Research Article Fuzzy Multilevel Image Thresholding Based...

Research ArticleFuzzy Multilevel Image Thresholding Based onModified Quick Artificial Bee Colony Algorithm andLocal Information Aggregation

Linguo Li12 Lijuan Sun13 Jian Guo13 Chong Han13 and Shujing Li2

1School of Computer Nanjing University of Posts and Telecommunications Nanjing 210003 China2College of Information Engineering Fuyang Normal University Fuyang 236041 China3Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks Nanjing University of Posts andTelecommunications Nanjing 210003 China

Correspondence should be addressed to Linguo Li llg-1212163com

Received 27 July 2016 Revised 26 November 2016 Accepted 28 November 2016

Academic Editor Erik Cuevas

Copyright copy 2016 Linguo Li et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Thresholding segmentation based on fuzzy entropy and intelligent optimization is one of the most commonly used and directmethods This paper takes fuzzy Kapurrsquos entropy as the best optimal objective function with modified quick artificial bee colonyalgorithm (MQABC) as the tool performs fuzzy membership initialization operations through Pseudo Trapezoid-Shaped (PTS)membership function and finally according to the imagersquos spacial location information conducts local information aggregationby way of median average and iterative average so as to achieve the final segmentation The experimental results show thatthe proposed FMQABC (fuzzy based modified quick artificial bee colony algorithm) and FMQABCA (fuzzy based modifiedquick artificial bee colony and aggregation algorithm) can search out the best optimal threshold very effectively precisely andspeedily and in particular show exciting efficiency in running timeThis paper experimentally compares the proposedmethod withKapurrsquos entropy-based Electromagnetism Optimization (EMO) method standard ABC and FDE (fuzzy entropy based differentialevolution algorithm) respectively and concludes that MQABCA is far more superior to the rest in terms of segmentation qualityiterations to convergence and running time

1 Introduction

Thresholding is one of the most direct simple and efficientapproaches to image segmentation which can discriminateobjects from the background through a set of thresholdsat pixel levels The automatic separation between objectsand background remains the most difficult yet intriguingdomain for image processing and pattern recognition [1] Itnot only can show well defined regions with the minimumoverlap and effectiveness of aggregation but also can provideinitial prediction or preprocessing for more complicatedsegmentation methods (such as level-sets or active contours)[2] Consequently thresholding segmentation is regarded asthe key link in image analysis and image understanding andtherefore is widely used in many areas like medical analysis[3] image classification [4] object recognition [5] and imagecopy detection [6] and so forth

Thresholding segmentation is the most commonly usedmethod for image segmentation [7] Bilevel thresholds aremainly used to separate the object from the backgroundnamely bilevel thresholding Multilevel thresholding (MT)can classify the pixels into different groups hence multipleregions are created which can lead to finer segmentationresults The thresholding methods proposed in the literaturecan be divided into two categories parametric and nonpara-metric [8ndash10] The parametric are usually time consumingand computationally expensive whereas the nonparametricaremore robust and precise and as a result getmore attentionas they can determine the optimal threshold by optimizingsome standards like between-class variance the entropy theerror rate and so forth [8ndash11]This paper from the perspectiveof nonparametric methods will be focused on the studyof MT methods through intelligent optimization and fuzzytheory

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5985616 18 pageshttpdxdoiorg10115520165985616

2 Mathematical Problems in Engineering

Of nonparametric methods gray level global thresh-olding adopting information entropy is the most popularone Such method takes the intelligent optimization as thetool and by optimizing some information entropy to getthe maximum (or minimum) objective function to achievethe final segmentation The most commonly used objectivefunctions include the maximization of the entropy (egKapurrsquos entropy) [12] maximization of the between-classvariance (eg Otsursquos method) [13] the use of the fuzzysimilarity measure [14] and minimization of the Bayesianerror [15] All of these techniques have been extended into thearea ofmultilevel image segmentation But the computationalcomplexity for the application of such methods in MTgrows dramatically As a consequence scholars in recentyears have extensively used evolutionary and swarm-basedintelligent computational techniques to promote multilevelcomputational efficiency for multilevel image segmentationand precision of threshold [11] But the global multilevelthresholds generally get the final segmentation results bymaximizing the separation between classes Yet it neglectsthe complexity in uneven illumination and soft transitionsbetween gray levels and fails to consider the spatial rela-tionships between pixels which lead to the imperfectionsof segmentation results [1] The biggest problem with theglobal thresholding lies in the fact that it is pixel orientedrather than region oriented Therefore pixels that have thesame gray levels are segmented into the same regions if suchfactors as connectivity of pixels and closeness of objects arenot considered As a result such methods are prone to createisolated pixels

Of all the research findings for the various thresholdingmethods each one of them is designed specifically to dealwith one particular kind of images Sezgin and Sankur [16]were the first to classify the image thresholding methods intosix categories Aja-Fernandez et al [1] further grouped theminto three kinds based on the six classifications (a) Methodsthat calculate a global threshold for the whole image (b)Methods that use an adaptive local threshold (c) Methodsthat use spatial local information for classifying the pixelsThe first kind was the earliest to be employed But due tothe drawbacks inherent in it currently scholars have directedtheir focus on the latter two The second kind is an improve-ment on and complement to the first kind which selects thethreshold locally through adaptability so as to avoid localnonadaptability but the computational complexity will growaccordingly The third kind is mainly employed to make upfor the deficiency in spatial relationship and introduces thelocal spatial information to heighten the effects of segmentedresults In this paper we introduce local spatial informationto solve segmentation problems and at the same time explorethe applications of fuzzy logic and fuzzy theory in imagethresholding The starting point of introducing local spatialinformation comes from the definition of one class or targetwhere the membership of a pixel is highly correlated withthe membership of the pixels of surrounding regions [1]In order to promote the quality of image segmentationfuzzy membership function is used to assign each pixel withfuzzy degrees And according to fuzzy theory the traditionalhard assigning regions or classes will be replaced with soft

assigning patternDetailedly the fuzzyKapurrsquos entropywhichis modified by Kapurrsquos entropy is adopted in this methodas the objective function It combines with the MQABCalgorithm to generate the optimal thresholds which are usedas the centroids for fuzzy membership assignment and localinformation aggregation

