SUFMACS: A machine learning-based robust image segmentation framework for COVID-19 radiological image interpretation

Graphical abstract

Abstract

The absence of dedicated vaccines or drugs makes the COVID-19 a global pandemic, and early diagnosis can be an effective prevention mechanism. RT-PCR test is considered as one of the gold standards worldwide to confirm the presence of COVID-19 infection reliably. Radiological images can also be used for the same purpose to some extent. Easy and no contact acquisition of the radiological images makes it a suitable alternative and this work can help to locate and interpret some prominent features for the screening purpose. One major challenge of this domain is the absence of appropriately annotated ground truth data. Motivated from this, a novel unsupervised machine learning-based method called SUFMACS (SUperpixel based Fuzzy Memetic Advanced Cuckoo Search) is proposed to efficiently interpret and segment the COVID-19 radiological images. This approach adapts the superpixel approach to reduce a large amount of spatial information. The original cuckoo search approach is modified and the Luus-Jaakola heuristic method is incorporated with McCulloch’s approach. This modified cuckoo search approach is used to optimize the fuzzy modified objective function. This objective function exploits the advantages of the superpixel. Both CT scan and X-ray images are investigated in detail. Both qualitative and quantitative outcomes are quite promising and prove the efficiency and the real-life applicability of the proposed approach.

1. Introduction

Machine learning and computer vision-based methods have widespread applications in various domains and they can explore different hidden relationships as well as some complicated patterns in a dataset without human intervention. The machine learning model is first applied in 1959 to replace humans in the checker games (Samuel, 2000). In general, machine learning systems can be broadly classified in two ways. One is supervised and another one is the unsupervised method. In the case of the supervised method, the machine learning model learns from the available annotated data, and in the case of unsupervised learning, there are no ground truth data available and the model is expected to explore the patterns from the dataset (Chakraborty and Mali, 2018, Chakraborty and Mali, 2020a). Machine learning can solve various complex problems where a human may not perform well. Machine learning can help the radiologists and physicians in different phases (Chakraborty and Mali, 2020b, Chakraborty and Mali, 2020c, Kahn et al., 2009, Sistrom et al., 2009). Apart from the interpretation, the machine learning and computer vision-based systems can help the radiologists in creating hanging and study protocols, deciding the dose of the radiation, enhance the quality of an image, save time to acquire an image, optimization, and parameter tuning of the scanning device, quality assessment, quantification, disease prevention and many more (Chakraborty and Mali, 2020a, Hore et al., 2016, Lakhani et al., 2018).

The COVID-19 pandemic is one of the biggest threats to mankind. It is incredibly crucial to detect and treat this deadly virus to overcome this global pandemic scenario. It is essential to consult a specialist if someone develops any symptoms of COVID-19 or is exposed to the COVID-19 virus anyhow. There are various factors like geographic locations and their operational laws, environment, etc. that decide whether some exposed person or someone with associated symptoms should undergo a test or not. Typically, nasopharyngeal swab or throat swab or both can be collected by a lab for testing. The RT-PCR test is considered one of the gold standards of the world to confirm the presence of the COVID-19 virus. Once detected, there are no such medicines available to treat this virus. Moreover, antibiotics are also not sufficient to treat this virus. A continuous effort can be observed to find a cure for this virus. Only supportive treatments for various symptoms can be provided. Some physicians also prescribe some immunity-boosting medicines and vitamins. Therefore, supportive medicines, rest, hygiene, fluid intake, isolation, etc. are the only ‘medicines’ to treat this virus. So, it is very essential to detect the presence of this virus as early as possible, and it can save many lives.

A computerized tomography scan or CT scan is a crucial tool to fight against the COVID 19 outbreak. CT scan can detect the COVID-19 (Kanne et al., 2020) and an early-stage diagnosis can save the patient as well as other people from getting infected accidentally (Zu et al., 2020). But it is not at all a substitute for the reverse-transcription polymerase chain reaction (RT-PCR) test. In fact, some cases are observed where the RT-PCR test confirms the presence of the COVID-19 infection but the radiological findings show nothing (Ai et al., 2020, Bernheim et al., 2020). Although radiological experiments cannot be used to make any decision regarding this infection still, radiological analysis can help in early detection and treatment. Chest CT scans are recommended in (Fang et al., 2020) to use it in the screening process. Some important features about the COVID-19 positive CT scan images are recorded in (Caruso et al., 2020). These features and the percentage of patient occurrence are given in Table 1 . Some important features are crazy paving (i.e. appearance of ground-glass opacity with superimposed interlobular septal thickening and intralobular septal thickening), and consolidation i.e., the air is completely replaced with fluid, etc. (Torkian et al., 2020). In most cases, properly annotated and ground truth images are not available for the COVID-19 radiographic images. Inspired by the above-mentioned points, an unsupervised machine learning method is proposed in this work namely SUFMACS (SUperpixel based Fuzzy Memetic Advanced Cuckoo Search) that can efficiently segment the radiological images. The proposed method is based on the superpixels and the memetic advanced cuckoo search procedure that optimize the proposed fuzzy objective function to get the optimal segmentation. This work can be helpful in the early screening of the COVID-19 disease and interpreting the radiological images automatically. Moreover, the analysis and screening will be possible without direct contact with the suspect. Therefore, machine learning can help to fight against the COVID-19 disease and also helps to save millions of lives by automated and intelligent screening systems.

Table 1.

Some important features and percentage of occurrence for the COVID-19 positive patients (Caruso et al., 2020).

Features of the CT scan image for the COVID-19 positive cases	Explanation/Meaning	Percentage of occurrence
ground-glass opacities (GGO)	A region of the lung with increased attenuation along with preserved bronchial and vascular markings	100%
multilobe and posterior involvement	Impact on both lobes and posterior region	93%
bilateral pneumonia	Both lungs are affected with pneumonia. Also called double pneumonia	91%
subsegmental vessel enlargement (>3 mm)	defined as vessel diameter > 3 mm	89%

The proposed SUFMACS approach contributes to the existing literature in three different ways. First of all, the proposed modified superpixel based approach is helpful in processing high-quality CT scan images to assess the COVID-19 disease conveniently. This approach is certainly helpful in reducing the computational burden of processing a large amount of spatial information. Secondly, the modified and advanced cuckoo search approach is proposed to determine the optimal clusters. This meta-heuristic procedure is hybridized with the type 2 fuzzy system to get some realistic segmented outcomes, where, the fuzzy objective function is modified to adapt the advantages of the superpixels, which is the third major contribution. The advanced extension of the cuckoo search method tries to optimize the modified version of the type 2 fuzzy objective function to get the optimal clusters. Although this threefold contribution is significant from the technical perspective, the main success of the proposed approach can be considered from the diagnostic perspective and the contribution of this approach can only be measured from the real life of benefits that can be gained by applying the proposed approach in this pandemic scenario. Again, the proposed approach is designed to work in a completely unsupervised environment i.e., without bothering about any expert delineations and without any training dataset. The proposed approach is compared with the similar kind of state-of-the-art approaches to establish the effectiveness of the proposed approach. The qualitative experimental results show obvious differences in the scope and density of the lesions. Although the segmentation results of the lesions are not specific, so different signs of infection could not be distinguished but the proposed approach can act as a third eye for the physicians to interpret the radiological images precisely and also helpful for further analysis. Apart from this, the proposed segmentation approach is a significant contribution to the present literature that will certainly enrich the concerned domain.

The clinical significance of the proposed work belongs to the early screening of the COVID-19 infection. The proposed approach can serve as a tool to fight against the COVID-19. The radiological images (i.e. the CT scans of the chest region) can be easily explicated to screen and isolate the COVID-19 infected patients that can be helpful to stop the spread of the COVID-19 virus in society. The proposed machine learning-based segmentation approach can be helpful for the computer-aided diagnostics systems to efficiently and reliably process the biomedical images and to produce timely diagnostic results. The experiments are performed on the radiological images, which are collected from a different point of view. It ensures the applicability of the proposed approach to different types of relevant images. Moreover, the proposed approach can also process the CT scan stacks to get finer details about a disease which can be beneficial for a clear and quick interpretation of the condition of a patient. This approach can help physicians to effectively analyze the CT scan images.

The proposed approach is compared with four standard approaches namely efficient GA (Kadri & Boctor, 2018), adaptive PSO (Taherkhani & Safabakhsh, 2016), beam-ACO (Blum, 2005), MCS method (Chakraborty et al., 2017). The efficient GA is a modification to the original genetic algorithm that uses a two-point crossover mechanism. The adaptive PSO approach computes the inertia weight in different directions using adaptive technique. Roles of the particles are not same in different dimension. Moreover, this approach determines the acceleration parameters adaptively. The beam-ACO technique hybridizes the ant colony optimization method and the beam search technique. The MCS approach uses McCulloch's method to generate the Lévy flight. The obtained experimental results are validated with the help of the four standard cluster validity indices namely Davies–Bouldin index (Davies & Bouldin, 1979), Xie-Beni index (Davies & Bouldin, 1979), Dunn index (Dunn, 1974), and $\beta$ index (Pal et al., 2000). The description of these indices is given in subsection 7.2.

This article is aimed at helping the physicians in easy interpretation of the chest CT scans so that they can effectively and quickly diagnose and get some hints about the early-stage infection of COVID-19 disease. Most of the state-of-the-art articles are based on supervised learning including CNNs, and other deep learning-based approaches. These approaches demand some ground truth data to get trained. But, one of the major problems associated with COVID-19 chest image dataset is the non-availability or non-sufficiency of the ground truth data. Therefore, the proposed approach is not dependent and hence, not using any ground truth data.

The proposed approach can be useful not only in the hospitals but in any radiological imaging institutes for quick and early diagnosis of the COVID-19 infection. In hospitals, doctors can quickly investigate the suspected patients and can isolate them from other patients if necessary. Moreover, it is also helpful for the common public to get quickly diagnosed. The cost of the RT-PCR test is around 1200–1700 INR in India whereas the chest X-Ray can be performed with 250–500 INR and the chest CT scan can cost 2000–2500 INR in India. The RT-PCR test is reliable but it is time-consuming complicated. The proposed approach helps physicians and other domain experts in better interpretation of the radiological images that can lead to an easy diagnosis of the COVID-19.

The rest of the article is organized as follows: Section 2 gives a brief overview of the cuckoo search approach based on the Lévy flight and Section 3 describes the cuckoo search procedure using McCulloch’s approach. Section 4 describes the segmentation procedure using the type 2 fuzzy clustering system. Section 5 briefly illustrates the Luus–Jaakola heuristic. Section 6 describes the proposed SUFMACS method in detail and the obtained results and their analysis is given in Section 7. Section 8 concludes the article.

2. A brief description of the cuckoo search algorithm

The Cuckoo search method is a simple and fast approach to find the global optimum solution to a problem and was first proposed by Yang and Deb in 2009 (Yang & Deb, 2009). Cuckoo search is known and frequently used due to the ease of implementation, faster convergence, and efficiency to get the optimal results. In this work, the traditional cuckoo search method is modified and applied to get the optimal segmented outcome. One of the major advantages of stochastic optimization approaches is that these approaches can efficiently handle various multimodal and non-convex optimization functions. Moreover, there is no need to assume any specific property of the objective functions like continuity, differentiability, etc. (Fateen & Bonilla-Petriciolet, 2014). In this section, the traditional cuckoo search algorithm is explained in brief.

2.1. Typical behavior of the cuckoo bird

The cuckoo search procedure is a well-known optimization method which is often used in several applications (Linguraru et al., 2006). The cuckoo search procedure is a well-known optimization method that is often used in several applications (Shehab et al., 2017). This method is based on the general reproduction behavior of some of the cuckoo birds. They possess parasitic behavior in the case of reproduction. They lay their eggs in the nest of some other host birds. The parasitic behavior is of three types such that (i) intraspecific, (ii) cooperative breeding, and (iii) nest conquer (Yang & Deb, 2009). Sometimes, the host bird can explore the eggs of these parasitic birds and can take necessary measures upon detection. They can throw the alien eggs or they can leave their old nest and can build a new one. Some species of the cuckoo birds are expert in mimicking the nature of the eggs of the host bird and creates some difficulty for the host species to distinguish between their eggs and the eggs of the parasites. The chicks of the cuckoo birds, sometimes, can throw the eggs of the host species to get more food and other resources.

2.2. Overview of the Lévy flight

Lévy flight is inspired by the food and other resource finding behavior of the several natural species and it is a ‘Markovian stochastic process’ (Brown et al., n.d.). The nature of the flight is sometimes compared with the behavior of the light (Barthelemy et al., 2008). Eq. (1) shows the probability distribution function $\rho \left(x\right)$ which is followed by the Lévy flight to generate the leaps and the stability index which is also known as the Lévy index of this probability distribution function ($\mathit{pdf}$) is denoted by $\theta$ and the range is denoted by $0<\theta <2$. This value is also equivalent to the fractal dimension $d_{\mathit{frac}}$ of the trajectory of the Lévy flight. Therefore, it can be written that $d_{\mathit{frac}}=\theta$. The variance of the random variables of this type generally diverges. But the convergence of the Lévy stable pdf which is given over the domain $\kappa \gg \upsilon$ in the characteristic Eq. (2) (Samoradnitsky & Taqqu, 1994), can be proved using the central limit theorem (Hughes, 1998). In this equation, $\gamma$ controls the skewness and the range can be [−1,+1]. The shift and the scale property are represented by the $\upsilon$ and $\varphi$ respectively and $\varphi >0$.

$$\rho \left(x\right)={\left|x\right|}^{-(1+\theta )}$$ (1)

$$\mathit{pd}{f}_{\theta ,\gamma }\left(\kappa ;\upsilon ,\varphi \right)=FT\left(\mathit{pd}{f}_{\theta ,\gamma }\left(x;\upsilon ,\varphi \right)\right)$$ (2)

$FT\left(·\right)$ computes the Fourier transform and Eq. (3) can be used to compute corresponding value of the $\mathit{pdf}$.

$$FT\left(\mathit{pd}{f}_{\theta ,\gamma }\left(x;\upsilon ,\varphi \right)\right)={\int }_{-\infty }^{+\infty }\mathit{dx}{e}^{i\kappa x}pd{f}_{\theta ,\gamma }\left(x;\upsilon ,\varphi \right)=exp\left[i\upsilon \kappa -{\varphi }^{\theta }{\left|\kappa \right|}^{\theta }\left(1-i\gamma \frac{\kappa }{\left|\kappa \right|}\mathrm{\Psi }\left(\kappa ,\theta \right)\right)\right]$$ (3)

The value for the function $\mathrm{\Psi }(\kappa ,\theta )$ can be determined using the Eq. (4).

$$\mathrm{\Psi }(\kappa ,\theta )=\left\{\begin{array}{cc}tan\frac{\pi \theta }{2},\hfill & \mathit{if}\theta \ne 1,0<\theta <2\\ -\frac{2}{\pi }ln\left|\kappa \right|,\hfill & \mathit{if}\theta =1\end{array}\right)$$ (4)

From the above discussion, it can be observed that the Lévy stable probability distribution function is controlled by the four following parameters $\theta ,\gamma ,\upsilon ,\varphi$. The cuckoo search using Lévy flight method is discussed next.

