Research Article A Swarm Optimization Algorithm for...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 948303 22 pageshttpdxdoiorg1011552013948303

Research ArticleA Swarm Optimization Algorithm for Multimodal Functionsand Its Application in Multicircle Detection

Erik Cuevas Daniel Zaldiacutevar and Marco Peacuterez-Cisneros

Departamento de Ciencias Computacionales Universidad de Guadalajara CUCEI Avenida Revolucion 1500CP 44430 Guadalajara Jal Mexico

Correspondence should be addressed to Erik Cuevas erikcuevascuceiudgmx

Received 3 September 2012 Accepted 25 December 2012

Academic Editor Baozhen Yao

Copyright copy 2013 Erik Cuevas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In engineering problems due to physical and cost constraints the best results obtained by a global optimization algorithm cannotbe realized always Under such conditions if multiple solutions (local and global) are known the implementation can be quicklyswitched to another solution without much interrupting the design process This paper presents a new swarm multimodal opti-mization algorithm named as the collective animal behavior (CAB) Animal groups such as schools of fish flocks of birds swarmsof locusts and herds of wildebeest exhibit a variety of behaviors including swarming about a food source milling around a centrallocation or migrating over large distances in aligned groupsThese collective behaviors are often advantageous to groups allowingthem to increase their harvesting efficiency to follow bettermigration routes to improve their aerodynamic and to avoid predationIn the proposed algorithm searcher agents emulate a group of animals which interact with each other based on simple biologicallaws that are modeled as evolutionary operators Numerical experiments are conducted to compare the proposed method with thestate-of-the-art methods on benchmark functions The proposed algorithm has been also applied to the engineering problem ofmulti-circle detection achieving satisfactory results

1 Introduction

A large number of real-world problems can be consideredas multimodal function optimization subjects An objectivefunction may have several global optima that is severalpoints holding objective function values which are equal tothe global optimum Moreover it may exhibit some otherlocal optima points whose objective function values lay near-by a global optimum Since the mathematical formulation ofa real-world problem often produces a multimodal optimiza-tion issue finding all global or even these local optima wouldprovide to the decision makers multiple options to choosefrom [1]

Several methods have recently been proposed for solvingthe multimodal optimization problem They can be dividedinto twomain categories deterministic and stochastic (meta-heuristic) methods When facing complex multimodal opti-mization problems deterministic methods such as gradientdescent method the quasi-Newton method and the Nelder-Meadrsquos simplex method may get easily trapped into the local

optimum as a result of deficiently exploiting local informa-tionThey strongly depend on a priori information about theobjective function yielding few reliable results

Metaheuristic algorithms have been developed combin-ing rules and randomness mimicking several phenomenaThese phenomena include evolutionary processes (eg theevolutionary algorithm proposed by Fogel et al [2] de Jong[3] and Koza [4] and the genetic algorithms (GAs) proposedby Holland [5] and Goldberg [6]) immunological systems(eg the artificial immune systems proposed by de Castroet al [7]) physical processes (eg simulated annealingproposed by Kirkpatrick et al [8] electromagnetism-likeproposed by Birbil et al [9] and the gravitational searchalgorithm proposed by Rashedi et al [10]) and the musicalprocess of searching for a perfect state of harmony (proposedby Geem et al [11] Lee and Geem [12] Geem [13] and Gaoet al [14])

Traditional GAs perform well for locating a single opti-mum but fail to provide multiple solutions Several methodshave been introduced into the GArsquos scheme to achieve

2 Mathematical Problems in Engineering

multimodal function optimization such as sequential fitnesssharing [15 16] deterministic crowding [17] probabilisticcrowding [18] clustering-based niching [19] clearing pro-cedure [20] species conserving genetic algorithm [21] andelitist-population strategies [22] However algorithms basedon the GAs do not guarantee convergence to global optimabecause of their poor exploitation capability GAs exhibitother drawbacks such as the premature convergence whichresults from the loss of diversity in the population andbecomes a common problem when the search continues forseveral generations Such drawbacks [23] prevent the GAsfrom practical interest for several applications

Using a different metaphor other researchers haveemployed artificial immune systems (AIS) to solve the multi-modal optimization problems Some examples are the clonalselection algorithm [24] and the artificial immune network(AiNet) [25 26] Both approaches use some operators andstructures which attempt to algorithmically mimic the natu-ral immune systemrsquos behavior of human beings and animals

Several studies have been inspired by animal behaviorphenomena in order to develop optimization techniques suchas the particle swarm optimization (PSO) algorithm whichmodels the social behavior of bird flocking or fish schooling[27] In recent years there have been several attempts to applythe PSO to multimodal function optimization problems [2829] However the performance of such approaches presentsseveral flaws when it is compared to the other multi-modalmetaheuristic counterparts [26]

Recently the concept of individual organization [30 31]has been widely used to understand collective behavior ofanimals The central principle of individual organization isthat simple repeated interactions between individuals canproduce complex behavioral patterns at group level [30 3233] Such inspiration comes from behavioral patterns seen inseveral animal groups such as ant pheromone trail networksaggregation of cockroaches and themigration of fish schoolswhich can be accurately described in terms of individuals fol-lowing simple sets of rules [34] Some examples of these rules[33 35] include keeping current position (or location) forbest individuals local attraction or repulsion randommove-ments and competition for the space inside of a determineddistance On the other hand new studies have also shownthe existence of collective memory in animal groups [36ndash38]The presence of such memory establishes that the previoushistory of group structure influences the collective behaviorexhibited in future stages Therefore according to these newdevelopments it is possible to model complex collectivebehaviors by using simple individual rules and configuringa general memory

On the other hand the problem of detecting circularfeatures holds paramount importance in several engineeringapplications The circle detection in digital images has beencommonly solved through the circular Hough transform(CHT) [39] Unfortunately this approach requires a largestorage space that augments the computational complexityand yields a low processing speed In order to overcome thisproblem several approaches which modify the original CHThave been proposed One well-known example is the ran-domized Hough transform (RHT) [40] As an alternative to

Hough-transform-based techniques the problem of shaperecognition has also been handled through optimizationmethods In general they have demonstrated to deliver betterresults than those based on the HT considering accuracyspeed and robustness [41] Such approaches have producedseveral robust circle detectors using different optimizationalgorithms such as genetic algorithms (GAs) [41] harmonysearch (HSA) [42] electromagnetism-like (EMO) [43] dif-ferential evolution (DE) [44] and bacterial foraging opti-mization (BFOA) [45] Since such evolutionary algorithmsare global optimizers they detect only the global optimum(only one circle) of an objective function that is defined overa given search space However extracting multiple-circleprimitives falls into the category of multi-modal optimiza-tion where each circle represents an optimumwhichmust bedetected within a feasible solution spaceThe quality for suchoptima is characterized by the properties of their geometricprimitives Big and well-drawn circles normally representpoints in the search space with high fitness values (possibleglobal maximum) whereas small and dashed circles describepoints with fitness values which account for local maximaLikewise circles holding similar geometric properties suchas radius and size tend to represent locations with similarfitness values Therefore a multi-modal method must beapplied in order to appropriately solve the problem of multi-shape detection In this paper a new multimodal optimiza-tion algorithm based on the collective animal behavior isproposed and also applied to multicircle detection

This paper proposes a new optimization algorithminspired by the collective animal behavior In this algorithmthe searcher agents emulate a group of animals that interactwith each other based on simple behavioral rules whichare modeled as evolutionary operators Such operations areapplied to each agent considering that the complete grouphas a memory which stores its own best positions seen so farby applying a competition principle Numerical experimentshave been conducted to compare the proposed method withthe state-of-the-art methods on multi-modal benchmarkfunctions Besides the proposed algorithm is also applied tothe engineering problem of multicircle detection achievingsatisfactory results

This paper is organized as follows Section 2 introducesthe basic biological aspects of the algorithm In Section 3the proposed algorithm and its characteristics are describedA numerical study on different multi-modal benchmarkfunctions is presented in Section 4 Section 5 presents theapplication of the proposed algorithm to multi-circle detec-tion whereas Section 6 shows the obtained results Finally inSection 7 the conclusions are discussed

2 Biological Fundaments

The remarkable collective behavior of organisms such asswarming ants schooling fish and flocking birds has longcaptivated the attention of naturalists and scientists Despitea long history of scientific investigation just recently we arebeginning to decipher the relationship between individu-als and group-level properties [46] Grouping individualsoften have to make rapid decisions about where to move

Mathematical Problems in Engineering 3

or what behavior to perform in uncertain and dangerousenvironments However each individual typically has onlyrelatively local sensing ability [47] Groups are thereforeoften composed of individuals that differ with respect to theirinformational status and individuals are usually not aware ofthe informational state of others [48] such as whether theyare knowledgeable about a pertinent resource or of a threat

Animal groups are based on a hierarchic structure [49]which differentiates individuals according to a fitness princi-ple known as dominance [50] Such concept represents thedomain of some individuals within a group and occurs whencompetition for resources leads to confrontation Severalstudies [51 52] have found that such animal behavior leads tostable groups with better cohesion properties among individ-uals

Recent studies have illustrated how repeated interac-tions among grouping animals scale to collective behaviorThey have also remarkably revealed that collective decision-making mechanisms across a wide range of animal grouptypes ranging from insects to birds (and even among humansin certain circumstances) seem to share similar functionalcharacteristics [30 34 53] Furthermore at a certain levelof description collective decision-making in organismsshares essential common features such as general memoryAlthough some differences may arise there are good reasonsto increase communication between researchers working incollective animal behavior and those involved in cognitivescience [33]

Despite the variety of behaviors and motions of animalgroups it is possible that many of the different collectivebehavioral patterns are generated by simple rules followed byindividual groupmembers Some authors have developed dif-ferentmodels such as the self-propelled particle (SPP)modelwhich attempts to capture the collective behavior of animalgroups in terms of interactions between group membersfollowing a diffusion process [54ndash57]

On other hand following a biological approach Couzinet al [33 34] have proposed a model in which individualanimals follow simple rules of thumb (1) keep the position ofbest individuals (2) move from or to nearby neighbors (localattraction or repulsion) (3) move randomly and (4) competefor the space inside of a determined distance Each individualthus admits three different movements attraction repulsionor random while holding two kinds of states preserve theposition or compete for a determined position In the modelthe movement experimented by each individual is decidedrandomly (according to an internal motivation) meanwhilethe states are assumed according to fixed criteria

The dynamical spatial structure of an animal group canbe explained in terms of its history [54] Despite this themajority of the studies have failed in considering the existenceof memory in behavioral models However recent researches[36 58] have also shown the existence of collective memoryin animal groups The presence of such memory establishesthat the previous history of the group structure influences thecollective behavior exhibited in future stages Such memorycan contain the position of special group members (thedominant individuals) or the averaged movements producedby the group

According to these new developments it is possible tomodel complex collective behaviors by using simple individ-ual rules and setting a general memory In this work thebehavioral model of animal groups is employed for definingthe evolutionary operators through the proposed meta-heuristic algorithm A memory is incorporated to store bestanimal positions (best solutions) considering a competition-dominance mechanism