The rest of the paper is organized as follows Section 2introduces related work of fuzzy entropy intelligent opti-mization and local information aggregation in the field ofMT In Section 3 the nonfuzzy hard thresholding and fuzzysoft thresholding are formulated and then the correspondingKapurrsquos entropy for image segmentation and the ultimateobjective function are presented In Section 4 the standardbee colony algorithm is briefly described and the proposedMQABC is dealt with in detail Section 5 analyzes themultiplethresholds-based fuzzy membership initialization and fuzzyaggregationmethodsThe experiment results and discussionsare presented in Section 6 The last section presents theconclusions

2 Related Works

Fuzzy theory and fuzzy logic are well-known to be flexibletools where imprecise knowledge or not-well defined featureshave to be used [17] It has been extensively used in systemcontrol domain and applied in image processing to someextent [18ndash21] In the past 10 years thresholding segmenta-tion approaches have been proposed based on fuzzy logicand fuzzy calculations In such approaches fuzzy logic iscombined with clustering method [22] fuzzy compactness[23] the interpretation of thresholds as type 2 fuzzy logic[24] and the soft calculations (such as ant colonies beecolonies and bacteria migration heuristics) [11] respectivelymainly used to search for the optimal thresholds but theyfailed to take the pixel spatial relationship segmentationmethods into full account In these fuzzy based thresholdingmethods clustering methods are among the most popu-lar but they are highly complicated where computationaltime is concerned With the progress and upgradation ofoptimization methods fuzzy optimization in thresholdingsegmentation gets more and more attention and it has beenproved feasible to employ intelligent optimization to solvethresholding problems [10] A large number of literature showthat intelligent optimizations are far superior to traditionalmethods in processing speed precision and robustness [510] Oliva et al [7] introduced Electromagnetism Optimiza-tion into thresholding segmentation and compared it withKapurrsquos entropy and Otsursquos method The experiments showthat Kapurrsquos entropy is more efficient Ghamisi et al [25]have fully analyzed the performance in thresholding of suchmethods as Particle swarm optimization (PSO) DarwinianParticle Swarm Optimization (DPSO) and Fractional-orderDarwinianParticle SwarmOptimization (FODPSO) In com-parison with Bacteria Foraging algorithm (BF) and geneticalgorithms (GA) FOPDSO delivers better performance inovercoming local optimum trap and computational speedKurban et al [10] have conducted a comparative study ofthe applications of the computational techniques involvedin MT exhaustively According to the statistical analysis of

Mathematical Problems in Engineering 3

the objective value the swarm-based method is more able toprecisely solve segmentation problems In a further analysisswarm-based thresholding segmentation mainly takes Otsursquosmethod between-class variance Tsallis entropy and Kapurrsquosentropy as the objective functions the experimental resultsshow that the optimized processing by way of artificialbee colony could obtain better segmentation results Akay[8] took between-class variance and Kapurrsquos entropy as theobjective functions respectively and compared artificial beecolony method with the particle swarms one it shows thatKapurrsquos entropy-based artificial bee colony method showsbetter performance when the number of the thresholdsincreases and the complexity of computational time is alsoreduced Bhandari et al [26] conducted detailed comparativeanalysis between Kapurrsquos Otsu and Tsallis functions throughexperiments which shows that in the remote sensing satelliteimage segmentation Kapurrsquos entropy-based artificial beecolony method is far superior

The above-mentioned optimized methods have madegreat progress in precision efficiency and segmentationquality in solving thresholding problems But classifyingregions by hard thresholding will easily lead to edge fuzzinessand even isolated pixels [27] In more recent years fuzzytheory based thresholding method has become the focusof study as the proposed soft thresholding strategy caneffectively solve the problem Zhao et al [28] were theearliest to define the three-membership functions and thenapplied them to three-level thresholding problems Basedon that Tao et al [29] proposed three-level fuzzy entropybased thresholding methods the three functions namely I-function F-function and S-function are defined in detailThemethodmaximizes global fuzzy entropy throughGeneticAlgorithm (GA) to get the optimal thresholds These twomethods only show the visual effects failing to present aqualitative analysis Pratamasunu et al [30] assigned fuzzycoefficient to each I-function and then surveyed the functionto achieve the automation of thresholding by defining thedistance between the fuzzy set and its closest crisp setAlthough the method shows a comparative advantage overthe Lopes method and Otsu thresholding the experimentalresults are only applicable to bilevel thresholds and fail toshow the effects of MT Muppidi et al [31] used Triangularmembership function Trapezoidal membership functionand Bell-shaped membership function to define three kindsof fuzzy entropies respectively and search out the optimalparameters set throughGA Suchmethods are only comparedwith Otsu thresholding and failed to give adequate quanti-tative analysis Sarkar et al [27] based on image histogramand fuzzy entropy theory used Different Evolution methodto get the optimal threshold to obtainmultilevel thresholdingsegmentation Compared with PSO and GA such methoddelivers better performance in speed and precision and atthe same time the author offers a large number of numericalexperimental comparisons and data contrast Aja-Fernandezet al [1] have come up with the solution scheme for multire-gion thresholding segmentation by using fuzzy entropy andaggregation methodsThe scheme also provides solutions forhistogram and fuzzy entropy More importantly the schemenoted the drawbacks of pixel spatial relationship which the