2.3. The cuckoo search using the Lévy flight method

Some basic assumptions which are considered in this approach are stated below (Yang & Deb, 2009):

• 1.The nest of the host birds is selected randomly and the nests can be initialized with the help of Eq. (5).
• 2.At most one egg can be laid by a cuckoo bird at a particular instance.
• 3.Nests with better-quality eggs can take part in the consequent generations.
• 4.Probability of exploring the eggs of the parasite species is $\mathit{pro}{b}_{\epsilon }\in [0,1]$.
• 5.The total number of nests cannot be changed and a new nest can be constructed after abandoning a nest due to the discovery of an alien egg otherwise, the host species can simply destroy the egg.

The nest initialization process assigns some eggs i.e. the solutions to the nests and this step is nothing but generating the initial solutions or the population in case of other standard optimization processes. Here, $w_{i,j}$ denotes the population matrix.

$$w_{i,j}=w_j^{\mathit{low}}+random\left(0,1\right)\left(w_j^{\mathit{high}}-w_j^{\mathit{low}}\right)$$ (5)

The value of $i=1,2,3,4,....,nNest$ and the $\mathit{nNest}$ denotes the total number of nests and $j=1,2,3,4,....,nParam$ where $\mathit{nParam}$ represents the total number of optimization parameters. The dimensional upper and the lower bounds are represented by the $w_j^{\mathit{low}}$ and $w_j^{\mathit{high}}$ respectively and it corresponds to the $j^{\mathit{th}}$ parameter. The eggs can be checked by means of the value of the objective function. Depending on the fitness value and the Lévy distribution (which can be generated using Eq. (7)), new solutions can be generated using Eq. (6). The step size is denoted by $s_z$, and the generation or the present iteration is indicated using $g$. Here, the i^th egg of the cuckoo bird is randomly selected and the new solution $x_i\left(g+1\right)$ is generated using Eq. (6).

$$x_i\left(g+1\right)=x_i\left(g\right)+s_z\oplus Lévy\left(\lambda \right)$$ (6)

$$L{e}^{\prime }vy\left(\lambda \right)=g^{-\lambda }\text{where}1<\lambda ⩽3$$ (7)

The global optimum can be explored nonlinearly using the Lévy flight and it helps to find the optimal solution effectively and within the stipulated amount of time because of the nonlinear relationship of variance and is illustrated in Eq. (8).

$${\sigma }^2\left(g\right)~g^{2-a}\text{where}1⩽a⩽2$$ (8)

2.4. Random number generation (RNG) using Lévy flight

The cuckoo search procedure depends on the random walk to explore the solution space and this random walk is mimicked with the help of the Lévy flight. Random numbers can be generated using the Lévy flight method by obtaining the directional details of the Lévy flight with the help of a uniform distribution. These random numbers are used to implement the concept of random walk in the cuckoo search procedure by computing the step size that follows the Lévy distribution (Siswantoro, 2013) and the step size is computed by applying the Mategna’s method (Mantegna, 1994) to reduce the computational burden (Yang & Deb, 2009). There are four controlling parameters involved in this phase $p_1,p_2,p_3,p_4$. The value of the first parameter can be anything between $0.3⩽p_1⩽1.99$. $p_2$ is a constant and $p_2>0$. The iteration count and the count of the random points are represented by the $p_3$ and $p_4$ respectively. The following steps are involved in the generation of the random number using Lévy flight.

• 1st Step: Compute $\zeta$ using Eq. (9). Two stochastic variables $\alpha$ and $\beta$ are normally distributed and the accurate value of these parameters cannot be determined. Instead of that, statistical analysis is possible $\left(\alpha ~p_4\left(0,{\sigma }_{\alpha }^2\right),\beta ~p_4\left(0,{\sigma }_{\beta }^2\right)\right)$ where, the standard deviation should be computed first using Eqs. (10), (11) with the help of the standard gamma function $\mathrm{\Gamma }\left(·\right)$.

$$\zeta =\frac{\alpha }{{\left|\beta \right|}^{\frac{1}{p_1}}}$$ (9)

$${\sigma }_{\alpha }\left(p_1\right)=\left\{\frac{\mathrm{\Gamma }\left(1+p_1\right)sin\left(\frac{\pi p_1}{2}\right)}{\mathrm{\Gamma }\left(1+p_1\right)p_1{2}^{\frac{\left(p_1-1\right)}{2}}}\right\}$$ (10)

$${\sigma }_{\beta }=1$$ (11)

Lévy distribution can be achieved depending on the $\zeta$ provided that $\zeta \gg 0$.

• 2nd Step: A non-linear transformation function is incorporated to produce a stable Lévy distribution within a limited amount of time and it is defined in Eq. (12) where the generated distribution of the stochastic variable $\nu$ leads to the Lévy distribution. Eqs. (13), (14) are used to determine the value of $\mathrm{\Phi }\left(·\right)$ and $\mathrm{\Theta }\left(·\right)$ respectively and $\vartheta$ denotes the scale of the problem. The value of $\mathrm{\Theta }\left(·\right)$ is a little difficult to compute because Eq. (14) needs to be solved first to get the value of it. Eq. (14) contains integrals on both the right-hand side and left-hand side. To get the value of $\mathrm{\Theta }\left(·\right)$, one needs to solve these integrals. So, methods to solve integrals can be adopted in this case also. The values of $\mathrm{\Phi }\left(·\right)$ and $\mathrm{\Theta }\left(·\right)$ can be determined from (Mantegna, 1994) where these values are given corresponding to a given value of $p_1$ In these equations.

$$\xi =\left\{\left[\mathrm{\Phi }\left(p_1\right)-1\right]exp\left(-\frac{\left|\zeta \right|}{\mathrm{\Theta }\left(p_1\right)}\right)+1\right\}\nu$$ (12)

$$\mathrm{\Phi }\left(p_1\right)=\frac{p_1\vartheta \left(\frac{p_1+1}{2p_1}\right)}{\vartheta \left(\frac{1}{p_1}\right)}{\left[\frac{p_1\vartheta \left(\frac{p_1+1}{2p_1}\right)}{\vartheta \left(p_1+1\right)sin\frac{\pi p_1}{2}}\right]}^{\frac{1}{p_1}}$$ (13)

$$\frac{1}{\pi {\sigma }_{\alpha }}{\int }_0^{\infty }{q}^{\frac{1}{p_1}}exp⌊-\frac{q^2}{2}-\frac{q^{\frac{2}{p_1}}\mathrm{\Phi }{(p_1)}^2}{2{\sigma }_{\alpha }^2(p_1)}⌋dq=\frac{1}{\pi }{\int }_0^{\infty }cos\left[\left(\frac{\mathrm{\Theta }\left(p_1\right)-1}{e}+1\right)\mathrm{\Theta }\left(p_1\right)\right]exp\left(-q^{p_1}\right)dq$$ (14)

• 3rd step: A random variable is computed in this step with the help of the central limit theorem and the $p_3$ number of individual copies of $\xi$ which are computed using Eq. (12). The step size can be determined with the help of Eq. (15) and the value of $\eta$ can be determined by Eq. (16).

$$s_z={\vartheta }^{\frac{1}{p_1}}\eta$$ (15)

$$\eta =\frac{1}{p_3^{\frac{1}{p_1}}}\sum _{i=1}^{p_3}{\xi }_i$$ (16)

Algorithm 1 demonstrates the cuckoo search procedure and the flow diagram of the cuckoo search procedure is illustrated in Fig. 1 .

Algorithm 1: Cuckoo search optimization procedure using Lévy flight
Input: Initial parameters to control the optimization process and the problem to be optimized
Output: Optimal result along with its corresponding fitness value
1.Initialize the nests randomly and assign some eggs in it using Eq. (5) (dependent on the underlying problem being solved).
2.Set $\mathit{itrCounter}\leftarrow 1$
3.Repeat while $\mathit{itrCounter}⩽cntGeneration$
a.Get an egg using the Lévy flight and find the fitness value (the fitness function can be defined depending on the underlying problem being solved). Eq. (6) is used to generate the new solution of a generation.
b.Choose a nest in a random manner
c.Check if the value of the objective function for the obtained egg is better than the chosen nest then
i.Existing egg is to be replaced by the newly generated egg.
d.end if
e.When the host bird detects the alien egg then some of the nests are left by the cuckoo bird.
f.Preserve the best solution and move it to the following generation.
g.Allot a rank to the solutions depending on the fitness values.
end while
4.Return the optimum solution and its fitness

Fig. 1.

The flow diagram of the cuckoo search method using Lévy flight.

3. A brief description of the McCulloch’s approach based cuckoo search approach

The convergence of the conventional cuckoo search approach can be improved further with the application of McCulloch’s approach (Chambers et al., 1976) and this approach is named after J.H. McCulloch (Chakraborty et al., 2017). Faster convergence is required to analyze the radiological images for quick interpretation and a clear understanding of the COVID-19 infection status which can be helpful in treating the patients and preventing the community spread. Although the McCulloch’s approach can replace the Lévy flight method to generate the stable random numbers, a significant inaccuracy can be introduced by incorporating this method and it can be very costly for the optimization process and the exploration process of the search space becomes inefficient which will cause some problems in reaching the global optima. The amount of inaccuracy can significantly increase with the number of iterations (Suresh & Lal, 2016). This problem can be avoided by using the method described in (Chambers et al., 1976) and this method is used to adapt McCulloch’s approach in this proposed SUFMACS method. The performance of the McCulloch’s approach can be referred from (Leccardi & Scalas, n.d.). Four controlling parameters are given below which are responsible to generate a matrix of random numbers of dimension $d_1\times d_2$.

• a.The exponent $exp$.
• b.The scaling parameter $\tau$.
• c.A skewness controlling parameter $\omega$.
• d.A parameter $\stackrel{⌢}{\kappa }$ to denote the location.

The McCulloch’s approach can take different forms depending on the value of the exponent $exp$ and these cases are discussed below:

• 1.If the value of the exponent $exp$ is 1 then, it is similar as the Cauchy distribution where, $\stackrel{⌢}{\kappa }$ denotes the median. The step size can be computed using Eq. (17) and the value of $\psi$ is computed using Eq. (19).

$$s_z=\tau ·tan(\psi )+\stackrel{⌢}{\kappa }$$ (17)

• 2.If the value of $exp$ is equal to 2 then, it is similar to the Gaussian distribution where, $\stackrel{⌢}{\kappa }$ represents the mean, and $2{\tau }^2$ denotes the variance. The step size can be computed using the Eq. (18) and the value of $\chi$ can be computed using Eq. (20).

$$s_z=2\tau \sqrt{\chi }sin(\psi )+\stackrel{⌢}{\kappa }$$ (18)

• 3.Typically, when $exp>1$ then the $\stackrel{⌢}{\kappa }$ denotes the median and it is not dependent on the skewness controller.

$$\psi =\left(random\left(d_1,d_2\right)-\frac{1}{2}\right)\pi$$ (19)

$$\chi =-{log}_{10}\left(random\left(d_1,d_2\right)\right)$$ (20)

If $\omega =0$ and the $exp\ne 1$ then, the step size can be computed using Eq. (21) and it can also be computed using Eq. (22) which is nothing but a simplified version of Eq. (21).

$$s_z=\tau \frac{sin\left[exp·\psi +{tan}^{-1}\left(\omega tan\left(\frac{exp·\pi }{2}\right)\right)\right]{\left[cos\left(\left(1-exp\right)\psi -{tan}^{-1}\left(\omega tan\left(\frac{exp·\pi }{2}\right)\right)\right)\right]}^{\frac{1}{exp}-1}}{{\left[cos\left({tan}^{-1}\left(\omega tan\left(\frac{exp·\pi }{2}\right)\right)\right)\right]}^{\frac{1}{exp}}{\left(cos\left(\psi \right)\right)}^{\frac{1}{exp}}{\vartheta }^{\frac{1}{exp}-1}}+\stackrel{⌢}{\kappa }~S_{exp}\left(\tau ,\omega ,\stackrel{⌢}{\kappa }\right)$$ (21)

$$s_z=\tau {\left[\frac{cos\left(\left(1-exp\right)·\psi \right)}{\chi }\right]}^{\frac{1}{exp}-1}{\left[\frac{sin\left(exp·\psi \right)}{cos\left(\psi \right)}\right]}^{\frac{1}{exp}}+\stackrel{⌢}{\kappa }$$ (22)

In this work, the exponent value $exp$ is selected as 0.5 and it can be chosen from the range $\left[0,2\right]$ to bypass the overflow situation. Moreover, the value of $\omega$ is selected as 0 to avoid the skewness. The algorithm 2 illustrates the cuckoo search method based on the McCulloch’s approach (See Fig. 2 ) (Chakraborty et al., 2017, Hore et al., 2015).

