm was improved by adaptively adjusting the Lévy flight step size. An adaptive step size led to significantly improved solution quality, overcame premature convergence, and helped the algorithm to come out of local optima. This resulted in more reliable and stable optimization performance.

5.4. Comparison Using GLCM as an Objective Function

The average segmentation results by using MABC, MPSO, MDE, MFA, MBFO, and MCS with GLCM methods as a fitness function are quantitatively shown in Table 7, Table 8 and Table 9 and graphically shown in Figure 10, Figure 11 and Figure 12. Figure 13 shows a visual comparison of the results. Detailed tables indicating the results computed over each of the segmentation level (2, 5, 8, and 12) are shown in

Table A6. CPU time using different optimization algorithms with GLCM and Rényi’s entropy method.

Images	m	GLCM						Rényi’s Entropy
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	16.339	10.562	10.047	15.824	16.026	12.114	8.072	12.826	4.728	10.562	5.625	2.159
	5	20.135	12.654	11.111	16.327	17.042	15.821	6.149	12.44	4.79	12.654	6.541	2.635
	8	20.811	15.854	12.321	17.21	17.667	18.5	8.207	15.441	4.002	14.854	6.521	2.628
	12	23.697	14.512	17.152	18.867	18.57	18.214	9.807	16.511	7.933	14.854	7.175	2.109
2	2	15.867	10.224	10.15	11.845	11.709	14.54	6.145	12.006	3.429	12.115	8.224	4.62
	5	20.475	11.652	11.472	15.327	14.962	16.469	6.629	13.823	4.762	14.652	8.345	4.216
	8	26.972	15.521	12.14	15.21	16.922	15.403	6.151	15.28	6.366	15.502	9.098	4.549
	12	24.21	16.798	15.667	17.867	18.846	16.098	7.812	15.422	7.84	17.798	8.968	4.402
3	2	20.988	11.658	12.42	15.066	15.913	13.514	4.738	10.37	4.574	11.658	8.294	6.642
	5	20.994	12.851	12.31	15.262	16.154	16.915	4.266	13.854	4.657	12.851	5.106	6.284
	8	22.007	12.521	14.572	15.964	16.409	19.745	9.863	18.074	4.417	12.521	8.856	6.569
	12	24.006	16.425	19.245	17.847	18.739	20.632	7.125	19.198	8.521	14.425	8.227	6.512
4	2	16.137	21.652	20.089	13.468	15.14	10.981	9.145	11.751	5.844	11.652	9.492	2.254
	5	20.905	21.465	25.768	15.685	16.995	13.351	7.447	14.792	5.256	15.465	9.326	2.502
	8	26.643	24.487	25.721	15.51	17.824	16.009	8.534	19.098	5.854	15.487	10.429	3.201
	12	26.443	24.854	26.016	16.811	18.35	11.098	6.082	20.806	7.251	16.854	10.593	3.872
5	2	17.001	20.285	21.335	14.253	10.448	12.264	6.315	14.348	5.685	12.285	9.194	6.598
	5	20.143	20.285	21.389	15.586	12.248	17.662	7.723	14.371	5.286	13.285	9.476	6.625
	8	20.729	21.865	23.612	15.452	17.845	18.624	9.085	17.097	5.546	14.865	10.379	6.514
	12	22.749	22.825	23.971	18.869	19.541	20.241	9.962	20.993	7.512	18.825	10.399	6.486

,

Table A7. MSE and PSNR computed using different optimization algorithms with GLCM method.