thresholding segmentation failed to take into account Basedon this by local information aggregation it improves thesegmentation results and removes the isolated pixels Inlocal information aggregation methods the author presentsmedian aggregation average aggregation and iterative aver-age aggregation The experimental results testified to theeffectiveness of these aggregation methods Taken togetherthe fuzzy and nonfuzzy multilevel thresholding methods itcan be concluded that the proposed method based on fuzzyentropy and intelligent optimization in combination withlocal spatial information aggregation is a feasible solutionscheme for multilevel thresholding segmentation

Artificial bee colony method is a simple yet highlyefficient optimized pattern and its search iteration is mucheasier to be implemented compared with other heuristicsearch methods [32] More importantly artificial bee colonyemploys few predefined control parameters which reducesthe interference of artificiality set parameters But therestill exists some drawbacks in its solution search equationwhich is fine at exploration but inferior at exploitationprocess [26] In quick artificial bee colony methods [33] anadvanced solution equation was proposed In the processof searching out the optimal equation the onlooker beesearches only around the best solution of previous iterationsto improve exploitation Therefore the core is to determinethe search scope The scope if larger than necessary willgreatly heighten the complexity otherwise it will be reducedto local optimization However different problems call fordifferent distance strategies and patterns it is also true withMTThis paper will be focused on the study of the multilevelthresholding methods which are based on Kapurrsquos entropymodified quick artificial bee colony algorithm for imagesegmentation and local information aggregation

3 Formulation of MultilevelFuzzy Kapurrsquos Entropy

MT needs to set a set of thresholds 119905119894 and the image canbe segmented into different regions based on different graylevels Let image 119868 is be segmented in order to achieve thefinal segmentation the proposed method assumes that everypixel in the image 119868will have a degree of membership in eachof the 119873 regions That membership will be modeled usingfuzzy membership functions In what follows we will givethe optimal threshold obtained through fuzzy entropy andartificial bee colony and then assign each pixel with a fuzzydegree in different regions

31 Concept of Kapurrsquos Entropy for Hard Thresholding Basedon the above analyses the intelligent optimization whichtakes Kapurrsquos entropy as the objective function gets thebetter segmentation results Kapurrsquos method can be easilyextended from bilevel thresholding to multilevel threshold-ing Entropy-based thresholding takes the gray level his-togram as the basisWhen the entropy reaches themaximiza-tion the optimal threshold will be distributed normally

Entropy of a discrete information source can be obtainedby the probability distribution 119901 = 119901119894 where 119901119894 is theprobability of the system in possible state 119894 [7]Theprobability


for each gray level 119894 is its relative occurrence frequency Letthe number of gray levels for image 119868 be ℎ(119894) the probabilitydistribution can be described as follows

119901119894 = ℎ (119894)sum119871minus1119894=0 ℎ (119894) 119894 = 0 1 119871 minus 1 (1)

Kapurrsquos entropy is used to measure the compactnessand separability of classes For MT Kapurrsquos entropy can bedescribed in

1198670 = minus1199051minus1sum119894=0

1199011198941205960 ln1199011198941205960 1205960 =

1199051minus1sum119894=0

119901119894

1198671 = minus1199052minus1sum119894=1199051

1199011198941205961 ln1199011198941205961 1205961 =

1199052minus1sum119894=1199051

119901119894

119867119895 = minus119905119895+1minus1sum119894=119905119895

119901119894120596119895 ln119901119894120596119895 120596119895 =

119905119895+1minus1sum119894=119905119895

119901119894

119867119898 = minus119871minus1sum119894=119905119898

119901119894120596119898 ln119901119894120596119898 120596119898 =

119871minus1sum119894=119905119898

119901119894

(2)

Thus the function of (119879) can be obtained by (3) used asa parameter of MQABCrsquos fitness function in Section 4

119891 (119879) = 119898sum119894=0

minus119867119894 119879 = [1199050 1199051 sdot sdot sdot 119905119898] (3)

where 119879 represents a vector quantity of thresholds and 119898represents the stop value of the number of thresholds

32 Concept of Fuzzy Kapurrsquos Entropy for Soft ThresholdingLet the set119877 be a collection of elements that can either belongto or not belong to region 119903 according to the fuzzy theory itcan be defined as follows

119877 = (119894 120583119903 (119894)) | 119894 isin 119868 0 le 120583119903 (119894) le 1 119903 isin [1119873] (4)

where 119906119903(119894) is called membership function which is usedto measure the closeness between pixel 119894 and region 119903 Thehigher the value is the bigger the probability that the pixelwill be distributed into region 119903 is and 119873 represents thenumber of regions

Option of membership function is the key factor forfuzzy thresholding segmentation A large number of lit-erature adopts different membership functions to test itseffectiveness trapezoidal membership function is the mostcommonly used and also the most efficient method [31] Justas in literature [27] in the case that there are four fuzzyparameters the curve of the fuzzy degrees can be shown inFigure 1

The fuzzy trapezoidal membership function with fourparameters can be easily extended to the scenario of 119899 (119899 =2 lowast (119898 + 1)) which can be formulated as in