3 Collective Animal BehaviourAlgorithm (CAB)

The CAB algorithm assumes the existence of a set of oper-ations that resembles the interaction rules that model thecollective animal behavior In the approach each solutionwithin the search space represents an animal position Theldquofitness valuerdquo refers to the animal dominance with respect tothe groupThe complete processmimics the collective animalbehavior

The approach in this paper implements a memory forstoring best solutions (animal positions) mimicking theaforementioned biologic process Such memory is dividedinto two different elements one for maintaining the bestfound positions in each generation (M119892) and the other forstoring best history positions during the complete evolution-ary process (Mℎ)

31 Description of the CAB Algorithm Like other meta-heuristic approaches the CAB algorithm is also an iterativeprocess It starts by initializing the population randomlythat is generating random solutions or animal positionsThefollowing four operations are thus applied until the termi-nation criterion is met that is the iteration number 119873119868 isreached as follows

(1) Keep the position of the best individuals(2) Move from or nearby neighbors (local attraction and

repulsion)(3) Move randomly(4) Compete for the space inside of a determined distance

(updating the memory)

311 Initializing the Population The algorithm begins byinitializing a set A of 119873119901 animal positions (A = a1 a2 a119873119901) Each animal position a119894 is a 119863-dimensional vectorcontaining the parameter values to be optimized which arerandomly and uniformly distributed between the prespeci-fied lower initial parameter bound 119886low

119895and the upper initial

parameter bound 119886high119895

119886119895119894 = 119886low119895

+ rand (0 1) sdot (119886high119895

minus 119886low119895)

119895 = 1 2 119863 119894 = 1 2 119873119901

(1)

with 119895 and 119894 being the parameter and individual indexes res-pectively Hence 119886119895119894 is the 119895th parameter of the 119894th individual


All the initial positions A are sorted according to the fit-ness function (dominance) to form a new individual set X =

x1 x2 x119873119901 so that we can choose the best 119861 positionsand store them in the memoryM119892 andMℎThe fact that bothmemories share the same information is only allowed at thisinitial stage

312 Keep the Position of the Best Individuals Analogouslyto the biological metaphor this behavioral rule typical inanimal groups is implemented as an evolutionary operationin our approach In this operation the first 119861 elements ofthe new animal position set A(a1 a2 a119861) are generatedSuch positions are computed by the values contained in thehistoric memory Mℎ considering a slight random perturba-tion around themThis operation can bemodelled as follows

a119897 = m119897ℎ+ v (2)

where 119897 isin 1 2 119861 while m119897ℎrepresents the 119897-element of

the historic memoryMℎ and v is a random vector holding anappropriate small length

313 Move from or to Nearby Neighbours From the biolog-ical inspiration where animals experiment a random localattraction or repulsion according to an internal motivationwe implement the evolutionary operators that mimic themFor this operation a uniform randomnumber 119903119898 is generatedwithin the range [0 1] If 119903119898 is less than a threshold 119867a determined individual position is moved (attracted orrepelled) considering the nearest best historical value of thegroup (the nearest position contained inMℎ) otherwise it isconsidered the nearest best value in the group of the currentgeneration (the nearest position contained inM119892)Thereforesuch operation can be modeled as follows

a119894 =

x119894 plusmn 119903 sdot (mnearestℎ

minus x119894) with probability 119867

x119894 plusmn 119903 sdot (mnearest119892

minus x119894) with probability (1 minus 119867)

(3)

where 119894 isin 119861+1 119861+2 119873119901mnearestℎ

andmnearest119892

representthe nearest elements ofMℎ andM119892 to x119894 while 119903 is a randomnumber between [minus1 1] Therefore if 119903 gt 0 the individualposition x119894 is attracted to the position mnearest

ℎor mnearest119892

otherwise such movement is considered as a repulsion

314Move Randomly Following the biologicalmodel undersome probability 119875 an animal randomly changes its positionSuch behavioral rule is implemented considering the nextexpression

a119894 = r with probability 119875x119894 with probability (1 minus 119875)

(4)

where 119894 isin 119861 + 1 119861 + 2 119873119901 and r is a random vectordefined within the search space This operator is similar toreinitialize the particle in a random position as it is done by(1)

120588

Figure 1 Dominance concept presented when two animals con-front each other inside of a 120588 distance

315 Compete for the Space Inside of a Determined Distance(Updating the Memory) Once the operations to preserve theposition of the best individuals to move from or to nearbyneighbors and to move randomly have all been applied to allthe 119873119901 animal positions generating 119873119901 new positions it isnecessary to update the memoryMℎ

In order to update the memory Mℎ the concept ofdominance is used Animals that interact in a group keep aminimum distance among them Such distance 120588 depends onhow aggressive the animal behaves [50 58] Hence when twoanimals confront each other inside of such distance the mostdominant individual prevails as the other withdraws Figure 1shows this process

In the proposed algorithm the historic memory Mℎ isupdated considering the following procedure

(1) The elements of Mℎ and M119892 are merged intoM119880(M119880 = Mℎ cupM119892)

(2) Each element m119894119880of the memory M119880 is compared

pairwise with the remainder memory elements (m1119880

m2119880 m2119861minus1

119880) If the distance between both ele-

ments is less than 120588 the element holding a betterperformance in the fitness function will prevailmeanwhile the other will be removed

(3) From the resulting elements of M119880 (as they areobtained in Step 2) the 119861 best value is selected tointegrate the newMℎ

Unsuitable values of 120588 result in a lower convergence ratelonger computation time larger function evaluation numberconvergence to a localmaximum or unreliability of solutionsThe 120588 value is computed considering the following equation

120588 =

prod119863

119895=1(119886

high119895

minus 119886low119895)

10 sdot 119863

(5)

where 119886low119895

and 119886high119895

represent the prespecified lower boundand the upper bound of the 119895-parameter respectively withina119863-dimensional space


316 Computational Procedure The computational proce-dure for the proposed algorithm can be summarized asfollows

Step 1 Set the parameters119873119901 119861119867 119875 and119873119868

Step 2 Generate randomly the position set A = a1 a2 a119873119901 using (1)

Step 3 Sort A according to the objective function (domi-nance) building X = x1 x2 x119873119901

Step 4 Choose the first 119861 positions of X and store them intothe memoryM119892

Step 5 Update Mℎ according to Section 315 (for the firstiterationMℎ = M119892)

Step 6 Generate the first 119861 positions of the new solution setA = a1a2 a119861 Such positions correspond to elements ofMℎ making a slight random perturbation around them a119897 =m119897ℎ+v v being a random vector holding an appropriate small

length

Step 7 Generate the rest of the A elements using the attrac-tion repulsion and random movements

for 119894 = 119861 + 1 119873119901if (1199031 lt 1 minus 119875) thenattraction and repulsion movementif (1199032 lt 119867) thena119894 = x119894 plusmn 119903 sdot (mnearest

ℎminus x119894)

else if

a119894 = x119894 plusmn 119903 sdot (mnearest119892

minus x119894)

else ifrandom movement

a119894 = r

end forwhere 1199031 1199032 119903 isin rand(0 1)

Step 8 If 119873119868 is completed the process is thus completedotherwise go back to Step 3

317 Optima Determination Just after the optimization pro-cess has finished an analysis of the final Mℎ memory is exe-cuted in order to find the global and significant local minimaFor it a threshold value Th is defined to decide whichelements will be considered as a significant local minimumSuch threshold is thus computed as

Th =maxfitness (Mℎ)

6 (6)

where maxfitness(Mℎ) represents the best fitness value amongMℎ elements Therefore memory elements whose fitnessvalues are greater than Th will be considered as global andlocal optima as other elements are discarded

318 Capacities of CAB andDifferences with PSO Evolution-ary algorithms (EAs) have been widely employed for solvingcomplex optimization problemsThese methods are found tobe more powerful than conventional methods based on for-mal logics or mathematical programming [59] Exploitationand exploration are two main features of the EA [60] Theexploitation phase searches around the current best solutionsand selects the best candidates or solutions The explorationphase ensures that the algorithm seeks the search space moreefficiently in order to analyze potential unexplored areas

The EAs do not have limitations in using different sourcesof inspiration (eg music-inspired [11] or physic-inspiredcharged system search [9]) However nature is a principalinspiration for proposing newmetaheuristic approaches andthe nature-inspired algorithms have been widely used indeveloping systems and solving problems [61] Biologicallyinspired algorithms are one of the main categories of thenature-inspired metaheuristic algorithms The efficiency ofthe bio inspired algorithms is due to their significant abilityto imitate the best features in nature More specifically thesealgorithms are based on the selection of the most suitableelements in biological systems which have evolved by naturalselection

Particle swarm optimization (PSO) is undoubtedly one ofthemost employed EAmethods that use biologically inspiredconcepts in the optimization procedure Unfortunately likeother stochastic algorithms PSO also suffers from the prema-ture convergence [62] particularly in multi modal problemsPremature convergence in PSO is produced by the stronginfluence of the best particle in the evolution process Suchparticle is used by the PSO movement equations as a mainindividual in order to attract other particles Under suchconditions the exploitation phase is privileged by allowingthe evaluation of new search position around the best individ-ual However the exploration process is seriously damagedavoiding searching in unexplored areas

As an alternative to PSO the proposed scheme modifiessome evolution operators for allowing not only attracting butalso repelling movements among particles Likewise insteadof considering the best position as reference our algorithmuses a set of neighboring elements that are contained in anincorporated memory Such improvements allow increasingthe algorithmrsquos capacity to explore and to exploit the set ofsolutions which are operated during the evolving process

In the proposed approach in order to improve the balancebetween exploitation and exploration we have introducedthree new concepts The first one is the ldquoattracting andrepelling movementrdquo which outlines that one particle cannotbe only attracted but also repelled The application of thisconcept to the evolution operators (3) increases the capac-ity of the proposed algorithm to satisfactorily explore thesearch space Since the process of attraction or repulsionof each particle is randomly determined the possibility of


premature convergence is very low even for cases that holdan exaggerated number of local minima (excessive numberof multimodal functions)

The second concept is the use of the main individualIn the approach the main individual that is considered aspivot in the equations (in order to generate attracting andrepulsive movements) is not the best (as in PSO) but oneelement (mnearest


) of a set which is contained inmemories that store the best individual seen so far Suchpivot is the nearest element in memory with regard to theindividual whose position is necessary to evolve Under suchconditions the points considered to prompt themovement ofa new individual are multiple Such fact allows to maintain abalance between exploring new positions and exploiting thebest positions seen so-far

Finally the third concept is the use of an incorporatedmemory which stores the best individuals seen so far As ithas been discussed in Section 315 each candidate individualto be stored in the memory must compete with elementsalready contained in thememory in order to demonstrate thatsuch new point is relevant For the competition the distancebetween each individual and the elements in the memoryis used to decide pair-wise which individuals are actuallyconsidered Then the individual with better fitness valueprevails whereas its pair is discarded The incorporation ofsuch concept allows simultaneous registering and refining ofthe best-individual set seen so farThis fact guarantees a highprecision for final solutions of the multi-modal landscapethrough an extensive exploitation of the solution set

319 Numerical Example In order to demonstrate the algo-rithmrsquos step-by-step operation a numerical example has beenset by applying the proposed method to optimize a simplefunction which is defined as follows