Images	m	MSE						PSNR
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	865.232	886.415	956.218	945.524	967.732	926.266	20.613	20.232	20.146	21.246	23.81	23.262
	5	856.557	826.473	925.846	906.246	967.318	927.666	21.796	21.446	20.153	21.876	24.143	24.486
	8	875.129	869.014	956.581	952.516	965.902	918.006	22.485	22.569	21.569	22.252	22.979	25.126
	12	869.238	874.206	924.852	914.308	911.564	800.634	22.396	22.176	21.176	22.462	22.866	25.533
2	2	867.378	814.541	965.564	956.254	913.624	962.473	23.927	23.246	23.715	22.545	23.184	24.543
	5	876.218	829.543	995.147	965.658	961.89	914.235	23.626	24.252	24.585	23.459	23.919	25.519
	8	861.51	856.218	921.653	926.513	996.961	909.284	24.136	24.487	25.226	23.78	25.106	25.515
	12	864.609	889.812	977.524	974.521	967.213	996.329	24.173	25.114	25.366	24.597	25.439	25.636
3	2	878.573	814.879	916.73	927.842	916.531	969.429	21.326	21.533	19.124	20.835	22.654	25.136
	5	826.673	896.982	917.218	914.437	937.234	997.182	21.529	20.429	19.482	21.857	23.126	22.176
	8	876.657	876.463	976.743	996.945	951.356	909.779	21.063	20.565	20.446	21.897	25.939	25.487
	12	895.586	861.518	966.218	906.212	996.621	922.851	21.587	22.493	20.772	21.492	25.466	25.517
4	2	856.731	863.752	996.642	962.541	985.652	912.13	23.835	24.539	23.525	23.848	24.569	25.152
	5	966.759	806.656	909.657	934.752	909.943	935.102	24.052	24.153	24.563	24.217	25.416	25.522
	8	954.753	823.659	901.665	906.555	996.949	995.257	24.277	25.422	25.482	25.836	25.735	26.657
	12	941.484	885.547	979.527	956.527	906.021	906.524	25.956	25.866	26.215	25.482	25.976	26.256
5	2	910.658	889.436	978.436	997.547	997.534	987.813	24.159	25.229	26.016	27.246	24.411	20.366
	5	965.436	856.09	901.658	990.547	993.56	946.698	26.784	25.473	26.543	27.895	26.314	26.792
	8	911.656	878.517	919.214	956.867	980.81	996.611	18.265	18.566	18.486	19.549	19.059	19.863
	12	911.209	875.347	990.257	942.527	961.624	961.624	19.476	19.893	19.216	19.486	19.287	19.483

and

Table A8. SSIM and FSIM computed using different optimization algorithms with the GLCM method.

Images	m	SSIM						FSIM
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	0.7823	0.8457	0.8144	0.8661	0.9521	0.9521	0.8533	0.8622	0.8325	0.9724	0.9801	0.9825
	5	0.7834	0.8568	0.8276	0.8645	0.9536	0.9567	0.8644	0.8656	0.8312	0.9702	0.9813	0.9846
	8	0.7867	0.8687	0.838	0.869	0.9542	0.9534	0.8799	0.8699	0.838	0.977	0.9804	0.9871
	12	0.7885	0.8705	0.8388	0.8692	0.955	0.9601	0.8991	0.8715	0.8381	0.9781	0.9825	0.9597
2	2	0.8067	0.817	0.8298	0.8612	0.9504	0.9526	0.8755	0.8597	0.8555	0.9745	0.9867	0.9894
	5	0.8068	0.8456	0.8355	0.8641	0.9518	0.9601	0.8875	0.8799	0.8608	0.9788	0.9865	0.9873
	8	0.8089	0.8586	0.8446	0.8634	0.9526	0.9621	0.8844	0.8807	0.8723	0.9732	0.9827	0.9889
	12	0.8114	0.8595	0.8474	0.8625	0.9564	0.9622	0.8962	0.8899	0.8835	0.9735	0.9835	0.9895
3	2	0.7965	0.8083	0.8682	0.841	0.9587	0.9615	0.8254	0.8155	0.8355	0.9755	0.9803	0.9867
	5	0.7978	0.8244	0.8391	0.8529	0.9563	0.9624	0.8319	0.8477	0.8466	0.9766	0.9841	0.9808
	8	0.7989	0.8237	0.8264	0.8639	0.9546	0.9629	0.8389	0.8588	0.8577	0.9777	0.9864	0.9813
	12	0.7912	0.8246	0.8294	0.8724	0.9587	0.9734	0.8399	0.8622	0.8534	0.9734	0.9875	0.9835
4	2	0.7924	0.8075	0.8255	0.8255	0.9518	0.9645	0.9268	0.8497	0.8433	0.9713	0.986	0.9849
	5	0.7924	0.8232	0.8369	0.8377	0.9526	0.9635	0.9366	0.8523	0.8544	0.9743	0.984	0.9864
	8	0.7945	0.8369	0.8308	0.8455	0.9546	0.9621	0.9499	0.8587	0.8622	0.9732	0.9876	0.9884
	12	0.7912	0.8416	0.8422	0.8474	0.9554	0.9643	0.9527	0.8622	0.8658	0.9758	0.9832	0.9892
5	2	0.7967	0.8521	0.8672	0.8559	0.9515	0.9648	0.9824	0.8825	0.8855	0.9785	0.9849	0.9782
	5	0.7967	0.8523	0.8622	0.8595	0.964	0.9685	0.9845	0.8912	0.8808	0.9788	0.9845	0.9783
	8	0.7923	0.8512	0.8652	0.8623	0.9648	0.9684	0.9923	0.8991	0.8977	0.9777	0.9874	0.9883
	12	0.7952	0.8525	0.8712	0.8621	0.9679	0.9712	0.9969	0.8981	0.8981	0.9781	0.9877	0.9846