1205830 (119896) =

1 119896 le 1198861119896 minus 11988621198861 minus 1198862 1198861 lt 119896 le 11988620 119896 gt 1198862

1205831 (119896) =

0 119896 le 1198861119896 minus 11988611198862 minus 1198861 1198861 lt 119896 le 11988621 1198862 lt 119896 le 1198863119896 minus 11988641198863 minus 1198864 1198863 lt 119896 le 11988640 119896 gt 1198864

120583119898 (119896) =

0 119896 le 119886119899minus1119896 minus 119886119899minus1119886119899 minus 119886119899minus1 119886119899minus1 lt 119896 le 1198861198991 119896 gt 119886119899

(5)

where 1198861 1198862 119886119899 represent the fuzzy parameters of thetrapezoidal membership function and 119896 is the gray value of asingle pixel Based on the trapezoidal membership functionthe fuzzification of Kapurrsquos entropy in Section 31 can bedescribed as in

1198670 = minus119871minus1sum119894=0

1205830 (119894) lowast 1199011198941205960 ln(1205830 (119894) lowast 1199011198941205960 )

1205960 =119871minus1sum119894=0

(1205830 (119894) lowast 119901119894)

1198671 = minus119871minus1sum119894=0

1205831 (119894) lowast 1199011198941205961 ln(1205831 (119894) lowast 1199011198941205961 )

1205961 =119871minus1sum119894=0

(1205831 (119894) lowast 119901119894)

119867119895 = minus119871minus1sum119894=0

120583119895 (119894) lowast 119901119894120596119895 ln(120583119895 (119894) lowast 119901119894120596119895 )

120596119895 =119871minus1sum119894=0

(120583119895 (119894) lowast 119901119894)

119867119898 = minus119871minus1sum119894=0

120583119898 (119894) lowast 119901119894120596119898 ln(120583119898 (119894) lowast 119901119894120596119898 )

120596119898 =119871minus1sum119894=0

(120583119898 (119894) lowast 119901119894)

(6)


In order to get the specific119873 parameters in (5) this paperwill adopt MQABC to minimize the total Kapurrsquos entropy asshown in (7) By comparing (2) and (6) we can clearly see thatthe fuzzy entropy involves more complicated computationalprocedures In such a case it is of necessity to adopt intelligentoptimization

120601 (1198861 1198862 119886119899) = 119891 (119879) =119898sum119894minus0

minus119867119894119879 = [1199050 1199051 sdot sdot sdot 119905119898]

(7)

After119873 parameters are obtained through the solution thecorresponding thresholds can be represented as in

1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(8)

4 Brief Explanation of Quick Artificial BeeColony Algorithm

ABC algorithms have been widely used as they show highefficiency and better convergence performance and employfewer control parameters in particular when the tuningof control parameters is more difficult than the problemitself In fuzzy MT the fuzzy entropy involves much moreprocedures of computations the algorithm which employs asmall number of control parameters is advised to be appliedA brief description of the ABC and theQABC algorithmswillbe given in the following subsections

41 Standard Artificial Bee Colony Algorithm In standardABC according to the migration behavior artificial beescan be classified into three types namely employed beesonlooker bees and scout bees Employed bees are relatedto specific food sources onlooker bees decide to choose acertain food source by observing the dance of employedbees while scout bees will search food randomlyThe specificbee colony behavior and related simulation can be found inliterature [32]

In the initialization phase all the food sources are initial-ized by (9) The number of food sources is set by predefinedparameters

119905119895119894 = 119897119894 + random (0 1) lowast (119906119894 minus 119897119894) (9)

where 119905119895119894 is the 119894th dimensional data of the 119895th food source119897119894 and 119906119894 stand for the lower limit and upper limit of theparameter 119905119895 119905119895 is the 119895th food source

In the employed bee phase a search is conducted in hermemory at a specific speed 120593119895119894 The speed (as shown in (10))determines the change rate of the 119905119895119894 food source which to a

certain degree affects the convergence speed If the solutionproduced by (10) is better than the beersquos solution then thememory is updated by a greedy selection approach

V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (10)

where 119905119903 represents a randomly selected food source 119894 isa randomly selected parameter index and 120593119895119894 is a randomnumber within the range [minus1 1]

To compare the advantages and disadvantages the fitnessof the solution is produced by (11) The higher fitness valuerepresents a better objective function value thus maximizingthe fitness function which can reach the optimal thresholds

fit (119905119895) = 1

(1 + 119891 (119905119895)) 119891 (119905119895) ge 01 + abs (119891 (119905119895)) 119891 (119905119895) lt 0

(11)

where 119891(119905119895) can be calculated by (4)An employed bee shares information about food source

fitness with an onlooker beeThe onlooker bee by probabilityselects one source to investigateThe probability function canbe represented by

prob119895 = fit (119905119895)sumSN119895=1 fit (119905119895) (12)

where SN represents the number of food sources

42ModifiedQuickArtificial Bee ColonyAlgorithm (MQABC)for Image Thresholding In standard ABC the employed beeand the onlooker bee use (10) to search for food sources Inother words the employed bee and the onlooker bee adoptthe same strategies in the same area to search for a newand better food source But in real honey bee colonies theemployed bee and the onlooker bee adopt different methodsto search for new food sources [33] Therefore for onlookerbees when they search for the best food it is quite reasonablethat the pattern differs from the search method A brand newmodified onlooker bee behavior is adopted in the literaturewith the optimal fitness food source as the center