Algorithm 2: Advanced cuckoo search algorithm based on the McCulloch’s approach
Input: The values of the controlling parameters and the problem to be solved
Output: Computed optimal solution along with its fitness value
1.Apply Eq. (5) to randomly initialize the nests
2.Compute the fitness using the objective function $Obj\left(x\right)$ where $x={\left[x_1,x_2,......,x_n\right]}^T$
3.Initialize the iteration counter $\mathit{itrCounter}\leftarrow 1$
4.Check if $\mathit{itrCounter}⩽cntGeneration$ then
a.Find and store the present best.
b.Build a new solution space.
c.Calculate the fitness of the individual eggs.
d.Store the best nest
e.Check if $r<pro{b}_{\epsilon }$ then
i.Apply the McCulloch’s approach to determine the step size using Eq. (22).
ii.Replace the worst nest using the computed step size.
iii.Determine the fitness of the solutions.
iv.Find the best nest and store it
v.Update the iteration counter
vi.Determine the optimum solution till this point.
otherwise
vii.Store those nests
end if
end if
5.Return the computed global optimal solution

Fig. 2.

The flow diagram of the cuckoo search method with the McCulloch’s approach.

4. Fuzzy type-2 C-means clustering

The Fuzzy C-means clustering method is useful in those situations where the crisp methods fail and therefore, the fuzzy clustering systems gain tremendous popularity and applied to solve real-life problems of computer vision and image processing (Bezdek et al., 1984). In fuzzy clustering, a single point can belong to more than one cluster with some degree of memberships. Eq. (23) is the objective function which is optimized by the type 1 fuzzy system and the cluster centers are updated using Eq. (25).

$$\mathit{Ob}{j}_{\phi }=\sum _{i=1}^{\mathit{nP}}\sum _{j=1}^{\mathit{nC}}{\mu }_{\mathit{ij}}^{\phi }{∥x_i-c_j∥}^2,\text{where}1⩽\phi <\infty$$ (23)

Eq. (24) is a squared error function and it gives the membership value of the point $x_i$ to the $j^{\mathit{th}}$ cluster. $\phi$ represents the fuzzifier. The sum of the membership values or the degree of membership for a data point must be 1 i.e. ${\sum }_{j=1}^{\mathit{nC}}{\mu }_{\mathit{ij}}=1fori=1,2,3,........,nP$ where the number of data points are represented by $\mathit{nP}$ and the number of cluster centers are denoted by $\mathit{nC}$ (see Fig. 3 ).

$${\mu }_{\mathit{ij}}=\frac{1}{{\sum }_{t=1}^{\mathit{nC}}{\left(\frac{∥x_i-c_j∥}{∥x_i-c_t∥}\right)}^{\frac{2}{\phi -1}}}$$ (24)

$$c_j=\frac{{\sum }_{i=1}^{\mathit{nP}}{\mu }_{\mathit{ij}}^{\phi }{x}_i}{{\sum }_{i=1}^{\mathit{nP}}{\mu }_{\mathit{ij}}^{\phi }}$$ (25)

Fig. 3.

Functional diagram of the type 2 fuzzy system.

The noise sensitivity and relativity of the membership values (i.e. the identical models can be approximated relatively to differing behaviors) are the major constraints of the type 1 fuzzy clustering method. This method is suitable in certain circumstances because the presence of noise can significantly alter the final results of the segmentation. To overcome these problems, the type 2 fuzzy system-based clustering method is adopted in this work. The associated uncertainty of a point is high if the degree of membership for a certain cluster is low and vice-versa (Rhee & Cheul Hwang, n.d.). Prime advantages of using the type 2 clustering systems are mentioned below (Rhee & Cheul Hwang, n.d.):

• IEfficient in modeling uncertainties.
• IIA point can contribute more or it has a larger impact if it has lesser uncertainty.
• IIIMore realistic results can be obtained compared to the type 1 fuzzy systems in presence of noise.
• IVNoise can be effectively modeled by the type 2 fuzzy systems.

In this approach, the membership values are treated as the weights. Eq. (26) can give the membership value ${\chi }_{\mathit{ij}}$ of a point in the fuzzy type 2 system and this value is derived from Eq. (24). The cluster centers can be updated using Eq. (27) but, in this work, this equation is not used, and instead of that, the memetic advanced cuckoo search method is applied to update and guide the cluster centers. The segmentation process based on the type 2 fuzzy clustering system is illustrated in algorithm 3 and the type 2 fuzzy system is explained in Fig. 3.

$${\widehat{\mu }}_{\mathit{ij}}={\mu }_{\mathit{ij}}-\frac{1-{\mu }_{\mathit{ij}}}{2}$$ (26)

$${\widehat{c}}_j=\frac{{\sum }_{i=1}^{\mathit{nP}}{\widehat{\mu }}_{\mathit{ij}}^{\phi }{x}_i}{{\sum }_{i=1}^{\mathit{nP}}{\widehat{\mu }}_{\mathit{ij}}^{\phi }}$$ (27)

Algorithm 3: Segmentation based on Type 2 fuzzy C-means clustering
Input: The data points to be clustered and the number of clusters $\mathit{nC}$ where, $2⩽nC⩽nP$
Output: Calculated cluster centers and their corresponding groups of points
1.Initialize the cluster centers and the membership values for all points in a random manner.
2.Decide a small threshold $\delta$ and use it to terminate the process.
3.Update the cluster centers using Eq. (27).
4.Determine the fitness value using Eq. (23).
5.Check if $\mathit{improvement}⩾\delta$ then
a.Find the type 2 fuzzy membership values ${\chi }_{\mathit{ij}}$ using Eq. (26).
b.Goto step 2 and repeat again
end if
6.Return the optimal cluster centers and the corresponding member data points

5. A brief overview of the Luus–Jaakola heuristic

The Luus–Jaakola method is a well-known and popular heuristic method which is named after Luus and Jaakola (1973). It is frequently used to solve various optimization problems. Let us assume an objective function ${\mathbb{R}}^m\to \mathbb{R}$ for a certain optimization problem. Let us assume that a $\alpha$ is a solution (in the present context, it is a local solution) that belongs to the problem space i.e. $\alpha \in {\mathbb{R}}^m$. Under this assumption, the Luus–Jaakola heuristic method can be implemented using algorithm 4. The process can be terminated when it achieves the desired goal or upon completion of the maximum number of iterations (Liao & Luus, 2005). The $\mathit{samplingRange}$ is reduced with the help of a multiplicative factor $\epsilon$ and in this work, the value of this parameter is considered as 0.95. In algorithm 4, the $\alpha$ begins with a feasible point gets updated by the algorithm, and at the end of the algorithm, $\alpha$ contains the optimal solution. The value gets updated with the statement $\alpha =\beta$ that is given in statement 4.c.i. of algorithm 4.

Algorithm 4: The Luus–Jaakola heuristic method
Input: The optimization problem to be solved
Output: The optimal solution
1.Randomly initialize a point $\alpha$ by taking a position from the search space using a uniform distribution. The value should be taken from the range $\left[{\rho }_{min},{\rho }_{max}\right]$ where ${\rho }_{min}$ and ${\rho }_{max}$ are the lower and the upper boundary values respectively.
2.Set $\mathit{samplingRange}\leftarrow \left({\rho }_{max}-{\rho }_{min}\right)$
3.Set a multiplicative factor $\epsilon$ to reduce the search space or the $\mathit{samplingRange}$.
4.Repeat while !terminationCriteria
a.Chose a vector $\upsilon$ randomly and uniformly from the range $\left[-samplingRange,+samplingRange\right]$.
b.Find a new probable location $\beta =\alpha +\upsilon$
c.Check if the fitness improved then
i.Visit the new location by assigning $\alpha =\beta$.
Otherwise
ii.Update $\mathit{samplingRange}\leftarrow \epsilon \times samplingRange$
end if
end while
5.Return $\alpha$

The crisp input indicates that a certain element completely belongs to a set or may not. They possess precise and defined characteristics

6. Proposed method

In this proposed method, a superpixel based biomedical image segmentation process is proposed. In general, if an image contains a large number of pixels then, processing spatial information becomes a difficult job and sometimes, it is computationally not feasible. Superpixels can help in this context (Moore et al., 2008). Moreover, superpixels can reveal more spatial information in a better way, compared to the neighboring window approach where some small neighboring windows are considered which are of similar shape and size. There are various algorithms that can be found in the literature (Achanta et al., 2012, Comaniciu and Meer, 2002, Hu et al., 2015). These methods can be applied before some segmentation algorithms to come up with a better-segmented outcome (Kim et al., 2013). The SLIC (Achanta et al., 2012) is one of the popular methods from the context of superpixel. It acquires hexagonal regions and produces regular superpixels. Mean shift (Comaniciu & Meer, 2002) and watershed (Hu et al., 2015) based approaches produce irregular superpixels. In comparison, mean shift and watershed-based approaches are better than the SLIC method but, the watershed-based method is sensitive to noise, and therefore, this approach is not highly suitable for real-life applications. In most practical applications, the mean shift method is frequently used to avoid the over-segmentation problem but, its computationally complex compared to the watershed approach. The mean shift method is controlled by some parameters like spatial bandwidth ($b_{\mathit{spatial}}$), range of the bandwidth ($b_{\mathit{range}}$), and the smallest possible size of the regions ($s_{min}$) and the output is sensitive to the values of these parameters.

Therefore, a method is proposed which is computationally less complex than the mean shift method to achieve better results incurring lesser computational overhead before the segmentation phase. In this work, a novel method is proposed to generate the superpixels and this method is based on the watershed-based method. In general, the watershed-based method finds the region minima from the gradient images and therefore, this method is computationally efficient than the mean shift method. This property is exploited to develop a new method that is not sensitive to the parameters. After computing the superpixels, the histogram is computed and the computation of histogram after finding the superpixels is easier compared to find the histogram from the original image due to the fact that superpixels represent a group of pixels by a single superpixel. Superpixel image consists of a lesser number of colors compared to the corresponding color image. The computed histogram is incorporated to perform the segmentation using the fuzzy clustering method.

6.1. Foundation of the watershed based modified superpixel method

As discussed earlier, the watershed-based superpixel method captures irregular regions from the superpixels which are efficient than the SLIC method and faster than the mean shift approach but, the watershed-based method is sensitive to noise, and therefore, it is susceptible to the over-segmentation problem. The gradient image is computed using the method described in (Hore et al., 2015) to determine the local or the regional minima and therefore, it can produce the superpixels effectively. Now, the over-segmentation can produce an erroneous outcome which results in wrong diagnosis and inappropriate treatment. This problem can be avoided by preserving the details of the contour. The noise and not-so-important information of an image can be removed by preserving only the important gradient information. It can be achieved by the morphological erosion and the dilation operations based reconstruction process (Vincent, 1993), which is given in Eqs. (28), (29) respectively. The actual image and the marker image are denoted by the $I$ and $\widehat{I}$.

$${\mathrm{\Lambda }}_I^{\psi }\left(\widehat{I}\right)={\psi }_I^k\left(\widehat{I}\right)$$ (28)

$${\mathrm{\Lambda }}_I^{\zeta }\left(\widehat{I}\right)={\zeta }_I^k\left(\widehat{I}\right)$$ (29)

Here, $\psi$ and $\zeta$ represents the erosion and dilation operation respectively. ${\mathrm{\Lambda }}^{\psi }$ and ${\mathrm{\Lambda }}^{\zeta }$ represents the reconstruction process based on the erosion and the dilation operations respectively. ${\mathrm{\Lambda }}^{\psi }$ requires $\widehat{I}⩾I$ and for ${\mathrm{\Lambda }}^{\zeta }$, $\widehat{I}⩽I$. The morphological erosion $\psi$ and the dilation $\zeta$ operation can be expressed using Eqs. (30), (31) respectively. Here, $\vee$ and the $\wedge$ operator represents the point wise highest and the lowest values.

$${\psi }_I^k\left(\widehat{I}\right)=\psi \left({\psi }^{k-1}\left(I\right)\right)\vee \widehat{I}$$ (30)

$${\zeta }_I^k\left(\widehat{I}\right)=\zeta \left({\zeta }^{k-1}\left(I\right)\right)\wedge \widehat{I}$$ (31)

Erosion and dilation are the two morphological operations that are the duals of each other and therefore, they always expressed in pairs. This reconstruction process can also be achieved by the morphological opening and closing which are given in Eqs. (32), (33). Opening and closing are the two morphological operations that are more frequently used compared to erosion and dilation because these operations can extract the features more efficiently and can handle noise more efficiently. Opening and closing operations are also dual and expressed simultaneously. Here, $o$ and $\varsigma$ denotes the opening and the closing operation respectively.

$${\mathrm{\Lambda }}_I^o\left(\widehat{I}\right)={\mathrm{\Lambda }}^{\zeta }\left({\mathrm{\Lambda }}^{\psi }\right)$$ (32)

$${\mathrm{\Lambda }}_I^{\varsigma }\left(\widehat{I}\right)={\mathrm{\Lambda }}^{\psi }\left({\mathrm{\Lambda }}^{\zeta }\right)$$ (33)

The marker image $\widehat{I}$ can be represented by Eqs. (34), (35) for the ${\mathrm{\Lambda }}^{\psi }$ and ${\mathrm{\Lambda }}^{\zeta }$ respectively where $\gamma$ represents the structuring element. The over segmentation problem can be efficiently handled by the ${\mathrm{\Lambda }}^o$ or ${\mathrm{\Lambda }}^{\varsigma }$ by effectively locating and eliminating local minima from the gradient image.

$$\widehat{I}={\zeta }_{\gamma }\left(I\right)$$ (34)

$$\widehat{I}={\psi }_{\gamma }\left(I\right)$$ (35)

It is worth mentioning in this context that, the structuring element $\gamma$ plays a significant role to get the segmented output. Under segmentation can occur due to the smaller structuring element and similarly, large structuring elements can cause over-segmentation problems. An appropriate structuring element is essential to get the accurate number of the segmented regions alone with the precise contours. Therefore, a single structuring element may not be suitable always for all types of images and different structuring elements may be required for different types of images to achieve the same. This dependency can be overcome by applying different structuring elements to obtain the reconstructed gradient images and then these images are combined and the pointwise highest values are computed to remove the local minima and to preserve the important details of the contours. The number of structuring elements can be controlled depending on the range of the values for the controlling parameter $\vartheta$ of the corresponding structuring elements. For example, the value of the distance from the origin to the points of the diamond structuring elements, the width of a square structuring element, or the radius of the circular or octagonal structuring elements can be adjusted to get structuring elements of different sizes and in this way, the size of the regions can be adjusted. For example, if this method is applied with the opening operation then, Eq. (32) can be modified as Eq. (36) where ${\vartheta }_l$ and ${\vartheta }_h$ represents the lowest and the highest value of the controlling parameter of the structuring elements. If a structuring element is dependent on more than one controlling parameter then this equation can be replicated accordingly. The effect of the size of the structuring element on the superpixel image is illustrated in Fig. 4, Fig. 5 . The image under consideration is collected from (COVID-19: Caso 47 | SIRM, n.d.). From Figs. 4(i) and 5(i), it can be clearly observed that the number of superpixels is decreasing with the increasing size of the structuring element. Fig. 4 is obtained with the help of the disk structuring element and Fig. 5 is obtained with the help of the square structuring element.

$${\tilde{\mathrm{\Lambda }}}_I^o\left(\widehat{I},{\vartheta }_l,{\vartheta }_h\right)=max\left\{{\mathrm{\Lambda }}_I^o{\left(\widehat{I}\right)}_{{\gamma }_{{\vartheta }_l}},{\mathrm{\Lambda }}_I^o{\left(\widehat{I}\right)}_{{\gamma }_{{\vartheta }_l+1}},{\mathrm{\Lambda }}_I^o{\left(\widehat{I}\right)}_{{\gamma }_{{\vartheta }_l+2}},.....,{\mathrm{\Lambda }}_I^o{\left(\widehat{I}\right)}_{{\gamma }_{{\vartheta }_h}}\right\}$$ (36)

Fig. 4.

