# Performance Comparison of Most Recently Proposed Evolutionary, Swarm Intelligence, and Physics-Based Metaheuristic Algorithms for Retinal Vessel Segmentation

Authors: Mehmet Bahadır Çetinkaya; Hakan Duran
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Biomedical image analysis based on metaheuristic algorithms is one of the most important research areas encountered in recent years. Due to the low contrast differences between the diseased areas and the image background in high-contrast biomedical images, effective methods are required to diagnose diseases with high accuracy. To overcome the difficulties encountered in this field, metaheuristic approaches may offer effective solutions due to their advantages such as the ability of converging to the global optimum, higher convergence rate, and having few control parameters. In this work, Jellyfish Search (JS), Marine Predators (MPA), Tunicate Swarm (TSA), Mayfly Optimization (MA), Chimp Optimization (ChOA), Slime Mould Optimization (SMA), Archimedes Optimization (AOA), and Equilibrium Optimizer (EO) algorithms, which are the most recently proposed metaheuristic algorithms in the literature, have been improved as clustering based in order to achieve vessel segmentation with high precision. Also, a detailed performance comparison of these algorithms has been realized for the rate of convergences, error values reached, CPU time, standard deviation, sensitivity, specificity, accuracy,F-score, and Wilcoxon rank sum-test. In order to present the compatibility of the results obtained with the literature, the performances of these novel algorithms have also been compared to that of Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), and Differential Evolution (DE) algorithms. The simulation results represent that each algorithm produces similar convergence and error performance. Also, it can be emphasized from the statistical analyses that the stability and robustness of each metaheuristic approach are quite adequate in separating the vessel pixels and the background pixels of a retinal image. In general, this paper proves that although having fewer number of control parameters, the JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms produce similar but a bit better results in terms of image segmentation when compared to PSO, GWO, and DE algorithms.


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## Body

## 1. Introduction

Although the conventional algorithms have frequently been used in literature for retinal image analysis, there are limited number of works based on heuristic and metaheuristic algorithms relevant to this area. In addition, to the best of our knowledge, the most recent algorithms analyzed in this work have not been used yet in biomedical image processing although they have been employed for a limited number of areas such as improvement of metaheuristic methods for reducing energy consumption (for JS algorithm) [1], developing metaheuristic based forecasting algorithms for pandemic diseases (for MPA algorithm) [2], improving artificial intelligence based approaches for optimization of distortions occurring in heat and power systems (for MPA algorithm) [3, 4], comparing the performances of PSO-type optimization algorithms (for MPA and SMA algorithms) [5], development of artificial intelligence based cyber security systems (for TSA algorithm) [6], developing a chaos mechanism based whale optimization algorithm (for TSA algorithm) [7], developing an emperor penguin optimizer for efficient ranking of cloud service providers (for TSA algorithm) [8], developing metaheuristic approaches in order to increase the efficiency of distribution system (for MA algorithm) [9], improving a Chebyshev based mayfly optimizer (for MA algorithm) [10], improving a metaheuristic optimizer modeling hierarchical structures (for ChOA algorithm) [11], developing advanced computational methods for agriculture machinery movement optimization (for ChOA algorithm) [12], development of metaheuristic based approaches in order to enhance the data path structures in data transfer applications (for ChOA algorithm) [13], improvement of optimum modeling approaches for solar panels (for SMA algorithm) [14], improving optimization algorithms for image segmentation problem encountered in chest X-ray images (for SMA algorithm) [15], in a detailed review work for monarch butterfly optimization (for SMA algorithm) [16], improving a golden eagle optimizer (for EO algorithm) [17], improving a multiobjective optimization algorithm using improvement based reference points approach (for EO algorithm) [18], and improving an artificial intelligence based optimizer for parameter extraction of a fuel cell dynamic model [19]. Finally, in the detailed literature review, it has been observed that the AOA algorithm has not been used yet for any application area.In order to diagnose and treat the important symptoms of retinal diseases, the retinal vessel segmentation is one of the most important areas of biomedical image processing [20]. Due to the difficulties and insufficiencies encountered in conventional methods, computer-aided diagnosis systems for ophthalmic disorders are developed [21]. Computer aided segmentation of retinal images provide higher accuracy in retinal image analysis, and so, more effective treatment methods are able to be developed for the abnormalities emerging as a result of obesity [22], hypertension [23], glaucoma [24], and diabetic retinopathy [25–27]. Computer based approaches also enable highly accurate diagnostics for applications such as prematurity retinopathy [28], determination of vessel sizes [29], regional analysis on the retinal image [30], arteriolar stenosis [31], surgical procedures [32], improving treatment methods for retinal diseases [33–35] and optic disk detection [36], and retinal image analyses by using the well-known swarm and population based approaches [37].In addition to the literature analyses given above, there are also several works in the literature about the application of metaheuristic algorithms to image segmentation. In these works, it is emphasized that image segmentation based on metaheuristic algorithms has the potential of obtaining near optimum solutions. In [38], a novel algorithm called self-adaptive moth-flame optimization TH (SAMFO-TH) has been improved for the aim of multilevel thresholding in color image segmentation, and its performance has been compared with other well-known eight metaheuristic algorithms in the literature. Xu and friends have introduced a novel algorithm that is called improved dragonfly algorithm (IDA) in [39] for color image segmentation and then compared its performance with that of differential evolution (DE) and dragonfly algorithms. In [40], a multilevel thresholding method for color image segmentation exploiting 1D OTSU cuckoo search (CS), 1D OTSU lightning search (LSA), and cuttlefish (CFA) algorithms has been proposed by Bhandari and friends. The work presented in [41] describes a novel bee foraging algorithm (BFA) based multilevel thresholding method for image segmentation. In [42], the task of designing an efficient methodology based on Harris Hawks Optimization (HHO) algorithm for multilevel image segmentation has been investigated. In [43], a detailed performance analysis between widely used evolutionary and swarm based optimization algorithms has been realized for multilevel color image thresholding. In work [44], a reliable retinal blood vessel segmentation approach consisting of a cascade connection of edge detection and shape analysis methods has been improved by the authors. In another work [45], a benchmark of thirteen metaheuristic optimization algorithms for image thresholding have been performed in high-dimensional search spaces. The authors have proposed a new hybrid algorithm called K-means Firefly algorithm (KFA) and then analyzed its performance in image segmentation in [46]. In [47], a water cycle algorithm (WCA) based efficient method has been proposed by Kandhway and Bhandari for multilevel thresholding in color image segmentation. Finally, an effective image clustering method based on human mental search (HMS) algorithm has been improved by Mousavirad et al. [48].The segmentation performances of the Jellyfish Search (JS), Marine Predators (MPA), Tunicate Swarm (TSA), Mayfly Optimization (MA), Chimp Optimization (ChOA), Slime Mould Optimization (SMA), Archimedes Optimization (AOA), and Equilibrium Optimizer (EO) algorithms improved in this work have been analyzed in detail for both healthy and diseased retinal images taken from the DRIVE and STARE databases. In Figures1 and 2, the images for each of the databases have been given, respectively.Figure 1
Healthy and diseased images taken from DRIVE database. (a) Healthy image. (b) Diseased image with bleeding.
(a)(b)Figure 2
Healthy and diseased images taken from STARE database. (a) Healthy image. (b) Diseased image with bleeding.
(a)(b)In order to distinguish the vessel and the background pixels with high accuracy, the retinal image should be enhanced before clustering by applying the band selection, bottom-hat transformation, and brightness correction preprocessing.When the red (R), green (G), and blue (B) layers of RGB retinal images have been analyzed separately in terms of illuminance, contrast, and brightness levels, it has been seen that the G layer provides the highest clustering performance due to its higher illuminance, optimal contrast, and optimal brightness levels.The green layers obtained after band selection preprocessing of the retinal images taken from the DRIVE database have been given in Figures3(a) and 3(c). The retinal images enhanced in terms of contrast difference by applying the bottom-hat transformation and brightness correction have also been shown in Figures 3(b) and 3(d).Figure 3
The resulting DRIVE database images after each preprocessing phase. (a, c) G layers obtained after band selection; (b, d) The resulting images after the bottom-hat filtering and brightness correction.
(a)(b)(c)(d)The green layers of the retinal images taken from the STARE database have been obtained as shown in Figures4(a) and 4(c). Also, the retinal images enhanced in terms of contrast difference obtained after the two subsequent preprocessing operations have been given in Figures 4(b) and 4(d).Figure 4
The resulting STARE database images after each preprocessing phase. (a, c) G layers obtained after band selection; (b, d) The resulting images after the bottom-hat filtering and brightness correction.
(a)(b)(c)(d)The main contributions of this paper to literature can be summarized as follows:(i)
The most recently proposed JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms have been improved as clustering based and then applied to retinal vessel segmentation for the first time in literature.(ii)
The performances of the improved algorithms have been compared to those of the well-known PSO, GWO, and DE algorithms in the literature.(iii)
The segmentation results obtained demonstrate that these algorithms produce similar or a bit better results than PSO, GWO, and DE, and they can successfully be used in segmentation of retinal images with high accuracy.(iv)
The statistical analyses demonstrate that these algorithms produce stable and reliable results despite their noncomplex algorithm structures.The rest of this paper is organized as follows. Section2 presents brief reviews about both the most recently proposed metaheuristic algorithms used in this work and the properties of the retinal images used. In Section 3, the simulation and the statistical analysis results obtained for DRIVE and STARE databases are given in detail and then compared with each other and literature. The relevant discussions are presented in Section 4. Finally, conclusions and future directions are given in Sections 5.


## 2. Material and Methods

Nature-inspired metaheuristic algorithms, which simulate biological or physical phenomena to solve optimization problems, can be classified into 4 major groups as evolutionary based, swarm intelligence based, physics based, and human based.In this work, the most novel evolutionary based, swarm intelligence based, and physics based metaheuristic algorithms in the literature have been applied to biomedical image analysis, and then their performances have been compared in detail.


### 2.1. Jellyfish Search Algorithm

Jellyfish search algorithm, which is proposed by Chou and Truong in 2020 [49], is a swarm based optimizer, which simulates the search behavior of jellyfishes. One of the most important advantages of JS algorithm is that it contains only two control parameters, namely, population size and cycle number. It is inspired from 3 main rules; (i) a jellyfish performs two types of movement, which can be defined as “moving through the ocean current” or “moving along with a swarm,” and the transition process between these movements is coordinated by a time control mechanism, (ii) while searching for food a jellyfish moves towards the locations having more amount of food, and (iii) the quality of a food source is determined by its location and its corresponding objective function.The detailed pseudocode of a bioinspired swarm intelligence based JS algorithm has been given as follows:Randomly create an initial population,xi∈1,2,…,NPOPCalculate the fitness value of eachxi solution in the population.Cycle = 1REPEATCalculate the value of time control parameter by using the following equation:(1)ct=1−1Max Cycle Number2.rand0,1−1.IFct≥0,5:follow ocean current(i)
Determine ocean current by(2)trend⟶=X∗−β.rand0,1.μ,whereX∗ represents the jellyfish currently with the best location in the swarm, μ is the mean location of all jellyfish and β>0 is determined as the distribution coefficient which is related to the length of trend⟶.(ii)
Determine a new location by(3)Xit+1=Xit+rand0,1.trend⟶.ELSE: jelly fish moves inside a swarmIfrand0,1>1−ct define a new position,(4)Xit+1=Xit+γ.rand0,1.Ub−Lb,where;Ub and Lb represent upper and lower bounds of search space, respectively and γ>0 is the motion coefficient.else: determine the direction d⟶ and new location of ith and jth jellyfishes by,(5)d⟶=Xjt−XitiffXi≥fXj,Xit−XjtiffXi<fXj,Xit+1=Xit+rand0,1.d⟶.end ifEND IFCheck boundary conditions and calculate the fitness value of food source at new locationUpdate the location ofXi and location of X∗Cycle = Cycle + 1UNTİL (termination criteria are met)


### 2.2. Marine Predators Algorithm

Marine Predators algorithm, which is proposed by Faramarzi et al. in 2020 [50], is a nature-inspired swarm intelligence based metaheuristic algorithm, which simulates foraging behavior of ocean predators by taking into account the relationship between the prey and predator. The most important metric of this algorithm is to optimize the search strategies used by the predators which is also called random walk strategy. The algorithm focuses on two main random walk strategies, namely, Brownian motion and Lévy motion.A detailed pseudocode of a nature-inspired swarm intelligence based MPA algorithm can be given as follows:Randomly create a uniformly distributed prey population,i=1,2,…,nCycle = 1WHILECycle≤Maximum CycleCalculate the fitness values and construct the Elite matrix which includes the best solutions (best positions of preys)(6)Elite=XI1,1,…,XI1,d,⋮,⋱,⋮XIn,1,⋯,XIn,d..IFCycle<Maximum Cycle Number/3:highvelocity ratioUpdate prey bypreyi⟶=preyi⟶+P.P⟶⊗stepsizei⟶where,stepsizei⟶=RB⟶⊗Elitei⟶−RB⟶⊗stepsizei⟶ is the velocity ratio for phase 1 in which the predator is moving faster than the prey; ⊗ notation represents the entry-wise multiplications; the multiplication of RB by prey simulates the movement of prey; P is a constant number which is equal to 0, 5 and finally, R is a vector of uniform random numbers in [0, 1]ELSE IF: unit velocity ratio(7)Maximum Cycle Number3<Cycle<2.Maximum Cycle Number3.For the first half of the populationi=1,2,…,n/2 update prey based on,(8)preyi⟶=preyi⟶+P.R⟶⊗stepsizei⟶,where,stepsizei⟶=RL⟶⊗Elitei⟶−RL⟶⊗preyi⟶ is the velocity ratio for phase 2 in which both predator and prey are moving at the same pace; RL⟶ represents a vector of random numbers produced from Lévy distribution;For the second half of the populationi=n/2+1,…,n update prey based on(9)preyi⟶=Elitei⟶+P.CF⊗stepsizei⟶,where,stepsizei⟶=RB⟶⊗RB⟶⊗Elitei⟶−preyi⟶ is the velocity ratio for phase 2 in which both predator and prey are moving at the same pace; the multiplication of RB and Elite simulates the movement of predator in Brownian manner and finally, CF=1−Cycle/Max Cycle2.Cycle/Max Cycle is an adaptive control parameter.ELSE IFCycle>2.Maximum Cycle Number/3:low velocity ratioUpdate prey,preyi⟶=Elitei⟶+P.CF⊗stepsizei⟶where,stepsizei⟶=RL⟶⊗RL⟶⊗Elitei⟶−preyi⟶ is the velocity ratio for phase 3 in which the predator is moving faster than the prey; the multiplication of RL⟶ and Elite simulates the movement of predator in Brownian mannerEND IFMemorize the best solutions so far and updateElite matrix Apply Fish Aggregating Devices (FADs) effect given below for r ≤ FADs and r > FADs, respectively, in order to prevent the algorithm to get stuck into the local minima.(10)preyi⟶=preyi⟶+CF.Xmin⟶+R⊗Xmax⟶−Xmin⟶⊗U⟶,preyi⟶+FADs.1−r+r.preyr1⟶−preyr2⟶,where,FADs = 0.2 is the probability of FADs effect on the optimization process; U⟶ is a binary vector constructed by generating a random vector in [0,1]; Xmin⟶ and Xmax⟶ are the vectors containing the lower and upper bounds of the dimensions; the r1 and r2 represent random indexes of prey matrix.END WHILE