. The analysis of the algorithms is discussed below.

5.4.1. Assessment Based on Computation Time (in Seconds)

The GLCM objective criterion is based on second-order statistics. This method was maximized to achieve thresholding results. Figure 10 shows a graphical analysis of the average CPU time. Table 7 shows a quantitative comparison of the average values. The results obtained using GLCM showed the efficiency of most of the optimization algorithms, as shown in Figure 10 and Table 7. In GLCM, in terms of increasing computation time, the algorithms could be arranged as JADE < MBFO < MCS < MABC < MPSO < MFA.

5.4.2. Assessment Based on PSNR, MSE, SSIM, and FSIM

Based on the average PSNR and MSE values in Figure 11 and Table 8, MCS had the best performance so far. In terms of the accuracy measured using PSNR, MABC, MBFO, JADE, and MFA showed satisfying performance. MPSO, however, was trapped into local minima, which affected its searching efficiency. MSE was the worst with the MCS algorithm. For GLCM, in terms of increasing PSNR and decreasing MSE values, the algorithms could be arranged as JADE < MBFO < MFA < MPSO < MABC < MCS.

Figure 12 and Table 9 report the average computed SSIM and FSIM. The complete analysis of the GLCM-based optimization techniques shows that MCS achieved optimal average values in comparison to other cases for most of the considered images. In terms of increasing SSIM and FSIM, the algorithms could be arranged as JADE < MFA < MBFO < MPSO < MABC < MCS.

5.4.3. Visual Analysis of the Results

A comparison of the segmented images in Figure 13 shows that MCS had the best segmented outputs, even though the segmented image looked under-segmented in some cases. MPSO resulted in poorly segmented images at lower and higher threshold levels. JADE was better; however, it was not as good as MCS. JADE showed the same results as the MCS for some cases. MBFO also presented poorly segmented results in a few cases.

5.5. Comparison Using Rényi’s Entropy as an Objective Function

In this section, the quantitative analysis of different optimization algorithms using Rényi’s entropy as a fitness function is shown. The performance was evaluated using the average values of the metrics over four segmentation levels (2, 5, 8, and 12), as shown in Table 7, Table 8 and Table 9 and Figure 14, Figure 15 and Figure 16. Figure 17 shows a visual comparison of the results for each segmentation technique. Detailed tables indicating the results computed over each of the segmentation levels (2, 5, 8, and 12) are shown in Table A6,

Table A9. MSE and PSNR computed using different optimization algorithms with the proposed Rényi-MCS.