119891 (1199091 1199092) = 119890minus((1199091minus4)

2minus(1199092minus4)

2)+ 119890minus((1199091+4)


2)

+ 2 sdot 119890minus((1199091)

2+(1199092)

2)+ 2 sdot 119890

minus((1199091)2minus(1199092+4)

2)

(7)

Considering the interval of minus5 le 1199091 1199092 le 5 the functionpossesses two global maxima of value 2 at (1199091 1199092) = (0 0)

and (0 minus4) Likewise it holds two local minima of value 1at (minus4 4) and (4 4) Figure 2(a) shows the 3D plot of thisfunction The parameters for the CAB algorithm are set as119873119901 = 10 119861 = 4119867 = 08 119875 = 01 120588 = 3 and119873119868 = 30

Like all evolutionary approaches CAB is a population-based optimizer that attacks the starting point problem bysampling the objective function at multiple randomly cho-sen initial points Therefore after setting parameter boundsthat define the problem domain 10 (119873119901) individuals (i1i2 i10) are generated using (1) Following an evaluationof each individual through the objective function (7) allare sorted decreasingly in order to build vector X =

(x1 x2 x10) Figure 2(b) depicts the initial individualdistribution in the search space Then both memoriesM119892 (m1119892 m

4

119892) and Mℎ (m1ℎ m

4

ℎ) are filled with the

first four (119861) elements present in X Such memory elementsare represented by solid points in Figure 2(c)

The new 10 individuals (a1 a2 a10) are evolved at eachiteration following three different steps (1) keep the positionof best individuals (2) move from or nearby neighbors and(3) move randomlyThe first new four elements (a1 a2 a3 a4)are generated considering the first step (keeping the positionof best individuals) Following such step new individual posi-tions are calculated as perturbed versions of all the elementswhich are contained in the Mℎ memory (that represent thebest individuals known so far) Such perturbation is doneby using a119897 = m119897

ℎ+ v (119897 isin 1 4) Figure 2(d) shows

a comparative view between the memory element positionsand the perturbed values of (a1 a2 a3 a4)

The remaining 6 new positions (a5 a10) are individ-ually computed according to Steps 2 and 3 of the numericalexample For such operation a uniform random number 1199031 isgenerated within the range [0 1] If 1199031 is less than 1 minus 119875 thenew position a119895 (119895 isin 5 10) is generated through Step 2otherwise a119895 is obtained from a random reinitialization (Step3) between search bounds

In order to calculate a new position a119895 at Step 2 a decisionmust be made on whether it should be generated by usingthe elements of Mℎ or M119892 For such decision a uniformrandom number 1199032 is generated within the range [0 1] If1199032 is less than 119867 the new position a119895 is generated by usingx119895plusmn119903 sdot (mnearest

ℎminusx119895) otherwise a119895 is obtained by considering


minus x119895) where mnearestℎ

and mnearest119892

representthe closest elements to x119895 inmemoryMℎ andM119892 respectivelyIn the first iteration since there is not available informationfrom previous steps both memories Mℎ and M119892 share thesame information which is only allowed at this initial stageFigure 2(e) shows graphically the whole procedure employedby Step 2 in order to calculate the new individual positiona8 whereas Figure 2(f) presents the positions of all newindividuals (a1 a2 a10)

Finally after all new positions (a1 a2 a10) have beencalculated memoriesMℎ andM119892 must be updated In orderto update Mℎ new calculated positions (a1 a2 a10) arearranged according to their fitness values by building vectorX = (x1 x2 x10) Then the elements of Mℎ are replacedby the first four elements in X (the best individuals of itsgeneration) In order to calculate the new elements of Mℎcurrent elements of Mℎ (the present values) and M119892 (theupdated values) are merged into M119880 Then by using thedominance concept (explained in Section 315) over M119880the best four values are selected to replace the elements inM119892 Figures 2(g) and 2(h) show the updating procedure forboth memories Applying the dominance (see Figure 2(g))since the distances 119886 = dist(m3

ℎm4119892) 119887 = dist(m2

ℎm3119892)

and 119888 = dist(m1ℎm1119892) are less than 120588 = 3 elements

with better fitness evaluation will build the new memoryMℎ Figure 2(h) depicts final memory configurations Thecircles and solid circles points represent the elements of M119892and Mℎ respectively whereas the bold squares perform aselements shared by bothmemoriesTherefore if the completeprocedure is repeated over 30 iterations the memoryMℎ willcontain the 4 global and localmaxima as elements Figure 2(i)depicts the final configuration after 30 iterations


1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)

Figure 2 CAB numerical example (a) 3D plot of the function used as example (b) Initial individual distribution (c) Initial configuration ofmemoriesM

119892andM

ℎ (d)The computation of the first four individuals (a

1 a2 a3 a4) (e) It shows the procedure employed by Step 2 in order

to calculate the new individual position a8 (f) Positions of all new individuals (a

1 a2 a

10) (g) Application of the dominance concept over

elements ofM119892andM

ℎ (h) Final memory configurations ofM

119892andM

ℎafter the first iteration (i) Final memory configuration ofM

ℎafter

30 iterations

4 Results on MultimodalBenchmark Functions

In this section the performance of the proposed algorithmis tested Section 41 describes the experiment methodologySections 42 and 43 report on a comparison between theCABexperimental results and other multimodal metaheuristicalgorithms for different kinds of optimization problems

41 ExperimentMethodology In this section wewill examinethe search performance of the proposed CAB by using a testsuite of 8 benchmark functions with different complexitiesThey are listed in Tables 1 and 2 The suite mainly containssome representative complicated and multimodal functionswith several local optima These functions are normallyregarded as difficult to be optimized as they are particularlychallenging to the applicability and efficiency of multimodal

metaheuristic algorithms The performance measurementsconsidered at each experiment are the following

(i) the consistency of locating all known optima(ii) the averaged number of objective function evalua-

tions that are required to find such optima (or therunning time under the same condition)

The experiments compare the performance of CAB againstthe deterministic crowding [17] the probabilistic crowding[18] the sequential fitness sharing [15] the clearing procedure[20] the clustering based niching (CBN) [19] the species con-serving genetic algorithm (SCGA) [21] the elitist-populationstrategy (AEGA) [22] the clonal selection algorithm [24] andthe artificial immune network (AiNet) [25]

Since the approach solves real-valued multimodal func-tions we have used in the GA approaches consistentreal coding variable representation uniform crossover and


Table 1 The test suite of multimodal functions for Experiment 42

Function Search space SketchDebrsquos function5 optima

1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1

Debrsquos decreasing function5 optima

1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1

Roots function6 optima

1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0

0 0minus1 minus1minus2minus2

Two dimensional multi-modal function100 optima

1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0

1 1 220 0minus1minus1

minus2minus2

minus2minus4

mutation operators for each algorithm seeking a fair com-parison The crossover probability 119875119888 = 08 and the muta-tion probability119875119898 = 01have been usedWeuse the standardtournament selection operator with a tournament size of 2in our implementation of sequential fitness sharing clearingprocedure CBN clonal selection algorithm and SCGA On

the other hand the parameter values for the AiNet algorithmhave been defined as suggested in [25] with the mutationstrength 120573 = 100 the suppression threshold 120590119904(119886119894119873119890119905) = 02and the update rate 119889 = 40

In the case of the CAB algorithm the parameters are setto119873119901 = 200 119861 = 100 119875 = 08 and119867 = 06 Once they have


Table 2 The test suite of multimodal functions used in the Experiment 43

Function Search space SketchRastringinrsquos function100 optima

1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500

10 105 50 0minus5 minus5minus10 minus10

Shubert function18 optima

1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]

minus100minus200minus300

0

200100


Griewank function100 optima

1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8

10010050 500 0minus50 minus50minus100 minus100

Modified Griewank function100 optima

1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0

been all experimentally determined they are kept for all thetest functions through all experiments

To avoid relating the optimization results to the choice ofa particular initial population and to conduct fair compar-isons we perform each test 50 times starting from various

randomly selected points in the search domain as it is com-monly given in the literature An optimum 119900119895 is consideredas found if exist119909119894 isin Pop(119896 = 119879) | 119889(119909119894 119900119895) lt 0005 wherePop(119896 = 119879) is the complete population at the end of the run119879 and 119909119894 is an individual in Pop(119896 = 119879)


All algorithms have been tested in MATLAB over thesameDell OptiPlexGX260 computerwith a Pentium4 266GHZ processor running Windows XP operating system over1 Gb of memory Next sections present experimental resultsfor multimodal optimization problems which have beendivided into two groups with different purposes The firstone consists of functions with smooth landscapes and well-defined optima (local and global values) while the secondgathers functions holding rough landscapes and complexlocation optima

42 Comparing CAB Performance for Smooth LandscapesFunctions This section presents a performance comparisonfor different algorithms solving multimodal problems 1198911ndash1198914in Table 1 The aim is to determine whether CAB is moreefficient and effective than other existing algorithms forfinding all multiple optima of 1198911ndash1198914 The stopping criterionanalyzes if the number-identified optima cannot be furtherincreased over 10 successive generations after the first 100generations then the execution will be stopped Four mea-surements have been employed to evaluate the performance

(i) the average of optima found within the final popula-tion (NO)

(ii) the average distance between multiple optima detect-ed by the algorithm and their closest individuals inthe final population (DO)

(iii) the average of function evaluations (FE)(iv) the average of execution time in seconds (ET)

Table 3 provides a summarized performance comparisonamong several algorithms Best results have been bold facedFrom the NO measure CAB always finds better or equallyoptimal solutions for the multimodal problems 1198911ndash1198914 It isevident that each algorithm can find all optima of 1198911 Forfunction 1198912 only AEGA clonal selection algorithm aiNetand CAB can eventually find all optima each time Forfunction1198913 clearing procedure SCGA AEGA and CAB canget all optima at each run For function 1198914 deterministiccrowding leads to premature convergence and all otheralgorithms cannot get any better results but CAB yet canfind all multiple optima 48 times in 50 runs and its averagesuccessful rate for each run is higher than 99 By analyzingthe DO measure in Table 3 CAB has obtained the best scorefor all the multimodal problems except for 1198913 In the caseof 1198913 the solution precision of CAB is only worse thanthat of clearing procedure On the other hand CAB hassmaller standard deviations in the NO and DO measuresthan all other algorithms and hence its solution is morestable

From the FEmeasure in Table 3 it is clear that CAB needsfewer function evaluations than other algorithms consideringthe same termination criterion Recall that all algorithms usethe same conventional crossover and mutation operators Itcan be easily deduced from results that the CAB algorithmis able to produce better search positions (better compromisebetween exploration and exploitation) in amore efficient andeffective way than other multimodal search strategies

To validate that CAB improvement over other algorithmsoccurs as a result of CAB producing better search positionsover iterations Figure 3 shows the comparison of CAB andother multimodal algorithms for 1198914 The initial populationsfor all algorithms have 200 individuals In the final pop-ulation of CAB the 100 individuals belonging to the Mℎmemory correspond to the 100 multiple optima while onthe contrary the final population of the other nine algo-rithms fail consistently in finding all optima despite that theyhave superimposed several times over some previously foundoptima