### 2.3. Tunicate Swarm Algorithm

Tunicate Swarm algorithm is a bioinspired metaheuristic optimization algorithm inspired from the swarm behaviors of tunicates during the foraging process. It was proposed by Kaur et al. in 2020 [51]. TSA algorithm has been constructed mathematically on two main behaviors of tunicates that are jet propulsion and swarm intelligence.A detailed pseudocode for a bioinspired swarm intelligence based TSA algorithm can be given as follows:Initialize the tunicate population,Ppx⟶Calculate the fitness value of each search agent in the initial populationCycle = 1REPEATDetermine the jet propulsion and swarm behaviors of tunicates by,(i)
Jet Propulsion Behavior(11)PD⟶=FS⟶−rand0,1.Ppx⟶,where;FS⟶ represents the fitness value vector of search agents;(ii)
Swarm Behavior(12)Ppx⟶=FS⟶+A⟶.PD⟶ifrand0,1≥0.5,FS⟶−A⟶.PD⟶ifrand0,1<0.5,where,A⟶ vector includes the variations to avoid conflicts among search agents.Update the position of each search agent by using(13)Ppx+1⟶=Ppx⟶+Ppx+1⟶2+c1,where,c1 is a randomly produced number in [0, 1].Calculate the fitness value for updated search agentsEliminate the search agents placing out of the search spaceCycle = Cycle + 1UNTİL(termination criteria are met)


### 2.4. Mayfly Optimization Algorithm

The basic philosophy of Mayfly optimization algorithm is to simulate the flight behavior and the mating process of mayflies. Mayfly is a hybrid metaheuristic algorithm combining the major advantages of swarm intelligence and evolutionary algorithms. It was proposed by Zervoudakis and Tsafarakis in 2020 [52] as an effective swarm intelligence based metaheuristic optimization algorithm in terms of convergence rate.The detailed pseudocode with mathematical expressions for a hybrid MA can be given as follows:Initialize the male mayfly population,xi∈1,2,…,N and velocities vmiInitialize the female mayfly population,yi∈1,2,…,N and velocities vfiCalculate the fitness value of each possible solution in the initial populationCycle = 1REPEATUpdate velocities and solutions of males and females by using,(i)
Male Mayflies(14)vijt+1=vijt+α1e−β.rp2pbestijxijt+α2.e−β.rg2gbestij−xijt,where,vijt represents the velocity of mayfly i in dimension j = 1, …, n at time step t; xijt is the position of mayfly i in dimension j at time step t;a1>0 and a2>0 are the attraction constants; pbesti is the best position of mayfly i so far; gbest represents the global best position so far; finally, β is a fixed visibility coefficient used to limit a mayfly’s visibility to others while rp is the Cartesian distance between xi and, and rg is the Cartesian distance between xi and gbest.(ii)
Female Mayflies(15)vijt+1=vijt+α2e−β.rmf2xijt−yijtiffyi>fxi,vijt+fl.riffyi≤fxi,where;yijt is the position of female mayfly i in dimension j at time step t; f l is a random walk coefficient, used when a female is not attracted by a male, so it flies randomly and r is a random value in the interval of [−1, 1]; finally, rmf represents the Cartesian distance between male and female.Calculate the fitness value of each new candidate solutionRank the mayfliesMate the mayflies by applying crossover operation as mentioned below,the crossover operation is realized mutually by using the following equations between the candidates selected from male and female populations.(16)offspring1=L.male+1−L.female,offspring2=L.female+1−L.male,where,male and female represent the male and female parents, respectively; L is a random value within a specific interval.Calculate the fitness value of each offspringSeparate offsprings to male and female populations randomlyReplace worst solutions with the new better onesUpdatepbest and gbestCycle = Cycle + 1UNTİL(termination criteria are met)


### 2.5. Chimp Optimization Algorithm

The Chimp optimization algorithm inspired by the individual intelligence and sexual motivation of chimps in their group hunting has been proposed by Khishe and Mosavi in 2020 [53]. The mathematical model of ChOA algorithm has been built on two main phases. While the intelligence phase includes four types of chimps entitled attacker, barrier, chaser, and driver, the hunting phase has been modelled by considering driving, chasing, blocking, and attacking behaviors of chimps.The detailed pseudocode of a nature-inspired swarm intelligence based ChOA algorithm can be given as follows:Initialize the chimp population,xi∈1,2,…,NCalculate the position of each chimp for driving by,(17)d=c.Xpreyt−m.Xchimpt.Calculate the position of each chimp for chasing by,(18)Xchimpt+1=Xpreyt−a.d,wherea, m and c are the coefficient vectors; Xchimp represents the position vector of a chimp while Xprey is the vector of prey position; also for the descriptions of a=2.f.r1−f and c=2.r2; f represents reduced nonlinearly from 2.5 to 0 through the cycles process, r1 and r2 are the random vectors in [0, 1]; finally, m represents a chaotic vector simulating the sexual motivation of chimps in the hunting phase.Divide chimps randomly into four independent groups as,XAttacker: the best search agentXChaser: the second best search agentXBarrier: the third best search agentXDriver: the fourth best search agentCycle = 1REPEATCalculate the fitness value of each chimpWHILE (Cycle < Maximum Cycles).For each chimp:Identify chimp groupsUse the group strategy to updatef, m and cUsef, m and c to calculate the new a and dEnd Forfor each search chimp:if (μ<0.5), μ is a randomly produced number in [0, 1]if (a<1)Update the position of the current search agent by using,Xchimpt+1=Xpreyt−a.delse if ( a>1 ).Select a random search agentend ifelse if ( μ>0.5),Update the position of the current search agent by using(19)Xchimpt+1,Xpreyt−a.difμ<0.5,mchaotic valueifμ≥0.5.end ifend forUpdatef, m, a and cUpdateXAttacker, XChaser, XBarrier and XDriverEND WHILECycle = Cycle + 1UNTIL (termination criteria are met).


### 2.6. Slime Mould Algorithm

The Slime Mould algorithm used in this work is a biological-evolution inspired algorithm, which simulates the diffusion and foraging behavior of slime mould. It was introduced by Li et al. in 2020 [54] as an effective physics-based metaheuristic algorithm with the detailed pseudocode given below:Initialize the positions of slime mould,Xi∈1,2,…,nCycle = 1WHILECycle≤Maximum CycleCalculate the fitness value of all slime mouldDetermine thebestFitness, XbCalculate the weight of slime mould (W) by using(20)WSmellIndext⟶=1+r.logbF−SibF−wF+1,condition,1−r.logbF−SibF−wF+1,others,where,r denotes a randomly produced value in the interval of [0,1]; bF represents the optimal fitness value obtained in the current cycle; wF is the worst fitness value obtained in the current cycle; S(i) represents the fitness value of the current location of slime mould; condition indicates that S(i) ranks first half of the population; finally, SmellIndex denotes the sequence of fitness values sorted.FOR each search portion,Update the values of control parametersp, vb and vc to realize variations in the population in the intervals of tanhSi−Best Fitness so far, [−a, a] and [−1, 1], respectively.Update positions of slime mould by(21)X∗⟶=randUB−LB+LB,rand<z,XBt⟶+vb⟶.W.XAt⟶−XBt⟶,r<p,vc⟶.Xt⟶,r≥p,where,LB and UB denote the lower and upper boundaries of search space; z is a value taken from a test table given in [52] according to problem type.END FORCycle = Cycle + 1END WHILE


### 2.7. Archimedes Optimization Algorithm

The Archimedes Optimization algorithm is a nondeterministic physics-based metaheuristic algorithm, which is established on Archimedes’ Principle. It was improved by Hashim et al. in 2020 [55] so as to simulate the relationship between an object immersed in a water and buoyant force applied on it.A detailed code of a physics-based AOA consisting mathematical explanations can be given as follows:Randomly create an initial population consisting of immersed objects with random positions, densities, volumes and accelerations by using(22)positions:Oi=lbi+rand.ubi−lbi,i=1,2,…,N,densityden:deni=rand,volumevol:voli=rand,accerelationacc:acci=lbi+rand.ubi−lbi,where;Oi is the ith object in the population; lbi and ubi represents the lower and upper bounds of the search space, respectively; rand is a randomly generated D-dimensional vector having values between [0,1]; the updated density, volume, acceleration determines the new position of an object.Evaluate initial population and determine the solution with the best fitness valueCycle = 1WHILECycle≤Maximum CycleFOR each object iUpdate density and volume by the using the equations(23)denit+1=denit+rand.denbest−denit,volit+1=volit+rand.volbest−volit,where;denbest and volbest are the volume and density associated with the best object found so far; and rand is a uniformly distributed random number.Update transfer and density decreasing factors TF andd,(24)TF=et−tmax/tmax,dt+1=et−tmax/tmax−ttmax.ifTF≤0.5: Exploration phaseUpdate acceleration by using(25)accit+1=denmr+volmr.accmrdenit+1.volit+1.Normalize acceleration by using(26)acci−normt+1=u.accit+1−minaccmaxacc−minacc+l,where;u and l represents the range of normalization and set to 0.9 and 0.1, respectively. acci−normt+1 determines the percentage of step that each agent will change.Update position by takingC1=1or2 ,(27)xit+1=xit+C1.rand.acci−normt+1.d.xrand−xit.elseTF>0.5: Exploitation phaseUpdate acceleration by using the following equation:(28)accit+1=denbest+volbest.accbestdenit+1.volit+1.Normalize acceleration by using(29)acci−normt+1=u.accit+1−minaccmaxacc−minacc+l.Update direction flagF by using the rule by taking C4=0.5or1 and P=2.rand−C4,(30)F=+1ifP≤0.5,−1ifP>0.5.Update position by using the equation given below forC2=6 and T is a constant which is directly proportional to TF,(31)xit+1=xbestt+F.C2.rand.acci−normt+1.d.T.xbest−xit.end ifEND FOREvaluate each object and select the one with the best fitness valueCycle = Cycle + 1END WHILE (until the termination criteria are met)


### 2.8. Equilibrium Optimizer

Equilibrium optimizer, which was proposed by Faramarzi in 2020, is a novel physics-based metaheuristic algorithm inspired by dynamic source and sink models used to estimate equilibrium states [56].A detailed pseudocode of a physics based EO algorithm can be given as follows:Randomly create an initial population consisting of particles by using(32)Ciinitial=Cmin+randi.Cmax−Cmin,i=1,2,…,n,where,Ciinitial is the initial concentration vector of the ith particle; Cmin and Cmax respectively represents the lower and upper values for the dimensions; finally, randi is a randomly generated vector in the interval of [0,1].Evaluate each particle and sort them according to fitness values to assign equilibrium candidates where, the particles nominated as equilibrium candidates construct an equilibrium pool vector,(33)Ceq.pool⟶=Ceq1⟶,Ceq2⟶,Ceq3⟶,Ceq4⟶,Ceqave⟶.Determine the free parameters asa1=2, a2=2 and GP=0.5; where, a1 parameter controls the exploration ability of the algorithm; a2 controls the exploitation ability of the algorithm; in all the simulations in this work and were taken as 2 and 1, respectively, as proposed in [54]; finally, GP is the generation probability used to control the update rate of the population.Cycle = 1WHILECycle≤Maximum CycleFORi=1,2,…,nCalculate the fitness value of ith particleiffitCİ⟶<fitCeq1⟶Replace,Ceq1⟶ with Ci⟶ and fitCeq1⟶ with fitCi⟶else iffitCi⟶>fitCeq1⟶and fitCi⟶<fitCeq2⟶Replace,Ceq2⟶ with Ci⟶ and fitCeq2⟶ with fitCeq2⟶else iffitCi⟶>fitCeq1⟶ and fitCi⟶>fitCeq2⟶andfitCi⟶>fitCeq3⟶Replace,Ceq3⟶ with Ci⟶ and fitCeq3⟶ with fitCi⟶else iffitCi⟶>fitCeq1⟶ and fitCi⟶>fitCeq2⟶fitCi⟶>fitCeq1⟶ and fitCi⟶<fitCeq4⟶Replace,Ceq4⟶ with Ci⟶ and fitCeq4⟶ with fitCi⟶End ifEND FORCalculateCave⟶=Ceq1⟶+Ceq2⟶+Ceq3⟶+Ceq4⟶/4 as in [56].Construct the equilibrium pool as shown below:(34)Ceq.pool⟶=Ceq1⟶,Ceq2⟶,Ceq3⟶,Ceq4⟶,Ceqave⟶.Accomplish memory saving (if Cycle >1).Assignt=1−Cycle/Max Cyclea2.Cycle/Max CycleFORRandomly choose one candidate from the equilibrium vectorGenerate random vectors ofλ⟶, r⟶ and construct F⟶ by(35)F⟶=a1.signr⟶−0.5.eλ⟶.t−1,where,λ⟶ is the control parameter called turnover rate; signr⟶−0.5 component effects on the direction of exploration and exploitation.Construct,GCP⟶=0.5r1ifr2≥GP,0ifr2<GP.Construct,G0⟶=GCP⟶.Ceq⟶−λ⟶.C⟶Construct generation rate,G⟶=G0⟶.F⟶Update concentrations by using the following expression,C⟶=Ceq⟶+C−Ceq⟶.F⟶+G⟶/λ⟶.v1−F⟶END FORCycle = Cycle + 1END WHILE


### 2.9. Retinal Vessel Segmentation

The Digital Retinal Images for Vessel Extraction (DRIVE) database [57] is the first open access database preferred in this work. The images in this database have been obtained by CanonCR5 3CCD camera under a 45° angle with a contrast of 565 × 584 pixels. The simulations have been realized on a total of 40 retinal images, half of which were used for training and the other half were used for testing.The Structured Analysis of the Retina (STARE) [58] database is the second database preferred in this work. While constructing this database, a TopCon TRV-50 fundus camera has been used under a 35° angle with a contrast of 700 × 605 pixels. This database also contains 10 healthy retinal images and 10 diseased retinal images.In the simulation, the pixels have been clustered as vessel pixels and nonvessel pixels. In order to create pixel intervals for both vessel pixels and nonvessel pixels, the number of cluster centers has been chosen ask = 2.The optimal pixel values, each of which represents an optimal cluster center, have been found by using the improved JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms.At the beginning of the optimization process, each pixel value is assigned to the closest clustering centers based on the fitness values obtained. The fitness value of each possible cluster center can be calculated by using the mean squared error (MSE) function given as follows:(36)MSE=1M∑i=1Mfi−yi2,where M is the total pixel number, fi is the nearest clustering center value to the ith pixel, and finally, the value of pixel i has been represented by yi. From (36), it can be concluded that the clustering performance depends directly on the distance between the pixel values and their related clustering centers.The encoding strategy used in the algorithms has a direct effect on optimization performance, and it should also be compatible with the optimization problem. During the metaheuristic based clustering processes improved in this work, the optimized cluster centers are obtained as real numbers. As a result of this, a Real-Valued Encoding Strategy has been applied to all algorithms. In this encoding scheme, for a clustering problem includingk cluster centers, possible solutions as shown below are produced by the algorithms at each cycle (Table 1).Table 1
Real-valued encoding.
CC1CC2…CCkwhereCCk is a real value representing an optimized clustering center and taking values in the interval of [0,255].The four common control parameters of the metaheuristic algorithms used in this work are the Population Size, Maximum Cycle Number,Xmax and Xmin. In the simulations, for each algorithm, the Population Sizes have been chosen as 10, and the Maximum Cycle Numbers have been chosen as 100. Also, Xmax and Xmin represent the highest and lowest pixel values of the relevant image, respectively.The other control parameter values of the algorithms used in the simulations are given in Table2.Table 2
Control parameter values used in the simulations.
AlgorithmControl parametersJS(i) No other control parameters except for four common control parameters given in the text.MPA(i) Fish Aggregating Devices (FADs) = 0, 2TSA(i) ParameterPmin = 1(ii) ParameterPmax  = 4MA(i) Inertia weight,g = 0,8(ii) Visibility coefficient,β = 0, 2(iii) Nuptial dance,d = 5(iv) Random flight,f1 = 1(v) Mutation rate, mu = 0, 01ChOA(i)m = chaotic vector(ii)f is reduced nonlinearly from 2,5 to 0 through the iteration process.SMA(i) Parameterz = 0,03AOA(i) Control variablesC1 = 2, C2 = 6, C3 = 1, C4 = 2EO(i) Generation rate, GP = 0,5(ii) Control parametersa1 = 2, a2 = 1PSO(i) Inertia factor,w = 0,72(ii) Cognitive factor,c1 = 1,49(iii) Social factor,c2 = 1,49GWO(i) The random parameters A and C assist candidate solutions to have hyper-spheres.(ii) Exploration and exploitation are guaranteed by the adaptive values of A and a.DE(i) Crossover rate = 0,8(ii) Scaling factor = 0,2


