SAR Image Segmentation Based on Improved Grey Wolf...

Research ArticleSAR Image Segmentation Based on Improved Grey WolfOptimization Algorithm and Fuzzy C-Means

M Q Li1 L P Xu 1 Na Xu2 Tao Huang1 and Bo Yan 1

1School of Aerospace Science and Technology XIDIAN University 266 Xinglong Section of Xifeng Road Xian China2School of Life Sciences and Technology XIDIAN University 266 Xinglong Section of Xifeng Road Xian China

Correspondence should be addressed to L P Xu lpxumailxidianeducn and Bo Yan 1152054491qqcom

Received 3 May 2018 Accepted 8 August 2018 Published 19 August 2018

Academic Editor Erik Cuevas

Copyright copy 2018 M Q Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An improved Grey Wolf Optimization (GWO) algorithm with differential evolution (DEGWO) combined with fuzzy C-meansfor complex synthetic aperture radar (SAR) image segmentation was proposed for the disadvantages of traditional optimizationand fuzzy C-means (FCM) in image segmentation precision In the process of image segmentation based on FCM algorithmthe number of clusters and initial centers estimation is regarded as a search procedure that searches for an appropriate value ina greyscale interval Hence an improved differential evolution Grey Wolf Optimization (DE-GWO) algorithm is introduced tosearch for the optimal initial centers then the image segmentation approach which bases its principle on FCM algorithm will geta better result Experimental results in this work infers that both the precision and efficiency of the proposed method are superiorto those of the state of the art

1 Introduction

Image segmentation plays a very important role in theinterpretation and understanding of SAR images It hasreceived an increasing amount of attention and thereforehundreds of approaches have been proposed over the lastfew decades [1] At present SAR images have been widelyused in hydrology remote sensing military and other fieldsto obtain accurate information of remote sensing imagewhich is the key for better application Among them SARimage segmentation is an important step to understand theimage information The SAR image is a coherent imagewith a complex background Because of the influence ofspeckle noise the image quality is reduced Some theorieshave been applied in image segmentation such as thelevel set [2] Markov random field [3] based on textons[4] multiscale [5] threshold method [6] validity-guided(re)clustering (VGC) algorithm [7] and fuzzy C-meansclustering (FCM) algorithm [8] which have achieved goodsegmentation results and have a good reference functionIn these theories Fuzzy C-means (FCM) algorithm is themost classical method of fuzzy clustering It has advantagesof conforming to humanrsquos cognitive characteristics easy

implementation simple description and good segmentationeffect [9] The FCM algorithm for improving the validity offuzzy clustering [10] and the semisupervised c-means algo-rithms [11] have also achieved good segmentation results inthe magnetic resonance image segmentation experiment Inrecent years many scholars have proposed lots of SAR imagesegmentation method combined with FCM algorithm Forinstance modified FCM SAR image segmentation methodis based on GLCM feature [12] multiresolution analysis ofwavelet [13] kernel theory [14] etc

In recent decades there is a growing significant attentionfor nature-inspired computation among which the twomost popular algorithms are swarm intelligence (SI) andEvolutionary Algorithms (EAs) SI like Ant Colony (ACO)algorithm [15] Artificial Fish Swarm (AFS) [16] algorithmArtificial Bee Colony (ABC) algorithm [17] and ParticleSwarm Optimization (PSO) [18] algorithm is enlightened byanimal foraging behavior EAs such as Genetic Algorithm(GA) [19] Evolutionary Programming (EP) [20 21] andEvolution Strategy (ES) [22 23] are inspired from naturalselection and survival of the fittest in the natural worldOwing to the simplicity and flexibility of EAs and SI variousmethods are developed for image engineering which almost

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4576015 11 pageshttpsdoiorg10115520184576015

2 Mathematical Problems in Engineering

cover all related fields including image enhancement imagedenoising super resolution restoration image registrationdigital watermarking edge detection image fusion imagecompression texture classification image retrieval imagerecognition and image segmentation [24ndash37] Similar tothe existing nature-inspired algorithms a new mimic algo-rithms on the basis of the behavior of grey wolves wasproposed in the last few years Grey Wolf Optimization(GWO) algorithm has been clearly proved to be better thanParticle Swarm Optimization (PSO) Gravitational SearchAlgorithm (GSA) Differential Evolution (DE) EvolutionaryProgramming (EP) and Evolution Strategy (ES) which arewell-knownmetaheuristics [38] As a powerful optimizationtoolGWOalgorithmshave beenutilized in complex functionoptimization parameter identification robot path planningclassical engineering design problems etc However its appli-cation in image segmentation is seldom studied Regardingthe insufficient diversity of the wolves in some cases theagents of GWO still may face the risk of stagnation inlocal extremum This problem may often appear when theconventional GWO cannot perform a smooth transitionfrom exploration to exploitation potential by more iteration[39] This paper employs DEGWO algorithm to estimate theFCM algorithm initial centers for SAR image segmentationAn improved modified GWO algorithm combined withdifferential evolution algorithm is proposed for solving theglobal problems

The remaining of this paper is organized as followsSection 2 makes a brief summary of the features of greywolves and describes the working mechanism of GWOalgorithm Section 3 gives the definition of FCM algorithmand introduces a useful method to estimate the number ofclusters Section 4 shows how to employ DE-GWO-FCMalgorithm to the segmentation of SAR images Some typicalexperiments on simulated image and real SAR images arecarried out in Section 5 where both segmented images andsegmenting precision are compared among some nature-inspired methods Finally Section 6 summarizes our workand the future prospects

2 GWO Algorithm

As a kind of social animal grey wolves live in colonies andexhibit many featuresThis algorithm is inspired by the socialhierarchy and hunting strategies of grey wolves in the wild Itcan be regarded as a robust swarm-based optimizer [40ndash45]The following discusses its working mechanism

In GWO A complete wolf pack consists of alpha (120572)beta (120573) delta (120575) and omega (120596) The best wolves should betreated as 120572 and 120573 and 120575 assist other wolves (120596) in exploringmore favorable regions of solution space The alphas areleaders of the pack and they are responsible for makingdecisions The alphas decisions are dictated to the packThe betas are subordinate wolves that can be either male orfemale and they help the alpha in decision making or otheractivities The best candidate to be the alpha mostly may bebetas The omega wolves are scapegoat of pack they haveto submit to all the other dominant wolves The deltas haveto submit alphas and betas but they dominate the omega

Scouts sentinels elders hunters and caretakers belong tothis category [46]