When comparing the execution time (ET) in Table 3CAB uses significantly less time to finish than other algo-rithmsThe situation can be registered by the reduction of theredundancy in the Mℎ memory due to competition (domi-nance) criterion All these comparisons show that CAB gen-erally outperforms all othermultimodal algorithms regardingefficacy and efficiency

43 Comparing CAB Performance in Rough Landscapes Func-tions This section presents the performance comparisonamong different algorithms solvingmultimodal optimizationproblems which are listed in Table 2 Such problems holdlots of local optima and very rugged landscapes The goal ofmultimodal optimizers is to find as many global optima aspossible and possibly good local optima Rastriginrsquos function1198915 and Griewankrsquos function 1198917 have 1 and 18 global optimarespectively becoming practical as to test whether a multi-modal algorithm can find a global optimum and at least 80higher fitness local optima to validate the algorithmsrsquo perfor-mance

Our main objective in these experiments is to determinewhether CAB is more efficient and effective than other exist-ing algorithms for finding the multiple high fitness optimaof functions 1198915ndash1198918 In the experiments the initial populationsize for all algorithms has been set to 1000 For sequentialfitness sharing clearing procedure CBN clonal selectionSCGA and AEGA we have set the distance threshold 120590119904

to 5 The algorithmsrsquo stopping criterion checks wheneverthe number of optima found cannot be further increased in50 successive generations after the first 500 generations Ifsuch condition prevails then the algorithm is halted We stillevaluate the performance of all algorithms using the afore-mentioned four measures NO DO FE and ET

Table 4 provides a summary of the performance compar-ison among different algorithms From the NO measure weobserve that CAB could always find more optimal solutionsfor the multimodal problems 1198915ndash1198918 For Rastriginrsquos function1198915 only CAB can find all multiple high fitness optima 49times out of 50 runs and its average successful rate for eachrun is higher than 97 On the contrary other algorithmscannot find all multiple higher fitness optima for any run For1198916 5 algorithms (clearing procedure SCGA AEGA clonalselection algorithm AiNet and CAB) can get all multiplehigher fitness maxima for each run respectively For Grie-wankrsquos function (1198917) only CAB can get all multiple higher fit-ness optima for each run In case of the modified Griewankrsquosfunction (1198918) it has numerous optima whose value is always


Table 3 Performance comparison among the multimodal optimization algorithms for the test functions 1198911ndash1198914 The standard unit in the

column ET is seconds (For all the parameters numbers in parentheses are the standard deviations) Bold faced letters represent best obtainedresults

Function Algorithm NO DO FE ET

1198911

Deterministic crowding 5 (0) 152 times 10minus4(138 times 10

minus4) 7153 (358) 0091 (0013)

Probabilistic crowding 5 (0) 363 times 10minus4(645 times 10

minus5) 10304 (487) 0163 (0011)

Sequential fitness sharing 5 (0) 476 times 10minus4(682 times 10

minus5) 9927 (691) 0166 (0028)

Clearing procedure 5 (0) 127 times 10minus4(213 times 10

minus5) 5860 (623) 0128 (0021)

CBN 5 (0) 294 times 10minus4(421 times 10

minus5) 10781 (527) 0237 (0019)

SCGA 5 (0) 116 times 10minus4(311 times 10

minus5) 6792 (352) 0131 (0009)

AEGA 5 (0) 46 times 10minus5(135 times 10

minus5) 2591 (278) 0039 (0007)

Clonal selection algorithm 5 (0) 199 times 10minus4(825 times 10

minus5) 15803 (381) 0359 (0015)

AiNet 5 (0) 128 times 10minus4(388 times 10

minus5) 12369 (429) 0421 (0021)

CAB 5 (0) 169 times 10minus5 (52 times 10minus6) 1776 (125) 0020 (0009)

1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)

Clearing procedure 6 (0) 743 times 10minus5 (407 times 10minus5) 12684 (1729) 159 (019)CBN 473 (114) 185 times 10

minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)

CAB 6 (0) 987 times 10minus5(169 times 10

minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)


the same However CAB still can find all global optima withan effectiveness rate of 95

From the FE and ET measures in Table 4 we can clearlyobserve that CAB uses significantly fewer function evalua-tions and a shorter running time than all other algorithmsunder the same termination criterion Moreover determin-istic crowding leads to premature convergence as CAB is atleast 25 38 4 31 41 37 14 79 and 49 times faster than

all others respectively according to Table 4 for functions1198915ndash1198918

5 Application of CAB in Multicircle Detection51 Individual Representation In order to detect circleshapes candidate images must be preprocessed first by thewell-known Canny algorithm which yields a single-pixel


Table 4 Performance comparison among multimodal optimization algorithms for the test functions 1198915ndash1198918 The standard unit of the column

ET is seconds (numbers in parentheses are standard deviations) Bold faced letters represent best results


1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)


edge-only image Then the (119909119894 119910119894) coordinates for each edgepixel 119901119894 are stored inside the edge vector 119875 = 1199011 1199012

119901119873119901 with 119873119901 being the total number of edge pixels Each

circle119862 uses three edge points as individuals in the optimiza-tion algorithm In order to construct such individuals threeindexes 119901119894 119901119895 and 119901119896 are selected from vector 119875 consideringthe circlersquos contour that connects them Therefore the circle

119862 = 119901119894 119901119895 119901119896 that crosses over such points may beconsidered as a potential solution for the detection problemConsidering the configuration of the edge points shown byFigure 4 the circle center (1199090 1199100) and the radius 119903 of 119862 canbe computed as follows

(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15

minus22151050minus05minus1minus15minus2

(a) Deterministic crowding

215

105

0minus05

minus1

minus15


(b) Probabilistic crowding

215

105

0minus05

minus1

minus15


(c) Sequential fitness sharing

215

105

0minus05

minus1

minus15


(d) Clearing procedure

215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15


(h) Clonal selction algorithm2

151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB

Figure 3 Typical results of the maximization of 1198914 (a)ndash(j) Local and global optima located by all ten algorithms in the performance

comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)

Figure 4 Circle candidate (individual) built from the combinationof points 119901

119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)

4 ((119909119895 minus 119909119894) (119910119896 minus 119910119894) minus (119909119896 minus 119909119894) (119910119895 minus 119910119894))

1199100 =det (B)


(10)

119903 = radic(1199090 minus 119909119889)2+ (1199100 minus 119910119889)

2 (11)

with det(sdot) being the determinant and 119889 isin 119894 119895 119896 Figure 2illustrates the parameters defined by (8) to (11)

52 Objective Function In order to calculate the error pro-duced by a candidate solution 119862 a set of test points iscalculated as a virtual shapewhich in turnmust be validatedthat is if it really exists in the edge image The test set isrepresented by 119878 = 1199041 1199042 119904119873119904

where 119873119904 is the numberof points over which the existence of an edge point corre-sponding to 119862 should be validated In our approach the set119878 is generated by the midpoint circle algorithm (MCA) [63]The MCA is a searching method which seeks the requiredpoints for drawing a circle digitallyThereforeMCAcalculatesthe necessary number of test points 119873119904 to totally draw thecomplete circle Such a method is considered the fastestbecause MCA avoids computing square-root calculations bycomparing the pixel separation distances among them

The objective function 119869(119862) represents thematching errorproduced between the pixels 119878 of the circle candidate 119862

(animal position) and the pixels that actually exist in the edgeimage yielding

119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)

where119864(119909119894 119910119894) is a function that verifies the pixel existence in(119909119907 119910119907) with (119909119907 119910119907) isin 119878 and119873119904 being the number of pixelslying on the perimeter corresponding to 119862 currently undertesting Hence function 119864(119909119907 119910119907) is defined as

119864 (119909119907 119910119907) =

1 if the pixel (119909119907 119910119907) is an edge point

0 otherwise(13)

A value near zero of 119869(119862) implies a better response fromthe ldquocircularityrdquo operator Figure 5 shows the procedure toevaluate a candidate solution 119862 with its representation as avirtual shape 119878 In Figure 5(b) the virtual shape is comparedto the original image point by point in order to findcoincidences between virtual and edge points The virtualshape is built from points 119901119894 119901119895 and 119901119896 shown by Figure 5(a)The virtual shape 119878 gathers 56 points (119873119904 = 56) with only 18of such points existing in both images (shown as blue pointsplus red points in Figure 5(c)) yielding sum119873119904

119907=1119864(119909119907 119910119907) = 18

and therefore 119869(119862) asymp 067

53 The Multiple-Circle Detection Procedure In order todetect multiple circles most detectors simply apply a one-minimum optimization algorithm which is able to detectonly one circle at a time repeating the same process severaltimes as previously detected primitives are removed fromthe image Such algorithms iterate until there are no morecandidates left in the image

On the other hand the method in this paper is able todetect single or multiples circles through only one optimiza-tion step The multidetection procedure can be summarizedas follows guided by the values of a matching function thewhole group of encoded candidate circles is evolved throughthe set of evolutionary operators The best circle candidate(global optimum) is considered to be the first detected circleover the edge-only image An analysis of the historical mem-oryMℎ is thus executed in order to identify other local optima(other circles)

In order to find other possible circles contained in theimage the historical memory Mℎ is carefully examined Theapproach aims to explore all elements one at a time assessingwhich of them represents an actual circle in the image Sinceseveral elements can represent the same circle (ie circlesslightly shifted or holding small deviations) a distinctivenessfactor119863119860119861 is required tomeasure themismatch between twogiven circles (119860 and 119861) Such distinctiveness factor is definedas follows

119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)

with (119909119860 119910119860) and 119903119860 being the central coordinates and radiusof the circle 119862119860 respectively while (119909119861 119910119861) and 119903119861 represent


119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)

Figure 5 Evaluation of candidate solutions 119862 the image in (a) shows the original image while (b) presents the virtual shape generatedincluding points 119901

119894 119901119895 and 119901

119896 The image in (c) shows coincidences between both images marked by blue or red pixels while the virtual

shape is also depicted in green

the corresponding parameters of the circle119862119861 One thresholdvalue 119864119904Th is also calculated to decide whether two circlesmust be considered different or not Th is computed as

Th =119903max minus 119903min

119889 (15)

where [119903min 119903max] is the feasible radiirsquos range and 119889 is a sen-sitivity parameter By using a high 119889 value two very similarcircles would be considered different while a smaller valuefor 119889 would consider them as similar shapes In this workafter several experiments the 119889 value has been set to 2

Thus since the historical memoryMℎ 119862M1 119862

M2 119862

M119861

groups the elements in descending order according to theirfitness values the first element 119862M

1 whose fitness value

represents the best value 119869(119862M1) is assigned to the first circle

Then the distinctiveness factor (119863119862M1119862M2) over the next

element 119862M2

is evaluated with respect to the prior 119862M1 If

119863119862M1119862M2gt Th then119862M

2is considered as a new circle otherwise

the next element 119862M3

is selected This process is repeateduntil the fitness value 119869(119862M

119894) reaches a minimum threshold

119869Th According to such threshold other values above 119869Threpresent individuals (circles) that are considered significantwhile other values lying below such boundary are consideredas false circles and hence they are not contained in the imageAfter several experiments the value of 119869Th is set to (119869(119862