## 2.1. Jellyfish Search Algorithm

Jellyfish search algorithm, which is proposed by Chou and Truong in 2020 [49], is a swarm based optimizer, which simulates the search behavior of jellyfishes. One of the most important advantages of JS algorithm is that it contains only two control parameters, namely, population size and cycle number. It is inspired from 3 main rules; (i) a jellyfish performs two types of movement, which can be defined as “moving through the ocean current” or “moving along with a swarm,” and the transition process between these movements is coordinated by a time control mechanism, (ii) while searching for food a jellyfish moves towards the locations having more amount of food, and (iii) the quality of a food source is determined by its location and its corresponding objective function.The detailed pseudocode of a bioinspired swarm intelligence based JS algorithm has been given as follows:Randomly create an initial population,xi∈1,2,…,NPOPCalculate the fitness value of eachxi solution in the population.Cycle = 1REPEATCalculate the value of time control parameter by using the following equation:(1)ct=1−1Max Cycle Number2.rand0,1−1.IFct≥0,5:follow ocean current(i)
Determine ocean current by(2)trend⟶=X∗−β.rand0,1.μ,whereX∗ represents the jellyfish currently with the best location in the swarm, μ is the mean location of all jellyfish and β>0 is determined as the distribution coefficient which is related to the length of trend⟶.(ii)
Determine a new location by(3)Xit+1=Xit+rand0,1.trend⟶.ELSE: jelly fish moves inside a swarmIfrand0,1>1−ct define a new position,(4)Xit+1=Xit+γ.rand0,1.Ub−Lb,where;Ub and Lb represent upper and lower bounds of search space, respectively and γ>0 is the motion coefficient.else: determine the direction d⟶ and new location of ith and jth jellyfishes by,(5)d⟶=Xjt−XitiffXi≥fXj,Xit−XjtiffXi<fXj,Xit+1=Xit+rand0,1.d⟶.end ifEND IFCheck boundary conditions and calculate the fitness value of food source at new locationUpdate the location ofXi and location of X∗Cycle = Cycle + 1UNTİL (termination criteria are met)


## 2.2. Marine Predators Algorithm

Marine Predators algorithm, which is proposed by Faramarzi et al. in 2020 [50], is a nature-inspired swarm intelligence based metaheuristic algorithm, which simulates foraging behavior of ocean predators by taking into account the relationship between the prey and predator. The most important metric of this algorithm is to optimize the search strategies used by the predators which is also called random walk strategy. The algorithm focuses on two main random walk strategies, namely, Brownian motion and Lévy motion.A detailed pseudocode of a nature-inspired swarm intelligence based MPA algorithm can be given as follows:Randomly create a uniformly distributed prey population,i=1,2,…,nCycle = 1WHILECycle≤Maximum CycleCalculate the fitness values and construct the Elite matrix which includes the best solutions (best positions of preys)(6)Elite=XI1,1,…,XI1,d,⋮,⋱,⋮XIn,1,⋯,XIn,d..IFCycle<Maximum Cycle Number/3:highvelocity ratioUpdate prey bypreyi⟶=preyi⟶+P.P⟶⊗stepsizei⟶where,stepsizei⟶=RB⟶⊗Elitei⟶−RB⟶⊗stepsizei⟶ is the velocity ratio for phase 1 in which the predator is moving faster than the prey; ⊗ notation represents the entry-wise multiplications; the multiplication of RB by prey simulates the movement of prey; P is a constant number which is equal to 0, 5 and finally, R is a vector of uniform random numbers in [0, 1]ELSE IF: unit velocity ratio(7)Maximum Cycle Number3<Cycle<2.Maximum Cycle Number3.For the first half of the populationi=1,2,…,n/2 update prey based on,(8)preyi⟶=preyi⟶+P.R⟶⊗stepsizei⟶,where,stepsizei⟶=RL⟶⊗Elitei⟶−RL⟶⊗preyi⟶ is the velocity ratio for phase 2 in which both predator and prey are moving at the same pace; RL⟶ represents a vector of random numbers produced from Lévy distribution;For the second half of the populationi=n/2+1,…,n update prey based on(9)preyi⟶=Elitei⟶+P.CF⊗stepsizei⟶,where,stepsizei⟶=RB⟶⊗RB⟶⊗Elitei⟶−preyi⟶ is the velocity ratio for phase 2 in which both predator and prey are moving at the same pace; the multiplication of RB and Elite simulates the movement of predator in Brownian manner and finally, CF=1−Cycle/Max Cycle2.Cycle/Max Cycle is an adaptive control parameter.ELSE IFCycle>2.Maximum Cycle Number/3:low velocity ratioUpdate prey,preyi⟶=Elitei⟶+P.CF⊗stepsizei⟶where,stepsizei⟶=RL⟶⊗RL⟶⊗Elitei⟶−preyi⟶ is the velocity ratio for phase 3 in which the predator is moving faster than the prey; the multiplication of RL⟶ and Elite simulates the movement of predator in Brownian mannerEND IFMemorize the best solutions so far and updateElite matrix Apply Fish Aggregating Devices (FADs) effect given below for r ≤ FADs and r > FADs, respectively, in order to prevent the algorithm to get stuck into the local minima.(10)preyi⟶=preyi⟶+CF.Xmin⟶+R⊗Xmax⟶−Xmin⟶⊗U⟶,preyi⟶+FADs.1−r+r.preyr1⟶−preyr2⟶,where,FADs = 0.2 is the probability of FADs effect on the optimization process; U⟶ is a binary vector constructed by generating a random vector in [0,1]; Xmin⟶ and Xmax⟶ are the vectors containing the lower and upper bounds of the dimensions; the r1 and r2 represent random indexes of prey matrix.END WHILE


## 2.3. Tunicate Swarm Algorithm

Tunicate Swarm algorithm is a bioinspired metaheuristic optimization algorithm inspired from the swarm behaviors of tunicates during the foraging process. It was proposed by Kaur et al. in 2020 [51]. TSA algorithm has been constructed mathematically on two main behaviors of tunicates that are jet propulsion and swarm intelligence.A detailed pseudocode for a bioinspired swarm intelligence based TSA algorithm can be given as follows:Initialize the tunicate population,Ppx⟶Calculate the fitness value of each search agent in the initial populationCycle = 1REPEATDetermine the jet propulsion and swarm behaviors of tunicates by,(i)
Jet Propulsion Behavior(11)PD⟶=FS⟶−rand0,1.Ppx⟶,where;FS⟶ represents the fitness value vector of search agents;(ii)
Swarm Behavior(12)Ppx⟶=FS⟶+A⟶.PD⟶ifrand0,1≥0.5,FS⟶−A⟶.PD⟶ifrand0,1<0.5,where,A⟶ vector includes the variations to avoid conflicts among search agents.Update the position of each search agent by using(13)Ppx+1⟶=Ppx⟶+Ppx+1⟶2+c1,where,c1 is a randomly produced number in [0, 1].Calculate the fitness value for updated search agentsEliminate the search agents placing out of the search spaceCycle = Cycle + 1UNTİL(termination criteria are met)


## 2.4. Mayfly Optimization Algorithm

The basic philosophy of Mayfly optimization algorithm is to simulate the flight behavior and the mating process of mayflies. Mayfly is a hybrid metaheuristic algorithm combining the major advantages of swarm intelligence and evolutionary algorithms. It was proposed by Zervoudakis and Tsafarakis in 2020 [52] as an effective swarm intelligence based metaheuristic optimization algorithm in terms of convergence rate.The detailed pseudocode with mathematical expressions for a hybrid MA can be given as follows:Initialize the male mayfly population,xi∈1,2,…,N and velocities vmiInitialize the female mayfly population,yi∈1,2,…,N and velocities vfiCalculate the fitness value of each possible solution in the initial populationCycle = 1REPEATUpdate velocities and solutions of males and females by using,(i)
Male Mayflies(14)vijt+1=vijt+α1e−β.rp2pbestijxijt+α2.e−β.rg2gbestij−xijt,where,vijt represents the velocity of mayfly i in dimension j = 1, …, n at time step t; xijt is the position of mayfly i in dimension j at time step t;a1>0 and a2>0 are the attraction constants; pbesti is the best position of mayfly i so far; gbest represents the global best position so far; finally, β is a fixed visibility coefficient used to limit a mayfly’s visibility to others while rp is the Cartesian distance between xi and, and rg is the Cartesian distance between xi and gbest.(ii)
Female Mayflies(15)vijt+1=vijt+α2e−β.rmf2xijt−yijtiffyi>fxi,vijt+fl.riffyi≤fxi,where;yijt is the position of female mayfly i in dimension j at time step t; f l is a random walk coefficient, used when a female is not attracted by a male, so it flies randomly and r is a random value in the interval of [−1, 1]; finally, rmf represents the Cartesian distance between male and female.Calculate the fitness value of each new candidate solutionRank the mayfliesMate the mayflies by applying crossover operation as mentioned below,the crossover operation is realized mutually by using the following equations between the candidates selected from male and female populations.(16)offspring1=L.male+1−L.female,offspring2=L.female+1−L.male,where,male and female represent the male and female parents, respectively; L is a random value within a specific interval.Calculate the fitness value of each offspringSeparate offsprings to male and female populations randomlyReplace worst solutions with the new better onesUpdatepbest and gbestCycle = Cycle + 1UNTİL(termination criteria are met)


## 2.5. Chimp Optimization Algorithm

The Chimp optimization algorithm inspired by the individual intelligence and sexual motivation of chimps in their group hunting has been proposed by Khishe and Mosavi in 2020 [53]. The mathematical model of ChOA algorithm has been built on two main phases. While the intelligence phase includes four types of chimps entitled attacker, barrier, chaser, and driver, the hunting phase has been modelled by considering driving, chasing, blocking, and attacking behaviors of chimps.The detailed pseudocode of a nature-inspired swarm intelligence based ChOA algorithm can be given as follows:Initialize the chimp population,xi∈1,2,…,NCalculate the position of each chimp for driving by,(17)d=c.Xpreyt−m.Xchimpt.Calculate the position of each chimp for chasing by,(18)Xchimpt+1=Xpreyt−a.d,wherea, m and c are the coefficient vectors; Xchimp represents the position vector of a chimp while Xprey is the vector of prey position; also for the descriptions of a=2.f.r1−f and c=2.r2; f represents reduced nonlinearly from 2.5 to 0 through the cycles process, r1 and r2 are the random vectors in [0, 1]; finally, m represents a chaotic vector simulating the sexual motivation of chimps in the hunting phase.Divide chimps randomly into four independent groups as,XAttacker: the best search agentXChaser: the second best search agentXBarrier: the third best search agentXDriver: the fourth best search agentCycle = 1REPEATCalculate the fitness value of each chimpWHILE (Cycle < Maximum Cycles).For each chimp:Identify chimp groupsUse the group strategy to updatef, m and cUsef, m and c to calculate the new a and dEnd Forfor each search chimp:if (μ<0.5), μ is a randomly produced number in [0, 1]if (a<1)Update the position of the current search agent by using,Xchimpt+1=Xpreyt−a.delse if ( a>1 ).Select a random search agentend ifelse if ( μ>0.5),Update the position of the current search agent by using(19)Xchimpt+1,Xpreyt−a.difμ<0.5,mchaotic valueifμ≥0.5.end ifend forUpdatef, m, a and cUpdateXAttacker, XChaser, XBarrier and XDriverEND WHILECycle = Cycle + 1UNTIL (termination criteria are met).


## 2.6. Slime Mould Algorithm

The Slime Mould algorithm used in this work is a biological-evolution inspired algorithm, which simulates the diffusion and foraging behavior of slime mould. It was introduced by Li et al. in 2020 [54] as an effective physics-based metaheuristic algorithm with the detailed pseudocode given below:Initialize the positions of slime mould,Xi∈1,2,…,nCycle = 1WHILECycle≤Maximum CycleCalculate the fitness value of all slime mouldDetermine thebestFitness, XbCalculate the weight of slime mould (W) by using(20)WSmellIndext⟶=1+r.logbF−SibF−wF+1,condition,1−r.logbF−SibF−wF+1,others,where,r denotes a randomly produced value in the interval of [0,1]; bF represents the optimal fitness value obtained in the current cycle; wF is the worst fitness value obtained in the current cycle; S(i) represents the fitness value of the current location of slime mould; condition indicates that S(i) ranks first half of the population; finally, SmellIndex denotes the sequence of fitness values sorted.FOR each search portion,Update the values of control parametersp, vb and vc to realize variations in the population in the intervals of tanhSi−Best Fitness so far, [−a, a] and [−1, 1], respectively.Update positions of slime mould by(21)X∗⟶=randUB−LB+LB,rand<z,XBt⟶+vb⟶.W.XAt⟶−XBt⟶,r<p,vc⟶.Xt⟶,r≥p,where,LB and UB denote the lower and upper boundaries of search space; z is a value taken from a test table given in [52] according to problem type.END FORCycle = Cycle + 1END WHILE


## 2.7. Archimedes Optimization Algorithm

The Archimedes Optimization algorithm is a nondeterministic physics-based metaheuristic algorithm, which is established on Archimedes’ Principle. It was improved by Hashim et al. in 2020 [55] so as to simulate the relationship between an object immersed in a water and buoyant force applied on it.A detailed code of a physics-based AOA consisting mathematical explanations can be given as follows:Randomly create an initial population consisting of immersed objects with random positions, densities, volumes and accelerations by using(22)positions:Oi=lbi+rand.ubi−lbi,i=1,2,…,N,densityden:deni=rand,volumevol:voli=rand,accerelationacc:acci=lbi+rand.ubi−lbi,where;Oi is the ith object in the population; lbi and ubi represents the lower and upper bounds of the search space, respectively; rand is a randomly generated D-dimensional vector having values between [0,1]; the updated density, volume, acceleration determines the new position of an object.Evaluate initial population and determine the solution with the best fitness valueCycle = 1WHILECycle≤Maximum CycleFOR each object iUpdate density and volume by the using the equations(23)denit+1=denit+rand.denbest−denit,volit+1=volit+rand.volbest−volit,where;denbest and volbest are the volume and density associated with the best object found so far; and rand is a uniformly distributed random number.Update transfer and density decreasing factors TF andd,(24)TF=et−tmax/tmax,dt+1=et−tmax/tmax−ttmax.ifTF≤0.5: Exploration phaseUpdate acceleration by using(25)accit+1=denmr+volmr.accmrdenit+1.volit+1.Normalize acceleration by using(26)acci−normt+1=u.accit+1−minaccmaxacc−minacc+l,where;u and l represents the range of normalization and set to 0.9 and 0.1, respectively. acci−normt+1 determines the percentage of step that each agent will change.Update position by takingC1=1or2 ,(27)xit+1=xit+C1.rand.acci−normt+1.d.xrand−xit.elseTF>0.5: Exploitation phaseUpdate acceleration by using the following equation:(28)accit+1=denbest+volbest.accbestdenit+1.volit+1.Normalize acceleration by using(29)acci−normt+1=u.accit+1−minaccmaxacc−minacc+l.Update direction flagF by using the rule by taking C4=0.5or1 and P=2.rand−C4,(30)F=+1ifP≤0.5,−1ifP>0.5.Update position by using the equation given below forC2=6 and T is a constant which is directly proportional to TF,(31)xit+1=xbestt+F.C2.rand.acci−normt+1.d.T.xbest−xit.end ifEND FOREvaluate each object and select the one with the best fitness valueCycle = Cycle + 1END WHILE (until the termination criteria are met)