Images	m	MSE						PSNR
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	754.122	575.149	745.428	845.252	815.625	256.377	22.262	22.421	22.515	23.525	25.289	27.662
	5	745.547	415.747	895.486	695.428	516.667	256.134	23.075	23.585	22.612	23.845	26.512	26.945
	8	564.125	658.105	645.857	641.156	408.008	254.098	24.944	24.758	23.758	24.621	25.898	27.315
	12	658.328	563.029	513.586	603.037	300.362	200.656	24.035	24.815	23.815	24.741	25.985	27.452
2	2	856.737	603.457	654.654	945.526	651.748	202.267	25.396	25.525	25.815	24.654	25.913	26.552
	5	765.123	718.456	684.418	654.564	103.327	250.888	25.365	26.621	26.954	25.648	25.298	27.258
	8	645.151	845.124	710.565	715.155	198.826	285.694	26.415	26.946	27.325	25.979	27.115	27.254
	12	453.068	878.009	766.254	563.252	285.234	256.126	26.812	27.213	27.735	26.056	27.448	27.465
3	2	567.754	803.783	805.373	416.485	658.248	205.357	23.335	23.452	21.313	22.484	24.663	29.415
	5	715.762	885.898	806.127	803.348	486.817	126.325	23.358	24.348	21.941	23.686	25.315	24.815
	8	665.565	865.643	865.475	785.497	398.779	140.537	23.702	24.754	22.545	23.086	28.498	27.946
	12	684.857	850.157	966.124	795.123	411.585	185.265	23.956	24.942	22.871	23.041	28.745	27.256
4	2	745.378	852.575	985.465	951.454	620.316	285.561	25.484	26.458	25.354	25.587	26.758	27.611
	5	645.57	895.564	798.456	523.575	224.094	298.492	26.601	26.612	26.752	26.226	27.245	27.351
	8	643.572	912.568	790.663	895.554	284.525	285.498	26.826	27.341	27.941	27.485	27.474	28.666
	12	630.843	874.456	868.256	745.256	295.254	295.2	27.695	27.785	28.224	27.941	27.895	28.625
5	2	709.567	878.345	767.345	686.456	276.187	297.353	26.518	27.228	28.105	27.425	26.14	28.635
	5	654.345	745.909	890.567	589.456	235.968	282.659	28.784	27.742	28.452	27.984	28.133	28.971
	8	500.565	567.156	808.123	445.686	885.162	289.189	28.265	28.655	28.845	27.458	28.508	28.682
	12	500.028	564.436	789.526	430.256	850.268	250.263	28.476	28.982	28.125	27.845	28.826	28.842

, and

Table A10. SSIM and FSIM computed by different algorithms using the proposed Rényi-MCS.

Images	m	SSIM						FSIM
		MFA	MBFO	JADE	MPSO	MABC	MCS	MFA	MBFO	JADE	MPSO	MABC	MCS
1	2	0.8523	0.8466	0.8244	0.8361	0.9806	0.9835	0.8567	0.9577	0.9623	0.9622	0.9814	0.9915
	5	0.8234	0.8678	0.8266	0.8355	0.9879	0.9878	0.8523	0.9513	0.9643	0.9655	0.9923	0.9966
	8	0.8267	0.8699	0.8499	0.838	0.9875	0.9834	0.8598	0.9588	0.9669	0.9699	0.9901	0.9927
	12	0.8315	0.8812	0.8399	0.8422	0.9832	0.9881	0.8525	0.9535	0.9671	0.9615	0.9822	0.9996
2	2	0.8267	0.8234	0.8208	0.8302	0.9837	0.9836	0.8634	0.9565	0.9655	0.9677	0.9867	0.9934
	5	0.8468	0.8567	0.8355	0.8381	0.9836	0.9864	0.8644	0.9584	0.9645	0.9699	0.9945	0.9982
	8	0.8389	0.8699	0.8466	0.8544	0.985	0.9839	0.8652	0.9579	0.9648	0.9677	0.9927	0.9949
	12	0.8414	0.868	0.8494	0.8532	0.9869	0.9883	0.8659	0.9542	0.9652	0.9689	0.9935	0.9965
3	2	0.8665	0.829	0.8502	0.858	0.9845	0.9883	0.8634	0.9524	0.9655	0.9635	0.9813	0.9968
	5	0.8678	0.8336	0.8901	0.8529	0.9846	0.9898	0.8645	0.9535	0.9619	0.9687	0.9861	0.9907
	8	0.8489	0.8348	0.8524	0.8639	0.9813	0.9836	0.8656	0.9546	0.9686	0.9687	0.9884	0.9912
	12	0.8412	0.8352	0.8254	0.8724	0.9894	0.9865	0.8626	0.9516	0.9635	0.9622	0.9935	0.9935
4	2	0.8824	0.8368	0.8255	0.8366	0.9833	0.9864	0.9246	0.9546	0.9967	0.9687	0.988	0.9949
	5	0.8824	0.8344	0.8469	0.8388	0.9834	0.9714	0.9245	0.9535	0.9666	0.9623	0.984	0.9964
	8	0.8845	0.8457	0.8408	0.8466	0.9873	0.9713	0.9254	0.9546	0.9678	0.9687	0.9896	0.9984
	12	0.8812	0.8402	0.8522	0.8465	0.9879	0.9712	0.9236	0.9536	0.9626	0.9622	0.9823	0.9901
5	2	0.8867	0.8202	0.8782	0.869	0.9823	0.9883	0.9278	0.9588	0.9623	0.9625	0.9849	0.9982
	5	0.8867	0.8301	0.8722	0.8698	0.983	0.9849	0.9267	0.9577	0.9644	0.9612	0.9864	0.9988
	8	0.8923	0.8329	0.8752	0.8821	0.9882	0.9904	0.9278	0.9568	0.9622	0.968	0.9884	0.9983
	12	0.8952	0.8395	0.8812	0.8732	0.9889	0.9939	0.9295	0.9589	0.9668	0.9672	0.9895	0.9986