The formulated search for new food source can berepresented by

Vbest119873119895119894 = 119905best119873119895119894 + 120593119905119894 lowast (119905best119873119905119894 minus 119905119903119894) (13)

where 119905best119873119895119894 is defined as the optimal solution of all theneighborhood food sources around the present food sourceand 119905119895 119873119895 represents all the neighborhood food sourcesincluding 119905119895 the rest of the parameters are the same as theones in (10) It can be clearly observed that the key is how theneighborhood food source is defined

From (13) the core of this pattern lies in how theneighboring food source is defined Karaboga and Gorkemli[33] only give a simple definition of neighboring food sourcesWith regard to different ways of defining problem solutionit needs to define different measurement for similarity ForMT every solution corresponds to a digital sequence In this


paper the neighboring food source distance is defined asfollows

119889119895119894 = sumSN119909=1 119905119909119894SN

119894 = 1 2 SL (14)

where 119889119895119894 is the search scope of the food source 119905119895 around theneighborhood SN stands for the number of the food sourcesand SL is the dimensions of a certain food source If theEuclidean distance of a solution to 119905119895 is shorter than 119889119895119894 thenit is regarded as the neighborhood of the present solution 119905119895which is different from the standard ABC algorithm Whenan onlooker bee reaches the food source 119905119895 firstly it willinvestigate all the neighborhood food sources choosing thebest food source 119905best119873119895 and improving her search by (13) In119873119895 the best food source is defined by

fit (119905best119873119895 ) = max (fit (1199051119873119895) fit (1199052119873119895) fit (119905119904119873119895)) (15)

The improved ABC algorithm can be represented asfollows

Algorithm 1 (main steps of the MQABC algorithm for imagethresholding)

Step 1 Initialization of the population size SN provide thenumber of thresholds 119872 and maximum cycle number CNinitialization of the population of source foods


Then evaluate the population via the specified optimizationfunction

Step 2 While (a termination criterion is not satisfied or CycleNumber lt CN) doStep 3 For 119895 = 1 to SN do

Produce new solutions (food source positions) V119895 in theneighborhood of 119905119895 for the employed bees using the formula

V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (17)

Step 4 Apply the greedy selection process between thecorresponding 119905119895 and V119895

Step 5 Calculate the probability values prob119895 for the solutions119905119895 by means of their fitness values using the formula


Step 6 Calculate the Euclidean distance 119889119895 chose the bestfood source 119905best119873119895 in119873119895

119889119895119894 = sumSN119909=1 119905119909119894SN

119894 = 1 2 SLfit (119905best119873119895 ) = max (fit (1199051119873119895) fit (1199052119873119895) fit (119905119904119873119895))

(19)

Step 7 Produce the new solutions (new positions) Vbest119873119895 for theonlookers from the solutions 119905best119873119895 using the formula


And then selected depending on prob119894 and evaluate them

Step 8 Apply the greedy selection process for the onlookersbetween 119905best119873119895 and Vbest119873119895

Step 9 Determine the abandoned solution (source food) if itexists and replace it with a new randomly produced solution119905119895 for the scoutStep 10 Memorize the best food source position (solution)achieved so far

Step 11 End for

Step 12 End while

5 Membership Initialization and LocalInformation Aggregation

After the optimized processing in the above Section 4 a setof thresholds is obtained through (8) and the subsequentprocessing will take the thresholds as the basis Based onfuzzy theory every pixel will be assigned with a fuzzymembershipThe process is calledmembership initializationOn the basis of the initialization value this paper proposes toconduct the local information aggregation based on the pixelspatial relationship so as to achieve the final segmentationresults

51 The Membership Initialization Using Pseudo Trapezoid-Shaped (PTS) Membership Function Literature [1] summa-rizes the initialization methods of image fuzzy sets intotwo kinds histogram based methods and centroid-basedmethods Histogram based methods are the most importantand themost commonly used for thresholding segmentationbut suchmethods need the prediction of probability distribu-tion of histograms (like the most commonly used Gaussiandistribution) Unfortunately the probability distribution ofthe histograms for different kinds of images follows differentprobability patterns like MR data is known to follow aRician distribution [34] ultrasound data have been describedmore accurately by a mixture of Gamma distributions [1]As a result applicability of histogram based methods isrelatively poor in contradistinction centroid-basedmethodsrender the shape and probability of histogram unnecessarywhat only needs to be done is to select a proper centroidAs previously described the thresholds which are obtainedthrough (8) are set as the centroids based on that the fuzzydegree of each pixel is distributed through a proper functionFigure 1 provides fuzzy membership function adopted by thefuzzy entropy In the process of initialization of image fuzzysets Pseudo Trapezoid-Shaped membership function (PTS)which is proposed in literature [1] will be adopted as shown


1

09

08

07

06

05

04

03

02

01

0

Trapezoidal membership function [a1 a2 a3 a4] = [500 100 150 200]

Mem

bers

hip

prob

abili

ty

250200150100500

Figure 1 Fuzzy trapezoidal membership function

1

09

08

07

06

05

04

03

02

01

0

Mem

bers

hip

prob

abili

ty

0 50 100 150 200 250

Pseudo Trapezoid-Shaped (PTS) membership function with three fuzzy sets

Figure 2 Fuzzy Pseudo Trapezoid-Shaped membership function

in Figure 2 Comparing Figures 2 and 1 PTS has less fuzzydegree values of 1 which is more conducive to the use of fuzzyspatial information aggregation in the following step