## 2.8. Equilibrium Optimizer

Equilibrium optimizer, which was proposed by Faramarzi in 2020, is a novel physics-based metaheuristic algorithm inspired by dynamic source and sink models used to estimate equilibrium states [56].A detailed pseudocode of a physics based EO algorithm can be given as follows:Randomly create an initial population consisting of particles by using(32)Ciinitial=Cmin+randi.Cmax−Cmin,i=1,2,…,n,where,Ciinitial is the initial concentration vector of the ith particle; Cmin and Cmax respectively represents the lower and upper values for the dimensions; finally, randi is a randomly generated vector in the interval of [0,1].Evaluate each particle and sort them according to fitness values to assign equilibrium candidates where, the particles nominated as equilibrium candidates construct an equilibrium pool vector,(33)Ceq.pool⟶=Ceq1⟶,Ceq2⟶,Ceq3⟶,Ceq4⟶,Ceqave⟶.Determine the free parameters asa1=2, a2=2 and GP=0.5; where, a1 parameter controls the exploration ability of the algorithm; a2 controls the exploitation ability of the algorithm; in all the simulations in this work and were taken as 2 and 1, respectively, as proposed in [54]; finally, GP is the generation probability used to control the update rate of the population.Cycle = 1WHILECycle≤Maximum CycleFORi=1,2,…,nCalculate the fitness value of ith particleiffitCİ⟶<fitCeq1⟶Replace,Ceq1⟶ with Ci⟶ and fitCeq1⟶ with fitCi⟶else iffitCi⟶>fitCeq1⟶and fitCi⟶<fitCeq2⟶Replace,Ceq2⟶ with Ci⟶ and fitCeq2⟶ with fitCeq2⟶else iffitCi⟶>fitCeq1⟶ and fitCi⟶>fitCeq2⟶andfitCi⟶>fitCeq3⟶Replace,Ceq3⟶ with Ci⟶ and fitCeq3⟶ with fitCi⟶else iffitCi⟶>fitCeq1⟶ and fitCi⟶>fitCeq2⟶fitCi⟶>fitCeq1⟶ and fitCi⟶<fitCeq4⟶Replace,Ceq4⟶ with Ci⟶ and fitCeq4⟶ with fitCi⟶End ifEND FORCalculateCave⟶=Ceq1⟶+Ceq2⟶+Ceq3⟶+Ceq4⟶/4 as in [56].Construct the equilibrium pool as shown below:(34)Ceq.pool⟶=Ceq1⟶,Ceq2⟶,Ceq3⟶,Ceq4⟶,Ceqave⟶.Accomplish memory saving (if Cycle >1).Assignt=1−Cycle/Max Cyclea2.Cycle/Max CycleFORRandomly choose one candidate from the equilibrium vectorGenerate random vectors ofλ⟶, r⟶ and construct F⟶ by(35)F⟶=a1.signr⟶−0.5.eλ⟶.t−1,where,λ⟶ is the control parameter called turnover rate; signr⟶−0.5 component effects on the direction of exploration and exploitation.Construct,GCP⟶=0.5r1ifr2≥GP,0ifr2<GP.Construct,G0⟶=GCP⟶.Ceq⟶−λ⟶.C⟶Construct generation rate,G⟶=G0⟶.F⟶Update concentrations by using the following expression,C⟶=Ceq⟶+C−Ceq⟶.F⟶+G⟶/λ⟶.v1−F⟶END FORCycle = Cycle + 1END WHILE


## 2.9. Retinal Vessel Segmentation

The Digital Retinal Images for Vessel Extraction (DRIVE) database [57] is the first open access database preferred in this work. The images in this database have been obtained by CanonCR5 3CCD camera under a 45° angle with a contrast of 565 × 584 pixels. The simulations have been realized on a total of 40 retinal images, half of which were used for training and the other half were used for testing.The Structured Analysis of the Retina (STARE) [58] database is the second database preferred in this work. While constructing this database, a TopCon TRV-50 fundus camera has been used under a 35° angle with a contrast of 700 × 605 pixels. This database also contains 10 healthy retinal images and 10 diseased retinal images.In the simulation, the pixels have been clustered as vessel pixels and nonvessel pixels. In order to create pixel intervals for both vessel pixels and nonvessel pixels, the number of cluster centers has been chosen ask = 2.The optimal pixel values, each of which represents an optimal cluster center, have been found by using the improved JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms.At the beginning of the optimization process, each pixel value is assigned to the closest clustering centers based on the fitness values obtained. The fitness value of each possible cluster center can be calculated by using the mean squared error (MSE) function given as follows:(36)MSE=1M∑i=1Mfi−yi2,where M is the total pixel number, fi is the nearest clustering center value to the ith pixel, and finally, the value of pixel i has been represented by yi. From (36), it can be concluded that the clustering performance depends directly on the distance between the pixel values and their related clustering centers.The encoding strategy used in the algorithms has a direct effect on optimization performance, and it should also be compatible with the optimization problem. During the metaheuristic based clustering processes improved in this work, the optimized cluster centers are obtained as real numbers. As a result of this, a Real-Valued Encoding Strategy has been applied to all algorithms. In this encoding scheme, for a clustering problem includingk cluster centers, possible solutions as shown below are produced by the algorithms at each cycle (Table 1).Table 1
Real-valued encoding.
CC1CC2…CCkwhereCCk is a real value representing an optimized clustering center and taking values in the interval of [0,255].The four common control parameters of the metaheuristic algorithms used in this work are the Population Size, Maximum Cycle Number,Xmax and Xmin. In the simulations, for each algorithm, the Population Sizes have been chosen as 10, and the Maximum Cycle Numbers have been chosen as 100. Also, Xmax and Xmin represent the highest and lowest pixel values of the relevant image, respectively.The other control parameter values of the algorithms used in the simulations are given in Table2.Table 2
Control parameter values used in the simulations.
AlgorithmControl parametersJS(i) No other control parameters except for four common control parameters given in the text.MPA(i) Fish Aggregating Devices (FADs) = 0, 2TSA(i) ParameterPmin = 1(ii) ParameterPmax  = 4MA(i) Inertia weight,g = 0,8(ii) Visibility coefficient,β = 0, 2(iii) Nuptial dance,d = 5(iv) Random flight,f1 = 1(v) Mutation rate, mu = 0, 01ChOA(i)m = chaotic vector(ii)f is reduced nonlinearly from 2,5 to 0 through the iteration process.SMA(i) Parameterz = 0,03AOA(i) Control variablesC1 = 2, C2 = 6, C3 = 1, C4 = 2EO(i) Generation rate, GP = 0,5(ii) Control parametersa1 = 2, a2 = 1PSO(i) Inertia factor,w = 0,72(ii) Cognitive factor,c1 = 1,49(iii) Social factor,c2 = 1,49GWO(i) The random parameters A and C assist candidate solutions to have hyper-spheres.(ii) Exploration and exploitation are guaranteed by the adaptive values of A and a.DE(i) Crossover rate = 0,8(ii) Scaling factor = 0,2


## 3. Results

The retinal images emerging after segmentation process applied to the images in Figures1(a) and 1(b) by using the clustering based JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms improved in this work have been given in Figure 5. Similarly, the resulting images obtained after PSO, GWO, and DE based segmentation have been given in Figure 6. As seen from the figures, some pixels belonging to the image background but having pixel value close to the vessels may incorrectly be defined as vessel pixels. In order to overcome these difficulties encountered in segmentation, some postprocessing methods have been proposed in the literature. However, these postprocessing methods have not been used in this work in order to find out the pure performance of the algorithms. In general, it can be concluded that all the algorithms have performances too close to each other in terms of clustering. The simulation results represent that JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms can distinguish the close pixel values with high accuracy similar to PSO, GWO, and DE algorithms. Figure 7 shows the clustering performance of the most recently proposed algorithms for the Figures 2(a) and 2(b) taken from the STARE database. Also, in Figure 8, the retinal images produced by PSO, GWO, and DE algorithms for STARE database have been demonstrated. The results represent that the algorithms produce poor performance in STARE database compared to DRIVE database. However, the simulation results prove that metaheuristic algorithms produce similar but a bit better performance than the conventional algorithms in literature in terms of clustering.Figure 5
Retinal images obtained after applying segmentation to the images in Figures1(a) and 1(b) by using the clustering based JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms. (a) JS. (b) MPA. (c) TSA. (d) MA. (e) ChOA. (f) SMA. (g) AOA. (h) EO. (i) JS. (j) MPA. (k) TSA. (l) MA. (m) ChOA. (n) SMA. (o) AOA. (p) EO.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)Figure 6
Retinal images obtained after applying segmentation to the images in Figures1(a) and 1(b)) by using the clustering based PSO, GWO, and DE algorithms. (a) PSO. (b) GWO. (c) DE. (d) PSO. (e) GWO. (f) DE.
(a)(b)(c)(d)(e)(f)Figure 7
Retinal images obtained after applying segmentation to the images in Figures2(a) and 2(b) by using the clustering based JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms. (a) JS. (b) MPA. (c) TSA. (d) MA. (e) ChOA. (f) SMA. (g) AOA. (h) EO. (i) JS. (j) MPA. (k) TSA. (l) MA. (m) ChOA. (n) SMA. (o) AOA. (p) EO.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)Figure 8
Retinal images obtained after applying segmentation to the images in Figures2(a) and 2(b) by using the clustering based PSO, GWO, and DE algorithms. (a) PSO. (b) GWO. (c) DE. (d) PSO. (e) GWO. (f) DE.
(a)(b)(c)(d)(e)(f)


### 3.1. Performance Measures

Firstly, a detailed statistical performance comparison based on SE, SP, ACC, and F-score has been realized for the algorithms improved. Then, the convergence rate, MSE, CPU time, and standard deviation performances of the algorithms have been compared at each other.The mathematical expressions of SE, SP, ACC, and F-score parameters can be given as follows:(37)SE=TPTP+FN,SP=TNTN+FP,ACC=TP+TNTP+FN+TN+FP,F−score=TPTP+1/2FP+FN,where TP is the correctly estimated vessel pixel number, FN is the number of vessel pixels incorrectly defined as background pixels, TN is the correctly estimated background pixel number, and finally FP is the number of background pixels incorrectly defined as vessel pixels.