Effect of the size of the disk structuring element on the superpixel image. (a)-(h) the superpixel images obtained using the circular structuring elements of size 3 to 10 respectively, (i) The relation between the size of the structuring element and the superpixel (on the X-axis the size of the structuring element, and on the Y-axis, the count of the superpixels is mapped).

Fig. 5.

Effect of the size of the square structuring element on the superpixel image. (a)-(h) the superpixel images obtained using the circular structuring elements of size 3 to 10 respectively, (i) The relation between the size of the structuring element and the superpixel (on the X-axis the size of the structuring element, and on the Y-axis, the count of the superpixels is mapped).

In this equation, ${\vartheta }_l⩽\vartheta ⩽{\vartheta }_h$ and $\left[{\vartheta }_l,{\vartheta }_h\right]\in {\mathbb{N}}^{+}$. If the value of ${\vartheta }_l$ is quite small then the final segmented image will consist of various small segmented regions. In contrast to this, a significant higher value of ${\vartheta }_l$ will significantly reduce the precision of the contours and important edge information (Chakraborty, 2020, Chakraborty and Mali, 2018) will be lost. Now the upper value ${\vartheta }_h$ can vary depending on the image under consideration. But it is not practical to decide this value depending on the image. Therefore, a small error value $\delta$ can be decided which can be treated as the threshold value to decide the upper bound and it is mathematically expressed by the non-equality 37.

$$\left\{{\tilde{\mathrm{\Lambda }}}_I^o\left(\widehat{I},{\vartheta }_l,{\vartheta }_h\right)-{\tilde{\mathrm{\Lambda }}}_I^o\left(\widehat{I},{\vartheta }_l,{\vartheta }_h+1\right)\right\}⩽\delta$$ (37)

One thing is obvious that, if the value of the threshold is very high then it represents the higher error rate but ${\vartheta }_h$ will be smaller. But if the value of the threshold is low then the error will be smaller but, the value of the ${\vartheta }_h$ will be comparatively larger. Increasing value of ${\vartheta }_h$ introduces larger computational overhead. Therefore, the value of $\delta$ is crucial to balance the computational overhead and the amount of error.

6.2. Fuzzy C-means based clustering for the image segmentation

Fuzzy C-means based clustering method is a well-known and frequently used clustering approach and has a widespread application domain (Chakraborty et al., 2020). This method forms clusters based on global features. The superpixel image has several regions consisting of some groups of pixels. Now, the fuzzy c-means objective function needs to be updated accordingly to exploit these advantages of this superpixel. The clusters are formed on the basis of the superpixels and therefore, distances can be measured from the centers of the clusters to the representative pixel value ${\vartheta }_k$ of the superpixel. The representative value of a superpixel can be computed by simply finding the arithmetic mean of the constituting pixel values which is given in Eq. (38). Here, $\mathit{nPi}{x}_k$ represents the number of pixels in the $k^{\mathit{th}}$ region $R_k$ and $p{x}_q$ denotes a pixel in the region $R_k$. Now, the objective function can be modified as given in Eq. (39).

$${\widehat{R}}_k=\frac{1}{\mathit{nPi}{x}_k}\sum _{q\in R_k}p{x}_q$$ (38)

$$\mathit{Ob}{j}_{\phi }=\sum _{k=1}^{\mathit{cntR}}\sum _{j=1}^{\mathit{cntC}}\mathit{nPi}{x}_k{\mu }_{\mathit{kj}}^{\phi }{∥{\widehat{R}}_k-c_j∥}^2,where1⩽\phi <\infty$$ (39)

In this equation, $\mathit{cntR}$ is the number of the total regions and the $\mathit{cntC}$ denotes the number of clusters. $\omega$ is the factor of fuzzification. This objective function will be helpful in reducing the computational overhead by reducing the number of points in each iteration. Here, ${\mu }_{\mathit{kj}}$ represents the degree of membership of the $k^{\mathit{th}}$ representative point ${\vartheta }_k$ to the $j^{\mathit{th}}$ cluster. The sum of ${\mu }_{\mathit{kj}}$ for a certain point must be 1 for all clusters i.e. ${\sum }_{j=1}^{\mathit{cntC}}{\mu }_{\mathit{kj}}^{\phi }=1$ where, $k=1,2,3,........,cntR$. The degree of the membership ${\mu }_{\mathit{kj}}$ can be computed using Eq. (40).

$${\mu }_{\mathit{kj}}=\frac{1}{{\sum }_{q=1}^{\mathit{cntC}}{\left(\frac{∥{\widehat{R}}_k-c_j∥}{∥{\widehat{R}}_k-c_q∥}\right)}^{\frac{2}{\phi -1}}}$$ (40)

As discussed earlier, the type 2 fuzzy system, can model the uncertainty better than the type 1 fuzzy system and that is why the type 2 fuzzy system is useful on many occasions and can perform well compared to the type 1 fuzzy system. Moreover, the type 2 fuzzy system-based clustering methods are also resilient against noise. Therefore, in this work, the type 2 fuzzy membership is used which can be computed using Eq. (41).

$${\tilde{\mu }}_{\mathit{kj}}={\mu }_{\mathit{kj}}-\frac{1-{\mu }_{\mathit{kj}}}{2}$$ (41)

There is no need to use the fuzzy cluster center modification equation to update the cluster centers. In this proposed work, the objective function which is given in Eq. (39) is optimized using the advanced cuckoo search method based on the McCulloch’s approach which is discussed earlier in Section 3.

6.3. Fuzzy memetic cuckoo search-based segmentation.

In this work, a memetic approach is proposed to modify the McCulloch’s approach based cuckoo search method. In this approach, the global search and the local search procedure is hybridized to improve the exploration of the search space. In this work, the Luus-Jaakola heuristic method (Luus & Jaakola, 1973) is incorporated with the McCulloch’s approach based cuckoo search method. The Luus-Jaakola method is not dependent on the gradient computation. Some lower and upper bounds are predefined from where a random uniform value is generated to perturb the solution. The newly generated solutions replace old solutions if fitness is better. If the solution space does not improve then the neighborhood sampling space is reduced by a multiplicative factor $\phi$. In this work, the value of $\phi$ is considered as 0.95. The objective function of Eq. (39) is used for the optimization purpose, to get the optimum segmentation output. The proposed approach combines both local and global optimization approach to get optimal segmentation output. The final segmented outcome is undoubtedly a global optimum. The local optimization approach is used to strengthen the segmentation approach. The proposed method begins by initializing the cluster centers randomly using Eq. (42) where the cluster centers are chosen from the intensity range of the superpixel image and it can be obtained from the histogram information. ${\gamma }_{min}$ and ${\gamma }_{max}$ represents the highest and the lowest value in the range and ${\widehat{C}}_j$ is the $j^{\mathit{th}}$ cluster center of the superpixel image. Eq. (43) can be used to express a nest in terms of its $m$ number of eggs i.e. the cluster centers.

$${\widehat{C}}_j={\gamma }_{min}+random\left(0,1\right)\left({\gamma }_{max}-{\gamma }_{min}\right)$$ (42)

$$\mathit{nes}{t}_k=\left\{{\widehat{C}}_1,{\widehat{C}}_2,..........,{\widehat{C}}_m\right\}$$ (43)

In this work, the number of nests is considered as 25. So $k=1,2,3,4,.........,30$. The controlling parameters and their corresponding values are presented in Table 2 . The memetic cuckoo search method, based on the McCulloch’s approach is used to control the total system. The cluster centers are randomly initialized and updated by this approach. Moreover, the cluster centers are initialized randomly, and therefore, there is no dependency on the initial choice of the cluster centers. The proposed approach helps is faster convergence and to get efficient segmented output due to the collaboration with the superpixel and the type 2 fuzzy clustering-based method. Therefore, the cluster centers can be guided by the proposed system to reach the global optima. The application of the Luus-Jaakola method in the local search procedure helps in exploring the search space more efficiently. The proposed approach is applied to segment the radiological images which are collected from the COVID-19 infected patients of different regions. The proposed method is tested and compared with some of the standard methods by both visual and numerical measures. The proposed method is illustrated in algorithm 5. Fig. 6 shows the flow diagram of the proposed method.

Algorithm 5: The proposed image segmentation approach
Input: The input image which is to be segmented and the value of the controlling parameters.
Output: The segmented image
6.Determine the gradient of the input image using the method described in (Hore et al., 2015).
7.Apply Eqs. (16) and (17) to determine the superpixel image corresponding to the input image.
8.Initialize the cluster centers in a random manner using Eqs. (22), and (23).
9.Assign the fuzzy type 2 membership value to the points i.e., superpixels using Eq. (21).
10.Compute the fitness of the nests by using the modified fuzzy objective function by using Eq. (19).
11.$\mathit{itrCounter}\leftarrow 1$
12.Check if $\mathit{itrCounter}⩽cntGeneration$ then
a.Find and store the current optimal solution
b.Perform a local search using algorithm 4.
c.Create a new solution space
d.Determine the value of the objective function using Eq. (19) for the current solution space.
e.Locate and preserve the optimal solution
f.Generate a random number $r$ and check if $r<pro{b}_{\epsilon }$ then
i.Apply McCulloch’s approach and find the step size using Eq. (22).
ii.Replace the worst nest with the help of the computed fitness values.
iii.Calculate the fitness values for the current solution space.
iv.Update the iteration counter.
v.Find the nest with the optimum fitness value.
otherwise
vi.Store those nests
end if
end if
13.Construct the segmented image depending on the optimal cluster centers.
14.Return the segmented image

Table 2.

The controlling parameters and their chosen values.

Name of the controlling parameter	Selected value
Total count of the nests	25
Alien egg discovery probability ($\mathit{pro}{b}_a$)	0.4
Maximum iteration count ($\mathit{cntGeneration}$)	250
Multiplicative factor ($\phi$)	0.95
Number of clusters	Subjective

Fig. 6.

Simple block diagram of the proposed method.

7. Results of the simulation

The proposed SUFMACS method is tested and compared by both quantitative and qualitative measures. As discusses in the introduction section, the ground truth images are not always available for the radiological images for early screening of the COVID-19 infection. Therefore, the prime objective of the proposed approach is to design an image segmentation framework that is robust in presence of noise and can generate the segmented image efficiently and quickly to combat the spread of this highly infectious virus in this pandemic scenario. The analysis of the proposed method is performed using the radiological images of the chest region which are collected from the COVID-19 infected patients. It is always better to compare the segmented images with some ground truth segmentations i.e., with some manual delineations. Manual delineations and markings must be done by some physicians and/or domain experts to get reliable and comparable ground truth images. But, as discussed in the manuscript, the ground truth data are quite limited and difficult to generate because it involved human experts and needs a significant amount of time. Although, the manual marking of irregularities can be helpful to enhance the quality of the overall system, but difficult to incorporate because no domain experts are associated with this work. The cluster validity indices that are used in this work will help in this scenario. The quantitative analysis is performed using four well-known cluster validity indices and the results are obtained for the different number of clusters.

7.1. Dataset description

Experiments are performed on the 250 CT images, and 250 X-Ray images that are collected from the COVID-19 positive patients from Italy, China, Iran, Taiwan, Korea etc. and these images are selected in a random manner. Among the 250 CT images, the results of 9 images are reported in this article. Similarly, Among the 250 X-Ray images, the results of 6 images are reported in this article. The solid white areas in the images typically indicate the ground glass opacity i.e. the partial filling of the airspaces and consolidation. In the segmented images, different regions are highlighted using different shades of the grayscale intensity. Apart from these images, some other images that belong to the normal category, as well as some other lung diseases, are also investigated. Among these images, experimental results on three images are reported. The details of these 18 images can be found in Table 3 . All 18 images are given in Fig. 7 .

Table 3.

Details of the dataset.