In conventional GWO in order to mathematically modelencircling behavior (1)ndash(4) are used [38]

997888rarr119863 = 100381610038161003816100381610038161003816997888rarr119862 sdot 119883119875 (119905) minus 997888rarr119883 (119905) 100381610038161003816100381610038161003816 (1)

997888rarr119883 (119905 + 1) = 119883119875 (119905) minus 997888rarr119860 sdot 997888rarr119863 (2)

where t is iteration 997888rarr119860 and 997888rarr119862 are random vectors 997888rarr119883 indicatesthe position vector of a grey wolf and 997888rarr119883119875 is location of theprey The random 997888rarr119860 and 997888rarr119862 vectors are calculated as [38]

997888rarr119860 = 2119886 sdot 1199031 minus 997888rarr119886 (3)997888rarr119862 = 2997888rarr119903 2 (4)

where components of 997888rarr119886 are a temporal parameter andlinearly decreased from 2 to 0 over the course of iterationsand r1 r2 are random vectors in [0 1] Grey wolves arecapable of identifying the position of the prey and to enclosethem Alpha is the guide in the hunting processThe beta anddelta might contribute to hunting as well in some conditionsAccording to the difference in the rank of the wolves in orderto have better knowledge about the potential location of preythe alpha beta and delta are assumed as the best the secondbest and the third best candidate solution respectively Thefirst three best candidate solutions obtained can lead otherhunters (including the omegas) to update their positionsaccording to the position of the best search agents [40] Sothe states of the updated solutions of wolves are determinedby [38]

997888rarr119883 (119905 + 1) =997888rarr1198831 + 997888rarr1198832 + 997888rarr1198833

3 (5)

where t shows recent iteration and997888rarr1198831997888rarr1198832997888rarr1198833 denote the finalstate of the updated solutions they are defined as in (6)ndash(8)respectively

997888rarr1198831 = 997888rarr119883120572 minus 997888rarr1198601 sdot (997888rarr119863120572) (6)



where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883120575 denote the locations of alpha betaand delta respectively in the swarm at a given iteration t 997888rarr1198601997888rarr1198602 and 997888rarr1198603 show random vectors and 997888rarr119863120572 997888rarr119863120573 and 997888rarr119863120575 aredefined using (9)-(11) respectively

997888rarr119863120572 =100381610038161003816100381610038161003816997888rarr1198621 sdot 997888rarr119883120572 minus 997888rarr119883100381610038161003816100381610038161003816 (9)



where 997888rarr1198621 997888rarr1198622 and 997888rarr1198623 are defined as representing randomvectors

Mathematical Problems in Engineering 3

InputMaxIter Number of iterations for optimizationn Number of grey wolves in the pack

1 Initialize a population of n grey wolves positions randomly2While Stopping criteria not met do3 Calculate the fitness values based on 120572 120573 120575 positions4 Update Alpha Beta and Delta5 Update a A and C6 Update the Position of search agents including omegas7 endOutput 119909120572 Optimal grey wolf position 119891119894119905(119909120572) Best fitness value

Algorithm 1 Algorithm GWO

Theupdating of parameter a controls the tradeoffbetweenexploration and exploitation in the grey wolf optimizer(GWO) Parameter a is linearly decreased in each iterationto range from 2 to 0 according to

119886 = 2 (1 minus 119905 1119872119886119909119868119905119890119903) (12)

whereMaxIter is the total number of iterations allowed for theoptimization and t is the iteration number Algorithm GWOoutlines the Grey Wolf Optimization (GWO) algorithm inAlgorithm 1

3 The Adaptive FCM Algorithm

Fuzzy c-mean proposed by Bezdek [47] is one of the maintechniques of unsupervised machine learning algorithmwhich is widely applied to the image segmentation [48]Fuzzy clustering has been proved to be very well suited todeal with the imprecise nature of geographical informationincluding remote sensing data [49] It has been effectivelyused in large-scale data analysis data mining vector quanti-zation image segmentation and pattern recognition and hasimportant theoretical and practical value According to thefuzzy clustering framework each cluster is a fuzzy set andeach pixel in the image has a membership value associatedwith each cluster ranging between 0 and 1 measuring howmuch the pixel belongs to that particular cluster [50] Inthe last decade many different new optimization methodsof fuzzy clustering algorithms have been proposed suchas using random projection and independent componentanalysis to improve fuzzy c-means clustering [51 52] and themetaheuristic algorithms combined with FCM algorithm toimprove the effect of clustering [53 54] etc

Suppose 119883 = 1198831 1198832 sdot sdot sdot 119883119899 which refers to a set of ndata points ( n pixels in an image ) and the objective functionof FCM algorithm is as follows

119869119898 (119880 119881) =119888

sum119894=1

119899

sum119896=1

(119906119894119896)119898 1198892119894119896 (119909119896 V119894) (13)

119889119894119896 = 1003817100381710038171003817119909119896 minus V1198941003817100381710038171003817 = (119909119896 minus V119894)119879 (119909119896 minus V119894) (14)

where c is number of clusters 119906119894119896 denotes the membershipdegree of 119909119896 in the 119894119905ℎ cluster Meanwhile the value of 119906119894119896

is inside [0 1] 119898 is the weighting exponent on each fuzzymembership and is generally a value of 2 V119894 is the 119894119905ℎ clustercenter 119889119894119896 is the Euclidean distance between cluster centerV119894 and object 119909119896 and sdot denotes the Euclidean norm Themembership function represents the probability that a pixelbelongs to a specific cluster when pixels far from the clustercenters possess lowmembership values and pixels in the localneighborhood of cluster centers possess high membershipvalue and a minimization criterion is accomplished [49]While the FCM algorithm is based on the initial parameterset determine the minimum objective function 119869119898(119880 119881) byiterative process U and V are defined as in

119906119894119896 =

1sum119888119895=1 (119889119894119896119889119895119896)

2(119898minus1) 119889119895119896

1 119889119895119896 = 0 119895 = 1198960 119889119895119896 = 0 119895 = 119896

(15)

V119894 = sum119899119896=1 (119906119894119896)119898 119909119896sum119899119896=1 (119906119894119896)119898

(16)

where 119906119894119896 V119894 denote the membership function and clustercenters respectively

FCM algorithm can effectively cluster analysis but thenumber of clusters needs to be given first The purpose ofclustering is to classify data and try to make the distancebetween classes as large as possible and the distance betweendata points in the class is as small as possible [55] In order toget the adaptive number of clusters c adaptive function of cis summarized below