M1)10)

Thefitness value of each detected circle is characterized byits geometric properties Big and well-drawn circles normallyrepresent points in the search space with higher fitnessvalues whereas small and dashed circles describe points withlower fitness values Likewise circles with similar geometricproperties such as radius and size tend to represent locationsholding similar fitness values Considering that the historicalmemoryMℎ groups the elements in descending order accord-ing to their fitness values the proposed procedure allows thecancelling of those circles which belong to the same circle andhold a similar fitness value

54 Implementation of CAB Strategy for Circle Detection Theimplementation of the proposed algorithm can be summa-rized in the following steps

Step 1 Adjust the algorithm parameters119873119901 119861119867 119875119873119868 and119889

Step 2 Randomly generate a set of 119873119901 candidate circles(position of each animal) C = 1198621 1198622 119862119873119901

set using (1)

Step 3 Sort C according to the objective function (domi-nance) to build X = x1 x2 x119873119901


Step 5 UpdateMℎ according to Section 315 (during the firstiterationMℎ = M119892)

Step 6 Generate the first 119861 positions of the new solutionset C(1198621 1198622 119862119861) Such positions correspond to theelements ofMℎ making a slight random perturbation aroundthem

119862119897 = m119897ℎ+ v with being v a random vector of

a small enough length(16)

Step 7 Generate the rest of the C elements using the attrac-tion repulsion and random movements

for 119894 = 119861 + 1 119873119901if (1199031 lt 119875) thenattraction and repulsion movementif (1199032 lt 119867) then119862119894 = x119894 plusmn 119903 sdot (mnearest

ℎminus x119894)

else if119862119894 = x119894 plusmn 119903 sdot (mnearest

119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)

Figure 6 Synthetic images and their detected circles for GA-based algorithm the BFOA method and the proposed CAB algorithm

end for where 1199031 1199032 119903 isin rand(0 1)

Step 8 If119873119868 is not completed the process goes back to Step 3Otherwise the best values inMℎ 119862M

1 119862

M2 119862

M119861 represent

the best solutions (the best found circles)

Step 9 The element with the highest fitness value 119869(119862M1) is

identified as the first circle 1198621

Step 10 The distinctiveness factor 119863119862M119898119862M119898minus1

of circle 119862M119898(ele-

ment 119898) with the next highest probability is evaluated with

respect to 119862M119898minus1

If119863119862M119898119862M119898minus1

gt Th then 119862M119898is considered as a

new circle otherwise the next action is evaluated

Step 11 Step 10 is repeated until the elementrsquos fitness valuereaches (119869(119862M

1)10)

The number of candidate circles 119873119901 is set considering abalance between the number of local minima to be detectedand the computational complexity In general terms a largevalue of 119873119901 suggests the detection of a great amount ofcircles at the cost of excessive computer time After exhaustiveexperimentation it has been found that a value of 119873119901 = 30

represents the best tradeoff between computational overhead


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)

Figure 7 Real-life images and their detected circles for GA-based algorithm the BFOA method and the proposed CAB algorithm

and accuracy and therefore such value is used throughout thestudy

6 Results on Multicircle Detection

In order to achieve the performance analysis the proposedapproach is compared to the BFAO detector the GA-basedalgorithm and the RHT method over an image set

The GA-based algorithm follows the proposal of Ayala-Ramirez et al [41] which considers the population size as 70the crossover probability as 055 the mutation probability as010 and the number of elite individuals as 2 The roulettewheel selection and the 1-point crossover operator are both

applied The parameter setup and the fitness function followthe configuration suggested in [41] The BFAO algorithmfollows the implementation from [45] considering the exper-imental parameters as 119878 = 50 119873119888 = 350 119873119904 = 4 119873ed = 1119875ed = 025 119889attract = 01 119908attract = 02 119908repellant = 10ℎrepellant = 01 120582 = 400 and 120595 = 6 Such values are foundto be the best configuration set according to [45] Both theGA-based algorithm and the BAFO method use the sameobjective function that is defined by (12) Likewise the RHTmethod has been implemented as it is described in [40]Finally Table 5 presents the parameters for the CAB algo-rithm used in this work They have been kept for all testimages after being experimentally defined


(a)

(b)

Original images CABRHT

(c)

Figure 8 Relative performance of the RHT and the CAB

Table 5 CAB detector parameters

119873119901

119867 119875 119861 NI30 05 01 12 200

Images rarely contain perfectly shaped circles Thereforewith the purpose of testing accuracy for a single circle thedetection is challenged by a ground-truth circle which isdetermined from the original edge map The parameters(119909true 119910true 119903true) representing the testing circle are computedusing (6)ndash(9) for three circumference points over the manu-ally drawn circle Considering the center and the radius of thedetected circle are defined as (119909119863 119910119863) and 119903119863 the error score(Es) can be accordingly calculated as

Es=120578 sdot (1003816100381610038161003816119909trueminus1199091198631003816100381610038161003816+1003816100381610038161003816119910true minus 119910119863

1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)

The central point difference (|119909true minus 119909119863| + |119910true minus 119910119863|)

represents the center shift for the detected circle as it is com-pared to a benchmark circleThe radiomismatch (|119903trueminus119903119863|)accounts for the difference between their radii 120578 and 120583

represent two weighting parameters which are to be appliedseparately to the central point difference and to the radiomismatch for the final error Es At this work they are chosenas 120578 = 005 and 120583 = 01 Such particular choice ensures thatthe radii difference would be strongly weighted in compari-son to the difference of central circular positions between themanually detected and themachine-detected circles Here weassume that if Es is found to be less than 1 then the algorithmgets a success otherwise we say that it has failed to detect theedge circle Note that for 120578 = 005 and 120583 = 01 Es lt 1meansthat the maximum difference of radius tolerated is 10 whilethe maximum mismatch in the location of the center can be20 (in number of pixels) In order to appropriately compare


Table 6 The averaged execution time detection rate and the averaged multiple error for the GA-based algorithm the BFOA method andthe proposed CAB algorithm considering six test images (shown by Figures 6 and 7)

Image Averaged execution time plusmn standard deviation (s) Success rate (DR) () Averaged ME plusmn standard deviationGA BFOA CAB GA BFOA CAB GA BFOA CAB

Synthetic images(a) 223 plusmn (041) 171 plusmn (051) 021 plusmn (022) 88 99 100 041 plusmn (0044) 033 plusmn (0052) 022 plusmn (0033)(b) 315 plusmn (039) 280 plusmn (065) 036 plusmn (024) 79 92 99 051 plusmn (0038) 037 plusmn (0032) 026 plusmn (0041)(c) 421 plusmn (011) 318 plusmn (036) 020 plusmn (019) 74 88 100 048 plusmn (0029) 041 plusmn (0051) 015 plusmn (0036)

Natural images(a) 511 plusmn (043) 345 plusmn (052) 110 plusmn (024) 90 96 100 045 plusmn (0051) 041 plusmn (0029) 025 plusmn (0037)(b) 633 plusmn (034) 411 plusmn (014) 161 plusmn (017) 83 89 100 081 plusmn (0042) 077 plusmn (0051) 037 plusmn (0055)(c) 762 plusmn (097) 536 plusmn (017) 195 plusmn (041) 84 92 99 092 plusmn (0075) 088 plusmn (0081) 041 plusmn (0066)

Table 7 119875 values produced by Wilcoxons test comparing CAB toGA and BFOA over the averaged ME from Table 2

Image 119875 valueCAB versus GA CAB versus BFOA

Synthetic images(a) 18061119890 minus 004 18288119890 minus 004

(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004

Natural images(a) 17788119890 minus 004 18698119890 minus 004

(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004

the detection results the detection rate (DR) is introducedas a performance index DR is defined as the percentage ofreaching detection success after a certain number of trials Forldquosuccessrdquo it does mean that the compared algorithm is able todetect all circles contained in the image under the restrictionthat each circle must hold the condition Es lt 1 Therefore ifat least one circle does not fulfil the condition of Es lt 1 thecomplete detection procedure is considered as a failure

In order to use an error metric for multiple-circle detec-tion the averaged Es produced from each circle in the imageis considered Such criterion defined as the multiple error(ME) is calculated as follows

ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)

where119873119862 represents the number of circles within the imageaccording to a human expert

Figure 6 shows three synthetic images and the resultingimages after applying theGA-based algorithm [41] the BFOAmethod [45] and the proposed approach Figure 7 presentsexperimental results considering three natural images Theperformance is analyzed by considering 35 different exe-cutions for each algorithm Table 6 shows the averagedexecution time the detection rate in percentage and theaveraged multiple error (ME) considering six test images(shown by Figures 6 and 7) The best entries are boldfaced in

Table 6 Close inspection reveals that the proposedmethod isable to achieve the highest success rate keeping the smallesterror still requiring less computational time for most cases

In order to statistically analyze the results in Table 6 anonparametric significance proof known as the Wilcoxonrsquosrank test [64ndash66] for 35 independent samples has been con-ducted Such proof allows assessing result differences amongtwo related methods The analysis is performed consideringa 5 significance level over multiple error (ME) data Table 7reports the 119875 values produced by Wilcoxonrsquos test for a pair-wise comparison of the multiple error (ME) consideringtwo groups gathered as CAB versus GA and CAB versusBFOA As a null hypothesis it is assumed that there is nodifference between the values of the two algorithmsThe alter-native hypothesis considers an existent difference betweenthe values of both approaches All 119875 values reported inTable 7 are less than 005 (5 significance level) which is astrong evidence against the null hypothesis indicating thatthe bestCABmean values for the performance are statisticallysignificant which has not occurred by chance

Figure 8 demonstrates the relative performance ofCAB incomparisonwith the RHT algorithm as it is described in [40]All images belonging to the test are complicated and con-tain different noise conditions The performance analysis isachieved by considering 35 different executions for each algo-rithmover the three imagesThe results exhibited in Figure 8present themedian-run solution (when the runs were rankedaccording to their finalME value) obtained throughout the 35runs On the other hand Table 4 reports the correspondingaveraged execution time detection rate (in ) and averagemultiple error (using (10)) for CAB and RHT algorithms overthe set of images (the best results are boldfaced) Table 8shows a decrease in performance of the RHT algorithm asnoise conditions change Yet the CAB algorithm holds itsperformance under the same circumstances

7 Conclusions

In recent years several metaheuristic optimization methodshave been inspired from nature-like phenomena In thispaper a new multimodal optimization algorithm known asthe collective animal behavior algorithm (CAB) has been


Table 8 Average time detection rate and averaged error for CAB and HT considering three test images

Image Average time plusmn standard deviation (s) Success rate (DR) () Average ME plusmn standard deviationRHT CAB RHT CAB RHT CAB

(a) 782 plusmn (034) 030 plusmn (010) 100 100 019 plusmn (0041) 011 plusmn (0017)(b) 865 plusmn (048) 022 plusmn (013) 64 100 047 plusmn (0037) 013 plusmn (0019)(c) 1065 plusmn (048) 025 plusmn (012) 11 100 121 plusmn (0033) 015 plusmn (0014)

introduced In CAB the searcher agents emulate a group ofanimals that interact with each other depending on simplebehavioral rules which are modeled as mathematical opera-tors Such operations are applied to each agent consideringthat the complete group hold a memory to store its own bestpositions seen so far using a competition principle