Similar to the formulation of trapezoidal membershipfunction in the case that there are 119873 centroids PTS mem-bership function can be formulated as follows

12058310158400 (119896) =


12058310158401 (119896) =

0 119896 le 1198861119896 minus 11988611198862 minus 1198861 1198861 lt 119896 le 1198862119896 minus 11988631198862 minus 1198863 1198862 lt 119896 le 11988630 119896 gt 1198863

1205831015840119898 (119896) =


(21)

where 12058310158400(119896) 12058310158401(119896) 1205831015840119898(119896) are initialization functions ofpixel fuzzy membership 1198861 1198862 119886119899 are the fuzzy parame-ters of the Pseudo trapezoidal membership 119896 is the gray scalevalue for each pixel in the test image

Figure 2 presents the distribution methods of fuzzydegree of each pixel when there are three centroids beingset The membership of any random pixel in image 119868 in119903 class is represented by 120583119903(119868(119894)) after the initialization ofthe imagersquos fuzzy membership by way of PTS membershipfunction every pixel corresponds to a membership vector120583119903(119868(119894)) as described in

120583 (119868 (119894)) = [1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894))] (22)

52 Membership Aggregation Based on Local Spatial Informa-tion After the initialization of fuzzy membership functionone pixel is likely to be distributed into two regions at most(in (22) there are two nonzero numbers at most) Simplyby adopting the biggest value it will be enough to distributethe pixel into the corresponding region but such methodsonly consider the single pixelrsquos fuzzy degree to the neglectof the pixel spatial relationship Therefore analyzing thelocal information aggregation of adjacent pixels will improvesegmentation precision so that the isolated pixels can beremoved

Image segmentation is in essence the aggregation ofadjacent pixels of the same features The biggest drawback ofthresholding segmentation lies in the fact that it only takesthe gray scale features of the test image to the neglect ofspatial relationship In the process of initialization of fuzzymembership function each pixel will be assigned with one ortwo fuzzy valuesThese indeterminate values provide optimalprerequisites for local aggregation By means of average andmedian aggregation of the fuzzy degrees of the pixels inneighborhood the quality of image segmentation can beimproved effectively

When there exists119873 surrounding regions for each pixelthe value of the fuzzy degree of each pixel can maximize thefuzzy relevance of pixels in surrounding regions by averageand median filtering As depicted in literature [1] there existthree methods for local fuzzy aggregation

(a) Median Aggregation The fuzzy membership degree ofeach pixel will get through median filter in 119873 surroundingregions which can be formulated as follows

120583aggr (119868 (119894))= median119868(119894)isin119873(119868(119894))

1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894)) (23)


where 120583aggr() represents fuzzy membership degree afteraggregation median represents the median operations ofvector quantity and119873(119868(119894)) is the number of119873 neighboringpixels for number 119894 pixel(b) Average Aggregation In the scope of the neighborhood foreach pixel (like 3 lowast 3) search for the solution which includesthe average values of all pixels (including itself) and take themas the fuzzy membership of the present pixel as describedin

120583aggr (119868 (119894)) = 1119873 (119868 (119894)) lowast sum119868(119894)isin119873(119868(119894))

120583 (119868 (119894)) (24)

(c) Iterative Averaging Aggregation Compared to medianaggregation average aggregation can make better use ofneighboring pixel information Therefore the latter performsbetter in removing isolated pixels But the aggregation capa-bility with one-time iteration is limited therefore iterativeaveraging aggregation can strengthen aggregation effects Atthe same time we should bear in mind that too large thenumber of iterations will make the segmentation boundaryblurry consequently it is necessary to set differentweights foradjacent pixels with different spatial distances as depicted in(25) Accordingly the iterative formulation is shown in (26)where 120583aggr(119868(119894)119905) represents the computational results of (24)when iterations occur once

119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)

After the fuzzy optimization and fuzzy aggregation eachpixel in the test image will get a vector by (22) In the setof vector quantities selecting the biggest fuzzy value willdetermine which region the pixel belongs to The wholeprocedures of computations are represented by Algorithm 2

Algorithm 2 (fuzzy multilevel image thresholding)

Step 1 Initialization of the number of region 119873 and convertthe input image to gray scale and calculate the histogram ofgrayscale image

Step 2 Set the number of fuzzy parameters 2lowast119873which is usedin fuzzy trapezoidal membership function (Figure 1) Providethe number of thresholds 119898 and maximum cycle numberCN

Step 3 Use QABC (as described in Algorithm 1) to computethe fuzzy parameters The objective function follows (7)which merges the fuzzy theory and Kapurrsquos entropy asdescribed in Section 32

Step 4 The soft threshold calculated by fuzzy parametersobtained in Step 3

1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)

Step 5 Regard the soft thresholds as the centroids assigningmembership to each pixel using fuzzy Pseudo Trapezoid-Shaped membership function (Figure 2)

Step 6 According to the local neighborhood informationperform aggregation operations on the membership degreeof each pixel as described in Section 52

Step 7 According to the results of aggregation determiningthe region for each pixel by way of maximum value

Step 8 According to the regional labels output the segmen-tation result

6 Experiments and Result Discussions

This paper on the basis of numerical comparative exper-iments through contrasts of segmented images data andgraph analysis testifies to the superiority of the proposedalgorithm From relevant literature it can be seen thatsome methods are superior to others The present paperwill compare the proposed algorithm with the state-of-the-art methods proven inferior ones will be sidelined In thefollowing sections the proposed algorithm will be comparedto Electromagnetism method standard ABC and the latestintelligent optimization that takes Kapurrsquos entropy as theobjective function proposed in literature [7 27 35] Electro-magnetism optimization with Kapurrsquos entropy as the objec-tive function is so far the newest intelligent optimizationPDE is one of the most effective fuzzy entropy thresholdingsegmentation methods The comparison with literature [35]is mainly to testify that MQABC outperforms standard ABC