119909 = sum119888119894=1sum119899119895=1 119906119898119894119895 119909119895119899 (17)

119871 (119888) = sum119888119894=1 (sum119899119895=1 119906119898119894119895 ) 1003817100381710038171003817V119894 minus 11990910038171003817100381710038172 (119888 minus 1)sum119888119894=1sum119899119895=1 119906119898119894119895 1003817100381710038171003817119909 minus V119894

10038171003817100381710038172 (119899 minus 119888) (18)

where 119909 is central vector of the total sample and 119871(119888) is adap-tive function of the number of clusters cThemolecule of 119871(119888)denotes the distance between classes and the denominatorrepresents the distance between data points in the class andthe center An appropriate classification usually obtains a high


Input Image data1 Initialization Parameter c=2 120576 gt0 119871(1)=0 k=0 119881(0)2While Stopping criteria not met do3 Calculate 119906(119896)119894119895 4 Calculate 119881(119896+1)5 If 119881(119896+1) minus 119881(119896) le 1205766 Break7 else8 119896 = 119896 + 19 end if10 Calculate 119871(119888)11 If 119871(119888 minus 1) gt 119871(119888 minus 2) ampamp 119871(119888 minus 1) gt 119871(119888)12 Break13 else14 119888 = 119888 + 115 end if16 end whileOutput119880 Partition matrix V Center matrix c Number of clusters 119871 Adaptive function of c

Algorithm 2 Algorithm improved FCM

value of function 119871(119888) The pseudocode of the algorithm ispresented in Algorithm 2

4 The Modified FCM withDE-GWO-FCM Algorithm

Differential Evolution (DE) algorithm is a heuristic randomsearch algorithm based on group differences Compared withthe evolutionary algorithm DE preserves the global searchstrategy based on population and reduces the complexity ofgenetic operation At the same time the unique memoryability ofDE enables it to dynamically track the current searchsituation to adjust its search strategy It has strong globalconvergence and robustness does not need the aid of thefeature information of the problem and is suitable for solvingsome optimization problems in the complex environmentwhich cannot be solved by conventional mathematical pro-gramming methods

The conventional GWO algorithm updates its hunterstowards the prey based on the condition of the alpha betaand delta (leader wolves) [39] However regarding the insuf-ficient diversity of the wolves in some cases the populationof GWO is still inclined to stagnate in local extremum andthe problems of immature convergence still exist To avoidthe above-mentioned concerns DE can assist GWO to obtainthe global optimal solution Using this concept it can beensured that GWO can perform global search more effect-ively

In order to achieve the best clustering effect the objectivefunction of fuzzy c-means should be minimum [56] but therandom initial clustering center has a great influence on thealgorithm in this process To solve this problem DE-GWOcan be used to search a set of global optimal centers Theaccuracy of FCM clustering can be significantly improved inthis way so as to achieve better clustering results

41 Fitness Function Setting Fitness function is a benchmarkset by objective function which is used to calculate the fitnessof individual wolves The smaller it means the better theindividual is and the bigger itmeans theworse the individualis Combining DE-GWO and FCM algorithm the fitnessfunction of DE-GWO is defined as in

119891119894119905119899119890119904119904 = 119869119865119862119872 (19)

The better the effect of clustering the smaller the value of119891119894119905119899119890119904119904 of DE-GWO By iterating the 120572 120573 and 120575 positionsin the algorithm the best fitness function 120572 can be obtainedand set 120572 as the initial centers of FCM

42 Population Initialization According to common meth-ods of swarm intelligence algorithm initialization in orderthat the population in the algorithm has diversity andrandomness the initialization formula is set as follows

119883119894 (0) | 119909119871119894119895 le 119909119894119895 (0) le 119909119880119894119895 119894 = 1 2 sdot sdot sdot 119873119875 119895

= 1 2 sdot sdot sdot 119863 (20)

119909119894119895 (0) = 119909119871119894119895 + rand (0 1) (119909119880119894119895 minus 119909119871119894119895) (21)

where NP represents the size of the grey wolf population Ddenotes the dimension of the grey wolf population rand(0 1)is a random value inside [0 1] and 119909119871119894119895 and 119909119880119894119895 are the lowerand upper bounds of the 119895 dimension respectively

43 Mutate The DE algorithm uses the difference strategyto realize the individual variation The common differencestrategy is to randomly select two different individuals inthe population and after the vector difference is scaled


the vectors are combined with the individuals which to bechanged

119881119894 (119892 + 1) = 1198831199031 (119892) + 119865 (1198831199032 (119892) minus 1198831199033 (119892)) (22)

where r1 r2 and r3 are random values in [1NP] 119865 is calledthe scaling factor which is a constant and 119892 denotes g-thgeneration

44 Crossover The purpose of cross operation is to selectindividuals randomly because differential evolution is alsoa stochastic algorithm The way of crossover operation isdefined as follows

119880119894119895 (119892 + 1) = 119881119894119895 (119892 + 1) 119894119891 rand (0 1) le 119862119877119909119894119895 (119892) 119900119905ℎ119890119903119908119894119904119890 (23)

where 119862119877 is cross probability and a new individual israndomly generated by a probability

45 Choice In DE greedy selection strategy is adopted thatis to choose better individuals as new individuals

119883119894 (119892 + 1)

= 119880119894 (119892 + 1) 119894119891 119891 (119880119894 (119892 + 1)) le 119891 (119883119894 (119892))119883119894 (119892) 119900119905ℎ119890119903119908119894119904119890

(24)

46 Update According to the search method of GWO weupdate the location of wolf by encircling hunting andattacking Mutation crossover and selection take place inthe position update process of wolves During the iterationprocess we get the best grey wolf position 119909120572 By summariz-ing the above process the update process of DE-GWO-FCMalgorithm flow is as follows

Step 1 Determine the number of clusters c the initial swarmsize NP number of iterations T lower and upper bound ofscaling factor and crossover probability

Step 2 Randomly generate the initial parent population themutant population and the offspring population of wolvesrespectively and initialize the parameters a A and C

Step 3 Compute the fitness of eachwolf determine the alphabeta and delta wolves in the parent population

Step 4 Update a A and C by (12) and (3)-(4)

Step 5 According to (5) update the position of currentwolves and compute the fitness of each wolf in the parentpopulation

Step 6 Generate mutated population

Step 7 Generate offspring population and crossover andcompute the fitness of each wolf in the offspring population