CAB has been experimentally evaluated over a testsuite consisting of 8 benchmark multimodal functions foroptimization The performance of CAB has been comparedto some other existing algorithms including deterministiccrowding [17] probabilistic crowding [18] sequential fitnesssharing [15] clearing procedure [20] clustering-based nich-ing (CBN) [19] species conserving genetic algorithm (SCGA)[21] elitist-population strategies (AEGA) [22] clonal selec-tion algorithm [24] and the artificial immune network(aiNet) [25] All experiments have demonstrated that CABgenerally outperforms all other multimodal metaheuristicalgorithms regarding efficiency and solution quality typicallyshowing significant efficiency speedupsThe remarkable per-formance of CAB is due to two different features (i) operatorsallow a better exploration of the search space increasing thecapacity to findmultiple optima (ii) the diversity of solutionscontained in the Mℎ memory in the context of multimodaloptimization ismaintained and even improved through of theuse of a competition principle (dominance concept)

The proposed algorithm is also applied to the engineeringproblem of multicircle detection Such a process is facedas a multimodal optimization problem In contrast to otherheuristic methods that employ an iterative procedure theproposed CAB method is able to detect single or multiplecircles over a digital image by running only one optimizationcycle The CAB algorithm searches the entire edge map forcircular shapes by using a combination of three noncollinearedge points as candidate circles (animal positions) in theedge-only image A matching function (objective function)is used to measure the existence of a candidate circle over theedge map Guided by the values of such matching functionthe set of encoded candidate circles is evolved using the CABalgorithm so that the best candidate can be fitted into anactual circle After the optimization has been completed ananalysis of the embedded memory is executed in order tofind the significant local minima (remaining circles) Theoverall approach generates a fast subpixel detector which caneffectively identify multiple circles in real images despite thatsome circular objects exhibit a significant occluded portion

In order to test the circle detection performance bothspeed and accuracy have been compared Score functions aredefined by (17) and (18) in order to measure accuracy andeffectively evaluate the mismatch between manually detectedandmachine-detected circlesWe have demonstrated that the

CABmethod outperforms both the GA (as described in [41])and the BFOA (as described in [45]) within a statisticallysignificant framework (Wilcoxon test) In contrast to theCAB method the RHT algorithm [40] shows a decrease inperformance under noisy conditions Yet the CAB algorithmholds its performance under the same circumstances FinallyTable 6 indicates that the CABmethod can yield better resultson complicated and noisy images compared with the GA andthe BFOA methods

References

[1] A Ahrari M Shariat-Panahi and A A Atai ldquoGEM a novelevolutionary optimization method with improved neighbor-hood searchrdquo Applied Mathematics and Computation vol 210no 2 pp 376ndash386 2009

[2] L J Fogel A J Owens and M J Walsh Artificial Intelligencethrough Simulated Evolution John Wiley amp Sons ChichesterUK 1966

[3] K de Jong Analysis of the behavior of a class of genetic adaptivesystems [PhD thesis] The University of Michigan Press AnnArbor Mich USA 1975

[4] J R Koza ldquoGenetic programming a paradigm for geneticallybreeding populations of computer programs to solve problemsrdquoTech Rep STAN-CS-90-1314 Stanford University Palo AltoCalif USA 1990

[5] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[6] D E Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Boston Mass USA 1989

[7] L N de Castro and F J von Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep TR-DCA0199 1999

[8] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[9] S I Birbil and S-C Fang ldquoAn electromagnetism-like mecha-nism for global optimizationrdquo Journal of Global Optimizationvol 25 no 3 pp 263ndash282 2003

[10] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[11] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[13] ZW Geem ldquoNovel derivative of harmony search algorithm fordiscrete design variablesrdquo Applied Mathematics and Computa-tion vol 199 no 1 pp 223ndash230 2008


[14] X Z Gao X Wang and S J Ovaska ldquoUni-modal and multi-modal optimization usingmodifiedHarmony SearchmethodsrdquoInternational Journal of Innovative Computing Information andControl vol 5 no 10 pp 2985ndash2996 2009

[15] D Beasley D R Bull and R R Matin ldquoA sequential nichetechnique for multimodal function optimizationrdquo EvolutionaryComputation vol 1 no 2 pp 101ndash125 1993

[16] B L Miller and M J Shaw ldquoGenetic algorithms with dynamicniche sharing for multimodal function optimizationrdquo in Pro-ceedings of the 3rd IEEE International Conference on Evolution-ary Computation (ICEC rsquo96) pp 786ndash791 May 1996

[17] S W Mahfoud Niching methods for genetic algorithms [PhDdissertation] Illinois Genetic Algorithm Laboratory Universityof Illinois Urbana Ill USA 1995

[18] O J Mengshoel and D E Goldberg ldquoProbability crowdingdeterministic crowding with probabilistic replacementrdquo inProceedings of the International Conference on Genetic andEvolutionary Computation Conferenc (GECCO rsquo99)W BanzhafEd pp 409ndash416 Orlando Fla USA 1999

[19] X Yin and N Germay ldquoA fast genetic algorithm with sharingscheme using cluster analysis methods in multimodal functionoptimizationrdquo in Proceedings of the International Conference onArtificial Neural Networks and Genetic Algorithms pp 450ndash4571993

[20] A Petrowski ldquoA clearing procedure as a niching method forgenetic algorithmsrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 798ndash803 IEEE Press Nagoya Japan May 1996

[21] J P Li M E Balazs G T Parks and P J Clarkson ldquoA speciesconserving genetic algorithm for multimodal function opti-mizationrdquoEvolutionary Computation vol 10 no 3 pp 207ndash2342002

[22] Y Liang and K S Leung ldquoGenetic Algorithm with adaptiveelitist-population strategies for multimodal function optimiza-tionrdquo Applied Soft Computing Journal vol 11 no 2 pp 2017ndash2034 2011

[23] L Y Wei and M Zhao ldquoA niche hybrid genetic algorithmfor global optimization of continuous multimodal functionsrdquoApplied Mathematics and Computation vol 160 no 3 pp 649ndash661 2005

[24] L N Castro and F J Zuben ldquoLearning and optimization usingthe clonal selection principlerdquo IEEE Transactions on Evolution-ary Computation vol 6 no 3 pp 239ndash251 2002

[25] L N Castro and J Timmis ldquoAn artificial immune network formultimodal function optimizationrdquo in Proceedings of the IEEEInternational Conference on Evolutionary Computation pp699ndash704 IEEE Press Honolulu Hawaii 2002

[26] Q Xu L Wang and J Si ldquoPredication based immune networkfor multimodal function optimizationrdquo Engineering Applica-tions of Artificial Intelligence vol 23 no 4 pp 495ndash504 2010

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[28] J J Liang A K Qin P N Suganthan and S Baskar ldquoCom-prehensive learning particle swarm optimizer for global opti-mization of multimodal functionsrdquo IEEE Transactions on Evo-lutionary Computation vol 10 no 3 pp 281ndash295 2006

[29] D B Chen and C X Zhao ldquoParticle swarm optimizationwith adaptive population size and its applicationrdquo Applied SoftComputing Journal vol 9 no 1 pp 39ndash48 2009

[30] D Sumper ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B vol 361 no1465 pp 5ndash22 2006

[31] O Petit and R Bon ldquoDecision-making processes the case ofcollective movementsrdquo Behavioural Processes vol 84 no 3 pp635ndash647 2010

[32] A Kolpas J Moehlis T A Frewen and I G KevrekidisldquoCoarse analysis of collective motion with different communi-cation mechanismsrdquoMathematical Biosciences vol 214 no 1-2pp 49ndash57 2008

[33] I D Couzin ldquoCollective cognition in animal groupsrdquo Trends inCognitive Sciences vol 13 no 1 pp 36ndash43 2008

[34] I D Couzin and J Krause ldquoSelf-organization and collectivebehavior in vertebratesrdquo Advances in the Study of Behavior vol32 pp 1ndash75 2003

[35] NW F Bode DW Franks andA JamieWood ldquoMaking noiseemergent stochasticity in collective motionrdquo Journal ofTheoret-ical Biology vol 267 no 3 pp 292ndash299 2010

[36] I D Couzin J Krause R James GD Ruxton andN R FranksldquoCollective memory and spatial sorting in animal groupsrdquoJournal of Theoretical Biology vol 218 no 1 pp 1ndash11 2002

[37] I D Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 pp715ndash728 2007

[38] S Bazazi J Buhl J J Hale et al ldquoCollective motion andcannibalism in locustmigratory bandsrdquoCurrent Biology vol 18no 10 pp 735ndash739 2008

[39] T J Atherton and D J Kerbyson ldquoUsing phase to representradius in the coherent circle Hough transformrdquo in Proceedingsof the IEE Colloquium on the Hough Transform IEE LondonUK 1993

[40] L Xu E Oja and P Kultanen ldquoA new curve detection methodrandomized Hough transform (RHT)rdquo Pattern RecognitionLetters vol 11 no 5 pp 331ndash338 1990

[41] V Ayala-Ramirez C H Garcia-Capulin A Perez-Garcia andR E Sanchez-Yanez ldquoCircle detection on images using geneticalgorithmsrdquo Pattern Recognition Letters vol 27 no 6 pp 652ndash657 2006

[42] E Cuevas N Ortega-Sanchez D Zaldivar and M Perez-Cisneros ldquoCircle detection by harmony search optimizationrdquoJournal of Intelligent and Robotic Systems vol 66 no 3 pp 359ndash376 2012

[43] E Cuevas D Oliva D Zaldivar M Perez-Cisneros and HSossa ldquoCircle detection using electro-magnetismoptimizationrdquoInformation Sciences vol 182 no 1 pp 40ndash55 2012

[44] E Cuevas D Zaldivar M Perez-Cisneros and M Ramırez-Ortegon ldquoCircle detection using discrete differential evolutionoptimizationrdquo Pattern Analysis and Applications vol 14 no 1pp 93ndash107 2011

[45] S Dasgupta S Das A Biswas and A Abraham ldquoAutomaticcircle detection on digital images with an adaptive bacterialforaging algorithmrdquo SoftComputing vol 14 no 11 pp 1151ndash11642010

[46] N W F Bode A J Wood and D W Franks ldquoThe impact ofsocial networks on animal collective motionrdquo Animal Behav-iour vol 82 no 1 pp 29ndash38 2011

[47] B H Lemasson J J Anderson and R A Goodwin ldquoCollectivemotion in animal groups from a neurobiological perspectivethe adaptive benefits of dynamic sensory loads and selectiveattentionrdquo Journal ofTheoretical Biology vol 261 no 4 pp 501ndash510 2009


[48] M Bourjade B Thierry M Maumy and O Petit ldquoDecision-making in przewalski horses (equus ferus przewalskii) is drivenby the ecological contexts of collective movementsrdquo Ethologyvol 115 no 4 pp 321ndash330 2009

[49] A Bang S Deshpande A Sumana and R Gadagkar ldquoChoos-ing an appropriate index to construct dominance hierarchiesin animal societies a comparison of three indicesrdquo AnimalBehaviour vol 79 no 3 pp 631ndash636 2010