Image Id	View	Modality	Source	Description
$\mathit{IM}{G}_{001}$	Axial	CT	(COVID-19. CASO 2 | SIRM, n.d.)	The image is collected from a 62 years old male COVID-19 positive patient from Italy. (Credit to UOC Radiology ASST Bergamo Est Director Dr Gianluigi Patelli). A few nuanced bilateral alveolar infiltrative thickens (it is a sign of interstitial lung disease where the natural process of the body to repair tissues is corrupted and air sacks of the lung are got thickened) are reported.
$\mathit{IM}{G}_{002}$	Coronal	CT	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-a)	This image is collected from 70 years old, female and COVID-19 positive patient of Ospedale Santo Spirito. Rome, Italy (Case courtesy of Dr Fabio Macori, Radiopaedia.org, rID: 74887). Some important features that can be observed from this image are bilateral ground-glass opacities (i.e. the area with increased attenuation).
$\mathit{IM}{G}_{003}$	Axial	CT	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-b)	This image is collected from 70 years old, male and COVID-19 positive patient from Riccione, Italy (Case courtesy of Dr Domenico Nicoletti, Radiopaedia.org, rID: 74724). Some oberservations and features which are reported for this image are ground glass opacities in the lower right and the upper lobes and Paraseptal emphysema (i.e. swelling and tissue damage to the tiny air sacks called alveoli) in the upper lobes.
$\mathit{IM}{G}_{004}$	Coronal	CT	(Case 001-61 Year Old, Female | Coronavirus Cases - 冠状病毒病例, n.d.)	This image is collected from 61 years old, female and COVID-19 positive patient from Wenzhou, China (Credit to Omir Antunes Paiva, Dr. Rodrigo Caruso Chate, Wenzhou Medical University, and coronacases.org). Some important features of this CT image are multiple ground-glass opacities and tiny foci of consolidation in all pulmonary lobes.
$\mathit{IM}{G}_{005}$	Sagittal	CT	(Case 004-41 Year Old, Male | Coronavirus Cases - 冠状病毒病例, n.d.)	This image is collected from 41 years old, male and COVID-19 positive patient from Wenzhou, China (Credit to Omir Antunes Paiva, Dr. Rodrigo Caruso Chate, Wenzhou Medical University, and coronacases.org). Some important features of this CT image are ground-glass opacities and linear opacities (that resembles a line, <=2mm) found in the left lower lobe.
$\mathit{IM}{G}_{006}$	Coronal	CT	(Case 009-64 Year Old, Female | Coronavirus Cases - 冠状病毒病例, n.d.)	This image is collected from 64 years old, female and COVID-19 positive patient from Wenzhou, China (Credit to Omir Antunes Paiva, Dr. Rodrigo Caruso Chate, Wenzhou Medical University, and coronacases.org). Some important features of this CT image are ground-glass opacities in the right upper, middle and lower lobes and in lingula, as well as in the right lower lobe, where some atelectasis can also be seen. There are other subtle ground-glass opacities in the right middle lobe and lingula.
$\mathit{IM}{G}_{007}$	Axial	CT	(Lim et al., 2020)	This image is collected from 54 years old, male and COVID-19 positive patient from Myongji Hospital, Goyang, Korea. Some important features of this CT image are consolidation in right upper lobe and ground-glass opacities in both lower lobes.
$\mathit{IM}{G}_{008}$	Coronal	CT	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-c)	This image is collected from 50 years old, female and COVID-19 positive patient from Iran. (Case courtesy of Dr Bahman Rasuli, Radiopaedia.org, rID: 74576). One important feature of this CT image is ground glass nodule is present at the left lower lobe.
$\mathit{IM}{G}_{009}$	Axial	CT	(Cheng et al., 2020)	This image is collected from 55 years old, female and COVID-19 positive patient from Taoyuan General Hospital, Taoyuan, Taiwan. Some important features of this CT image are ground-glass opacities, mild fibrotic change (i.e. lungs become scarred) at bilateral lungs, and two small irregular opacities at the right upper and middle lung.
$\mathit{IM}{G}_{010}$	Frontal	X-Ray	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-d)	This image is collected from 75 years old, male and COVID-19 positive patient from Italy. (Case courtesy of Dr Fabio Macori, Radiopaedia.org, rID: 74867) Observed the presence of extensive bilateral GGO.
$\mathit{IM}{G}_{011}$	Frontal	X-Ray	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-e)	These images are collected from 70 years old, female and COVID-19 positive patient from Northern Italy. (Case courtesy of Dr Fabio Macori, Radiopaedia.org, rID: 74887) Observed coarsening of lung markings (it is also a sign of interstitial lung disease) at the lower fields.
$\mathit{IM}{G}_{012}$	Lateral	X-Ray
$\mathit{IM}{G}_{013}$	Frontal	X-Ray	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-f)	These images are collected from an elderly, male and COVID-19 positive patient. (Case courtesy of Dr Ali Mashalla Åhre, Radiopaedia.org, rID: 75037). Peripheral opacifications (i.e., the filling of airspace in lungs) can be observed.
$\mathit{IM}{G}_{014}$	Lateral	X-Ray
$\mathit{IM}{G}_{015}$	Frontal	X-Ray	(COVID-19 pneumonia | Radiology Case | Radiopaedia.org., n.d.-g)	This image is collected from 75 years old, female and COVID-19 positive patient (Case courtesy of Dr Yair Glick, Radiopaedia.org, rID: 75137). Observed Bronchial wall thickening (some of the pathological entities can cause this situation i.e., abnormal thickening of bronchial walls).
$\mathit{IM}{G}_{016}$	Frontal	CT	(Normal CT Chest | Radiology Case | Radiopaedia.Org, n.d.)	This image is collected from 25 years old, male patient (Case courtesy of Dr Andrew Dixon, Radiopaedia.org, rID: 36676)
$\mathit{IM}{G}_{017}$	Frontal	X-Ray	(Sarcoidosis | Radiology Case | Radiopaedia.Org, n.d.)	This image is collected from male patient (Case courtesy of Assoc Prof Frank Gaillard, Radiopaedia.org, rID: 6546). Stage II sarcoidosis (abnormal set of inflammatory cells that form lumps) is diagnosed.
$\mathit{IM}{G}_{018}$	Axial	CT	(Usual Interstitial Pneumonia (UIP) | Radiology Case | Radiopaedia.Org, n.d.)	This image is collected from 55 years old, male patient (Case courtesy of Dr Hani Makky Al Salam, Radiopaedia.org, rID: 13199) Usual interstitial pneumonia (UIP) is diagnosed.

Fig. 7.

Test images under consideration.

7.2. Validity indices

The segmented output of the proposed method is evaluated by the four well-known cluster validity indices and these indices are briefly discussed below. Apart from the visual investigation, these validity indices allow us to analyze and compare the proposed SUFMACS method in a quantitative way.

• iDavies–Bouldin index ($\mathit{DBIndex}$): The Davies–Bouldin index is a well-known cluster validity index that measures the ratio of the intra-cluster and inter-cluster distances (Davies & Bouldin, 1979) and therefore, the minimum value of this index indicates a good clustering result. The value of this index can be computed using Eq. (44) where $c$ denotes the total number of clusters and $l\ne m$, $1⩽l⩽c$.

$$\mathit{DBIndex}=\frac{1}{c}\sum _{l=1}^{c}max\left(\frac{d_w\left(a_l\right)+d_w\left(a_m\right)}{d_b\left(a_l,a_m\right)}\right)$$ (44)

• iiXie-Beni index ($\mathit{XBIndex}$): It is one of the most popular and frequently used fuzzy cluster validity measures (Xie & Beni, 1991). It measures the ratio between the compactness and the separation of the clusters and its value can be computed using Eq. (45). The lower value of this index indicates a good clustering outcome.

$$\mathit{XBIndex}=\frac{{\sum }_{k=1}^c{{\sum }_{i=1}^n{U}_{\mathit{ki}}^2∥V_k-X_i∥}^2}{d_{min}{∥V_k-V_i∥}^2}$$ (45)

• iiiDunn index ($D{I}_n$): The value of the Dunn index is dependent on the inter-cluster distance $d\left(c_i,c_j\right)$ and the mean distance among all possible combination of the clusters ${\mathrm{\Delta }}_k$ (Dunn, 1974). The value of this index can be computed using Eq. (46) and higher values imply a good clustering outcome.

$$D{I}_n=\underset{1⩽i⩽n}{min}\left(\underset{1⩽j⩽n,j\ne i}{min}\left(\frac{d\left(c_i,c_j\right)}{\underset{1⩽k⩽m}{max}{\mathrm{\Delta }}_k}\right)\right)$$ (46)

• iv$\beta$index:$\beta$ index is another useful cluster validity index to evaluate the segmented image outcomes, that measures the ratio of the total and the intracluster variation (Pal et al., 2000). Eq. (47) can be used to find the value of this index where $\mathit{pi}{x}_m$ represents the number of pixels in the $m^{\mathit{th}}$ cluster, the intensity of these pixels is denoted by $I_{\mathit{ml}}$ and $x_m=\frac{1}{\mathit{pi}{x}_m}{\sum }_{i=1}^{\mathit{pi}{x}_m}{I}_{\mathit{im}}$. A higher value is desirable to get a good clustering outcome.

$$\beta =\frac{{\sum }_{m=1}^c{\sum }_{l=1}^{\mathit{pi}{x}_m}{\left(I_{\mathit{ml}}-x\right)}^2}{{\sum }_{m=1}^c{\sum }_{l=1}^{\mathit{pi}{x}_m}{\left(I_{\mathit{ml}}-x_m\right)}^2}$$ (47)

7.3. Results of the experiments

The simulation is carried out with the help of MatLab R2014a (windows version) and a computer system equipped with an Intel i3 processor (1.8 GHz Clock Speed) and 4 GB of main memory. The proposed SUFMACS method is compared with the beam-ACO (Blum, 2005), adaptive PSO (Taherkhani & Safabakhsh, 2016), efficient GA (Kadri & Boctor, 2018) and the modified cuckoo search (MCS) method (Chakraborty et al., 2017). Fig. 8 shows the visual comparison of the segmented outputs for $\mathit{IM}{G}_{009}$ (CT image), Fig. 9 depicts a visual comparison of the segmented output for image $\mathit{IM}{G}_{010}$ (X-Ray image), and Fig. 10 shows the results of the segmentation for the rest of the images, which are obtained by applying the proposed SUFMACS method for different numbers of clusters. Fig. 8, Fig. 9, Fig. 10 shows the clustering outcomes for a different number of clusters. It can be easily observed that the proposed approach will produce more realistic segmented outcomes compared to the other methods. Quantitative comparison of different algorithms using different indices and for different number of clusters are given in Table 4, Table 5, Table 6, Table 7 and the acceptable values are highlighted in the boldface. For the sake of conciseness, the qualitative comparison is reported for only one image but the results of the quantitative comparisons are reported for each of the nine images.

Fig. 8.

Segmented output of the CT Scan image $\mathit{IM}{G}_{009}$ obtained using different methods and for different no. of clusters.

Fig. 9.

Segmented output of the X-Ray image $\mathit{IM}{G}_{010}$ obtained using different methods and for different no. of clusters.

Fig. 10.

Segmented output for different number of clusters using the proposed SUFMACS method.

Table 4.

Quantitative comparison of different algorithms using the Davies–Bouldin index (The acceptable values are highlighted in boldface).