Step 8 If the offspring are superior to the parent the parentpopulation is updated

Step 9 Reconfirm the alpha beta and delta wolves in theparent population T+T+1

Step 10 If one gets the best 119909120572 end the search processotherwise continue executing Step 3simStep 9 until the end

After obtaining the best number and center of clustersSAR image can be segmented by FCM algorithm AlgorithmDE-GWO-FCM outlines the differential evolution GreyWolfOptimization algorithm To have a better description ofthe DE-GWO-FCM the pseudocode of the algorithm ispresented in Algorithm 3

47 Adaptive Fuzzy c-Means Clustering Algorithm Based onDE-GWO Optimization Through the analysis of (17)-(18)an adaptive image segmentation method is proposed Thealgorithm adaptively searches the optimal number of clustersand initial centers and it is not easy to fall into local extremepoints thus obtaining the optimal classification results Insummary the process of ADE-GWO-FCM algorithm flow isas follows

(1) Initialization determine the fuzzy exponent m lowerand upper bound of scaling factor crossover proba-bility initial swarm size NP and the number of initialclusters c=2 (default classification numbergt=2)

(2) Image clustering analysis by DE-GWO-FCMmethod(3) Calculate the cost function L based on (17)-(18) If the

value of a begins to become smaller turn to fourthstep otherwise set 119888 + 1 997888rarr 119888 and turn to third step

(4) Set 119888 minus 1 997888rarr 119888 calculate initial center by DE-GWO-FCM and get the final segmentation image

5 Experimental Results andPerformance Analysis

51 Segmentation Results on Simulated Image All the imagesand data utilized in this work are available [57] In orderto compare the efficiency of our method with others seg-mentation methods based on FCM GA-FCM and ABC-FCM algorithm are used to segment some typical imagesExperimental results are given in Figure 1 covering a noise-free optical image an optical image polluted by syntheticnoise (composed of salt and pepper noise with density002 speckle noise with variance 0005 and Gaussian noisewith mean 0 and variance 001) and a real SAR imageIn this experiment for GA algorithm the population sizeis 20 the maximum number of iterations is 100 binarydigits of variable are 16 the crossover probability is 07and the mutation probability is 001 In ABC algorithm thepopulation size is 20 the maximum number of iterations is100 and the number of restrictions to give up the search is 20and the lower and upper bounds are 0 and 255 respectivelyIn DE-GWO algorithm the lower bound of scaling factor is01 the upper bound of scaling factor is 09 the crossoverprobability is 01 the population size is 20 and the maximumiteration is 100 Because when the number of optimal clustersis not reached the more the number of clusters is the longer


Input119883 Image data1 Determine the initial swarm size NP the number of initial clusters c iterations T lower and upper bound ofscaling factor and crossover probability2 Randomly generate the initial the parent population mutant population and offspring population of wolvesrespectively and initialize the parameter a A and C3 Compute the fitness of each wolf in the parent population4 Set119883120572 to be the best wolf Set119883120573 to be the second best wolf Set119883120575 to be the third best wolf5 While (Stopping criteria not met) or (tltT) do6 for each wolf in the parent population7 Update a A and C8 Update the position of current wolves by Eq (5)9 Compute the fitness of each wolf10 end for11 Generate Mutated population12 Generate offspring population and crossover13 for each wolf in the offspring population14 Crossover15 Compute the fitness of each wolf16 end for17 If the offspring are superior to the parent18 Update the parent population19 end if20 Update119883120572119883120573 and11988312057521 t=t+122 end while23 Return119883120572119883120573 and11988312057524 119883120572 997888rarr FCMOutput119880 Partition matrix target image

Algorithm 3 Algorithm DE-GWO-FCM

the algorithm runs the clearer the segmented image is Inorder to facilitate the analysis and comparison of algorithmperformance we set the same number of clusters in severalalgorithms

In Figure 1 it is clear to see that under the same numberof clusters the FCM algorithm has the fastest convergencerate because of its simple structure however because theinitial center is randomly generated the accuracy is relativelylowThe accuracy of the GA-FCM andABC-FCMalgorithmsis better than FCM but effects of their convergence are notstable The proposed algorithm is relatively complex it is ahybrid algorithm combining differential evolution andGWObut the speed of the proposed algorithm is similar to that ofABC-FCM the algorithm has been steadily converging andthe variant machine can avoid the algorithm falling into thelocal minimum

In Figure 2 this is the convergent graph of the fouralgorithms Because these four algorithms use the valuefunction of FCM algorithm as the search basis under thesame clustering number the smaller the value of FCM valuefunction the better the search performance of the algorithmIt is clear to see that under the same number of clusterscompared with the contrast algorithm the value of the FCMfunction obtained by the proposed method is the small-est

In the field of image segmentation there are many eval-uation criteria Precision recall and F-measure are widely

used and approved Precision rate represents the proportionof pixels detected by the segmentation algorithm in thewhole region Recall rate indicates the degree of agreementbetween the number of pixels detected by the segmentationalgorithm and the area in the artificial annotation truevalue (GT) The F-measure value is the harmonic meansynthesized by precision and recall which can reflect thecomprehensive quality of image segmentationThe area of thegiven annotation is represented by GThe pixel area detectedby the algorithm is represented by S The formula is definedas follows

119875119903119890119888119894119904119894119900119899 = 119866 cap 119878119878 (25)

119877119890119888119886119897119897 = 119866 cap 119878119866 (26)

Most of the time not only the high accuracy rate but alsothe high recall rate is needed so F-measure is used as theevaluation mechanism of the overall performance

119865120573 =(1 + 1205732) ∙ 119875119903119890119888119894119904119894119900119899 ∙ 119877119890119888119886119897119897

1205732 ∙ 119875119903119890119888119894119904119894119900119899 + 119877119890119888119886119897119897 (27)

Parameter119865120573 is themost commonF1-measure standardwhena equals 1

According to the commonly used image segmentationevaluation criteria the proposed algorithm and existing


(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e

ABC-FCMDE-GWOFCMGA-FCM

(g)

Figure 1 Comparative experiments on image segmentation (a) Noise-free optical image (b) optical image polluted by synthetic noise (c)segmented image by DE-GWO (d) segmented image by ABC-FCM (e) segmented image by GA-FCM (f) segmented image by FCM (g)best value of the four approaches

Table 1

category Precision Recall F1-measureDE-GWO-FCM 09382 07770 08500ABC-FCM 09253 07654 08378GA-FCM 09296 07385 08231FCM 08998 07414 08130

segmentation algorithms are evaluated from the aspectsof segmentation accuracy recall rate and overall accuracyindex The results of the several experiments are shown inTable 1