[50] Y Hsu R L Earley and L L Wolf ldquoModulation of aggressivebehaviour by fighting experience mechanisms and contestoutcomesrdquo Biological Reviews of the Cambridge PhilosophicalSociety vol 81 no 1 pp 33ndash74 2006

[51] M Broom A Koenig and C Borries ldquoVariation in dominancehierarchies among group-living animals modeling stability andthe likelihood of coalitionsrdquo Behavioral Ecology vol 20 no 4pp 844ndash855 2009

[52] K L Bayly C S Evans and A Taylor ldquoMeasuring social struc-ture a comparison of eight dominance indicesrdquo BehaviouralProcesses vol 73 no 1 pp 1ndash12 2006

[53] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[54] A Okubo ldquoDynamical aspects of animal groupingrdquo Advancesin Biophysics vol 22 pp 1ndash94 1986

[55] C W Reynolds ldquoFlocks herds and schools a distributedbehavioural modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash33 1987

[56] S Gueron S A Levin and D I Rubenstein ldquoThe dynamics ofherds from individuals to aggregationsrdquo Journal of TheoreticalBiology vol 182 no 1 pp 85ndash98 1996

[57] A Czirok and T Vicsek ldquoCollective behavior of interacting self-propelled particlesrdquo Physica A vol 281 no 1 pp 17ndash29 2000

[58] M Ballerini ldquoInteraction ruling collective animal behaviordepends on topological rather than metric distance evidencefrom a field studyrdquo Proceedings of the National Academy ofSciences of the United States of America vol 105 pp 1232ndash12372008

[59] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Beckington UK 2008

[60] A H Gandomi X S Yang and A H Alavi ldquoCuckoo searchalgorithm a metaheuristic approach to solve structural opti-mization problemsrdquo Engineering with Computers vol 29 no 1pp 17ndash35 2013

[61] H Zang S Zhang andKHapeshi ldquoA review of nature-inspiredalgorithmsrdquo Journal of Bionic Engineering vol 7 pp S232ndashS2372010

[62] A Gandomi and A Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 pp 4831ndash4845 2012

[63] J E Bresenham ldquoA linear algorithm for incremental digitalsisplay of circular arcsrdquo Communications of the ACM vol 20no 2 pp 100ndash106 1977

[64] F Wilcoxon ldquoIndividual comparisons by ranking methodsrdquoBiometrics Bulletin vol 1 no 6 pp 80ndash83 1945

[65] S Garcia D Molina M Lozano and F Herrera ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 specialsession on real parameter optimizationrdquo vol 15 no 6 pp 617ndash644 2009

[66] J Santamarıa O Cordon S Damas J M Garcıa-Torres and AQuirin ldquoPerformance evaluation of memetic approaches in 3D

reconstruction of forensic objectsrdquo Soft Computing vol 13 no8-9 pp 883ndash904 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


multimodal function optimization such as sequential fitnesssharing [15 16] deterministic crowding [17] probabilisticcrowding [18] clustering-based niching [19] clearing pro-cedure [20] species conserving genetic algorithm [21] andelitist-population strategies [22] However algorithms basedon the GAs do not guarantee convergence to global optimabecause of their poor exploitation capability GAs exhibitother drawbacks such as the premature convergence whichresults from the loss of diversity in the population andbecomes a common problem when the search continues forseveral generations Such drawbacks [23] prevent the GAsfrom practical interest for several applications

Using a different metaphor other researchers haveemployed artificial immune systems (AIS) to solve the multi-modal optimization problems Some examples are the clonalselection algorithm [24] and the artificial immune network(AiNet) [25 26] Both approaches use some operators andstructures which attempt to algorithmically mimic the natu-ral immune systemrsquos behavior of human beings and animals

Several studies have been inspired by animal behaviorphenomena in order to develop optimization techniques suchas the particle swarm optimization (PSO) algorithm whichmodels the social behavior of bird flocking or fish schooling[27] In recent years there have been several attempts to applythe PSO to multimodal function optimization problems [2829] However the performance of such approaches presentsseveral flaws when it is compared to the other multi-modalmetaheuristic counterparts [26]

Recently the concept of individual organization [30 31]has been widely used to understand collective behavior ofanimals The central principle of individual organization isthat simple repeated interactions between individuals canproduce complex behavioral patterns at group level [30 3233] Such inspiration comes from behavioral patterns seen inseveral animal groups such as ant pheromone trail networksaggregation of cockroaches and themigration of fish schoolswhich can be accurately described in terms of individuals fol-lowing simple sets of rules [34] Some examples of these rules[33 35] include keeping current position (or location) forbest individuals local attraction or repulsion randommove-ments and competition for the space inside of a determineddistance On the other hand new studies have also shownthe existence of collective memory in animal groups [36ndash38]The presence of such memory establishes that the previoushistory of group structure influences the collective behaviorexhibited in future stages Therefore according to these newdevelopments it is possible to model complex collectivebehaviors by using simple individual rules and configuringa general memory

On the other hand the problem of detecting circularfeatures holds paramount importance in several engineeringapplications The circle detection in digital images has beencommonly solved through the circular Hough transform(CHT) [39] Unfortunately this approach requires a largestorage space that augments the computational complexityand yields a low processing speed In order to overcome thisproblem several approaches which modify the original CHThave been proposed One well-known example is the ran-domized Hough transform (RHT) [40] As an alternative to

Hough-transform-based techniques the problem of shaperecognition has also been handled through optimizationmethods In general they have demonstrated to deliver betterresults than those based on the HT considering accuracyspeed and robustness [41] Such approaches have producedseveral robust circle detectors using different optimizationalgorithms such as genetic algorithms (GAs) [41] harmonysearch (HSA) [42] electromagnetism-like (EMO) [43] dif-ferential evolution (DE) [44] and bacterial foraging opti-mization (BFOA) [45] Since such evolutionary algorithmsare global optimizers they detect only the global optimum(only one circle) of an objective function that is defined overa given search space However extracting multiple-circleprimitives falls into the category of multi-modal optimiza-tion where each circle represents an optimumwhichmust bedetected within a feasible solution spaceThe quality for suchoptima is characterized by the properties of their geometricprimitives Big and well-drawn circles normally representpoints in the search space with high fitness values (possibleglobal maximum) whereas small and dashed circles describepoints with fitness values which account for local maximaLikewise circles holding similar geometric properties suchas radius and size tend to represent locations with similarfitness values Therefore a multi-modal method must beapplied in order to appropriately solve the problem of multi-shape detection In this paper a new multimodal optimiza-tion algorithm based on the collective animal behavior isproposed and also applied to multicircle detection

This paper proposes a new optimization algorithminspired by the collective animal behavior In this algorithmthe searcher agents emulate a group of animals that interactwith each other based on simple behavioral rules whichare modeled as evolutionary operators Such operations areapplied to each agent considering that the complete grouphas a memory which stores its own best positions seen so farby applying a competition principle Numerical experimentshave been conducted to compare the proposed method withthe state-of-the-art methods on multi-modal benchmarkfunctions Besides the proposed algorithm is also applied tothe engineering problem of multicircle detection achievingsatisfactory results

This paper is organized as follows Section 2 introducesthe basic biological aspects of the algorithm In Section 3the proposed algorithm and its characteristics are describedA numerical study on different multi-modal benchmarkfunctions is presented in Section 4 Section 5 presents theapplication of the proposed algorithm to multi-circle detec-tion whereas Section 6 shows the obtained results Finally inSection 7 the conclusions are discussed

2 Biological Fundaments

The remarkable collective behavior of organisms such asswarming ants schooling fish and flocking birds has longcaptivated the attention of naturalists and scientists Despitea long history of scientific investigation just recently we arebeginning to decipher the relationship between individu-als and group-level properties [46] Grouping individualsoften have to make rapid decisions about where to move



















119886119895119894 = 119886low119895

+ rand (0 1) sdot (119886high119895

minus 119886low119895)

119895 = 1 2 119863 119894 = 1 2 119873119901

(1)






a119897 = m119897ℎ+ v (2)




a119894 =





(3)


andmnearest119892






(4)


120588








m2119880 m2119861minus1





120588 =

prod119863

119895=1(119886

high119895

minus 119886low119895)

10 sdot 119863

(5)

where 119886low119895

and 119886high119895










length



ℎminus x119894)

else if


minus x119894)


a119894 = r





6 (6)














119891 (1199091 1199092) = 119890minus((1199091minus4)


2)+ 119890minus((1199091+4)


2)

+ 2 sdot 119890minus((1199091)

2+(1199092)

2)+ 2 sdot 119890

minus((1199091)2minus(1199092+4)

2)

(7)





4

119892) and Mℎ (m1ℎ m

4











and mnearest119892



ℎm4119892) 119887 = dist(m2

ℎm3119892)




1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)


119892andM




1 a2 a




119892andM


ℎafter

30 iterations












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























119886119895119894 = 119886low119895

+ rand (0 1) sdot (119886high119895

minus 119886low119895)

119895 = 1 2 119863 119894 = 1 2 119873119901

(1)






a119897 = m119897ℎ+ v (2)




a119894 =





(3)


andmnearest119892






(4)


120588








m2119880 m2119861minus1





120588 =

prod119863

119895=1(119886

high119895

minus 119886low119895)

10 sdot 119863

(5)

where 119886low119895

and 119886high119895










length



ℎminus x119894)

else if


minus x119894)


a119894 = r





6 (6)














119891 (1199091 1199092) = 119890minus((1199091minus4)


2)+ 119890minus((1199091+4)


2)

+ 2 sdot 119890minus((1199091)

2+(1199092)

2)+ 2 sdot 119890

minus((1199091)2minus(1199092+4)

2)

(7)





4

119892) and Mℎ (m1ℎ m

4











and mnearest119892



ℎm4119892) 119887 = dist(m2

ℎm3119892)




1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)


119892andM




1 a2 a




119892andM


ℎafter

30 iterations












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













a119897 = m119897ℎ+ v (2)




a119894 =





(3)


andmnearest119892






(4)


120588








m2119880 m2119861minus1





120588 =

prod119863

119895=1(119886

high119895

minus 119886low119895)

10 sdot 119863

(5)

where 119886low119895

and 119886high119895










length



ℎminus x119894)

else if


minus x119894)


a119894 = r





6 (6)














119891 (1199091 1199092) = 119890minus((1199091minus4)


2)+ 119890minus((1199091+4)


2)

+ 2 sdot 119890minus((1199091)

2+(1199092)

2)+ 2 sdot 119890

minus((1199091)2minus(1199092+4)

2)

(7)





4

119892) and Mℎ (m1ℎ m

4











and mnearest119892



ℎm4119892) 119887 = dist(m2

ℎm3119892)




1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)


119892andM




1 a2 a




119892andM


ℎafter

30 iterations












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















length



ℎminus x119894)

else if


minus x119894)


a119894 = r





6 (6)














119891 (1199091 1199092) = 119890minus((1199091minus4)


2)+ 119890minus((1199091+4)


2)

+ 2 sdot 119890minus((1199091)

2+(1199092)