Image Id	Algorithm	No. of Clusters
		3	5	7	9
$\mathit{IM}{G}_{001}$	efficient GA (Kadri & Boctor, 2018)	1.55479151	1.98368604	2.26846147	0.967247997
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.17214821	1.65914975	2.43787454	2.093061798
	beam-ACO (Blum, 2005)	1.49043209	1.2422774	0.66940905	1.13123168
	MCS method (Chakraborty et al., 2017)	1.26311145	1.64951221	0.4447157	1.05115426
	SUFMACS (Proposed)	1.58519391	0.50973766	0.72450084	0.870031184
$\mathit{IM}{G}_{002}$	efficient GA (Kadri & Boctor, 2018)	1.37398389	1.56117374	2.11994856	1.413140206
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.514081	1.85458132	2.11506967	1.757333248
	beam-ACO (Blum, 2005)	3.05769215	2.191229	2.00861842	1.800988018
	MCS method (Chakraborty et al., 2017)	1.83079569	1.87000353	2.19750437	1.211065816
	SUFMACS (Proposed)	1.3011462	1.17945849	1.17915455	2.468921133
$\mathit{IM}{G}_{003}$	efficient GA (Kadri & Boctor, 2018)	1.19242097	0.66067109	0.5934101	1.19369104
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.015606	1.08660969	1.2708262	1.039265784
	beam-ACO (Blum, 2005)	1.64556936	1.45435548	2.11888994	1.99536551
	MCS method (Chakraborty et al., 2017)	0.55115933	0.93696416	1.35817229	0.442500929
	SUFMACS (Proposed)	1.03577276	1.59392865	1.02464713	0.754485337
$\mathit{IM}{G}_{004}$	efficient GA (Kadri & Boctor, 2018)	1.47898392	2.23456907	2.30859711	1.750180709
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.32122911	1.93188294	1.91581813	1.942071652
	beam-ACO (Blum, 2005)	1.82771469	1.19569297	1.5555629	0.971450681
	MCS method (Chakraborty et al., 2017)	2.44818796	1.1687552	1.23136925	1.321599632
	SUFMACS (Proposed)	2.25154247	1.1516908	1.27441012	2.294149102
$\mathit{IM}{G}_{005}$	efficient GA (Kadri & Boctor, 2018)	2.26275296	1.81753513	1.6856485	2.123757304
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.40627082	1.54318548	1.86874369	2.287884464
	beam-ACO (Blum, 2005)	1.36549916	2.30898973	1.56865777	1.882146627
	MCS method (Chakraborty et al., 2017)	1.08734619	2.55675195	2.39993229	2.392363007
	SUFMACS (Proposed)	1.54312225	2.04648899	1.06711119	1.528749902
$\mathit{IM}{G}_{006}$	efficient GA (Kadri & Boctor, 2018)	1.08215918	1.33270038	0.64031244	0.780034081
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.33194699	1.28471985	1.9428484	2.351300317
	beam-ACO (Blum, 2005)	0.67923652	0.7451336	1.46472229	1.508461266
	MCS method (Chakraborty et al., 2017)	0.51229317	1.05271042	0.61131401	0.610643116
	SUFMACS (Proposed)	1.03508187	0.49451386	0.63349679	0.990483514
$\mathit{IM}{G}_{007}$	efficient GA (Kadri & Boctor, 2018)	1.22722611	1.48717486	1.98374914	1.872445325
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.79086059	1.22456373	0.96052337	2.47035332
	beam-ACO (Blum, 2005)	1.98189917	1.83656211	1.39739053	1.395654116
	MCS method (Chakraborty et al., 2017)	1.5359324	1.58292756	1.59942769	1.997748567
	SUFMACS (Proposed)	1.09750421	1.57422001	1.01513912	1.559944992
$\mathit{IM}{G}_{008}$	efficient GA (Kadri & Boctor, 2018)	2.62984415	2.57794722	3.20418654	2.345888276
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.74749563	1.27498539	1.72772781	2.213275766
	beam-ACO (Blum, 2005)	1.9169715	2.01724481	1.54560725	2.120151606
	MCS method (Chakraborty et al., 2017)	2.05518495	1.755137	0.96366438	1.82595824
	SUFMACS (Proposed)	1.07490406	1.7681992	2.2385635	1.937915028
$\mathit{IM}{G}_{009}$	efficient GA (Kadri & Boctor, 2018)	1.74652535	2.53290509	1.14993605	2.692438051
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.01526332	2.42673342	0.60702854	2.819545218
	beam-ACO (Blum, 2005)	1.2954329	1.79127496	1.98918478	1.688857448
	MCS method (Chakraborty et al., 2017)	2.0452042	1.10428871	0.56870942	2.231041951
	SUFMACS (Proposed)	0.15340193	1.03775227	1.34553862	2.349675985
$\mathit{IM}{G}_{010}$	efficient GA (Kadri & Boctor, 2018)	1.159959155	1.572293723	2.049048397	2.369487471
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.481967425	0.992617494	1.46654405	2.716646267
	beam-ACO (Blum, 2005)	2.186222532	1.559266381	1.300098406	1.774370126
	MCS method (Chakraborty et al., 2017)	0.915216295	1.322270743	1.07412764	2.56406602
	SUFMACS (Proposed)	1.401495941	1.509172745	0.982112837	1.111322941
$\mathit{IM}{G}_{011}$	efficient GA (Kadri & Boctor, 2018)	1.960796	1.263238	2.553092	2.198831
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.360829	1.990291	1.314942	2.707776
	beam-ACO (Blum, 2005)	1.915767	1.598651	2.251792	0.93635
	MCS method (Chakraborty et al., 2017)	1.880177	2.409802	1.974296	2.23445
	SUFMACS (Proposed)	1.010382	0.687246	1.713563	1.371933
$\mathit{IM}{G}_{012}$	efficient GA (Kadri & Boctor, 2018)	1.483966	1.683302	2.500869	1.876279
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.779125	0.915693	1.157111	1.749074
	beam-ACO (Blum, 2005)	2.626314	2.390078	1.173226	0.995537
	MCS method (Chakraborty et al., 2017)	1.925726	2.284318	1.097542	2.324067
	SUFMACS (Proposed)	1.355346	1.422174	0.885344	0.922402
$\mathit{IM}{G}_{013}$	efficient GA (Kadri & Boctor, 2018)	1.786761	1.737199	2.019286	2.580888
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.019829	1.098247	0.978279	2.927639
	beam-ACO (Blum, 2005)	2.285756	1.8769	0.816931	0.743269
	MCS method (Chakraborty et al., 2017)	1.795155	1.243073	1.537979	2.26101
	SUFMACS (Proposed)	1.598513	2.096478	1.243392	1.21976
$\mathit{IM}{G}_{014}$	efficient GA (Kadri & Boctor, 2018)	1.565769	1.434748	1.717284	1.737907
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.475295	1.728619	0.904168	2.704765
	beam-ACO (Blum, 2005)	1.773972	2.459344	0.996768	1.124991
	MCS method (Chakraborty et al., 2017)	1.23986	1.686725	1.627618	1.346573
	SUFMACS (Proposed)	1.064897	1.657952	0.777975	0.910011
$\mathit{IM}{G}_{015}$	efficient GA (Kadri & Boctor, 2018)	1.423537	2.007593	2.039646	2.376893
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.329055	1.054688	0.630083	2.452387
	beam-ACO (Blum, 2005)	1.819639	2.063003	1.070184	1.310508
	MCS method (Chakraborty et al., 2017)	1.358109	1.98053	0.873202	2.208312
	SUFMACS (Proposed)	1.383092	1.661142	1.774125	2.405794
$\mathit{IM}{G}_{016}$	efficient GA (Kadri & Boctor, 2018)	1.677098	1.500814	1.623586	2.429711
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.732348	0.911604	0.671013	2.687133
	beam-ACO (Blum, 2005)	2.208789	2.309481	1.44189	1.757942
	MCS method (Chakraborty et al., 2017)	0.826681	1.938968	1.067422	2.08513
	SUFMACS (Proposed)	1.262862	2.003538	0.95592	2.688271
$\mathit{IM}{G}_{017}$	efficient GA (Kadri & Boctor, 2018)	1.374163	1.591125	2.867268	2.662051
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.088007	1.106915	1.052627	2.057302
	beam-ACO (Blum, 2005)	1.624201	3.020656	1.507298	1.00198
	MCS method (Chakraborty et al., 2017)	1.403158	1.137044	1.539086	1.853181
	SUFMACS (Proposed)	0.569208	0.999981	1.400653	1.792843
$\mathit{IM}{G}_{018}$	efficient GA (Kadri & Boctor, 2018)	1.455346	1.481315	1.200555	2.826705
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.070765	1.398769	1.23125	2.621128
	beam-ACO (Blum, 2005)	1.974946	2.023246	0.73931	1.651843
	MCS method (Chakraborty et al., 2017)	1.705055	2.521398	0.64855	1.575533
	SUFMACS (Proposed)	2.031454	2.152765	0.817271	2.178575

Table 5.

Quantitative comparison of different algorithms using the Xie-Beni index (The acceptable values are highlighted in bold face).

Image Id	Algorithm	No. of Clusters
		3	5	7	9
$\mathit{IM}{G}_{001}$	efficient GA (Kadri & Boctor, 2018)	2.52685677	2.18401314	1.85634911	1.200268899
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.92240898	1.7758955	2.07850825	1.660881813
	beam-ACO (Blum, 2005)	1.12726544	1.47708713	1.10542839	2.364954794
	MCS method (Chakraborty et al., 2017)	1.40040064	2.08372272	1.38569651	1.762615341
	SUFMACS (Proposed)	2.08567042	0.85372989	1.28923336	0.557084754
$\mathit{IM}{G}_{002}$	efficient GA (Kadri & Boctor, 2018)	2.36233211	3.40346567	1.51295704	2.759644862
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.0991102	2.24554797	1.29598923	2.111346769
	beam-ACO (Blum, 2005)	2.75405243	2.6439685	2.99968832	2.513821833
	MCS method (Chakraborty et al., 2017)	3.36370349	2.52588612	1.99845274	2.275100747
	SUFMACS (Proposed)	1.84081404	1.69690035	1.0872016	2.089323662
$\mathit{IM}{G}_{003}$	efficient GA (Kadri & Boctor, 2018)	4.40568139	3.45931104	2.80750816	3.019538561
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.7043356	4.5178127	2.70556497	2.658747238
	beam-ACO (Blum, 2005)	3.91784798	3.74675439	2.74947471	3.321669423
	MCS method (Chakraborty et al., 2017)	2.76559288	2.59694782	2.55060726	2.965200253
	SUFMACS (Proposed)	2.68631049	2.48584721	2.97641601	1.63971928
$\mathit{IM}{G}_{004}$	efficient GA (Kadri & Boctor, 2018)	1.84177562	1.56596076	2.85368921	1.902136499
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.08616902	1.76942747	1.8502289	2.353248497
	beam-ACO (Blum, 2005)	2.05567383	1.14866923	1.43200409	2.42965351
	MCS method (Chakraborty et al., 2017)	1.55543853	0.98513063	1.61748395	1.681781694
	SUFMACS (Proposed)	1.08630098	1.60706127	1.76838166	2.19222821
$\mathit{IM}{G}_{005}$	efficient GA (Kadri & Boctor, 2018)	2.77819863	1.36892334	1.38336791	1.161588175
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.25869174	2.21186504	1.99487358	2.647165165
	beam-ACO (Blum, 2005)	2.94123781	2.02330793	2.20816425	1.912234237
	MCS method (Chakraborty et al., 2017)	2.4273985	1.42202729	0.88278745	1.06840477
	SUFMACS (Proposed)	2.12101338	1.89028251	0.70300561	1.768682015
$\mathit{IM}{G}_{006}$	efficient GA (Kadri & Boctor, 2018)	1.62712145	0.85035428	0.45865802	1.775284225
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.04305525	0.58246934	1.39004354	2.189833714
	beam-ACO (Blum, 2005)	0.72811754	0.88530667	1.1721333	1.282353891
	MCS method (Chakraborty et al., 2017)	1.37733711	0.60781875	0.88605973	0.528830478
	SUFMACS (Proposed)	1.25992686	0.4085281	0.65801747	2.082544599
$\mathit{IM}{G}_{007}$	efficient GA (Kadri & Boctor, 2018)	3.36616015	4.43348032	2.5537189	4.003129374
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.93826388	3.61161779	2.84400403	2.400627947
	beam-ACO (Blum, 2005)	3.535417	4.11789487	2.44074506	2.702760392
	MCS method (. Chakraborty et al., 2017)	3.09992289	3.32954489	3.57596612	2.732061326
	SUFMACS (Proposed)	2.34822513	3.73226035	2.13967403	3.337715973
$\mathit{IM}{G}_{008}$	efficient GA (Kadri & Boctor, 2018)	2.68481737	2.50082958	1.48419563	1.397930034
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.52439234	2.24050031	2.64903547	2.23877869
	beam-ACO (Blum, 2005)	2.47452895	2.0411861	1.0791494	2.30077142
	MCS method (. Chakraborty et al., 2017)	1.0476316	2.52047294	2.33108764	1.683884834
	SUFMACS (Proposed)	1.51571309	1.11331545	1.92239761	1.712257334
$\mathit{IM}{G}_{009}$	efficient GA (Kadri & Boctor, 2018)	1.3947934	1.07280059	0.98131095	2.129336354
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.63664031	1.69090607	1.56929904	1.685466434
	beam-ACO (Blum, 2005)	1.79712018	2.30118806	2.03902643	1.833713323
	MCS method (Chakraborty et al., 2017)	1.93283998	0.64129523	1.34406921	0.702427629
	SUFMACS (Proposed)	0.67589946	0.63937478	0.58636326	1.229247823
$\mathit{IM}{G}_{010}$	efficient GA (Kadri & Boctor, 2018)	0.671509	1.744179	0.923772	1.979253
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.945205	0.868609	1.011834	2.090718
	beam-ACO (Blum, 2005)	0.943716	0.866512	1.84464	0.873554
	MCS method (Chakraborty et al., 2017)	1.019129	0.620031	0.970231	0.612015
	SUFMACS (Proposed)	1.0691	0.84037	0.948285	2.027637
$\mathit{IM}{G}_{011}$	efficient GA (Kadri & Boctor, 2018)	2.367396	0.967872	0.772598	2.656482
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.999127	1.016013	1.004203	2.29621
	beam-ACO (Blum, 2005)	0.60045	0.585852	1.627528	1.641013
	MCS method (Chakraborty et al., 2017)	2.180073	0.957864	1.485118	0.80518
	SUFMACS (Proposed)	0.780851	0.938434	0.859453	1.839339
$\mathit{IM}{G}_{012}$	efficient GA (Kadri & Boctor, 2018)	1.365546	0.869596	0.81675	1.121872
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.342647	1.096372	1.620871	1.535056
	beam-ACO (Blum, 2005)	0.899003	1.564101	0.742408	1.346221
	MCS method (Chakraborty et al., 2017)	1.043839	0.70359	1.245132	0.81226
	SUFMACS (Proposed)	1.482057	0.601223	0.997735	2.204383
$\mathit{IM}{G}_{013}$	efficient GA (Kadri & Boctor, 2018)	1.551348	0.775163	0.662083	1.592869
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.885174	0.692624	0.984438	2.335705
	beam-ACO (Blum, 2005)	1.207277	1.320022	0.726982	1.351773
	MCS method (Chakraborty et al., 2017)	1.65709	1.240389	0.869034	0.793294
	SUFMACS (Proposed)	1.597312	1.32548	0.769052	1.686971
$\mathit{IM}{G}_{014}$	efficient GA (Kadri & Boctor, 2018)	1.140882	1.090185	0.96606	1.959196
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.503407	0.990181	1.142317	2.650642
	beam-ACO (Blum, 2005)	1.382789	0.87776	0.973237	1.081854
	MCS method (Chakraborty et al., 2017)	0.656826	0.627675	0.620359	0.622231
	SUFMACS (Proposed)	1.441876	0.811331	0.874653	1.878663
$\mathit{IM}{G}_{015}$	efficient GA (Kadri & Boctor, 2018)	1.178133	0.567809	0.534498	1.337996
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.479272	0.93335	0.934253	2.292727
	beam-ACO (Blum, 2005)	0.605351	0.825761	0.728386	1.270704
	MCS method (Chakraborty et al., 2017)	1.215022	1.217728	1.213068	1.279283
	SUFMACS (Proposed)	0.957758	0.705139	0.615537	1.932139
$\mathit{IM}{G}_{016}$	efficient GA (Kadri & Boctor, 2018)	0.886243	0.562286	0.488716	0.745659
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.695329	1.134685	1.021853	2.343821
	beam-ACO (Blum, 2005)	1.216232	1.554659	0.810095	1.172306
	MCS method (Chakraborty et al., 2017)	1.17033	1.660234	0.638595	1.259871
	SUFMACS (Proposed)	1.725499	0.928691	0.586035	1.992016
$\mathit{IM}{G}_{017}$	efficient GA (Kadri & Boctor, 2018)	0.80458	0.672848	0.453557	0.837357
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.441602	1.05936	0.704048	2.322052
	beam-ACO (Blum, 2005)	0.662796	1.10381	0.808296	0.544734
	MCS method (Chakraborty et al., 2017)	2.0185	0.416521	0.874094	1.483268
	SUFMACS (Proposed)	0.705414	0.890879	0.493189	1.905358
$\mathit{IM}{G}_{018}$	efficient GA (Kadri & Boctor, 2018)	1.401385	1.0148	0.926574	1.100244
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.370169	0.886205	1.089041	1.669752
	beam-ACO (Blum, 2005)	0.83541	0.715873	0.502379	1.349703
	MCS method (Chakraborty et al., 2017)	0.949702	0.947707	0.975885	0.924699
	SUFMACS (Proposed)	0.792323	1.291956	0.505907	1.884662

Table 6.

Quantitative comparison of different algorithms using the Dunn index (The acceptable values are highlighted in boldface).