Combined with the experimental results when the imagecontains synthetic noise the overall accuracy of the algorithmis better than other comparison algorithms

52 Segmentation Results on Real SAR Images This sectiondescribes the application of our method to real SAR imagesFigure 3 shows the segmentation results of the proposedalgorithm and the contrast algorithms for real SAR ima-ges

According to Figure 3 all the four algorithms can seg-ment the edge accurately but the proposed algorithm is betterbecause of the better global search ability

6 Conclusion

In this paper a robust FCM algorithm based on improvedadaptive differential evolution Grey Wolf Optimizationis proposed In essence the segmentation effect of ourmethod owes to GWO algorithm which has an outstandingconvergence performance First in order to get the bestclustering number we classify the data and maximize thedistance between the classes as far as possible and thedistance between the data points within the class is assmall as possible In the GWO algorithm the differentialevolution theory is introduced and the population variationis used to avoid the local optimal solution of the GWOalgorithmThrough the adaptive differential evolution GWOalgorithm we get the initial centers and the number ofclusters and put them into the FCM algorithm to complete


(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)

Figure 2 Comparative experiments on two images (a) The first noise-free optical image (b) the first optical image which is polluted bysynthetic noise (c) the second noise-free optical image (d) the second optical image which is polluted by synthetic noise (e) the best valueof the four approaches with the first image (f) the best value of the four approaches with the second image

the image segmentation Experimental results indicatethat the efficiency of the proposed algorithm is higherthe misclassification rate is smaller and the segmentationaccuracy and the overall accuracy are higher which provesthe validity and correctness of the algorithm

The feasibility of GWO-based image segmentation isdemonstrated in the paper and it offers a new option tothe conventional methods with the merit of simplicity andefficiency However as a new heuristic model in swarmintelligence GWO algorithm is not perfect some controlparameters of the mixed Grey Wolf Optimization algorithm(DE-GWO) have to be specified by experiences and FCM

algorithm is sensitive to noise It is necessary for us to paymore attention to noise reduction and other new fitnessfunctions in the future work

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest


(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)

Figure 3 Comparative experiments on real SAR image segmentation (a) A real SAR image (b) local magnified image of (a) (c) segmentedimage of (a) by our method (d) local magnified image of (c) (e) segmented image of (a) by ABC-FCM (f) local magnified image of (e) (g)segmented image of (a) by GA-FCM (h) local magnified image of (g) (i) segmented image of (a) by FCM (j) local magnified image of (i)

References

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[2] A Ebrahiminia M S Helfroush H Danyali and S BazrafkanldquoContourlet-based levelset SAR image segmentationrdquo Commu-nications in Computer and Information Science vol 427 pp 51ndash59 2014

[3] P C Smits and S G Dellepiane ldquoDiscontinuity-adaptivemarkov random field model for the segmentation of intensitySAR imagesrdquo IEEE Transactions on Geoscience and RemoteSensing vol 37 no 1 pp 627ndash631 1999

[4] F Seixas M Silveira and S Heleno ldquoSegmentation of SARimages using textonsrdquo inProceedings of the Joint 2014 IEEE Inter-national Geoscience and Remote Sensing Symposium IGARSS2014 and the 35th Canadian Symposium on Remote SensingCSRS 2014 pp 1600ndash1603 Canada July 2014

[5] A Schmitt AWendleder and S Hinz ldquoTheKennaugh elementframework for multi-scale multi-polarized multi-temporaland multi-frequency SAR image preparationrdquo ISPRS Journal ofPhotogrammetry and Remote Sensing vol 102 pp 122ndash139 2015

[6] K Zeng Y Ma X Ding and M He ldquoAn adaptive threshodsegmentation algorithm to extract dark targets from SAR

imagesrdquo in Proceedings of the Joint 2014 IEEE InternationalGeoscience and Remote Sensing Symposium IGARSS 2014 andthe 35th Canadian Symposium on Remote Sensing CSRS 2014pp 1765ndash1768 Canada July 2014

[7] A M Bensaid L O Hall J C Bezdek et al ldquoValidity-guided(re)clustering with applications to image segmentationrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 112ndash123 1996

[8] J C Bezdek R Ehrlich and W Full ldquoFCM the fuzzy c-meansclustering algorithmrdquo Computers amp Geosciences vol 10 no 2-3pp 191ndash203 1984

[9] M A Balafar A R Ramli S Mashohor and A FarzanldquoCompare different spatial based fuzzy-c mean (FCM) exten-sions for MRI image segmentationrdquo in Proceedings of the2nd International Conference on Computer and AutomationEngineering (ICCAE rsquo10) vol 5 pp 609ndash611 IEEE SingaporeFebruary 2010

[10] A Bensaid M L Hall O and L Clarke P ldquoFuzzy cluster validityin magnetic resonance images[J]rdquo in Proceedings of SPIE - TheInternational Society for Optical Engineering p 2167 1994

[11] A M Bensaid L O Hall J C Bezdek and L P Clarke ldquoPar-tially supervised clustering for image segmentationrdquo PatternRecognition vol 29 no 5 pp 859ndash871 1996


[12] J Liu Y Cheng L and D Sun J ldquoModified FCM SAR imagesegmentation method based on GLCM featurerdquo ComputerEngineering amp Design vol 33 no 9 pp 3502ndash3506 2012

[13] X Xue HWang F Xiang and J Wang ldquoA newmethod of SARImage Segmentation Based on FCM and wavelet transformrdquoin Proceedings of the 2012 5th International Congress on Imageand Signal Processing (CISP) pp 621ndash624 Chongqing SichuanChina October 2012

[14] X Zhang T Shan S Wang and L Jiao ldquoSAR image segmenta-tion using kernel based spatial FCMrdquo LectureNotes in ComputerScience (including subseries LectureNotes inArtificial Intelligenceand Lecture Notes in Bioinformatics) Preface vol 3656 pp 48ndash54 2005

[15] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006

[16] LM NieArtificial fish swarm algorithm and its application [D]Guangxi University For Nationalities 2009

[17] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numericfunction optimizationrdquo in Proceedings ofthe IEEE swarm intelligence symposium p 12 2006

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[19] J H Holland ldquoGenetic algorithmsrdquo Scientific American vol267 no 1 pp 66ndash72 1992