2)+ 2 sdot 119890

minus((1199091)2minus(1199092+4)

2)

(7)





4

119892) and Mℎ (m1ℎ m

4











and mnearest119892



ℎm4119892) 119887 = dist(m2

ℎm3119892)




1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)


119892andM




1 a2 a




119892andM


ℎafter

30 iterations












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119891 (1199091 1199092) = 119890minus((1199091minus4)


2)+ 119890minus((1199091+4)


2)

+ 2 sdot 119890minus((1199091)

2+(1199092)

2)+ 2 sdot 119890

minus((1199091)2minus(1199092+4)

2)

(7)





4

119892) and Mℎ (m1ℎ m

4











and mnearest119892



ℎm4119892) 119887 = dist(m2

ℎm3119892)




1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)


119892andM




1 a2 a




119892andM


ℎafter

30 iterations












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1199092

05

1199091

05

119909 0119909 00

Fitn

ess v

alue 2

1

0

minus5minus5

(a)

0 5

0

5

1199091

1199092

1199095

11990941199098

1199092

11990910

1198942

1199099

1199091

1198949

1199093

11989481199097

1198944

1199096

1198943

1198947

11989410

11989411198946

1198945

minus5minus5

(b)

0 5

0

5

1199091

1199092

11990910

1199092

11990951199099

1199093

1199096

1199097 1199098 1199094

1199091

1199504ℎ

1199502ℎ

1199501ℎ

1199503ℎ 1199503

119892

1199501119892

1199502119892

1199504119892

minus5minus5

(c)

0 5

0

5

1199091

1199092

1199095

11990910

1199099

1199096

1199097 1199098

1199383

1199384

1199381

11993821199503ℎ

1199504ℎ

1199502ℎ

1199501ℎ

119907

119907

119907

119907

minus5minus5

(d)

0 5

0

5

1199091

1199092

11990910

11990951199099

1199096

1199097 1199098

1199382

1199388

1199383

1199381

1199384

1199501ℎ

1199503ℎ

1199502ℎ

1199504ℎ

minus119903(1199504ℎ minus 1199098)

minus5minus5

(e)

0

5

1199092

0 51199091

1199384

1199387

1199386

1199383

119938 1

11993891199503ℎ 1199382

1199385

11993810

1199388

1199502ℎ

1199504ℎ

1199501ℎ

minus5minus5

(f)

0

5

1199092

0 51199091

1199093

1199092

1199091

1199094119887

119938

1199501ℎ

1199504ℎ

1199503ℎ

1199502ℎ

1199504119892

1199503119892

1199501119892

1199502119892

119888

minus5minus5

(g)

0

5

1199092

0 51199091

1199502ℎ

1199501ℎ

1199504ℎ 1199503

ℎ

1199501119892

1199503119892

1199504119892

1199502g

minus5minus5

(h)

0

5

1199092

0 51199091

1199503ℎ

1199501ℎ

1199502ℎ

1199504ℎ

minus5minus5

(i)


119892andM




1 a2 a




119892andM


ℎafter

30 iterations












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198911= sin6(5120587119909) 119909 isin [0 1]

121

08060402

00

01 02 03 04 05 06 07 08 09 1


1198912(119909) = 2

minus2((119909minus01)09)2

sdot sin(5120587119909) 119909 isin [0 1]

121

08060402

00 01 02 03 04 05 06 07 08 09 1


1198913(119911) =

1

1 +10038161003816100381610038161199116 + 1

1003816100381610038161003816

119911 isin 119862 119911 = 1199091+ 1198941199092

1199091 1199092isin [minus2 2] 1 1 22

108060402

0



1198914(1199091 1199092) = 1199091sin(4120587119909

1) minus 1199092sin(4120587119909

2+ 120587) + 1 119909

1 1199092isin [minus2 2]

246

0


minus2minus2

minus2minus4







1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198915(1199091 1199092) = minus(20 + 119909

2

1+ 1199092

2minus 10(cos(2120587119909

1) + cos(2120587119909

2))) 119909

1 1199092isin [minus10 10]

minus100

minus200minus250

minus150

minus500



1198916(1199091 1199092) = minus

2

prod

119894=1

5

sum

119895=1

cos((119895 + 1)119909119894+ 119895) 119909

1 1199092isin [minus10 10]


0

200100



1198917(1199091 1199092) =

1

4000

2

sum

119894=1

1199092

119894minus

2

prod

119894=1

cos(119909119894

radic2) + 1 119909

1 1199092isin [minus100 100]

0minus2

minus4minus6minus8



1198918(1199091 1199092) =

cos(051199091) + cos(05119909

2)

4000+ cos(10119909

1) cos(10119909

2) 119909

1 1199092isin [0 120]

minus2

02

120

12080

8040

400 0






















1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























1198911


minus4) 7153 (358) 0091 (0013)


minus5) 10304 (487) 0163 (0011)


minus5) 9927 (691) 0166 (0028)


minus5) 5860 (623) 0128 (0021)


minus5) 10781 (527) 0237 (0019)


minus5) 6792 (352) 0131 (0009)


minus5) 2591 (278) 0039 (0007)


minus5) 15803 (381) 0359 (0015)


minus5) 12369 (429) 0421 (0021)


1198912


minus4) 6026 (832) 0271 (006)


minus4) 10940 (9517) 0392 (007)


minus4) 12796 (1430) 0473 (011)


minus3) 8465 (773) 0326 (005)


minus4) 14120 (2187) 0581 (014)


minus4) 10548 (1382) 0374 (009)


minus5) 3605 (426) 0102 (004)


minus4) 21922 (746) 0728 (006)


minus4) 18251 (829) 0664 (008)


1198913


minus4) 11009 (1137) 107 (013)


minus4) 16391 (1204) 172 (012)


minus3) 14424 (2045) 184 (026)


minus3(542 times 10

minus4) 18755 (2404) 203 (031)


minus5) 13814 (1382) 175 (021)


minus5) 6218 (935) 053 (007)


minus3) 25953 (2918) 255 (033)


minus4) 20335 (1022) 215 (010)


minus5) 4359 (75) 011 (0023)

1198914


minus3) 1861707 (329254) 2163 (201)


minus4) 2638581 (597658) 3124 (532)


minus4) 2498257 (374804) 2847 (351)


minus4) 2257964 (742569) 2531 (624)


minus3) 2978385 (872050) 3527 (841)


minus3) 2845789 (432117) 3215 (485)


minus5) 1202318 (784114) 1217 (229)


minus3) 3752136 (191849) 4595 (156)


minus4) 2745967 (328176) 3818 (377)










1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1198915


minus3) 1760199 (254341) 1462 (283)


minus4) 2631627 (443522) 3439 (520)


minus3) 2726394 (562723) 3655 (713)


minus3) 2107962 (462622) 2861 (647)


minus3) 2835119 (638195) 3705 (823)


minus4) 2518301 (643129) 3027 (704)


minus4) 978435 (71135) 1056 (481)


minus3) 5075208 (194376) 5802 (219)


minus3) 3342864 (549452) 5165 (691)


1198916


minus4) 832546 (75413) 458 (057)


minus4) 1823774 (265387) 1292 (201)


minus4) 1767562 (528317) 1412 (351)


minus4) 1875729 (265173) 1120 (269)


minus3) 2049225 (465098) 1826 (441)


minus4) 2261469 (315727) 1371 (184)


minus4) 656639 (84213) 312 (112)


minus3) 4989856 (618759) 3385 (536)


minus3) 3012435 (332561) 2632 (254)


1198917


minus3) 2386960 (221982) 1910 (226)


minus3) 3861904 (457862) 4353 (438)


minus3) 3619057 (565392) 4298 (635)


minus3) 3746325 (594758) 4542 (764)


minus3) 4155209 (465613) 4823 (542)


minus3) 3629461 (373382) 4784 (021)


minus4) 1723342 (121043) 1254 (131)


minus3) 5423739 (231004) 4784 (609)


minus4) 4329783 (167932) 4164 (265)


1198918


minus3) 2843452 (353529) 2314 (385)


minus3) 4325469 (574368) 4951 (672)


minus3) 4416150 (642415) 5443 (126)


minus3) 4172462 (413537) 5239 (721)


minus3) 4711925 (584396) 6107 (814)


minus3) 3964491 (432117) 5387 (846)


minus4) 2213754 (412538) 1621 (319)


minus3) 5835452 (498033) 7426 (547)


minus4) 4123342 (213864) 6038 (521)






(119909 minus 1199090)2+ (119910 minus 1199100)

2= 1199032 (8)


215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15



215

105

0minus05

minus1

minus15


(e) CBN

215

105

0minus05

minus1

minus15


(f) SCGA2

151

050

minus05

minus1

minus15


(g) AEGA

215

105

0minus05

minus1

minus15



151

050

minus05

minus1

minus15


(i) AiNet

215

105

0minus05

minus1

minus15


(j) CAB


comparison


119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119903

119901119894

119901119896

119901119895

(1199090 1199100)


119894 119901119895 and 119901

119896

Consider

A = [

[

1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894) 2 sdot (119910119895 minus 119910119894)

1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894) 2 sdot (119910119896 minus 119910119894)

]

]

B = [

[

2 sdot (119909119895 minus 119909119894) 1199092

119895+ 1199102

119895minus (1199092

119894+ 1199102

119894)

2 sdot (119909119896 minus 119909119894) 1199092

119896+ 1199102

119896minus (1199092

119894+ 1199102

119894)

]

]

(9)

1199090 =det (A)


1199100 =det (B)


(10)


2 (11)






119869 (119862) = 1 minussum119873119904

119907=1119864 (119909119907 119910119907)

119873119904

(12)


119864 (119909119907 119910119907) =


0 otherwise(13)


119907=1119864(119909119907 119910119907) = 18





119863119860119861 =1003816100381610038161003816119909119860 minus 119909119861

1003816100381610038161003816 +1003816100381610038161003816119910119860 minus 119910119861

1003816100381610038161003816 +1003816100381610038161003816119903119860 minus 119903119861

1003816100381610038161003816 (14)



119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119901119894 119901119896

119901119895

(a) (b)

119901119894 119901119896

119901119895

(c)


119894 119901119895 and 119901





119889 (15)



M2 119862

M119861





element 119862M2


119863119862M1119862M2gt Th then119862M






M1)10)





set using (1)









ℎminus x119894)


119892minus x119894)


119862119894 = r


(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)




1 119862

M2 119862

M119861 represent








If119863119862M119898119862M119898minus1




1)10)




(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

Original images

GA-based algorithm

BFOA

CAB

(b) (c)








(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

(b)


(c)



119873119901

119867 119875 119861 NI30 05 01 12 200



1003816100381610038161003816) + 120583 sdot1003816100381610038161003816119903true minus 119903119863

1003816100381610038161003816 (17)












(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















(b) 17454119890 minus 004 19011119890 minus 004

(c) 17981119890 minus 004 18922119890 minus 004


(b) 16989119890 minus 004 19124119890 minus 004

(c) 17012119890 minus 004 19081119890 minus 004



ME = ( 1

119873119862) sdot

119873119862

sum

119877=1

Es119877 (18)






7 Conclusions











References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Swarm Optimization Algorithm for...

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