61 The Discussion of Initialization Parameters and Conver-gence in MQABC The proposed algorithm has been testedunder a set of benchmark images Some of these images arewidely used in themultilevel image segmentation literature totest different methods (Lena Baboon and so forth) [7 8 35]Others are chosen purposefully from the Berkeley Segmen-tation Data Set and Benchmarks 500 (BSDS500 for short see[36]) as shown in Figure 3 The experiments were carriedout on a LENOVO Laptop with Intel(R) Core i3 processorand 4GB memory And the algorithm was developed viathe signal processing toolbox image processing toolbox andglobal optimization toolbox of MatlabR2011b the parametersused for ABC algorithms [35] were presented in Table 1In order to test the specific effects of these parameters inMQABC many experiments have been conducted Figure 4presents the convergence curve of objective functions with


Table 1 Parameters used for MQABC

Parameters Swam size Number of iterations Lower bound Upper bound Max trial limitValue 30 200 1 256 10

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 3 The original images and their histograms (a) Baboon (b) Lena (c) Aeroplane (d) Swan (e) Ladyhand (f) Starfish (g) Mountainand (h) Building


Table 2 Comparison of image segmentation quality with different swarm size

Swam size 10 20 30 40 50PSNR 203650 204675 207193 204243 204071

304

302

30

298

296

294

292

Number of iterations

Obj

ectiv

e fun

ctio

n va

lue

Obj

ectiv

e fun

ctio

n va

lue

Function values versus iterations Function values versus iterations

50

50454035302520151050

Number of iterations100

1009080706050403020100

304

306

302

30

298

296

294

292


100 120 140 160 180 200806040200


100 150 200 250 300 350 400 500450500

Function values versus iterations Function values versus iterations308

306

304

302

30

298

296

294

292

29

288

Obj

ectiv

e fun

ctio

n va

lue

308

31

306

304

302

30

298

296

294

292

Obj

ectiv

e fun

ctio

n va

lue

Figure 4 The comparison of convergence with different number of iterations

iterations ranging from 50 to 500 When the iterations rangefrom 50 to 150 the convergence is not so apparent Butwhen the iterations reach 200 the convergence is evidentenough However when the number of iterations increasesfurther the convergence speed is hardly improved Table 2presents the assessment parameters for the segmentationquality of image Baboon with different swarm-sizes (detailedPSNR is provided at Section 64) From the table it can beseen that when swarm-size reaches 30 the quality of imagesegmentation reaches the highest At the same time we alsotest the parameter changes of max trial limit In the followingtests the biggest parameter value is 7 but most of them arebelow 5 Therefore the parameters presented in Table 1 canbe applied directly to MQABC

In order to highlight the convergence performance ofMQABC visually Figure 5 shows the convergence curves ofMQABC and ABC for test images Baboon Lena and starfishwhen the thresholds are 5 respectively From the curvelines MQABC not only reaches the optimal objective earlier

but also is speedier toward the optima More importantlycompared with ABC algorithm the objective value obtainedby MQABC is equal or even bigger which means MQABCcan get better thresholds Therefore MQABC shows betterconvergence performance and processing results

62 The Fuzzy Image Segmentation Result Based on MQABCwith Different Thresholds In order to make comparisonswith other thresholding segmentation algorithms one of themost popular performance indicators PSNR (the peak-to-signal ratio) is employed to compare the segmentation resultsbetween the original image and the segmented image It isdefined as in

PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)

where RMSE is the root mean-squared errorIn order to compare the influence of different thresholds

and different aggregation methods we conducted numerical


Obj

ectiv

e fun

ctio

n va

lue

Number of iterations100 120 140 160 180 200806040200

304

306

308

31

302

30

298

296

294

292

Obj

ectiv

e fun

ctio

n va

lue

Function values versus iterations

ABCMQABC

(a) Baboon with 5 th


ABCMQABC

Function values versus iterations302

30

298

296

294

292

29

288

286

284

282

Obj

ectiv

e fun

ctio

n va

lue

(b) Lena with 5 th


ABCMQABC


316

315

314

313

312

311

31

309

308

Obj

ectiv

e fun

ctio

n va

lue

(c) Starfish with 5 th

Figure 5 The comparison of convergence (a)ndash(c) with 5 thresholds respectively

experimental tests Visual effects and qualitative data indicatethat with the increase of the number of thresholds imagesegmentation results will get finer Note that the numberof thresholds cannot increase without limit The number ofinformation units in each image is finite toomuch thresholdscannot improve segmentation results on the contrary it willadd to the complexity Based on the data sets adopted bythis paper we test the segmentation results when the numberof thresholds ranges from 2 to 5 and adopt three differentmembership aggregations to conduct neighboring aggrega-tions The experimental results indicate that in the case ofthe same number of thresholds the median aggregation getsthe best segmentation results When the average aggregationis compared to iterative average aggregation in most casesiterative average aggregation performs better except for someextremely rare occasions Table 3 presents different PSNRvalue of Baboon image with different thresholds and aggrega-tion Figure 6 shows the segmentation results for the images

in Table 3 From Figure 6 we can see that with the increaseof the number of thresholds the segmentation results getfiner gradually but the visual effects are not obvious enoughfor different aggregation methods To compensate for theinsufficiency of visual comparisons Table 3 clearly indicatesthat the median aggregation is far superior to the otheraggregation methods