[20] X Yao Y Liu and G Lin ldquoEvolutionary programming madefasterrdquo IEEE Transactions on Evolutionary Computation vol 3no 2 pp 82ndash102 1999

[21] L J Fogel A J Owens and M J Walsh Artificial Intelligencethrough Simulated Evolution John Wiley New York NY USA1966

[22] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003

[23] I Rechenberg ldquoEvolution strategyrdquo Comput Intel Imitat Life p1 1994

[24] Y Q Wang and W Y Liu ldquoApplication of swarm intelligencein image processingrdquo Computer Applications vol 27 no 7 pp1647ndash1650 2007

[25] Y Tian and W Q Yuan ldquoApplication of the genetic algorithmin image processingrdquo Journal of Image and Graphics vol 12 no3 pp 389ndash396 2007

[26] C Lai and D Tseng ldquoA Hybrid Approach Using GaussianSmoothing and Genetic Algorithm for Multilevel Threshold-ingrdquo International Journal of Hybrid Intelligent Systems vol 1no 3-4 pp 143ndash152 2005

[27] AMishra P K Dutta andM K Ghosh ldquoAGA based approachfor boundary detection of left ventricle with echocardiographicimage sequencesrdquo Image and Vision Computing vol 21 no 11pp 967ndash976 2003

[28] W-B Tao J-W Tian and J Liu ldquoImage segmentation bythree-level thresholding based on maximum fuzzy entropy andgenetic algorithmrdquo Pattern Recognition Letters vol 24 no 16pp 3069ndash3078 2003

[29] P-Y Yin ldquoA fast scheme for optimal thresholding using geneticalgorithmsrdquo Signal Processing vol 72 no 2 pp 85ndash95 1999

[30] G Chen and H Zuo ldquo2-Dmaximum entropy method of imagesegmentation based on genetic algorithmrdquo Jisuanji Fuzhu ShejiYu Tuxingxue XuebaoJournal of Computer-Aided Design andComputer Graphics vol 14 no 6 pp 530ndash534 2002

[31] M Ma Y J Lu Y N Zhang and X L He ldquoFast SARimage segmentation method based on the two-dimensionalgrey entropy modelrdquo Journal of Xidian University vol 36 no6 pp 1114ndash1119 2009

[32] S Ouadfel and M Batouche ldquoAnt colony system with localsearch for Markov random field image segmentationrdquo in Pro-ceedings of the International Conference on Image Processing ppIndash133-6 Barcelona Spain

[33] C Li L Wang and S Wu ldquoAnt colony fuzzy clusteringalgorithm applied to SAR image segmentationrdquo in Proceedingsof the 2006 CIE International Conference on Radar ICR 2006China October 2006

[34] X-N Wang Y-J Feng and Z-R Feng ldquoAnt colony opti-mization with active contour models for image segmentationrdquoControl Theory and Applications vol 23 no 4 pp 515ndash5222006

[35] Z Rubo and L Jing ldquoUnderwater image segmentation withmaximum entropy based on particle swarm optimization(PSO)rdquo in Proceedings of the First International Multi- Sympo-siums on Computer and Computational Sciences IMSCCSrsquo06pp 360ndash362 China April 2006

[36] D Feng S Wenkang C Liangzhou D Yong and Z ZhenfuldquoInfrared image segmentation with 2-D maximum entropymethod based on particle swarm optimization (PSO)rdquo PatternRecognition Letters vol 26 no 5 pp 597ndash603 2005

[37] Z Pan and Y Q Wu ldquoThe two-dimensional otsu thresholdingbased on fish swarm algorithmrdquo Acta Optica Sinica vol 29 no8 pp 2115ndash2121 2009

[38] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[39] A A Heidari and P Pahlavani ldquoAn efficient modified grey wolfoptimizer with Levy flight for optimization tasksrdquo Applied SoftComputing vol 60 pp 115ndash134 2017

[40] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[41] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016

[42] S A Medjahed T Ait Saadi A Benyettou andM Ouali ldquoGrayWolf Optimizer for hyperspectral band selectionrdquo Applied SoftComputing vol 40 pp 178ndash186 2016

[43] SMirjalili S Saremi SMMirjalili andLD S Coelho ldquoMulti-objective grey wolf optimizer A novel algorithm for multi-criterion optimizationrdquo Expert Systems with Applications vol47 pp 106ndash119 2016

[44] X H Song L Tang S T Zhao et al ldquoGrey Wolf Optimizerfor parameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015

[45] M Niu Y Wang S Sun and Y Li ldquoA novel hybriddecomposition-and-ensemble model based on CEEMD andGWO for short-term PM25 concentration forecastingrdquo Atmo-spheric Environment vol 134 pp 168ndash180 2016

[46] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[47] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Plenum New York NY USA 1981


[48] W Chumsamrong P Thitimajshima and Y RangsanserildquoSynthetic Aperture Radar (SAR) image segmentation using anew modified fuzzy c-means algorithmrdquo in Proceedings of the2000 International Geoscience and Remote Sensing Symposium(IGARSS 2000) pp 624ndash626 July 2000

[49] Y Xie ldquoHybrid image segmentation using fuzzy c-meansand gravitational search algorithmrdquo Proceedings of SPIE - TheInternational Society for Optical Engineering vol 8334 p 1022012

[50] F A Lootsma ldquoFuzzy set theory and its applications 3rdeditionrdquo European Journal of Operational Research vol 101 no1 pp 227-228 1997

[51] M Ye W Liu J Wei and X Hu ldquoFuzzy c -Means and ClusterEnsemble with Random Projection for Big Data ClusteringrdquoMathematical Problems in Engineering vol 2016 Article ID6529794 13 pages 2016

[52] Y-T Chen ldquoMedical Image Segmentation Using IndependentComponent Analysis-Based Kernelized Fuzzy c -Means Clus-teringrdquoMathematical Problems in Engineering vol 2017 ArticleID 5892039 21 pages 2017

[53] M Ma J Liang M Guo Y Fan and Y Yin ldquoSAR imagesegmentation based on artificial bee colony algorithmrdquo AppliedSoft Computing vol 11 no 8 pp 5205ndash5214 2011

[54] C Li W Lingzhi and W Shunjun ldquoAnt Colony Fuzzy Clus-tering Algorithm Applied to SAR Image Segmentationrdquo inProceedings of the 2006 CIE International Conference on Radarpp 1ndash4 Shanghai China October 2006

[55] G Y Gong FCM algorithm parameter research and its applica-tion [D] Xidian University 2004