. The analysis of the algorithms is discussed below.

5.5.1. Assessment Based on Computation Time (in Seconds)

Based on the average computation time results recorded in Table 7 and shown in Figure 14, MCS was the most suitable algorithm to use with Rényi’s entropy for producing an output in less time. The MCS more efficiently obtained results compared to other algorithms. JADE showed satisfactory performance, and the computation complexity of MABC was almost similar to that of JADE. On the other hand, the performance of MBFO was inferior to that of MPSO and MFA. For Rényi’s entropy, in terms of increasing CPU time, the algorithms could be arranged as MCS < JADE < MABC < MFA < MPSO < MBFO.

5.5.2. Assessment Based on PSNR, MSE, SSIM, and FSIM

Figure 15 shows a graphical comparison of the average PSNR and MSE values computed using Rényi’s entropy based on MCS, MABC, MPSO, JADE, and MFA. Table 10 shows the quantitative results. For each of the algorithms, it can be seen that the PSNR value improved as the thresholding level increased. On the other hand, the MSE value decreased. This indicates that the segmented results better resembled the original image when increasing the thresholding level. For Rényi’s entropy, in terms of increasing PSNR and decreasing MSE values, the algorithms could be arranged as MFA < MBFO < JADE < MPSO < MABC < MCS.

Figure 16 shows a graphical comparison of the average SSIM and FSIM values computed using the Rényi’s entropy-based algorithms. Table 11 shows the quantitative results. The MCS-based results were superior to those of other compared algorithms. Here, MABC again showed better performance than the rest of the optimization algorithms, which are compared in Table 10. For Rényi’s entropy, in terms of increasing SSIM and FSIM values, the algorithms could be arranged as MFA < MBFO < JADE < MPSO < MABC < MCS.

The proposed REMCS algorithm obtained superior results in most of the cases than other recently developed modified metaheuristics algorithms (MABC, MFA, MBFO, and MPSO). This occurred because every image had diverse features characterizing a specific optimization problem. In addition, the random nature of these algorithms generated some fluctuations in the segmentation outcomes. For instance, if a thresholding value was obtained using a metaheuristic that was not suitable, it generated a segmented image that was not optimal. Therefore, in terms of Rényi’s entropy, it can be seen that MCS produced the best results. Unlike other optimization algorithms, MCS increased the probability of obtaining the global optimum because of its well-balanced exploration and exploitation stages.