Image Id	Algorithm	No. of Clusters
		3	5	7	9
$\mathit{IM}{G}_{001}$	efficient GA (Kadri & Boctor, 2018)	2.08335998	2.32520506	3.39505389	2.261532289
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.62666592	3.95085495	3.69140329	3.497444024
	beam-ACO (Blum, 2005)	4.02053177	1.60460455	2.79340813	3.288950032
	MCS method (Chakraborty et al., 2017)	3.17123195	3.73165344	2.20394834	3.389165179
	SUFMACS (Proposed)	1.88811413	3.90746936	1.42054496	1.70166278
$\mathit{IM}{G}_{002}$	efficient GA (Kadri & Boctor, 2018)	0.8887936	1.07409716	0.98758632	0.482367705
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.76542272	0.31000091	2.35572819	0.87270862
	beam-ACO (Blum, 2005)	0.38132524	1.31452453	1.35000955	1.445684538
	MCS method (Chakraborty et al., 2017)	0.80687584	0.65940617	0.92244884	2.035489605
	SUFMACS (Proposed)	1.04616611	1.98770638	2.897906	0.60469652
$\mathit{IM}{G}_{003}$	efficient GA (Kadri & Boctor, 2018)	0.51704978	1.95552077	1.9428017	1.602738784
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.30066727	0.84262009	1.9121285	1.574231672
	beam-ACO (Blum, 2005)	0.47017388	1.04283179	2.49903397	0.25635234
	MCS method (Chakraborty et al., 2017)	1.90055086	0.7870909	1.69600557	2.677387518
	SUFMACS (Proposed)	1.96066739	0.9985229	2.41147026	2.695152777
$\mathit{IM}{G}_{004}$	efficient GA (Kadri & Boctor, 2018)	0.69600456	0.45097131	0.74466136	1.349350862
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.44800505	0.88494491	1.84523152	2.494550285
	beam-ACO (Blum, 2005)	1.25070544	0.11649073	0.94942353	0.2767343
	MCS method (Chakraborty et al., 2017)	1.41124552	0.64732118	1.48998584	2.098470116
	SUFMACS (Proposed)	2.78352028	1.28867042	1.34779004	1.135858939
$\mathit{IM}{G}_{005}$	efficient GA (Kadri & Boctor, 2018)	0.67592709	1.60028766	−0.2623357	0.746087118
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.14533188	0.5913468	1.46715135	0.316867025
	beam-ACO (Blum, 2005)	1.00715834	1.0615136	1.83242994	2.023880726
	MCS method (Chakraborty et al., 2017)	1.75083597	1.12624742	2.11227717	1.975211569
	SUFMACS (Proposed)	3.07476178	1.68513903	1.31080669	0.770706118
$\mathit{IM}{G}_{006}$	efficient GA (Kadri & Boctor, 2018)	1.00702755	1.127624	1.85093056	1.277256547
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.3417881	3.03047011	2.88588264	1.582574025
	beam-ACO (Blum, 2005)	2.68818192	1.93496041	1.96961439	2.290428962
	MCS method (Chakraborty et al., 2017)	1.92048497	1.13472871	1.95413319	2.515440198
	SUFMACS (Proposed)	1.04875517	1.6607005	2.57035718	1.570068095
$\mathit{IM}{G}_{007}$	efficient GA (Kadri & Boctor, 2018)	0.1210708	0.37214974	1.86614085	2.098947018
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.05098832	1.3596237	2.20028228	0.690423766
	beam-ACO (Blum, 2005)	0.38008116	1.02759561	1.38450175	0.694369995
	MCS method (Chakraborty et al., 2017)	0.0667757	0.544294	1.10477165	2.118018394
	SUFMACS (Proposed)	1.40051355	1.77481802	1.1897956	2.946617271
$\mathit{IM}{G}_{008}$	efficient GA (Kadri & Boctor, 2018)	0.90385027	0.8106123	0.46520982	1.777865309
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.27821603	0.94581004	1.48406266	0.982762622
	beam-ACO (Blum, 2005)	1.35869295	2.52499648	0.32488558	1.522170221
	MCS method (Chakraborty et al., 2017)	1.03533553	0.7524562	0.17791001	1.54185478
	SUFMACS (Proposed)	2.41381629	3.00720965	1.27233345	1.472802086
$\mathit{IM}{G}_{009}$	efficient GA (Kadri & Boctor, 2018)	1.71639361	1.03062709	1.4774164	2.885606995
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.09691081	2.51480927	1.71265403	0.694974333
	beam-ACO (Blum, 2005)	0.78235409	2.90553952	3.37411914	2.243581086
	MCS method (Chakraborty et al., 2017)	3.97876815	2.99970477	3.68375115	4.838316206
	SUFMACS (Proposed)	2.90993747	3.86981501	3.25331667	2.014079494
$\mathit{IM}{G}_{010}$	efficient GA (Kadri & Boctor, 2018)	1.926312	1.190401	1.249185	2.478174
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.368255	2.762778	1.742779	0.877326
	beam-ACO (Blum, 2005)	0.940972	2.585184	3.88169	2.447928
	MCS method (. Chakraborty et al., 2017)	3.67858	3.041665	3.455766	4.821794
	SUFMACS (Proposed)	2.757923	4.19606	3.021206	2.571978
$\mathit{IM}{G}_{011}$	efficient GA (Kadri & Boctor, 2018)	1.558866	1.881619	2.129268	3.298512
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.556064	1.868165	2.097383	0.373858
	beam-ACO (Blum, 2005)	1.054784	2.792089	2.999097	1.476081
	MCS method (Chakraborty et al., 2017)	4.300068	3.219115	3.912612	4.677951
	SUFMACS (Proposed)	2.611327	2.981608	4.039419	2.262326
$\mathit{IM}{G}_{012}$	efficient GA (Kadri & Boctor, 2018)	2.065147	0.671573	0.88079	2.762007
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.303688	2.063408	1.555371	1.019777
	beam-ACO (Blum, 2005)	1.414983	3.579358	2.781708	2.184043
	MCS method (Chakraborty et al., 2017)	3.441812	2.838719	3.806946	3.933333
	SUFMACS (Proposed)	2.26046	3.917694	2.95925	2.178155
$\mathit{IM}{G}_{013}$	efficient GA (Kadri & Boctor, 2018)	1.660949	1.13149	1.523319	2.864289
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.822297	3.019048	2.212642	1.363021
	beam-ACO (Blum, 2005)	0.503023	3.30804	3.018466	2.732243
	MCS method (Chakraborty et al., 2017)	4.504922	2.944836	2.882331	3.51782
	SUFMACS (Proposed)	3.054921	3.934504	3.278456	2.107407
$\mathit{IM}{G}_{014}$	efficient GA (Kadri & Boctor, 2018)	1.557998	0.508429	1.872015	3.298924
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.839968	2.434693	2.345428	2.00048
	beam-ACO (Blum, 2005)	1.16717	2.234873	2.407618	2.314919
	MCS method (Chakraborty et al., 2017)	3.481479	2.417493	3.578893	2.637046
	SUFMACS (Proposed)	3.118551	3.20922	3.259197	2.608554
$\mathit{IM}{G}_{015}$	efficient GA (Kadri & Boctor, 2018)	1.674839	1.358943	2.300374	3.103451
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.609361	2.953384	2.087327	2.684347
	beam-ACO (Blum, 2005)	1.438625	2.244327	3.583091	2.537237
	MCS method (Chakraborty et al., 2017)	3.555614	3.65412	4.067408	3.368569
	SUFMACS (Proposed)	2.715284	4.433092	3.225229	2.079237
$\mathit{IM}{G}_{016}$	efficient GA (Kadri & Boctor, 2018)	2.350359	1.791013	2.179383	3.918785
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.688224	2.779497	1.293425	2.659883
	beam-ACO (Blum, 2005)	1.129553	2.623741	2.863123	1.797293
	MCS method (Chakraborty et al., 2017)	3.765014	4.2684	3.96961	3.668268
	SUFMACS (Proposed)	2.099074	4.84109	3.491666	2.009523
$\mathit{IM}{G}_{017}$	efficient GA (Kadri & Boctor, 2018)	1.794146	1.470607	2.695213	2.93387
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.431808	2.47149	2.616826	2.550327
	beam-ACO (Blum, 2005)	1.407331	2.582904	4.303966	3.089016
	MCS method (Chakraborty et al., 2017)	2.761357	3.550739	3.58071	2.800829
	SUFMACS (Proposed)	1.908147	4.542297	3.19099	1.732512
$\mathit{IM}{G}_{018}$	efficient GA (Kadri & Boctor, 2018)	0.793841	1.782326	2.308078	2.473275
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.412433	3.396034	2.388098	3.452825
	beam-ACO (Blum, 2005)	1.924267	1.415964	3.368604	2.592132
	MCS method (Chakraborty et al., 2017)	3.384676	3.869108	3.522317	4.001938
	SUFMACS (Proposed)	2.944476	3.809717	3.298318	1.339301

Table 7.

Quantitative comparison of different algorithms using $\beta$ index (The acceptable values are highlighted in boldface).

Image Id	Algorithm	No. of Clusters
		3	5	7	9
$\mathit{IM}{G}_{001}$	efficient GA (Kadri & Boctor, 2018)	0.93047281	2.28130783	2.4200175	2.458787129
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.68038179	1.81298255	3.17014825	2.654608179
	beam-ACO (Blum, 2005)	1.23830871	1.16363667	1.15236996	2.574909802
	MCS method (Chakraborty et al., 2017)	3.30107739	2.54874472	1.69486857	1.968878803
	SUFMACS (Proposed)	1.27621712	1.46000578	3.52908003	2.402996019
$\mathit{IM}{G}_{002}$	efficient GA (Kadri & Boctor, 2018)	1.53831084	2.99243615	2.08288206	0.745554548
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.10440634	1.41830193	1.54199458	1.824417709
	beam-ACO (Blum, 2005)	0.33007752	3.04781307	1.31028981	2.755679078
	MCS method (Chakraborty et al., 2017)	2.06563862	2.84236865	1.83420856	2.613016749
	SUFMACS (Proposed)	1.64955203	1.62151639	1.90050555	2.981803457
$\mathit{IM}{G}_{003}$	efficient GA (Kadri & Boctor, 2018)	1.25031772	1.30396523	1.8269896	2.704643281
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.8536208	2.32136195	2.98659332	1.878205728
	beam-ACO (Blum, 2005)	1.53048244	1.03776429	2.12830465	1.896175157
	MCS method (Chakraborty et al., 2017)	1.64968215	0.86879211	1.66393498	2.112497471
	SUFMACS (Proposed)	1.09226951	2.929977	2.75557257	1.824431161
$\mathit{IM}{G}_{004}$	efficient GA (Kadri & Boctor, 2018)	0.26315552	1.7370115	1.99725146	1.721767327
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	1.29228784	2.57428543	3.07515894	2.036390346
	beam-ACO (Blum, 2005)	2.82046333	2.5030317	2.58139708	2.270031671
	MCS method (Chakraborty et al., 2017)	2.2281125	2.29012426	1.96696473	2.06763696
	SUFMACS (Proposed)	1.87899668	1.83913997	3.36728469	1.446875213
$\mathit{IM}{G}_{005}$	efficient GA (Kadri & Boctor, 2018)	0.55680493	1.43822333	1.59071162	1.685672746
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.5150899	2.75498041	3.80674117	1.673736021
	beam-ACO (Blum, 2005)	2.30763295	3.01075331	2.40020842	1.71718978
	MCS method (Chakraborty et al., 2017)	1.6444183	2.71621506	2.84243339	3.74244153
	SUFMACS (Proposed)	2.36682966	4.52618127	2.69561565	2.312132462
$\mathit{IM}{G}_{006}$	efficient GA (Kadri & Boctor, 2018)	1.57784584	2.4467198	1.47198831	2.055727007
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.47671534	2.23949143	1.59293845	2.904007455
	beam-ACO (Blum, 2005)	1.2529781	1.53131482	2.20179772	2.974517863
	MCS method (Chakraborty et al., 2017)	2.46337089	2.43404172	3.09973551	3.200142722
	SUFMACS (Proposed)	1.44010661	2.85070702	4.02102389	1.714519052
$\mathit{IM}{G}_{007}$	efficient GA (Kadri & Boctor, 2018)	1.0187963	0.55602669	2.13207745	2.416024694
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.24666115	2.66777331	4.01612988	3.48262208
	beam-ACO (Blum, 2005)	1.20205269	1.26356439	2.68895151	0.931324726
	MCS method (Chakraborty et al., 2017)	3.57290575	3.26613196	2.83808811	1.14310012
	SUFMACS (Proposed)	1.31994386	3.84587036	1.79917933	2.09707359
$\mathit{IM}{G}_{008}$	efficient GA (Kadri & Boctor, 2018)	2.51426903	2.32639584	2.46880708	1.457353634
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	0.10330027	1.20300968	1.83654505	1.235057549
	beam-ACO (Blum, 2005)	1.17425724	2.73234865	2.77489363	2.805632346
	MCS method (Chakraborty et al., 2017)	2.43072508	3.6814858	3.09909169	3.001298406
	SUFMACS (Proposed)	2.02132665	4.43297205	1.86031772	2.472550254
$\mathit{IM}{G}_{009}$	efficient GA (Kadri & Boctor, 2018)	0.93517503	2.02987876	1.95500223	3.343449018
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.68674784	1.10847743	1.93992629	1.579812318
	beam-ACO (Blum, 2005)	2.74726358	2.82358208	3.47480738	1.842501754
	MCS method (Chakraborty et al., 2017)	2.24069175	2.44436824	2.85660783	2.872937964
	SUFMACS (Proposed)	1.5750965	2.52240892	3.86978531	3.17286345
$\mathit{IM}{G}_{010}$	efficient GA (Kadri & Boctor, 2018)	0.967114	1.582421	1.918011	3.68922
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.727972	1.336132	2.080632	1.664572
	beam-ACO (Blum, 2005)	3.057892	3.080276	3.976019	1.645924
	MCS method (Chakraborty et al., 2017)	2.055247	2.315963	2.827663	2.795451
	SUFMACS (Proposed)	2.130795	2.609342	3.645853	2.919436
$\mathit{IM}{G}_{011}$	efficient GA (Kadri & Boctor, 2018)	1.295187	1.689744	1.83195	3.23864
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.788331	1.703366	2.264678	1.735064
	beam-ACO (Blum, 2005)	2.630898	2.397755	4.131748	1.588874
	MCS method (Chakraborty et al., 2017)	2.420508	2.185359	2.08597	3.23354
	SUFMACS (Proposed)	1.096166	3.088883	3.706678	2.220064
$\mathit{IM}{G}_{012}$	efficient GA (Kadri & Boctor, 2018)	1.283903	2.057142	2.049281	3.077072
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.019856	1.136661	1.913912	0.737852
	beam-ACO (Blum, 2005)	2.356268	3.216396	3.138161	1.122671
	MCS method (Chakraborty et al., 2017)	2.024194	2.942996	3.135802	2.76436
	SUFMACS (Proposed)	1.656903	1.972191	3.693269	2.713049
$\mathit{IM}{G}_{013}$	efficient GA (Kadri & Boctor, 2018)	1.336807	1.968276	1.989995	2.608285
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.478075	1.314553	1.592016	2.172581
	beam-ACO (Blum, 2005)	3.128278	2.658721	3.487506	1.478773
	MCS method (Chakraborty et al., 2017)	1.985579	2.850567	3.042927	3.20563
	SUFMACS (Proposed)	1.602228	2.330009	3.402566	3.736042
$\mathit{IM}{G}_{014}$	efficient GA (Kadri & Boctor, 2018)	1.173781	2.229015	2.025946	3.540327
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.930074	1.084106	2.15638	1.72502
	beam-ACO (Blum, 2005)	2.297096	3.274123	3.464483	2.216525
	MCS method (Chakraborty et al., 2017)	2.849159	2.362549	2.098505	2.831332
	SUFMACS (Proposed)	1.325939	3.191802	4.203894	3.482025
$\mathit{IM}{G}_{015}$	efficient GA (Kadri & Boctor, 2018)	0.499157	2.181652	1.820285	3.523981
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.954136	1.601824	2.322025	1.529449
	beam-ACO (Blum, 2005)	2.615026	3.240616	3.12386	1.610758
	MCS method (Chakraborty et al., 2017)	1.627528	2.136822	2.666278	3.305198
	SUFMACS (Proposed)	1.673612	2.907563	3.935184	3.076986
$\mathit{IM}{G}_{016}$	efficient GA (Kadri & Boctor, 2018)	0.090813	1.781247	1.432998	3.363032
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.022681	0.96996	2.440515	1.334247
	beam-ACO (Blum, 2005)	3.202519	2.753586	3.441416	1.41356
	MCS method (Chakraborty et al., 2017)	1.858133	2.079266	2.188201	3.566132
	SUFMACS (Proposed)	2.283997	3.374073	4.001367	3.362313
$\mathit{IM}{G}_{017}$	efficient GA (Kadri & Boctor, 2018)	1.267164	3.052179	1.103045	3.169363
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	3.24286	1.351714	1.451006	0.997094
	beam-ACO (Blum, 2005)	2.078783	3.430719	2.533635	1.825645
	MCS method (Chakraborty et al., 2017)	1.314205	1.942686	2.959747	3.358512
	SUFMACS (Proposed)	2.414454	2.592589	3.916247	2.5707
$\mathit{IM}{G}_{018}$	efficient GA (Kadri & Boctor, 2018)	0.61265	1.966463	1.925426	3.931988
	adaptive PSO (Taherkhani & Safabakhsh, 2016)	2.608791	1.707394	2.006927	1.571506
	beam-ACO (Blum, 2005)	2.496877	2.565924	2.614735	1.759885
	MCS method (Chakraborty et al., 2017)	1.383944	2.154332	2.313664	2.38145
	SUFMACS (Proposed)	1.878646	3.361145	3.257262	4.021743