[56] P Pu and G B Wang ldquoLiu TANImproved algorithm of fuzzyG means clustering based on particle swarm optimizationrdquoComputer Engineering andDesign vol 29 no 16 pp 4277ndash42792008

[57] website httpwwwpudncom

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cover all related fields including image enhancement imagedenoising super resolution restoration image registrationdigital watermarking edge detection image fusion imagecompression texture classification image retrieval imagerecognition and image segmentation [24ndash37] Similar tothe existing nature-inspired algorithms a new mimic algo-rithms on the basis of the behavior of grey wolves wasproposed in the last few years Grey Wolf Optimization(GWO) algorithm has been clearly proved to be better thanParticle Swarm Optimization (PSO) Gravitational SearchAlgorithm (GSA) Differential Evolution (DE) EvolutionaryProgramming (EP) and Evolution Strategy (ES) which arewell-knownmetaheuristics [38] As a powerful optimizationtoolGWOalgorithmshave beenutilized in complex functionoptimization parameter identification robot path planningclassical engineering design problems etc However its appli-cation in image segmentation is seldom studied Regardingthe insufficient diversity of the wolves in some cases theagents of GWO still may face the risk of stagnation inlocal extremum This problem may often appear when theconventional GWO cannot perform a smooth transitionfrom exploration to exploitation potential by more iteration[39] This paper employs DEGWO algorithm to estimate theFCM algorithm initial centers for SAR image segmentationAn improved modified GWO algorithm combined withdifferential evolution algorithm is proposed for solving theglobal problems

The remaining of this paper is organized as followsSection 2 makes a brief summary of the features of greywolves and describes the working mechanism of GWOalgorithm Section 3 gives the definition of FCM algorithmand introduces a useful method to estimate the number ofclusters Section 4 shows how to employ DE-GWO-FCMalgorithm to the segmentation of SAR images Some typicalexperiments on simulated image and real SAR images arecarried out in Section 5 where both segmented images andsegmenting precision are compared among some nature-inspired methods Finally Section 6 summarizes our workand the future prospects

2 GWO Algorithm

As a kind of social animal grey wolves live in colonies andexhibit many featuresThis algorithm is inspired by the socialhierarchy and hunting strategies of grey wolves in the wild Itcan be regarded as a robust swarm-based optimizer [40ndash45]The following discusses its working mechanism

In GWO A complete wolf pack consists of alpha (120572)beta (120573) delta (120575) and omega (120596) The best wolves should betreated as 120572 and 120573 and 120575 assist other wolves (120596) in exploringmore favorable regions of solution space The alphas areleaders of the pack and they are responsible for makingdecisions The alphas decisions are dictated to the packThe betas are subordinate wolves that can be either male orfemale and they help the alpha in decision making or otheractivities The best candidate to be the alpha mostly may bebetas The omega wolves are scapegoat of pack they haveto submit to all the other dominant wolves The deltas haveto submit alphas and betas but they dominate the omega

Scouts sentinels elders hunters and caretakers belong tothis category [46]

In conventional GWO in order to mathematically modelencircling behavior (1)ndash(4) are used [38]

997888rarr119863 = 100381610038161003816100381610038161003816997888rarr119862 sdot 119883119875 (119905) minus 997888rarr119883 (119905) 100381610038161003816100381610038161003816 (1)

997888rarr119883 (119905 + 1) = 119883119875 (119905) minus 997888rarr119860 sdot 997888rarr119863 (2)

where t is iteration 997888rarr119860 and 997888rarr119862 are random vectors 997888rarr119883 indicatesthe position vector of a grey wolf and 997888rarr119883119875 is location of theprey The random 997888rarr119860 and 997888rarr119862 vectors are calculated as [38]

997888rarr119860 = 2119886 sdot 1199031 minus 997888rarr119886 (3)997888rarr119862 = 2997888rarr119903 2 (4)

where components of 997888rarr119886 are a temporal parameter andlinearly decreased from 2 to 0 over the course of iterationsand r1 r2 are random vectors in [0 1] Grey wolves arecapable of identifying the position of the prey and to enclosethem Alpha is the guide in the hunting processThe beta anddelta might contribute to hunting as well in some conditionsAccording to the difference in the rank of the wolves in orderto have better knowledge about the potential location of preythe alpha beta and delta are assumed as the best the secondbest and the third best candidate solution respectively Thefirst three best candidate solutions obtained can lead otherhunters (including the omegas) to update their positionsaccording to the position of the best search agents [40] Sothe states of the updated solutions of wolves are determinedby [38]

997888rarr119883 (119905 + 1) =997888rarr1198831 + 997888rarr1198832 + 997888rarr1198833

3 (5)

where t shows recent iteration and997888rarr1198831997888rarr1198832997888rarr1198833 denote the finalstate of the updated solutions they are defined as in (6)ndash(8)respectively




where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883120575 denote the locations of alpha betaand delta respectively in the swarm at a given iteration t 997888rarr1198601997888rarr1198602 and 997888rarr1198603 show random vectors and 997888rarr119863120572 997888rarr119863120573 and 997888rarr119863120575 aredefined using (9)-(11) respectively




where 997888rarr1198621 997888rarr1198622 and 997888rarr1198623 are defined as representing randomvectors






119886 = 2 (1 minus 119905 1119872119886119909119868119905119890119903) (12)





119869119898 (119880 119881) =119888

sum119894=1

119899

sum119896=1

(119906119894119896)119898 1198892119894119896 (119909119896 V119894) (13)




119906119894119896 =

1sum119888119895=1 (119889119894119896119889119895119896)

2(119898minus1) 119889119895119896

1 119889119895119896 = 0 119895 = 1198960 119889119895119896 = 0 119895 = 119896

(15)

V119894 = sum119899119896=1 (119906119894119896)119898 119909119896sum119899119896=1 (119906119894119896)119898

(16)



119909 = sum119888119894=1sum119899119895=1 119906119898119894119895 119909119895119899 (17)


10038171003817100381710038172 (119899 minus 119888) (18)











119891119894119905119899119890119904119904 = 119869119865119862119872 (19)




= 1 2 sdot sdot sdot 119863 (20)

119909119894119895 (0) = 119909119871119894119895 + rand (0 1) (119909119880119894119895 minus 119909119871119894119895) (21)





119881119894 (119892 + 1) = 1198831199031 (119892) + 119865 (1198831199032 (119892) minus 1198831199033 (119892)) (22)