5.5.3. Visual Analysis of the Results

Figure 17 shows the visual results of the proposed and other compared techniques. The segmentation was visually best for the proposed algorithm. The performances of the algorithms are visually shown by their qualitative results. MABC presented better results than MPSO. The JADE and MBFO algorithms under-segmented the outputs because of their poor capability of accurately locating thresholding levels so that they could separate the pixels into homogenous regions. Figure 6 shows that as the thresholding level increased, the segmentation quality also improved.

5.6. Comparison between Rényi’s Entropy, Energy-Otsu Method, MCE, and GLCM

In this section, different objective criteria are compared on the basis of the quantitative outcomes reported in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10 and Table 11 with graphical and visual representations in Figures 19–22. Qualitative results are shown in Figure 23.

5.6.1. Assessment Based on Computation Time (in Seconds)

The average computation times obtained using Rényi’s entropy, the EC-Otsu method, MCE, and GLCM are given in Table 2 and Table 7 and shown in Figure 18. Rényi’s entropy uses global information obtained from the gray-level histogram and local information. Because the objective criteria had strong significance in locating the thresholds, the mathematical modeling of Rényi’s entropy provided good results. After Rényi’s entropy, GLCM presented the fastest results due to the use of the second-order statistics. Energy-Otsu appeared to be the most inefficient algorithm in terms of determining the optimal thresholds. The major drawback of the EC-Otsu method is the high processing time it requires to perform segmentation due to the time required for the computation of the energy function. In Table 2 and Table 7, it is clear that the computation time of each test image via the proposed approach was minimum. The algorithms based on EC-Otsu require large computation times.

5.6.2. Assessment Based on PSNR, MSE, SSIM, and FSIM

Table 3, Table 5, Table 8 and Table 10 report the PSNR and MSE for each of the four objective functions used. Graphically, the quantitative results are shown in Figure 19 and Figure 20. Rényi’s entropy works with the information of the image’s histogram, which usually presents multi-modality and local sub-optima configurations. Moreover, the architecture of Rényi’s entropy more efficiently explores the search space. Under such a scenario, the population-based MCS algorithm generated accurate and near-optimal threshold values compared to all other optimization techniques. In contrast, GLCM, EC-Otsu, and MCE only offered information about how the intensity values were distributed in different regions. These could be verified from qualitative metrics as reported in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10 and Table 11 and shown graphically in Figure 19, Figure 20, Figure 21 AND Figure 22. The EC-Otsu model is good when accuracy is of major concern due to the mathematical model of EC-Otsu and the properties of EF. The MCE approach can also be considered for multilevel thresholding, but this objective function requires a large number of evaluations and iterations to give optimal values. GLCM is an average method when accuracy is the major concern; the method integrates intensity and edge magnitude information. Table 4, Table 6, Table 9 and Table 11 compare the performances of objective functions in terms of similarity measures such as SSIM and FSIM. The results are shown graphically in Figure 21 and Figure 22.

5.6.3. Visual Analysis of the Results

A comparison of the segmented test images in Figure 23 shows the proposed algorithm presented much better results than the other compared algorithms. Comparing the results reveals that in the case of the MCS algorithm, Rényi’s entropy showed the best performance due to its balanced exploration–exploitation and noteworthy optimization capability. This also indicates that the segmented output using REMCS was of high quality in terms of details and information, as the entropy provided the average information content of the image; in other words, the reason for its high precision was the use of a powerful hybrid algorithm. The results of MCE followed Rényi’s entropy. EC-Otsu and GLCM performed fairly as well at classifying pixels for higher levels. In the cases of MABC and MPSO, Rényi’s entropy exhibited better segmentation. GLCM and EC-Otsu improperly segmented pixels. MCE showed somewhat better outputs than when using both MCS and JADE. EC-Otsu showed poor results with MFA and MBFO.