### 3.2. Performance Analysis for DRIVE and STARE Databases

In Tables3 and 4, the results relevant to sensitivity, specificity, and accuracy produced by the novel algorithms for 20 different images are given for DRIVE and STARE databases, respectively. Similarly, the results obtained for PSO, GWO, and DE algorithms have been given in Table 5 for DRIVE database and in Table 6 for STARE database.Table 3
The performance measures of the different images in the DRIVE database in terms of sensitivity, specificity, and accuracy.
SensitivitySpecificityAccuracyImageJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEO10.89130.89130.89130.89130.89130.89130.9960.89130.98840.98840.98840.98840.98840.98840.99960.98840.97910.97910.97910.97910.97910.97910.99930.979120.78570.78570.78570.78570.78570.78570.83020.78570.97030.97030.97030.97030.97030.97030.97760.97030.94790.94790.94790.94790.94790.94790.96040.947930.87870.87870.87870.87870.87870.87870.87870.87870.98670.98670.98670.98670.98670.98670.98670.98670.9760.9760.9760.9760.9760.9760.9760.97640.7230.7230.7230.7230.7230.7230.7230.7230.96450.96450.96450.96450.96450.96450.96450.96450.93710.93710.93710.93710.93710.95470.93710.937150.95140.95140.95140.95140.95140.95140.83260.95140.99550.99550.99550.99550.99550.99550.98250.99550.99180.99180.99180.99180.99180.99180.96840.991860.95720.95720.95720.95720.95720.95720.95720.95720.99540.99540.99540.99540.99540.99540.99540.99540.99170.99170.99170.99170.99170.99170.99170.991770.88740.88740.88740.88740.95930.88740.88740.88740.99030.99030.99030.99030.99670.99030.99030.99030.98220.98220.98220.98220.9940.98220.98220.982280.55060.55060.55060.55060.55060.55060.55060.55060.94490.94490.94490.94490.94490.94490.94490.94490.90190.90190.90190.90190.90190.90190.90190.901990.78180.78180.78180.78180.78180.78180.78180.78180.97610.97610.97610.97610.97610.97610.97610.97610.95680.95680.95680.95680.95680.95680.95680.9568100.89320.89320.89320.89320.89320.93740.89320.89320.99080.99080.99080.99080.99080.99490.99080.99080.98310.98310.98310.98310.98310.99050.98310.9831110.7870.7870.7870.7870.74990.7870.83630.7870.9760.9760.9760.9760.97060.9760.98250.9760.95690.95690.95690.95690.94740.95690.96850.9569120.66240.66240.66240.66240.86090.66240.66240.66240.95740.95740.95740.95740.98610.95740.95740.95740.92430.92430.92430.92430.97470.92430.92430.9243130.79920.79920.79920.79920.74070.79920.79920.79920.97250.97250.97250.97250.96210.97250.97250.97250.95160.95160.95160.95160.93380.95160.95160.9516140.74390.74390.74390.74390.91370.74390.74390.74390.9730.9730.9730.9730.99240.9730.9730.9730.95110.95110.95110.95110.98610.95110.95110.9511150.94440.94440.94440.94440.94440.94440.89660.94440.99530.99530.99530.99530.99530.99530.99080.99530.99130.99130.99130.99130.99130.99130.98310.9913160.83870.83870.83870.83870.83870.83870.83870.83870.98240.98240.98240.98240.98240.98240.98240.98240.96830.96830.96830.96830.96830.96830.96830.9683170.70170.70170.70170.70170.8740.70170.70170.70170.96720.96720.96720.96720.98860.96720.96720.96720.94090.94090.94090.94090.97910.94090.94090.9409180.89720.89720.89720.89720.89720.89720.89720.89720.98850.98850.98850.98850.98850.98850.98850.98850.97940.97940.97940.97940.97940.97940.97940.9794190.81610.81610.81610.81610.81610.81610.93120.81610.97580.97580.97580.97580.97580.97580.99190.97580.95720.95720.95720.95720.95720.95720.98560.9572200.74770.74770.74770.74770.96460.74770.74770.74770.96670.96670.96670.96670.99630.96670.96670.96670.94120.94120.94120.94120.99330.94120.94120.9412Mean0.81190.81190.81190.81190.84860.81410.81920.81190.97780.97780.97780.97780.98230.97800.97900.97780.96040.96040.96040.96040.96850.96170.962540.9604Bold values represent the best values obtained for the relevant metric.Table 4
The performance measures of the different images in the STARE database in terms of sensitivity, specificity, and accuracy.
ImageSensitivitySpecificityAccuracyJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEO10.74980.74980.74980.74980.74980.74980.74980.74980.97420.97420.97420.97420.97420.97420.97420.97420.95330.95330.95330.95330.95330.95330.95330.953320.68760.68760.68760.68760.68760.68760.68760.68760.97010.97010.97010.97010.97010.97010.97010.97010.94540.94540.94540.94540.94540.94540.94540.945430.64230.64230.64230.64230.64230.64230.64230.64230.95680.95680.95680.95680.95680.95680.95680.95680.92290.92290.92290.92290.92290.92290.92290.922940.35130.35130.35130.35130.35130.35130.35130.35130.85080.85080.85080.85080.85080.85080.85080.85080.75750.75750.75750.75750.75750.75750.75750.757550.84070.84070.84070.84070.84070.84070.91930.84070.98250.98250.98250.98250.98250.98250.99180.98250.96850.96850.96850.96850.96850.96850.98520.968560.82110.82110.82110.82110.82110.82110.95240.82110.9810.9810.9810.9810.9810.9810.99560.9810.96560.96560.96560.96560.96560.96560.99190.965670.93240.99450.99450.99450.93240.93240.93010.99450.99370.99950.99950.99950.99370.99370.99340.99950.98840.99910.99910.99910.98840.98840.9880.999180.99870.89550.89550.89550.89550.89550.89550.89550.9990.99060.99060.99060.99060.99060.99060.99060.99980.98280.98280.98280.98280.98280.98280.982890.92810.92810.92810.92810.92810.92810.92810.92810.99320.99320.99320.99320.99320.99320.99320.99320.98760.98760.98760.98760.98760.98760.98760.9876100.81430.81430.81430.81430.81430.81430.81430.81430.97960.97960.97960.97960.97960.97960.97960.97960.96330.96330.96330.96330.96330.96330.96330.9633110.93770.93770.93770.93770.73010.93770.93770.93770.9950.9950.9950.9950.97260.9950.9950.9950.99070.99070.99070.99070.95030.99070.99070.9907120.81630.81630.81630.81630.89340.81630.81630.81630.97850.97850.97850.97850.98850.97850.97850.97850.96150.96150.96150.96150.97920.96150.96150.9615130.88130.88130.88130.88130.88130.88130.87520.88130.98830.98830.98830.98830.98830.98830.98760.98830.97870.97870.97870.97870.97870.97870.97750.9787140.98220.98220.98220.89180.98220.98220.89180.98220.99860.99860.99860.99050.99860.99860.99050.99860.99730.99730.99730.98250.99730.99730.98250.9973150.79680.79680.79680.79680.79680.79680.66210.79680.97940.97940.97940.97940.97940.97940.95960.97940.96250.96250.96250.96250.96250.96250.92780.9625160.90950.90950.90950.90950.90950.90950.90950.90950.99130.99130.99130.99130.99130.99130.99130.99130.98410.98410.98410.98410.98410.98410.98410.9841170.94490.94490.94490.94490.94490.94490.96120.94490.99550.99550.99550.99550.99550.99550.99690.99550.99160.99160.99160.99160.99160.99160.99420.9916180.75930.75930.75930.75930.75930.75930.95470.75930.97480.97480.97480.97480.97480.97480.99610.97480.95440.95440.95440.95440.95440.95440.99290.9544190.90550.90550.90550.90550.90550.90550.90550.90550.98960.98960.98960.98960.98960.98960.98960.98960.98130.98130.98130.98130.98130.98130.98130.9813200.88170.88170.88170.90140.88170.90140.90140.88170.98690.98690.98690.98130.98690.98930.98930.98690.97640.97640.97640.98070.97640.98070.98070.9764Mean0.82900.82700.82700.82340.81730.82490.83430.82700.97790.97780.97780.97710.97690.97760.97850.97780.96150.96120.96120.96070.95950.96090.96250.9612Bold values represent the best values obtained for the relevant metric.Table 5
The performance measures of the different images in the DRIVE database in terms of sensitivity, specificity and accuracy for PSO, GWO and DE.
ImageSensitivitySpecificityAccuracyPSOGWODEPSOGWODEPSOGWODE10.89130.86060.89130.98840.98470.98840.97910.97240.979120.78570.83020.83020.97030.97760.97760.94790.96040.960430.87870.53920.87870.98670.9230.98670.9760.8680.97640.7230.77930.7230.96450.97350.96450.93710.95270.937150.95140.95140.95140.99550.99550.99550.99180.99180.991860.95720.86050.95720.99540.98360.99540.99170.97070.991770.88740.73310.88740.99030.97270.99030.98220.95040.982280.55060.86720.55060.94490.98920.94490.90190.980.901990.78180.82510.78180.97610.98170.97610.95680.96690.9568100.89320.89320.7180.99080.99080.97050.98310.98310.9467110.7870.98790.7870.9760.99890.9760.95690.9980.9569120.66240.68240.66240.95740.96090.95740.92430.93050.9243130.79920.64890.79920.97250.94260.97250.95160.90130.9516140.74390.92920.74390.9730.99390.9730.95110.98870.9511150.94440.55510.94440.99530.93930.99530.99130.89320.9913160.83870.70840.83870.98240.96310.98240.96830.93450.9683170.70170.70170.70170.96720.96720.96720.94090.94090.9409180.89720.74380.89720.98850.96630.98850.97940.94050.9794190.81610.50840.81610.97580.90370.95780.95720.8390.9572200.74770.74640.96460.99670.96650.99630.94120.94080.9933Mean0.81190.76760.81620.97930.96870.97780.96040.94510.9619Bold values represent the best values obtained for the relevant metric.Table 6
The performance measures of the different images in the STARE database in terms of sensitivity, specificity, and accuracy for PSO, GWO, and DE.
ImageSensitivitySpecificityAccuracyPSOGWODEPSOGWODEPSOGWODE10.72050.96110.72050.97020.99680.97020.94610.99410.946120.68760.84610.68760.97010.98780.97010.94540.97740.945430.64230.97610.64230.95680.9980.95680.92290.99630.922940.66340.66340.35130.9540.9540.85080.91910.91910.757550.84070.75720.84070.98250.97080.98250.96850.94790.968560.82110.71040.87890.9810.96490.98790.96560.93740.977970.99450.79990.99450.99950.97850.99950.99910.96120.999180.89550.79470.89550.99060.97950.99060.98280.96270.982890.92810.92810.92810.99320.99320.99320.98760.98760.9876100.81430.9250.81430.97960.99270.97960.96330.98660.9633110.93770.76540.93770.9950.97720.9950.99070.95840.9907120.81630.81630.78620.97850.97850.97420.96150.96150.9539130.88130.70340.88130.98830.96430.98830.97870.93620.9787140.98220.71020.89180.99860.96870.99050.99730.94340.9825150.79680.93930.79680.97940.99470.97940.96250.99020.9625160.99050.90950.90950.99130.99130.99130.98410.98410.9841170.94490.96120.94490.99550.99690.99550.99160.99420.9916180.75930.75930.75930.97480.97480.97480.95440.95440.9544190.90950.90550.90950.98960.98960.98960.98130.98130.9813200.88170.64380.76210.98690.94820.97010.97640.90950.9468Mean0.84540.82370.81660.98270.98000.97640.96890.96410.9588Bold values represent the best values obtained for the relevant metric.In addition to the SE, SP, and ACC parameters, theF-score values obtained for each algorithms have also been given in Tables 7 and 8, respectively, for DRIVE and STARE databases.Table 7
The performance measures of the different images in the DRIVE database in terms ofF-score.
ImageF-scoreJSMPATSAMAChOASMAAOAEOPSOGWODE10.78310.78310.78310.78310.78310.78310.78530.78310.78310.76380.783120.79040.79040.79040.79040.79040.79040.80650.79040.79040.80650.806530.70430.70430.70430.70430.70430.70430.70430.70430.70430.60550.704340.72210.72210.72210.72210.72210.76320.72210.72210.72210.71620.722150.7660.7660.7660.7660.7660.7660.73220.7660.7660.7660.76660.7110.7110.7110.7110.7110.7110.7110.7110.7110.7230.71170.70890.70890.70890.70890.7220.70890.70890.70890.70890.67570.708980.5870.5870.5870.5870.5870.5870.5870.5870.5870.70290.58790.65630.65630.65630.65630.65630.65630.65630.65630.65630.69120.6563100.75740.75740.75740.75740.75740.73920.75740.75740.75740.75740.6835110.74180.74180.74180.74180.72440.74180.75550.74180.74180.75460.7418120.65910.65910.65910.65910.71610.65910.65910.65910.65910.65750.6591130.72460.72460.72460.72460.70550.72460.72460.72460.72460.60580.7246140.7060.7060.7060.7060.74710.7060.7060.7060.7060.76230.706150.70370.70370.70370.70370.70370.70370.68860.70370.70370.58250.7037160.78430.78430.78430.78430.78430.78430.78430.78430.78430.70460.7843170.67010.67010.67010.67010.71010.67010.67010.67010.67010.67010.6701180.75380.75380.75380.75380.75380.75380.75380.75380.75380.69810.7538190.76910.76910.76910.76910.76910.76910.76610.76910.76910.59670.7691200.66960.66960.66960.66960.70830.66960.66960.66960.66960.70550.7083Mean0.71840.71840.71840.71840.72610.71950.71740.71840.71840.69720.7174Bold values represent the best values obtained for the relevant metric.Table 8
The performance measures of the different images in the STARE database in terms ofF-score.
ImageF-scoreJSMPATSAMAChOASMAAOAEOPSOGWODE10.6010.6010.6010.6010.6010.6010.6010.6010.6010.58970.60120.55940.55940.55940.55940.55940.55940.55940.55940.55940.57080.559430.5680.5680.5680.5680.5680.5680.5680.5680.5680.61020.56840.43850.43850.43850.43850.43850.43850.43850.43850.52330.52330.438550.64180.64180.64180.64180.64180.64180.67110.64180.64180.61370.641860.65220.65220.65220.65220.65220.65220.66810.65220.65220.59060.665270.72310.72480.72480.72480.72310.72310.7240.72480.72480.70150.724880.71230.70980.70980.70980.70980.70980.70980.70980.70980.69860.709890.72710.72710.72710.72710.72710.72710.72710.72710.72710.72710.7271100.61910.61910.61910.61910.61910.61910.61910.61910.61910.59920.6191110.71310.71310.71310.71310.69030.71310.71310.71310.71310.66820.7131120.69960.69960.69960.69960.72270.69960.69960.69960.69960.69960.6842130.63670.63670.63670.63670.63670.63670.65820.63670.63670.56420.6367140.69690.69690.69690.69760.69690.69690.69760.69690.69690.67830.6976150.66860.66860.66860.66860.66860.66860.63350.66860.66860.68650.6686160.69380.69380.69380.69380.69380.69380.69380.69380.69380.69380.6938170.70820.70820.70820.70820.70820.70820.70970.70820.70820.70970.7082180.55250.55250.55250.55250.55250.55250.56350.55250.55250.55250.5525190.46780.46780.46780.46780.46780.46780.46780.46780.46780.46780.4678200.55520.55520.55520.57880.55520.57880.57880.55520.55520.40920.516Mean0.63170.63170.63170.63290.63160.63280.63500.63170.63590.61770.6296Bold values represent the best values obtained for the relevant metric.For a fair comparison and the results to be more understandable, the mean values have also been calculated and added to the Tables. The results obtained for SE prove that all the algorithms can successfully classify the vessel pixels. On the other hand, higher SP values obtained in the simulations correspond to a successful distinguishing of the vessel pixels and background pixels. Finally, each algorithm can be expressed as successful enough in classifying both the vessel pixels and background due to the ACC values obtained. The F-score statistical parameter provides a single score that balances both the concerns of sensitivity and precision. The higher and similar F-score values produced by the algorithms prove the classifying success on whole image.As mentioned before, for the statistical results to be more understandable, the mean values of the all statistical parameters have been combined in Tables9 and 10. As seen from the statistical results given in these two tables, ChOA and PSO algorithms have produced a bit better results, respectively, for DRIVE and STARE databases. It can generally be said that the metaheuristic algorithms show similar clustering performances and can successfully be used for clustering based biomedical image processing.Table 9
The statistical performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms for the retinal images taken from DRIVE database.
MethodSensitivitySpecificityAccuracyF-scoreJS0.811930.9778850.960490.71843MPA0.811930.9778850.960490.71843TSA0.811930.9778850.960490.71843MA0.811930.9778850.960490.71843ChOA0.848620.982370.96850.7261SMA0.814140.978090.961740.719575AOA0.819280.9790650.9625450.717435EO0.811930.9778850.960490.71843PSO0.811930.9793850.960490.71843GWO0.76760.9687350.945190.697295DE0.816240.9778150.96190.717475Bold values represent the best values obtained for the relevant metric.Table 10
The statistical performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms for the retinal images taken from STARE database.
MethodSensitivitySpecificityAccuracyF-scoreJS0.8290750.977940.961540.631745MPA0.827020.977810.9612250.631705TSA0.827020.977810.9612250.631705MA0.8234850.9771250.96070.63292ChOA0.817390.97690.9595550.631635SMA0.82490.977640.9609050.6328AOA0.8343050.9785250.9625550.635085EO0.827020.977810.9612250.631705PSO0.845410.982770.9689450.635945GWO0.8237950.980020.9641750.617725DE0.816640.9764950.958880.62966Bold values represent the best values obtained for the relevant metric.In Tables11 and 12, a detailed statistical performance comparison has been realized between the metaheuristic approaches analyzed in this work and the studies published in literature [59–78]. As seen from Table 11 given for DRIVE database, while the algorithms other than ChOA produce close results to [61] in terms of sensitivity, the ChOA algorithm produces a bit better sensitivity value than all the other algorithms. Also, due to their higher SE values, the performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms can be stated as too similar to [61, 76] for STARE database. On the other hand, the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms can provide high SP performance as in the other studies in literature for both databases. Finally, in both databases, the ACC values obtained for each of the metaheuristic approaches seem similar to [61] but a bit better than the other algorithms. When the accuracy values obtained are evaluated, it can clearly be seen that the performance of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms in terms of pixel classification is higher than the conventional gradient based algorithms.Table 11
Performance comparison of JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms and other methods for DRIVE database.
MethodSensitivitySpecificityAccuracyYou et al. [59]0.74100.97510.9434Roychowdhury et al. [60]0.7250.9830.952Wang et al. [61]0.81730.97330.9767Mendonca and Campilho [62]0.73440.97640.9452Azzopardi et al. [63]0.7650.9700.944Fraz et al. [64]0.74060.98070.9480Odstrcilik et al. [65]0.7060.96930.934Marin et al. [66]0.70680.98010.9452Kaba et al. [67]0.74660.9680.941Dash and Bhoi [68]0.7190.9760.955Argüello et al. [69]0.72090.97580.9431Asad et al. [70]0.73880.92880.9028Zhao et al. [71]0.7350.9780.947Zhang et al. [72]0.71200.97240.9382Imani et al. [73]0.75240.97530.9523Metaheuristic algorithmsJS0.811930.9778850.96049MPA0.811930.9778850.96049TSA0.811930.9778850.96049MA0.811930.9778850.96049ChOA0.848620.982370.9685SMA0.814140.978090.96174AOA0.819280.9790650.962545EO0.811930.9778850.96049PSO0.811930.9793850.96049GWO0.76760.9687350.94519DE0.816240.9778150.9619Bold values represent the best values obtained for the relevant metric.Table 12
Performance comparison of JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms and other methods for STARE database.
MethodSensitivitySpecificityAccuracyYou et al. [59]0.7260.9760.950Roychowdhury et al. [60]0.7720.9730.952Wang et al. [61]0.81040.97910.9813Fraz et al. [64]0.7550.9760.953Odstrcilik et al. [65]0.78470.95120.9341Kaba et al. [67]0.76190.9670.9456Argüello et al. [69]0.73050.96880.9448Zhao et al. [71]0.7190.9770.951Zhang et al. [72]0.7180.9750.948Imani et al. [73]0.75020.97450.959Mendonca et al. [74]0.7180.9730.946Xiao et al. [75]0.7150.9740.948Yin et al. [76]0.8540.9420.933Lázár and Hajdu [77]0.7250.9750.949Strisciuglio et al. [78]0.8010.9720.954Metaheuristic algorithmsJS0.8290750.977940.96154MPA0.827020.977810.961225TSA0.827020.977810.961225MA0.8234850.9771250.9607ChOA0.817390.97690.959555SMA0.82490.977640.960905AOA0.8343050.9785250.962555EO0.827020.977810.961225PSO0.845410.982770.968945GWO0.8237950.980020.964175DE0.816640.9764950.95888Bold values represent the best values obtained for the relevant metric.The retinal images contained in the DRIVE and STARE databases differ from each other in terms of contrast, brightness, vessel structures, and image sizes. Therefore, the analyses of the statistical performances of the improved algorithms on each database also become important [79]. The performance of each algorithm for DRIVE and STARE databases has been given in terms of mean SE, SP, ACC, and F-score in Figure 9. It can be seen from the figure that the algorithms can produce similar statistical performances on two separate databases containing different images and having different structures.Figure 9
Statistical performance analyzes of the improved algorithms for DRIVE and STARE databases. (a) DRIVE database. (b) STARE database.
(a)(b)The convergence rates of the algorithms have been compared in Figure10 for the diseased retinal image taken from DRIVE database and the healthy retinal image taken from STARE database. As seen from Figure 10(a) obtained for the diseased retinal image, the MA algorithm converges to the minimum error value at highest convergence rate. Similarly, when Figure 10(b) obtained for the healthy retinal image is examined, it can clearly be seen that MA reaches to the minimum error value at the highest convergence rate. In general, all the metaheuristic algorithms have reached their optimal MSE values approximately at 15 cycles and MA has produced the minimum MSE value for both databases.Figure 10
Convergence speeds of the algorithms. (a) Convergence rates to minimum MSE error values obtained in the simulations realized for Figure1(b). (b) Convergence rates to minimum MSE error values obtained in the simulations realized for Figure 2(a).
(a)(b)Due to the different optimization phases they include, the Convergence Rate metric alone will be inadequate in the performance analysis of metaheuristic algorithms. In order to reflect the actual computational requirements, another performance criterion calledNumber of Function Evaluations (NFEs) has also to be analyzed. The NFEs defines the number of times the objective function is evaluated during the optimization process, and it also provides more fair and objective results in terms of the performance analysis. In addition to the NFEs of each algorithm, the Elapsed CPU times for NFEs and the Percentage of these times in the total CPU times have also been calculated. The results related with the NFEs based analyses have been presented in Table 13. From the results, it can be concluded that the value of the NFEs parameter has a directly proportional effect on the CPU time. However, when Figure 10 and Table 13 have been evaluated simultaneously, it is also seen that the NFEs value does not cause an important variation on the minimum MSE values reached.Table 13
Number of function evolutions (NFEs) based convergence analyzes of the metaheuristic algorithms.
AlgorithmTotal NFEsElapsed time for NFEs (seconds)Figure1(b)Figure2(a)JS10103.192 s (%80.90 of the total CPU time)3.988 s (%82.10 of the total CPU time)MPA20006.346 s (%81.53 of the total CPU time)7.886 s (%82.67 of the total CPU time)TSA10003.264 s (%82.19 of the total CPU time)4.011 s (%82.88 of the total CPU time)MA31209.752 s (%80.21 of the total CPU time)12.043 s (%81.23 of the total CPU time)ChOA10003.244 s (%81.30 of the total CPU time)4.060 s (%82.58 of the total CPU time)SMA10003.238 s (%81.14 of the total CPU time)4.073 s (%82.61 of the total CPU time)AOA10103.419 s (%82.25 of the total CPU time)4.112 s (%82.26 of the total CPU time)EO10003.151 s (%81.40 of the total CPU time)3.976 s (%82.22 of the total CPU time)PSO10103.230 s (%80.79 of the total CPU time)4.083 s (%82.15 of the total CPU time)GWO10103.313 s (%82.05 of the total CPU time)4.105 s (%82.83 of the total CPU time)DE10103.221 s (%80.01 of the total CPU time)4.067 s (%81.71 of the total CPU time)The standard deviation, which represents the stability of the algorithms, is one of the other important performance criteria that must be analyzed. If an algorithm can reach approximately similar MSE values at each cycle, the low standard deviation value of that algorithm will also be an indicator of its stability. The standard deviations of the algorithms for 20 random runs have been given in Table14 for DRIVE and STARE databases.Table 14
Performance comparison of the JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms for DRIVE and STARE databases.
DRIVESTAREJSMinimum MSE0.74210.7507Standard deviation1.03908e − 053.11027e − 07CPU time (second)3.94524.8571MPAMinimum MSE0.72150.7218Standard deviation7.72127e − 094.84936e − 09CPU time (second)7.78339.5382TSAMinimum MSE0.76450.7782Standard deviation0.0207773590.01876178CPU time (second)3.97104.8395MAMinimum MSE0.70580.71Standard deviation2.54232e − 102.5175e − 10CPU time (second)12.156614.8251ChOAMinimum MSE0.77260.7928Standard deviation0.0136456480.308106679CPU time (second)3.98974.9164SMAMinimum MSE0.75120.7619Standard deviation0.0002239777.81482e − 05CPU time (second)3.99064.9299AOAMinimum MSE0.79330.8Standard deviation0.1122974120.150191448CPU time (second)4.15674.9382EOMinimum MSE0.73280.7416Standard deviation2.51e − 079.04214e − 06CPU time (second)3.87064.8354Minimum MSE0.7520.7337PSOStandard deviation2.58125e − 083.06854e − 08CPU time (second)3.99764.9700Minimum MSE0.72530.719GWOStandard deviation5.45881e − 069.51344e − 06CPU time (second)4.03754.9556Minimum MSE0.82120.7732DEStandard deviation2.15721e − 053.41578e − 05CPU time (second)4.02574.9771Bold values represent the best values obtained for the relevant metric.For a more detailed analysis, the minimum MSE values reached and the CPU times required have also been added to Table14. The close MSE values obtained for the algorithms correspond to a similar convergence and clustering performances for each algorithm. On the other hand, due to their lower standard deviation values for the images taken from both DRIVE and STARE databases, the MA and MPA algorithms seem a little more stable in terms of clustering when compared to other algorithms. Another important performance metric analyzed in this work is the CPU time value.For a fair and realistic comparison, the convergence speed in addition to the CPU time parameter should also be analyzed. In this work, the simulations have been realized on an Intel i7-6700 CPU with 3,4 GHz frequency and 16 GB RAM. However, the operating system was chosen as 64 bit Windows 10 Pro. When the CPU time values obtained for the algorithms in both databases are examined, it is seen that although the MA is the algorithm with the highest convergence speed, it is obtained as the slowest algorithm in terms of CPU time. Similarly, the MPA algorithm, which is the second best algorithm (almost same as GWO) in terms of convergence speed, was obtained as the second worst algorithm in terms of the CPU time.In order to statistically validate the numerical results obtained, theWilcoxon rank sum-test which indicates the significant differences among algorithms has been applied [80]. The test results obtained on the RStudio-Software for the confidence interval of 95% (p<0.05) have been given in Table 15. This table includes only the algorithms of which significant differences are (p<0.05) obtained according to Wilcoxon rank sum-test results. The results obtained for DRIVE database indicate that the statistical performance of the MA algorithm is better than the other algorithms that produce statistically significant results. Similarly, it has also been observed for STARE database that the JS, MA, and MPA algorithms produce higher statistically significant results when compared to other algorithms. Finally, it is also seen for both databases that the AOA algorithm does not produce statistically significant results compared to other algorithms.Table 15
The Wilcoxon sum-rank test results of the metaheuristic algorithms (p<0.05).
AlgorithmBetter thanDRIVE database (Figure1(b))AOAChOAAOA (6.6063e − 08)DEAOA (3.3675e − 08) ChOA (3.4729e − 08) SMA (2.7754e − 06) TSA (3.4729e − 08) GWO (3.3254e − 09)EOAOA (2.3615e − 08) ChOA (2.4387e − 08) SMA (4.6093e − 08) TSA (2.4387e − 08) GWO (2.0841e − 09)JSAOA (2.8636e − 08) ChOA (2.9550e − 08) SMA (1.0764e − 06) TSA (2.9550e − 08) GWO (2.6874e − 09)MAAOA (7.7176e − 09) ChOA (8.0065e − 09) DE (9.5756e − 03) EO (4.0127e − 02) JS (1.9799e − 02) PSO (3.9817e − 02) SMA (8.0065e − 09) TSA (8.0065e − 09) GWO (4.6827e − 10)MPAAOA (1.0876e − 08) ChOA (1.1267e − 08) DE (3.5354e − 02) SMA (1.1267e − 08) TSA (1.1267e − 08) GWO (7.4275e − 10)PSOAOA (2.3274e − 08) ChOA (2.4037e − 08) SMA (2.4037e − 08) TSA (2.4037e − 08) GWO (2.0445e − 09)SMAAOA (6.6063e − 08) ChOA (1.4509e − 11) TSA (2.9018e − 11) GWO (2.1025e − 07)TSAAOA (6.6063e − 08) ChOA (1.4509e − 11)GWOAOA (7.7176e − 09) ChOA (8.0065e − 09) TSA (8.0065e − 09)STARE database (Figure2(a))AOAChOAAOA (6.7860e − 08)DEAOA (4.9511e − 08) ChOA (1.3011e − 05) SMA (1.9188e − 03) TSA (4.9511e − 08) GWO (4.7792e − 08)EOAOA (3.4782e − 08) ChOA (1.0345e − 05) SMA (9.2244e − 07) TSA (3.4782e − 08) GWO (3.3520e − 08)JSAOA (1.1267e − 08) ChOA (4.9771e − 06) DE (3.3383e − 03) EO (4.3208e − 02) SMA (1.5720e − 08) TSA (1.1267e − 08) GWO (1.0799e − 08)MAAOA (8.0065e − 09) ChOA (3.9879e − 06) DE (9.3030e − 04) EO (9.5844e − 03) SMA (7.9919e − 09) TSA (8.0065e − 09) GWO (7.6609e − 09)MPAAOA (8.0065e − 09) ChOA (3.9879e − 06) DE (9.3030e − 04) EO (9.5844e − 03) SMA (7.9919e − 09) TSA (8.0065e − 09) GWO (7.6609e − 09)PSOAOA (1.5124e − 08) ChOA (6.0247e − 06) DE (6.9581e − 03) SMA (1.5098e − 08) TSA (1.5124e − 08) GWO (1.4517e − 08)SMAAOA (6.7860e − 08) ChOA (1.5968e − 05) TSA (6.7860e − 08) GWO (1.5634e − 05)TSAAOA (1.4509e − 11) ChOA (1.5983e − 05)GWOAOA (6.5690e − 08) ChOA (1.5634e − 05) TSA (6.5690e − 08)