119880119894119895 (119892 + 1) = 119881119894119895 (119892 + 1) 119894119891 rand (0 1) le 119862119877119909119894119895 (119892) 119900119905ℎ119890119903119908119894119904119890 (23)



119883119894 (119892 + 1)

= 119880119894 (119892 + 1) 119894119891 119891 (119880119894 (119892 + 1)) le 119891 (119883119894 (119892))119883119894 (119892) 119900119905ℎ119890119903119908119894119904119890

(24)




























119875119903119890119888119894119904119894119900119899 = 119866 cap 119878119878 (25)

119877119890119888119886119897119897 = 119866 cap 119878119866 (26)


119865120573 =(1 + 1205732) ∙ 119875119903119890119888119894119904119894119900119899 ∙ 119877119890119888119886119897119897

1205732 ∙ 119875119903119890119888119894119904119894119900119899 + 119877119890119888119886119897119897 (27)




(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(g)


Table 1






6 Conclusion



(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018













119886 = 2 (1 minus 119905 1119872119886119909119868119905119890119903) (12)





119869119898 (119880 119881) =119888

sum119894=1

119899

sum119896=1

(119906119894119896)119898 1198892119894119896 (119909119896 V119894) (13)




119906119894119896 =

1sum119888119895=1 (119889119894119896119889119895119896)

2(119898minus1) 119889119895119896

1 119889119895119896 = 0 119895 = 1198960 119889119895119896 = 0 119895 = 119896

(15)

V119894 = sum119899119896=1 (119906119894119896)119898 119909119896sum119899119896=1 (119906119894119896)119898

(16)



119909 = sum119888119894=1sum119899119895=1 119906119898119894119895 119909119895119899 (17)


10038171003817100381710038172 (119899 minus 119888) (18)











119891119894119905119899119890119904119904 = 119869119865119862119872 (19)




= 1 2 sdot sdot sdot 119863 (20)

119909119894119895 (0) = 119909119871119894119895 + rand (0 1) (119909119880119894119895 minus 119909119871119894119895) (21)





119881119894 (119892 + 1) = 1198831199031 (119892) + 119865 (1198831199032 (119892) minus 1198831199033 (119892)) (22)



119880119894119895 (119892 + 1) = 119881119894119895 (119892 + 1) 119894119891 rand (0 1) le 119862119877119909119894119895 (119892) 119900119905ℎ119890119903119908119894119904119890 (23)



119883119894 (119892 + 1)

= 119880119894 (119892 + 1) 119894119891 119891 (119880119894 (119892 + 1)) le 119891 (119883119894 (119892))119883119894 (119892) 119900119905ℎ119890119903119908119894119904119890

(24)




























119875119903119890119888119894119904119894119900119899 = 119866 cap 119878119878 (25)

119877119890119888119886119897119897 = 119866 cap 119878119866 (26)


119865120573 =(1 + 1205732) ∙ 119875119903119890119888119894119904119894119900119899 ∙ 119877119890119888119886119897119897

1205732 ∙ 119875119903119890119888119894119904119894119900119899 + 119877119890119888119886119897119897 (27)




(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(g)


Table 1






6 Conclusion



(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018

















119891119894119905119899119890119904119904 = 119869119865119862119872 (19)




= 1 2 sdot sdot sdot 119863 (20)

119909119894119895 (0) = 119909119871119894119895 + rand (0 1) (119909119880119894119895 minus 119909119871119894119895) (21)





119881119894 (119892 + 1) = 1198831199031 (119892) + 119865 (1198831199032 (119892) minus 1198831199033 (119892)) (22)



119880119894119895 (119892 + 1) = 119881119894119895 (119892 + 1) 119894119891 rand (0 1) le 119862119877119909119894119895 (119892) 119900119905ℎ119890119903119908119894119904119890 (23)



119883119894 (119892 + 1)

= 119880119894 (119892 + 1) 119894119891 119891 (119880119894 (119892 + 1)) le 119891 (119883119894 (119892))119883119894 (119892) 119900119905ℎ119890119903119908119894119904119890

(24)




























119875119903119890119888119894119904119894119900119899 = 119866 cap 119878119878 (25)

119877119890119888119886119897119897 = 119866 cap 119878119866 (26)


119865120573 =(1 + 1205732) ∙ 119875119903119890119888119894119904119894119900119899 ∙ 119877119890119888119886119897119897

1205732 ∙ 119875119903119890119888119894119904119894119900119899 + 119877119890119888119886119897119897 (27)




(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(g)


Table 1






6 Conclusion



(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018










119881119894 (119892 + 1) = 1198831199031 (119892) + 119865 (1198831199032 (119892) minus 1198831199033 (119892)) (22)



119880119894119895 (119892 + 1) = 119881119894119895 (119892 + 1) 119894119891 rand (0 1) le 119862119877119909119894119895 (119892) 119900119905ℎ119890119903119908119894119904119890 (23)



119883119894 (119892 + 1)

= 119880119894 (119892 + 1) 119894119891 119891 (119880119894 (119892 + 1)) le 119891 (119883119894 (119892))119883119894 (119892) 119900119905ℎ119890119903119908119894119904119890

(24)




























119875119903119890119888119894119904119894119900119899 = 119866 cap 119878119878 (25)

119877119890119888119886119897119897 = 119866 cap 119878119866 (26)


119865120573 =(1 + 1205732) ∙ 119875119903119890119888119894119904119894119900119899 ∙ 119877119890119888119886119897119897

1205732 ∙ 119875119903119890119888119894119904119894119900119899 + 119877119890119888119886119897119897 (27)




(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(g)


Table 1






6 Conclusion



(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018
















119875119903119890119888119894119904119894119900119899 = 119866 cap 119878119878 (25)

119877119890119888119886119897119897 = 119866 cap 119878119866 (26)


119865120573 =(1 + 1205732) ∙ 119875119903119890119888119894119904119894119900119899 ∙ 119877119890119888119886119897119897

1205732 ∙ 119875119903119890119888119894119904119894119900119899 + 119877119890119888119886119897119897 (27)




(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(g)


Table 1






6 Conclusion



(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c) (d)

(e) (f)

600

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(g)


Table 1






6 Conclusion



(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c)

(d)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration


Best

Valu

e

(e)

600

700

500

400

300

200

0 20 40 60 80 100

Iteration

Best

Valu

e


(f)





Data Availability





(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)


References




































































Journal of












Journal of







Volume 2018






Volume 2018































































Journal of












Journal of







Volume 2018






Volume 2018








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