5.7. Statistical Analysis Test

This section discusses a rank-based statistical analysis of the performance assessment of the presented multilevel thresholding techniques and included optimization algorithms. The assessment technique for the non-parametric statistical hypotheses—namely, the Wilcoxon signed-rank test—measures the capability of the proposed approach compared to other considered approaches. These statistical results were obtained for 20 cases used in the experiment (five images and four different threshold levels). The analysis was performed using a 5% significance level over PSNR values to check the substantial variance between the proposed approach and other algorithms. In the Wilcoxon test, null hypotheses indicate no considerable change between the PSNR values of compared techniques, while the alternative hypothesis indicates a remarkable change. An h value of 1 means the null hypothesis can be rejected at a 5% level of significance, while an h value of 0 indicates that the null hypothesis cannot be rejected. If p < 0.05 (5% significance level), there is strong evidence against the null hypothesis, demonstrating that the better final objective function values achieved by the best algorithm in each case are statistically significant and have not occurred by chance. In Table 12, the p-values produced by Wilcoxon’s test results for Rényi entropy and other entropy functions (GLCM, MCE, and EC-Otsu) using MCS as a control algorithm are compared over the PSNR. The p-values were less than 0.05 (5% significance level) for the majority of cases. In the experiments using Rényi’s entropy as an objective function, the MCS algorithm presented better results in 18 out of 20 total cases when compared to the MPSO, MFA, and JADE algorithms, and it produced better results in 20 out of 20 total cases when compared to the MABC and MBFO algorithms.

6. Conclusions and Future Work

6.1. Conclusions

In this paper, we present a new color image multilevel thresholding technique using a multilevel Rényi’s entropy function and MCS algorithm. This method is a hybrid between the MCS algorithm and Rényi’s entropy model for remote sensing image segmentation. Remote sensing images are by nature multi-dimensional and multimodal with dense characteristics and uncertainties, which increases the computation complexity of determining the best thresholds with an exhaustive search procedure. Since the parameter estimation in the segmentation algorithm is typically a nonlinear optimization problem, the parameters used in Rényi’s entropy are determined using MCS. The results of the proposed REMCS algorithm were compared with modified versions of different popular bio-inspired optimizations (MFA, MBFO, JADE, MPSO, and MABC). To justify the performance of the proposed algorithm, other popular entropy models (GLCM, EC-Otsu, and MCE) were also included.

Multilevel thresholding is an extremely difficult problems to overcome because, upon increasing segmentation levels, the difficulty exponentially increases. In terms of the accuracy measured by PSNR and MSE values, REMCS showed significantly better results than the other methods. REMCS was successful in achieving high SSIM and FSIM values at all segmentation levels, while other algorithms failed when the level increased. To assess the significant differences between the methods, comprehensive statistical tests (Wilcoxon’s rank sum test) were used, indicating the significant differences between the proposed algorithms. The experimental results based on the evaluation of the satellite images verified the performance of the proposed algorithm on the low-level and high-level thresholds. The complete analysis revealed that the proposed REMCS produced the best value for a maximum number of fidelity parameters compared to the other techniques. REMCS preserved the fine details in the segmented images, as is necessary for the analysis of remote sensing imageries. A remarkable characteristic of the proposed algorithm was due to Rényi’s entropy, which was derived from the gray-level distribution of an image and hence provided better results. The numerical and visual analyses of the segmentation outcomes revealed the proficiency, fast convergence, and robustness of the proposed algorithm compared to the other meta-heuristic-based segmentation algorithms.

6.2. Future Work

The proposed method was found to significantly increases segmentation accuracy without affecting the original color and details of the input image. However, a limitation is that the histogram fails to consider the spatial contextual information of the image, which affects the accuracy of segmentation. To overcome such drawbacks, future works in this area will focus on improving the proposed algorithm by integrating contextually fused objective criteria. On the other hand, the approach does not need any training or learning phases, which generalizes its applicability to a diversified set of images. Hence, it can be explored for complex image processing and practical engineering problems. The proposed algorithm can be used to solve several real-time complex applications related to image processing such as the enhancement and denoising of remote sensing images, optimization-based remote sensing image classification and analysis, and various computer vision applications. Moreover, the segmented images can be used in the feature extraction process for machine learning-based classification and for deep learning models, which would further boost their accuracy and performance.