## 3.1. Performance Measures

Firstly, a detailed statistical performance comparison based on SE, SP, ACC, and F-score has been realized for the algorithms improved. Then, the convergence rate, MSE, CPU time, and standard deviation performances of the algorithms have been compared at each other.The mathematical expressions of SE, SP, ACC, and F-score parameters can be given as follows:(37)SE=TPTP+FN,SP=TNTN+FP,ACC=TP+TNTP+FN+TN+FP,F−score=TPTP+1/2FP+FN,where TP is the correctly estimated vessel pixel number, FN is the number of vessel pixels incorrectly defined as background pixels, TN is the correctly estimated background pixel number, and finally FP is the number of background pixels incorrectly defined as vessel pixels.


## 3.2. Performance Analysis for DRIVE and STARE Databases

In Tables3 and 4, the results relevant to sensitivity, specificity, and accuracy produced by the novel algorithms for 20 different images are given for DRIVE and STARE databases, respectively. Similarly, the results obtained for PSO, GWO, and DE algorithms have been given in Table 5 for DRIVE database and in Table 6 for STARE database.Table 3
The performance measures of the different images in the DRIVE database in terms of sensitivity, specificity, and accuracy.
SensitivitySpecificityAccuracyImageJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEO10.89130.89130.89130.89130.89130.89130.9960.89130.98840.98840.98840.98840.98840.98840.99960.98840.97910.97910.97910.97910.97910.97910.99930.979120.78570.78570.78570.78570.78570.78570.83020.78570.97030.97030.97030.97030.97030.97030.97760.97030.94790.94790.94790.94790.94790.94790.96040.947930.87870.87870.87870.87870.87870.87870.87870.87870.98670.98670.98670.98670.98670.98670.98670.98670.9760.9760.9760.9760.9760.9760.9760.97640.7230.7230.7230.7230.7230.7230.7230.7230.96450.96450.96450.96450.96450.96450.96450.96450.93710.93710.93710.93710.93710.95470.93710.937150.95140.95140.95140.95140.95140.95140.83260.95140.99550.99550.99550.99550.99550.99550.98250.99550.99180.99180.99180.99180.99180.99180.96840.991860.95720.95720.95720.95720.95720.95720.95720.95720.99540.99540.99540.99540.99540.99540.99540.99540.99170.99170.99170.99170.99170.99170.99170.991770.88740.88740.88740.88740.95930.88740.88740.88740.99030.99030.99030.99030.99670.99030.99030.99030.98220.98220.98220.98220.9940.98220.98220.982280.55060.55060.55060.55060.55060.55060.55060.55060.94490.94490.94490.94490.94490.94490.94490.94490.90190.90190.90190.90190.90190.90190.90190.901990.78180.78180.78180.78180.78180.78180.78180.78180.97610.97610.97610.97610.97610.97610.97610.97610.95680.95680.95680.95680.95680.95680.95680.9568100.89320.89320.89320.89320.89320.93740.89320.89320.99080.99080.99080.99080.99080.99490.99080.99080.98310.98310.98310.98310.98310.99050.98310.9831110.7870.7870.7870.7870.74990.7870.83630.7870.9760.9760.9760.9760.97060.9760.98250.9760.95690.95690.95690.95690.94740.95690.96850.9569120.66240.66240.66240.66240.86090.66240.66240.66240.95740.95740.95740.95740.98610.95740.95740.95740.92430.92430.92430.92430.97470.92430.92430.9243130.79920.79920.79920.79920.74070.79920.79920.79920.97250.97250.97250.97250.96210.97250.97250.97250.95160.95160.95160.95160.93380.95160.95160.9516140.74390.74390.74390.74390.91370.74390.74390.74390.9730.9730.9730.9730.99240.9730.9730.9730.95110.95110.95110.95110.98610.95110.95110.9511150.94440.94440.94440.94440.94440.94440.89660.94440.99530.99530.99530.99530.99530.99530.99080.99530.99130.99130.99130.99130.99130.99130.98310.9913160.83870.83870.83870.83870.83870.83870.83870.83870.98240.98240.98240.98240.98240.98240.98240.98240.96830.96830.96830.96830.96830.96830.96830.9683170.70170.70170.70170.70170.8740.70170.70170.70170.96720.96720.96720.96720.98860.96720.96720.96720.94090.94090.94090.94090.97910.94090.94090.9409180.89720.89720.89720.89720.89720.89720.89720.89720.98850.98850.98850.98850.98850.98850.98850.98850.97940.97940.97940.97940.97940.97940.97940.9794190.81610.81610.81610.81610.81610.81610.93120.81610.97580.97580.97580.97580.97580.97580.99190.97580.95720.95720.95720.95720.95720.95720.98560.9572200.74770.74770.74770.74770.96460.74770.74770.74770.96670.96670.96670.96670.99630.96670.96670.96670.94120.94120.94120.94120.99330.94120.94120.9412Mean0.81190.81190.81190.81190.84860.81410.81920.81190.97780.97780.97780.97780.98230.97800.97900.97780.96040.96040.96040.96040.96850.96170.962540.9604Bold values represent the best values obtained for the relevant metric.Table 4
The performance measures of the different images in the STARE database in terms of sensitivity, specificity, and accuracy.
ImageSensitivitySpecificityAccuracyJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEOJSMPATSAMAChOASMAAOAEO10.74980.74980.74980.74980.74980.74980.74980.74980.97420.97420.97420.97420.97420.97420.97420.97420.95330.95330.95330.95330.95330.95330.95330.953320.68760.68760.68760.68760.68760.68760.68760.68760.97010.97010.97010.97010.97010.97010.97010.97010.94540.94540.94540.94540.94540.94540.94540.945430.64230.64230.64230.64230.64230.64230.64230.64230.95680.95680.95680.95680.95680.95680.95680.95680.92290.92290.92290.92290.92290.92290.92290.922940.35130.35130.35130.35130.35130.35130.35130.35130.85080.85080.85080.85080.85080.85080.85080.85080.75750.75750.75750.75750.75750.75750.75750.757550.84070.84070.84070.84070.84070.84070.91930.84070.98250.98250.98250.98250.98250.98250.99180.98250.96850.96850.96850.96850.96850.96850.98520.968560.82110.82110.82110.82110.82110.82110.95240.82110.9810.9810.9810.9810.9810.9810.99560.9810.96560.96560.96560.96560.96560.96560.99190.965670.93240.99450.99450.99450.93240.93240.93010.99450.99370.99950.99950.99950.99370.99370.99340.99950.98840.99910.99910.99910.98840.98840.9880.999180.99870.89550.89550.89550.89550.89550.89550.89550.9990.99060.99060.99060.99060.99060.99060.99060.99980.98280.98280.98280.98280.98280.98280.982890.92810.92810.92810.92810.92810.92810.92810.92810.99320.99320.99320.99320.99320.99320.99320.99320.98760.98760.98760.98760.98760.98760.98760.9876100.81430.81430.81430.81430.81430.81430.81430.81430.97960.97960.97960.97960.97960.97960.97960.97960.96330.96330.96330.96330.96330.96330.96330.9633110.93770.93770.93770.93770.73010.93770.93770.93770.9950.9950.9950.9950.97260.9950.9950.9950.99070.99070.99070.99070.95030.99070.99070.9907120.81630.81630.81630.81630.89340.81630.81630.81630.97850.97850.97850.97850.98850.97850.97850.97850.96150.96150.96150.96150.97920.96150.96150.9615130.88130.88130.88130.88130.88130.88130.87520.88130.98830.98830.98830.98830.98830.98830.98760.98830.97870.97870.97870.97870.97870.97870.97750.9787140.98220.98220.98220.89180.98220.98220.89180.98220.99860.99860.99860.99050.99860.99860.99050.99860.99730.99730.99730.98250.99730.99730.98250.9973150.79680.79680.79680.79680.79680.79680.66210.79680.97940.97940.97940.97940.97940.97940.95960.97940.96250.96250.96250.96250.96250.96250.92780.9625160.90950.90950.90950.90950.90950.90950.90950.90950.99130.99130.99130.99130.99130.99130.99130.99130.98410.98410.98410.98410.98410.98410.98410.9841170.94490.94490.94490.94490.94490.94490.96120.94490.99550.99550.99550.99550.99550.99550.99690.99550.99160.99160.99160.99160.99160.99160.99420.9916180.75930.75930.75930.75930.75930.75930.95470.75930.97480.97480.97480.97480.97480.97480.99610.97480.95440.95440.95440.95440.95440.95440.99290.9544190.90550.90550.90550.90550.90550.90550.90550.90550.98960.98960.98960.98960.98960.98960.98960.98960.98130.98130.98130.98130.98130.98130.98130.9813200.88170.88170.88170.90140.88170.90140.90140.88170.98690.98690.98690.98130.98690.98930.98930.98690.97640.97640.97640.98070.97640.98070.98070.9764Mean0.82900.82700.82700.82340.81730.82490.83430.82700.97790.97780.97780.97710.97690.97760.97850.97780.96150.96120.96120.96070.95950.96090.96250.9612Bold values represent the best values obtained for the relevant metric.Table 5
The performance measures of the different images in the DRIVE database in terms of sensitivity, specificity and accuracy for PSO, GWO and DE.
ImageSensitivitySpecificityAccuracyPSOGWODEPSOGWODEPSOGWODE10.89130.86060.89130.98840.98470.98840.97910.97240.979120.78570.83020.83020.97030.97760.97760.94790.96040.960430.87870.53920.87870.98670.9230.98670.9760.8680.97640.7230.77930.7230.96450.97350.96450.93710.95270.937150.95140.95140.95140.99550.99550.99550.99180.99180.991860.95720.86050.95720.99540.98360.99540.99170.97070.991770.88740.73310.88740.99030.97270.99030.98220.95040.982280.55060.86720.55060.94490.98920.94490.90190.980.901990.78180.82510.78180.97610.98170.97610.95680.96690.9568100.89320.89320.7180.99080.99080.97050.98310.98310.9467110.7870.98790.7870.9760.99890.9760.95690.9980.9569120.66240.68240.66240.95740.96090.95740.92430.93050.9243130.79920.64890.79920.97250.94260.97250.95160.90130.9516140.74390.92920.74390.9730.99390.9730.95110.98870.9511150.94440.55510.94440.99530.93930.99530.99130.89320.9913160.83870.70840.83870.98240.96310.98240.96830.93450.9683170.70170.70170.70170.96720.96720.96720.94090.94090.9409180.89720.74380.89720.98850.96630.98850.97940.94050.9794190.81610.50840.81610.97580.90370.95780.95720.8390.9572200.74770.74640.96460.99670.96650.99630.94120.94080.9933Mean0.81190.76760.81620.97930.96870.97780.96040.94510.9619Bold values represent the best values obtained for the relevant metric.Table 6
The performance measures of the different images in the STARE database in terms of sensitivity, specificity, and accuracy for PSO, GWO, and DE.
ImageSensitivitySpecificityAccuracyPSOGWODEPSOGWODEPSOGWODE10.72050.96110.72050.97020.99680.97020.94610.99410.946120.68760.84610.68760.97010.98780.97010.94540.97740.945430.64230.97610.64230.95680.9980.95680.92290.99630.922940.66340.66340.35130.9540.9540.85080.91910.91910.757550.84070.75720.84070.98250.97080.98250.96850.94790.968560.82110.71040.87890.9810.96490.98790.96560.93740.977970.99450.79990.99450.99950.97850.99950.99910.96120.999180.89550.79470.89550.99060.97950.99060.98280.96270.982890.92810.92810.92810.99320.99320.99320.98760.98760.9876100.81430.9250.81430.97960.99270.97960.96330.98660.9633110.93770.76540.93770.9950.97720.9950.99070.95840.9907120.81630.81630.78620.97850.97850.97420.96150.96150.9539130.88130.70340.88130.98830.96430.98830.97870.93620.9787140.98220.71020.89180.99860.96870.99050.99730.94340.9825150.79680.93930.79680.97940.99470.97940.96250.99020.9625160.99050.90950.90950.99130.99130.99130.98410.98410.9841170.94490.96120.94490.99550.99690.99550.99160.99420.9916180.75930.75930.75930.97480.97480.97480.95440.95440.9544190.90950.90550.90950.98960.98960.98960.98130.98130.9813200.88170.64380.76210.98690.94820.97010.97640.90950.9468Mean0.84540.82370.81660.98270.98000.97640.96890.96410.9588Bold values represent the best values obtained for the relevant metric.In addition to the SE, SP, and ACC parameters, theF-score values obtained for each algorithms have also been given in Tables 7 and 8, respectively, for DRIVE and STARE databases.Table 7
The performance measures of the different images in the DRIVE database in terms ofF-score.
ImageF-scoreJSMPATSAMAChOASMAAOAEOPSOGWODE10.78310.78310.78310.78310.78310.78310.78530.78310.78310.76380.783120.79040.79040.79040.79040.79040.79040.80650.79040.79040.80650.806530.70430.70430.70430.70430.70430.70430.70430.70430.70430.60550.704340.72210.72210.72210.72210.72210.76320.72210.72210.72210.71620.722150.7660.7660.7660.7660.7660.7660.73220.7660.7660.7660.76660.7110.7110.7110.7110.7110.7110.7110.7110.7110.7230.71170.70890.70890.70890.70890.7220.70890.70890.70890.70890.67570.708980.5870.5870.5870.5870.5870.5870.5870.5870.5870.70290.58790.65630.65630.65630.65630.65630.65630.65630.65630.65630.69120.6563100.75740.75740.75740.75740.75740.73920.75740.75740.75740.75740.6835110.74180.74180.74180.74180.72440.74180.75550.74180.74180.75460.7418120.65910.65910.65910.65910.71610.65910.65910.65910.65910.65750.6591130.72460.72460.72460.72460.70550.72460.72460.72460.72460.60580.7246140.7060.7060.7060.7060.74710.7060.7060.7060.7060.76230.706150.70370.70370.70370.70370.70370.70370.68860.70370.70370.58250.7037160.78430.78430.78430.78430.78430.78430.78430.78430.78430.70460.7843170.67010.67010.67010.67010.71010.67010.67010.67010.67010.67010.6701180.75380.75380.75380.75380.75380.75380.75380.75380.75380.69810.7538190.76910.76910.76910.76910.76910.76910.76610.76910.76910.59670.7691200.66960.66960.66960.66960.70830.66960.66960.66960.66960.70550.7083Mean0.71840.71840.71840.71840.72610.71950.71740.71840.71840.69720.7174Bold values represent the best values obtained for the relevant metric.Table 8
The performance measures of the different images in the STARE database in terms ofF-score.
ImageF-scoreJSMPATSAMAChOASMAAOAEOPSOGWODE10.6010.6010.6010.6010.6010.6010.6010.6010.6010.58970.60120.55940.55940.55940.55940.55940.55940.55940.55940.55940.57080.559430.5680.5680.5680.5680.5680.5680.5680.5680.5680.61020.56840.43850.43850.43850.43850.43850.43850.43850.43850.52330.52330.438550.64180.64180.64180.64180.64180.64180.67110.64180.64180.61370.641860.65220.65220.65220.65220.65220.65220.66810.65220.65220.59060.665270.72310.72480.72480.72480.72310.72310.7240.72480.72480.70150.724880.71230.70980.70980.70980.70980.70980.70980.70980.70980.69860.709890.72710.72710.72710.72710.72710.72710.72710.72710.72710.72710.7271100.61910.61910.61910.61910.61910.61910.61910.61910.61910.59920.6191110.71310.71310.71310.71310.69030.71310.71310.71310.71310.66820.7131120.69960.69960.69960.69960.72270.69960.69960.69960.69960.69960.6842130.63670.63670.63670.63670.63670.63670.65820.63670.63670.56420.6367140.69690.69690.69690.69760.69690.69690.69760.69690.69690.67830.6976150.66860.66860.66860.66860.66860.66860.63350.66860.66860.68650.6686160.69380.69380.69380.69380.69380.69380.69380.69380.69380.69380.6938170.70820.70820.70820.70820.70820.70820.70970.70820.70820.70970.7082180.55250.55250.55250.55250.55250.55250.56350.55250.55250.55250.5525190.46780.46780.46780.46780.46780.46780.46780.46780.46780.46780.4678200.55520.55520.55520.57880.55520.57880.57880.55520.55520.40920.516Mean0.63170.63170.63170.63290.63160.63280.63500.63170.63590.61770.6296Bold values represent the best values obtained for the relevant metric.For a fair comparison and the results to be more understandable, the mean values have also been calculated and added to the Tables. The results obtained for SE prove that all the algorithms can successfully classify the vessel pixels. On the other hand, higher SP values obtained in the simulations correspond to a successful distinguishing of the vessel pixels and background pixels. Finally, each algorithm can be expressed as successful enough in classifying both the vessel pixels and background due to the ACC values obtained. The F-score statistical parameter provides a single score that balances both the concerns of sensitivity and precision. The higher and similar F-score values produced by the algorithms prove the classifying success on whole image.As mentioned before, for the statistical results to be more understandable, the mean values of the all statistical parameters have been combined in Tables9 and 10. As seen from the statistical results given in these two tables, ChOA and PSO algorithms have produced a bit better results, respectively, for DRIVE and STARE databases. It can generally be said that the metaheuristic algorithms show similar clustering performances and can successfully be used for clustering based biomedical image processing.Table 9