In Table 8 , a comparative overview of the running times of different algorithms corresponding to the different number of clusters and different validity indices.

Table 8.

Comparison of the running time (in seconds) concerning $\mathit{IM}{G}_{002}$

Validity Index	No. of clusters	efficient GA (Kadri & Boctor, 2018)	adaptive PSO (Taherkhani & Safabakhsh, 2016)	beam-ACO (Blum, 2005)	MCS method (Chakraborty et al., 2017)	SUFMACS (Proposed)
Davies–Bouldin	3	5.02362	6.15033	5.96364	6.003692	8.025365
	5	5.20053773	8.059774971	7.283463635	6.155470007	9.211810612
	7	8.687193939	9.41009445	8.319782859	9.231996594	9.94167873
	9	10.496845003	12.128954877	12.905376714	11.226114235	10.62646407
Xie-Beni	3	5.386404149	7.84973186	7.676206235	6.918436579	8.275284827
	5	7.706106824	6.082308305	9.771738778	8.191909494	8.878155935
	7	9.635925316	8.871990636	10.244350443	11.19679946	10.89713627
	9	14.86361925	11.18578921	11.46178597	13.51716117	12.90402758
Dunn	3	4.728459412	5.146740405	5.730107997	6.343859862	7.12403458
	5	5.381881867	5.424536622	6.360624788	7.310659158	8.119558804
	7	11.719787099	12.295617369	11.507700579	12.753516849	9.14274987
	9	15.53638599	14.82501584	13.57978028	15.65883255	12.81412843
$\beta$	3	6.132593592	5.241800821	5.952910422	6.328640419	7.204568618
	5	8.749542248	9.599830664	8.189885756	7.164894266	7.626018985
	7	13.425369045	15.06851261	16.53562997	10.77168572	11.61072519
	9	14.59800255	15.39611637	15.24954022	13.08306328	12.72053915

As discussed earlier, one of the major advantages of the proposed approach is that the proposed approach exploits the advantages of superpixels that reduce the huge amount of spatial information. The fuzzy objective function is also modified in such a way so that the advantages of the superpixels can be completely exploited. Moreover, both global and local optimization approaches are hybridized in such a way so that the overall search space can be explored effectively and reflects in the performance of the proposed system. From the above qualitative and quantitative analysis, it can be easily observed that the proposed approach outperforms the other competitive standard approaches in most of the cases.

7.4. Analysis of the convergence rate

The convergence of any proposed technique is an important topic to be discussed. It is also essential to compare any proposed technique with some other standard methods. Therefore, in this subsection, the study of the convergence is discussed in this subsection and also compared with other methods for different number of clusters. $\mathit{IM}{G}_{009}$ is used for this analysis. The Xie-Beni index is considered in the y-axis and the number of iterations is considered in the x-axis. The graphical representation of the convergence is given in Fig. 11 .

Fig. 11.

The convergence curves for different number of clusterThe Xie-Beni index for the image $\mathit{IM}{G}_{001}$ is plotted in the Y-axis and the number of iterations is plotted in the X-axis of each curve. The curves are generated by applying the (a) efficient genetic algorithm, (b) adaptive PSO, (c) Beam-ACO, (d) MCS, (e) SUFMACS (proposed).

One important point that can be observed from the above curves that the proposed SUFMACS approach outperforms most of the standard approaches in the majority of the cases. Moreover, the proposed approach performs well while dealing with a higher number of clusters. It can be observed that the proposed approach converges faster compared to the other standard optimization approaches and this property can help apply the proposed approach in several real-life scenarios.

8. Conclusion

In this work, a novel framework is proposed to segment the radiological images. This method is tested using the radiological images which are collected from the COVID-19 positive patients from different geographic regions. The proposed approach is designed to help the physicians and act as a helping hand to the physicians and other domain experts. This approach is targeted to deploy at the hospitals and other healthcare facility centers so that they can quickly screen the suspected patients. So, both COVID-19 positive and negative patients can be examined with the help of the proposed approach. The segmented images are helpful in the diagnosing process and can act as the third eye to the physicians. The segmentation procedure cannot detect or isolate the virus from the radiological images (maybe some microscopic images can serve this purpose). However, the proposed approach can help domain experts in easy interpretation and diagnosing the presence of the COVID-19 virus by highlighting different regions of the chest CT scans or X-ray images. The proposed method may work on different modalities of the medical images as well as on other images too. However, the current work focuses on radiological images (i.e., CT images, and X-Ray images) only. This approach is based on the superpixel and memetic advanced fuzzy cuckoo search. From the detailed analysis and comparison, it can be observed that the proposed method can outperform many standard algorithms and can produce clear and well-segmented images that can help the physicians to study the radiological images. This method can help in automated screening the COVID-19 patients without collecting any sample from the patient’s body and without directly interacting with the patients which will be beneficial to combat the spread of the COVID-19 disease. Along with the proposed SUFMACS method, general human intelligence can also be incorporated to make the diagnosis process efficient and to improve the radiological practices for the benefit of society. In our future works, we shall try to eradicate the dependency on the number of clusters (which is to be supplied externally in this case) and try to develop a model that can estimate the number of clusters automatically. Our approach does not address this issue in its present form and supplies the number of clusters externally. It is interesting to determine the number of clusters automatically from a given image. In the future, the proposed approach can be modified further so that it can be capable to compute the number of clusters from the input image. A summary of the used symbols is provided (see Appendinx A) in Table A1.

CRediT authorship contribution statement

Shouvik Chakraborty: Conceptualization, Investigation, Formal Analysis, Methodology, Writing - original draft, Writing - review & editing, Software. Kalyani Mali: Data curation, Supervision, Validation, Project administration, Resources.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A.

Table A1.

Summary of the Symbols used in this manuscript.

Symbol	Description	Remarks
$\rho \left(x\right)$	Probability distribution function which is followed by the Lévy flight to generate the leaps.
$\theta$	The stability index which is also known as the Lévy index of the probability distribution function.	$0<\theta <2$
$d_{\mathit{frac}}$	Fractal dimension of the trajectory of the Lévy flight	$d_{\mathit{frac}}=\theta$
$\gamma$	Skewness controlling parameter	The range is [−1,+1]. Please refer Eq. (2).
$\upsilon$	The shift property	Please refer Eq. (2).
$\varphi$	The scale property	$\varphi >0$. Please refer Eq. (2).
$\kappa$	The domain of the probability density function of the Lévy distribution	$\kappa ⩾\upsilon$
$\mathit{pro}{b}_{\epsilon }$	Probability of exploring the eggs of the parasite species	$\mathit{pro}{b}_{\epsilon }\in [0,1]$
$w_{i,j}$	Population matrix	Please refer Eq. (5).
$w_j^{\mathit{high}}$	The dimensional upper bound	Please refer Eq. (5).
$w_j^{\mathit{high}}$	The dimensional upper bound	Please refer Eq. (5).
$\mathit{nNest}$	Denotes the total number of nests
$\mathit{nParam}$	Represents the total number of optimization parameters
$s_z$	Step size of the Lévy flight	$s_z>0$. Please refer Eqs. (6), (15).
$g$	the generation or the present iteration is indicated using.
$p_1,p_2,p_3,p_4$	Four controlling parameters involved in in the generation of the random number using Lévy flight. The iteration count and the count of the random points are represented by $p_3$, and $p_4$ respectively.	$0.3⩽p_1⩽1.99$ $p_2$ is a constant and $p_2>0$
$\alpha$ and $\beta$	Normally distributed stochastic parameters and the accurate value of these	the value of these parameters cannot be accurately computed but, possible to analyse statistically $\left(\alpha ~p_4\left(0,{\sigma }_{\alpha }^2\right),\beta ~p_4\left(0,{\sigma }_{\beta }^2\right)\right)$.
$\zeta$	Lévy distribution can be achieved depending on the $\zeta$ provided that $\zeta \gg 0$	Please refer Eq. (9).
$\vartheta$	The scaling attribute	Please refer Eq. (13).
$exp$	The exponent	Controlling parameters to adapt McCulloch’s approach using Chamber’s method and these parameters are responsible to generate a matrix of random numbers of dimension $d_1\times d_2$.
$\tau$	The scaling parameter
$\omega$	A skewness controlling parameter
$\stackrel{⌢}{\kappa }$	denote the location
$\mathit{Ob}{j}_{\phi }$	Fuzzy objective function	Please refer Eq. (23).
$\mathit{nP}$	Number of data points	Please refer Eq. (23).
$\mathit{nC}$	Number of cluster centers	Please refer Eq. (23).
$\phi$	Represents the fuzzifier	Please refer Eq. (23).
${\mu }_{\mathit{ij}}$	Represents the membership value of the point $x_i$ to the $j^{\mathit{th}}$ cluster.	Please refer Eq. (24).
$\epsilon$	Multiplicative factor to reduce the $\mathit{samplingRange}$ in the Luus–Jaakola method.	$\epsilon =0.95$
$\psi$	Erosion operation	Please refer Eq. (28).
$\zeta$	Dilation operation	Please refer Eq. (29).
${\mathrm{\Lambda }}^{\psi }$	Reconstruction process based on erosion operation	Please refer Eq. (28).
${\mathrm{\Lambda }}^{\zeta }$	Reconstruction process based on dilation operation	Please refer Eq. (29).
$o$	Morphological opening operation	Please refer Eq. (32).
$\varsigma$	Morphological closing operation	Please refer Eq. (33).
${\vartheta }_l$ and ${\vartheta }_h$	Represents the lowest and the highest value of the controlling parameter of the structuring elements	$\left[{\vartheta }_l,{\vartheta }_h\right]\in {\mathbb{N}}^{+}$. Please refer Eq. (36).
$\mathit{nPi}{x}_k$	represents the number of pixels in the $k^{\mathit{th}}$ region	Please refer Eq. (38).
$R_k$	The $k^{\mathit{th}}$ region	Please refer Eq. (38).
$p{x}_q$	A pixel in the $k^{\mathit{th}}$ region	Please refer Eq. (38).
${\widehat{R}}_k$	Representative value of the $k^{\mathit{th}}$ superpixel.	Please refer Eq. (38).
$\tilde{\mu }$	Type-2 fuzzy membership function	Please refer Eq. (41).
${\gamma }_{min}$ and ${\gamma }_{max}$	The highest and the lowest value in the intensity range of the superpixel image.	Please refer Eq. (42)
${\widehat{C}}_j$	The $j^{\mathit{th}}$ cluster center of the superpixel image	Please refer Eq. (42).
$\mathit{DBIndex}$	Davies–Bouldin index
$\mathit{XBIndex}$	Xie-Beni index
$D{I}_n$	Dunn index
$\beta$ index	$\beta$ index