The statistical performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms for the retinal images taken from DRIVE database.
MethodSensitivitySpecificityAccuracyF-scoreJS0.811930.9778850.960490.71843MPA0.811930.9778850.960490.71843TSA0.811930.9778850.960490.71843MA0.811930.9778850.960490.71843ChOA0.848620.982370.96850.7261SMA0.814140.978090.961740.719575AOA0.819280.9790650.9625450.717435EO0.811930.9778850.960490.71843PSO0.811930.9793850.960490.71843GWO0.76760.9687350.945190.697295DE0.816240.9778150.96190.717475Bold values represent the best values obtained for the relevant metric.Table 10
The statistical performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms for the retinal images taken from STARE database.
MethodSensitivitySpecificityAccuracyF-scoreJS0.8290750.977940.961540.631745MPA0.827020.977810.9612250.631705TSA0.827020.977810.9612250.631705MA0.8234850.9771250.96070.63292ChOA0.817390.97690.9595550.631635SMA0.82490.977640.9609050.6328AOA0.8343050.9785250.9625550.635085EO0.827020.977810.9612250.631705PSO0.845410.982770.9689450.635945GWO0.8237950.980020.9641750.617725DE0.816640.9764950.958880.62966Bold values represent the best values obtained for the relevant metric.In Tables11 and 12, a detailed statistical performance comparison has been realized between the metaheuristic approaches analyzed in this work and the studies published in literature [59–78]. As seen from Table 11 given for DRIVE database, while the algorithms other than ChOA produce close results to [61] in terms of sensitivity, the ChOA algorithm produces a bit better sensitivity value than all the other algorithms. Also, due to their higher SE values, the performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms can be stated as too similar to [61, 76] for STARE database. On the other hand, the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms can provide high SP performance as in the other studies in literature for both databases. Finally, in both databases, the ACC values obtained for each of the metaheuristic approaches seem similar to [61] but a bit better than the other algorithms. When the accuracy values obtained are evaluated, it can clearly be seen that the performance of the JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms in terms of pixel classification is higher than the conventional gradient based algorithms.Table 11
Performance comparison of JS, MPA, TSA, MA, ChOA, SMA, AOA, EO, PSO, GWO, and DE algorithms and other methods for DRIVE database.
MethodSensitivitySpecificityAccuracyYou et al. [59]0.74100.97510.9434Roychowdhury et al. [60]0.7250.9830.952Wang et al. [61]0.81730.97330.9767Mendonca and Campilho [62]0.73440.97640.9452Azzopardi et al. [63]0.7650.9700.944Fraz et al. [64]0.74060.98070.9480Odstrcilik et al. [65]0.7060.96930.934Marin et al. [66]0.70680.98010.9452Kaba et al. [67]0.74660.9680.941Dash and Bhoi [68]0.7190.9760.955Argüello et al. [69]0.72090.97580.9431Asad et al. [70]0.73880.92880.9028Zhao et al. [71]0.7350.9780.947Zhang et al. [72]0.71200.97240.9382Imani et al. [73]0.75240.97530.9523Metaheuristic algorithmsJS0.811930.9778850.96049MPA0.811930.9778850.96049TSA0.811930.9778850.96049MA0.811930.9778850.96049ChOA0.848620.982370.9685SMA0.814140.978090.96174AOA0.819280.9790650.962545EO0.811930.9778850.96049PSO0.811930.9793850.96049GWO0.76760.9687350.94519DE0.816240.9778150.9619Bold values represent the best values obtained for the relevant metric.Table 12
Performance comparison of JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms and other methods for STARE database.
MethodSensitivitySpecificityAccuracyYou et al. [59]0.7260.9760.950Roychowdhury et al. [60]0.7720.9730.952Wang et al. [61]0.81040.97910.9813Fraz et al. [64]0.7550.9760.953Odstrcilik et al. [65]0.78470.95120.9341Kaba et al. [67]0.76190.9670.9456Argüello et al. [69]0.73050.96880.9448Zhao et al. [71]0.7190.9770.951Zhang et al. [72]0.7180.9750.948Imani et al. [73]0.75020.97450.959Mendonca et al. [74]0.7180.9730.946Xiao et al. [75]0.7150.9740.948Yin et al. [76]0.8540.9420.933Lázár and Hajdu [77]0.7250.9750.949Strisciuglio et al. [78]0.8010.9720.954Metaheuristic algorithmsJS0.8290750.977940.96154MPA0.827020.977810.961225TSA0.827020.977810.961225MA0.8234850.9771250.9607ChOA0.817390.97690.959555SMA0.82490.977640.960905AOA0.8343050.9785250.962555EO0.827020.977810.961225PSO0.845410.982770.968945GWO0.8237950.980020.964175DE0.816640.9764950.95888Bold values represent the best values obtained for the relevant metric.The retinal images contained in the DRIVE and STARE databases differ from each other in terms of contrast, brightness, vessel structures, and image sizes. Therefore, the analyses of the statistical performances of the improved algorithms on each database also become important [79]. The performance of each algorithm for DRIVE and STARE databases has been given in terms of mean SE, SP, ACC, and F-score in Figure 9. It can be seen from the figure that the algorithms can produce similar statistical performances on two separate databases containing different images and having different structures.Figure 9
Statistical performance analyzes of the improved algorithms for DRIVE and STARE databases. (a) DRIVE database. (b) STARE database.
(a)(b)The convergence rates of the algorithms have been compared in Figure10 for the diseased retinal image taken from DRIVE database and the healthy retinal image taken from STARE database. As seen from Figure 10(a) obtained for the diseased retinal image, the MA algorithm converges to the minimum error value at highest convergence rate. Similarly, when Figure 10(b) obtained for the healthy retinal image is examined, it can clearly be seen that MA reaches to the minimum error value at the highest convergence rate. In general, all the metaheuristic algorithms have reached their optimal MSE values approximately at 15 cycles and MA has produced the minimum MSE value for both databases.Figure 10
Convergence speeds of the algorithms. (a) Convergence rates to minimum MSE error values obtained in the simulations realized for Figure1(b). (b) Convergence rates to minimum MSE error values obtained in the simulations realized for Figure 2(a).
(a)(b)Due to the different optimization phases they include, the Convergence Rate metric alone will be inadequate in the performance analysis of metaheuristic algorithms. In order to reflect the actual computational requirements, another performance criterion calledNumber of Function Evaluations (NFEs) has also to be analyzed. The NFEs defines the number of times the objective function is evaluated during the optimization process, and it also provides more fair and objective results in terms of the performance analysis. In addition to the NFEs of each algorithm, the Elapsed CPU times for NFEs and the Percentage of these times in the total CPU times have also been calculated. The results related with the NFEs based analyses have been presented in Table 13. From the results, it can be concluded that the value of the NFEs parameter has a directly proportional effect on the CPU time. However, when Figure 10 and Table 13 have been evaluated simultaneously, it is also seen that the NFEs value does not cause an important variation on the minimum MSE values reached.Table 13
Number of function evolutions (NFEs) based convergence analyzes of the metaheuristic algorithms.
AlgorithmTotal NFEsElapsed time for NFEs (seconds)Figure1(b)Figure2(a)JS10103.192 s (%80.90 of the total CPU time)3.988 s (%82.10 of the total CPU time)MPA20006.346 s (%81.53 of the total CPU time)7.886 s (%82.67 of the total CPU time)TSA10003.264 s (%82.19 of the total CPU time)4.011 s (%82.88 of the total CPU time)MA31209.752 s (%80.21 of the total CPU time)12.043 s (%81.23 of the total CPU time)ChOA10003.244 s (%81.30 of the total CPU time)4.060 s (%82.58 of the total CPU time)SMA10003.238 s (%81.14 of the total CPU time)4.073 s (%82.61 of the total CPU time)AOA10103.419 s (%82.25 of the total CPU time)4.112 s (%82.26 of the total CPU time)EO10003.151 s (%81.40 of the total CPU time)3.976 s (%82.22 of the total CPU time)PSO10103.230 s (%80.79 of the total CPU time)4.083 s (%82.15 of the total CPU time)GWO10103.313 s (%82.05 of the total CPU time)4.105 s (%82.83 of the total CPU time)DE10103.221 s (%80.01 of the total CPU time)4.067 s (%81.71 of the total CPU time)The standard deviation, which represents the stability of the algorithms, is one of the other important performance criteria that must be analyzed. If an algorithm can reach approximately similar MSE values at each cycle, the low standard deviation value of that algorithm will also be an indicator of its stability. The standard deviations of the algorithms for 20 random runs have been given in Table14 for DRIVE and STARE databases.Table 14
Performance comparison of the JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms for DRIVE and STARE databases.
DRIVESTAREJSMinimum MSE0.74210.7507Standard deviation1.03908e − 053.11027e − 07CPU time (second)3.94524.8571MPAMinimum MSE0.72150.7218Standard deviation7.72127e − 094.84936e − 09CPU time (second)7.78339.5382TSAMinimum MSE0.76450.7782Standard deviation0.0207773590.01876178CPU time (second)3.97104.8395MAMinimum MSE0.70580.71Standard deviation2.54232e − 102.5175e − 10CPU time (second)12.156614.8251ChOAMinimum MSE0.77260.7928Standard deviation0.0136456480.308106679CPU time (second)3.98974.9164SMAMinimum MSE0.75120.7619Standard deviation0.0002239777.81482e − 05CPU time (second)3.99064.9299AOAMinimum MSE0.79330.8Standard deviation0.1122974120.150191448CPU time (second)4.15674.9382EOMinimum MSE0.73280.7416Standard deviation2.51e − 079.04214e − 06CPU time (second)3.87064.8354Minimum MSE0.7520.7337PSOStandard deviation2.58125e − 083.06854e − 08CPU time (second)3.99764.9700Minimum MSE0.72530.719GWOStandard deviation5.45881e − 069.51344e − 06CPU time (second)4.03754.9556Minimum MSE0.82120.7732DEStandard deviation2.15721e − 053.41578e − 05CPU time (second)4.02574.9771Bold values represent the best values obtained for the relevant metric.For a more detailed analysis, the minimum MSE values reached and the CPU times required have also been added to Table14. The close MSE values obtained for the algorithms correspond to a similar convergence and clustering performances for each algorithm. On the other hand, due to their lower standard deviation values for the images taken from both DRIVE and STARE databases, the MA and MPA algorithms seem a little more stable in terms of clustering when compared to other algorithms. Another important performance metric analyzed in this work is the CPU time value.For a fair and realistic comparison, the convergence speed in addition to the CPU time parameter should also be analyzed. In this work, the simulations have been realized on an Intel i7-6700 CPU with 3,4 GHz frequency and 16 GB RAM. However, the operating system was chosen as 64 bit Windows 10 Pro. When the CPU time values obtained for the algorithms in both databases are examined, it is seen that although the MA is the algorithm with the highest convergence speed, it is obtained as the slowest algorithm in terms of CPU time. Similarly, the MPA algorithm, which is the second best algorithm (almost same as GWO) in terms of convergence speed, was obtained as the second worst algorithm in terms of the CPU time.In order to statistically validate the numerical results obtained, theWilcoxon rank sum-test which indicates the significant differences among algorithms has been applied [80]. The test results obtained on the RStudio-Software for the confidence interval of 95% (p<0.05) have been given in Table 15. This table includes only the algorithms of which significant differences are (p<0.05) obtained according to Wilcoxon rank sum-test results. The results obtained for DRIVE database indicate that the statistical performance of the MA algorithm is better than the other algorithms that produce statistically significant results. Similarly, it has also been observed for STARE database that the JS, MA, and MPA algorithms produce higher statistically significant results when compared to other algorithms. Finally, it is also seen for both databases that the AOA algorithm does not produce statistically significant results compared to other algorithms.Table 15
The Wilcoxon sum-rank test results of the metaheuristic algorithms (p<0.05).
AlgorithmBetter thanDRIVE database (Figure1(b))AOAChOAAOA (6.6063e − 08)DEAOA (3.3675e − 08) ChOA (3.4729e − 08) SMA (2.7754e − 06) TSA (3.4729e − 08) GWO (3.3254e − 09)EOAOA (2.3615e − 08) ChOA (2.4387e − 08) SMA (4.6093e − 08) TSA (2.4387e − 08) GWO (2.0841e − 09)JSAOA (2.8636e − 08) ChOA (2.9550e − 08) SMA (1.0764e − 06) TSA (2.9550e − 08) GWO (2.6874e − 09)MAAOA (7.7176e − 09) ChOA (8.0065e − 09) DE (9.5756e − 03) EO (4.0127e − 02) JS (1.9799e − 02) PSO (3.9817e − 02) SMA (8.0065e − 09) TSA (8.0065e − 09) GWO (4.6827e − 10)MPAAOA (1.0876e − 08) ChOA (1.1267e − 08) DE (3.5354e − 02) SMA (1.1267e − 08) TSA (1.1267e − 08) GWO (7.4275e − 10)PSOAOA (2.3274e − 08) ChOA (2.4037e − 08) SMA (2.4037e − 08) TSA (2.4037e − 08) GWO (2.0445e − 09)SMAAOA (6.6063e − 08) ChOA (1.4509e − 11) TSA (2.9018e − 11) GWO (2.1025e − 07)TSAAOA (6.6063e − 08) ChOA (1.4509e − 11)GWOAOA (7.7176e − 09) ChOA (8.0065e − 09) TSA (8.0065e − 09)STARE database (Figure2(a))AOAChOAAOA (6.7860e − 08)DEAOA (4.9511e − 08) ChOA (1.3011e − 05) SMA (1.9188e − 03) TSA (4.9511e − 08) GWO (4.7792e − 08)EOAOA (3.4782e − 08) ChOA (1.0345e − 05) SMA (9.2244e − 07) TSA (3.4782e − 08) GWO (3.3520e − 08)JSAOA (1.1267e − 08) ChOA (4.9771e − 06) DE (3.3383e − 03) EO (4.3208e − 02) SMA (1.5720e − 08) TSA (1.1267e − 08) GWO (1.0799e − 08)MAAOA (8.0065e − 09) ChOA (3.9879e − 06) DE (9.3030e − 04) EO (9.5844e − 03) SMA (7.9919e − 09) TSA (8.0065e − 09) GWO (7.6609e − 09)MPAAOA (8.0065e − 09) ChOA (3.9879e − 06) DE (9.3030e − 04) EO (9.5844e − 03) SMA (7.9919e − 09) TSA (8.0065e − 09) GWO (7.6609e − 09)PSOAOA (1.5124e − 08) ChOA (6.0247e − 06) DE (6.9581e − 03) SMA (1.5098e − 08) TSA (1.5124e − 08) GWO (1.4517e − 08)SMAAOA (6.7860e − 08) ChOA (1.5968e − 05) TSA (6.7860e − 08) GWO (1.5634e − 05)TSAAOA (1.4509e − 11) ChOA (1.5983e − 05)GWOAOA (6.5690e − 08) ChOA (1.5634e − 05) TSA (6.5690e − 08)


## 4. Discussion

Computer-aided biomedical image analysis provides effective approaches in diagnosing and treatment of diseases with high accuracy. Metaheuristic algorithms, as one of the most important computer aided approaches, have advantages such as global search ability, higher convergence rates, robustness, and flexibility.The simulations have been realized by using JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms, which are the most recently proposed metaheuristic algorithms in the literature. From the simulation results obtained for segmentation, it is seen that the metaheuristic algorithms produce similar and effective results for both healthy and diseased images in terms of convergence speed and error values reached. Furthermore, the results obtained in the detailed statistical analyses realized in this work prove the inherently stable, robust, and flexible behavior of the metaheuristic algorithms.When the segmentation and statistical results obtained with high accuracy have been evaluated together, it can clearly be expressed that JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms provide more effective and stable solutions in image analysis regardless of the image properties.


## 5. Conclusion

In this work, JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms, which are among the most recently proposed metaheuristic algorithms in literature, have been improved as clustering based and applied to the retinal vessel segmentation. From the simulation results, it has been observed that the performances of the algorithms in terms of convergence speed and the MSE value are close to each other. As a result of the detailed statistical analyses, including the sensitivity, specificity, and accuracy parameters, it is seen that JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms present high robustness and stability in retinal image analyses. If the algorithms are evaluated separately, the MA and MPA algorithms seem a bit more stable than the other algorithms in terms of clustering based retinal vessel segmentation due to their lower standard deviation values. Consequently, it is clearly seen from the results obtained that the performances of the JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms in terms of clustering are too similar, and they can efficiently be used in retinal vessel segmentation.In future research, firstly, the novel algorithms used in this work will be improved so as to obtain the optimal threshold values in segmentation of biomedical images with high accuracy. Secondly, the performance of each algorithm will be analyzed in functional diffusion magnetic resonance imaging in the area of ear diseases. Then, improvement strategies will be applied to each algorithm for the purpose of enhancing the sensitivity, specificity, accuracy, andF-score of the algorithms. Finally, hardware implementations for retinal vessel segmentation will be developed on a selected microprocessor.



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*Source: 4639208-2022-02-22.xml*