Figures 7 and 8 show the segmentation results of theoriginal images in Figure 3 after median aggregation is donewith the thresholds being 2 3 4 and 5 respectively Visuallyit is adequate enough to discriminate the object from itsbackground when the thresholds are 2 With the increase ofthe number of thresholds the segmentation results get finerIn actual applications we can vary the number of thresholdsto obtain the desired output in response to different functionsand scenes Surely if the size of the image is much largeror needs dividing into more regions to that end it is quitenecessary to increase the number of thresholds What needs


Table 3 Comparison of segmentation quality with different thresholds and aggregation

M 2 3 4 5Median aggregation 188793 199228 204195 210405Average aggregation 182370 187917 193352 197773Iterative averaging 186391 195356 200112 205206

M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging

Figure 6 The results of image segmentation with different thresholds and aggregation

to be done in the proposed method is just to set the numberof thresholds 119872 to get the desired results Compared withother segmentation methods it is hard to tell visually thedifference of segmentation results As for more detailedand comprehensive assessment concerning the segmentationresults and performance we will provide qualitative analysisin the subsequent sections

63 Comparison of Best Objective Function Values and TheirCorresponding Thresholds In this section the results of bestobjective function values and their corresponding thresholdsacquired by various images are discussed Table 4 depicts thenumber of thresholds objective values and the correspond-ing optimal thresholds obtained by EMO ABC andMQABCmethods From the tables it can be observed that all theimages when processed by different methods have obtainedrelatively high objective function values Notably with the

increase of the number of thresholds the three methodsall get high objective function values Generally MQABCand EMO obtain similar and even higher objective functionvalues while ABC appears a little inferior but the differencerange is less than 01

Table 4 provides only the comparisons of the bestobjective function values and their corresponding thresholdsbetween EMO ABC and MQABC After the introduction offuzzy entropy we also tested the optimized results in termsof thresholds and objective function values based on DE(differential evolution) and FMQABC

Tables 3 and 4 indicate that MQABC and FMQABCare able to obtain relatively good objective function valuesAlthough in some cases these objective function valuesappear a little smaller in comparison with EMQ and FDEthe difference is insignificantly trivial This paper lays more


Table 4 Comparison of best objective function values and their corresponding thresholds between EMO ABC and MQABC

Image M Optimum thresholds Best objective function valuesEMO ABC MQABC EMO ABC MQABC

Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983

emphasis on time efficiency of MQABC and FMQABCdetailed comparisons are presented in Section 65

64 Comparison of Fuzzy Multilevel Image SegmentationResult and Quality Assessment by PSNR Figures 7 and 8demonstrate the segmentation results of FMQABCA Inorder to further compare the proposedmethodwithMQABCand other fuzzy optimized methods Figure 9 presents thesegmentation results of FDE FMQABC and FMQABCAwith different thresholds For the sake of convenience theresults in Figure 7 will have to be grayscaled the gray levelin the same region will be replaced by the average valueAnalyzed visually all the three methods get relatively finesegmentation results but it is hard to tell visually which oneis superior or inferior

In order to further compare the advantages and disad-vantages between EMO ABC MQABC FDE FMQABCand FQMABCA Table 5 demonstrates PSNR values these

methods can achieve on fuzzy entropy which indicate thatMQABC gets the high assessment value testifying to thesuperiority of MQABC in segmentation quality resultsFurthermore when the number of thresholds increasesMQABC gets better PSNR values The comparison of fuzzyentropy (FDE FMQABC and FQMBCA) with nonfuzzyentropy (EMO ABC and MQABC) shows that in most ofthe cases PSNR value is higher when fuzzy entropy is appliednoticeably FMABCA is far superior to the rest of themethodsThe FMQABCA method gets the highest increase of 42 forthe Swan image when 119872 = 3 the lowest also increases by08 for the Aeroplane image when119872 = 4 On average theFMQABCA method shows more than 12 increase on thePSNR value for the eight figures

65 Comparison of the Characteristics of Running Time Onthe prior condition that segmentation quality is guaranteedwe examine the running time to test the performance


M = 2 M = 3 M = 4 M = 5

Figure 7 The segmentation results of (a)ndash(d) in Figure 1

This is the biggest highlight of this paper As shown inTable 1 the experiments are set to the same number ofiterations for testing But in order to obtain the operatingtime for convergence as described in Table 6 when the fitnessfunction remains unchanged for 20 consecutive times theexperimental iteration is forced to be terminated Moreoverin order to better ensure the effectiveness of MQABC itsconsecutive times are set to 50 specifically Table 6 presentsdata concerning running time to convergence The runningtime that EMO takes is much longer than the others Firstwhen comparing ABC with MQABC we find that MQABCmethod gets much less running time than the ABC And thehighest increase of 12 is for the Lena image with 119872 = 5the average also increases by 77 with 119872 = 5 and 48with 119898 = 4 Then the comparison of fuzzy methods tononfuzzy methods shows that the running time for fuzzymethods takes a little longer but if we take the segmentationquality assessment (PSNR) into account as shown in Table 5the segmentation result is much better All taken togetherfuzzy methods are much more valuable

In the three fuzzy methods when FDE is compared withFMQABC and FBCA it shows the latter two enjoy highertime efficiency significantly The FMQABC method gets the

highest increase of 53 for the Swan image when119872 = 4 thelowest also increases by 28 for the Lena image when119872 = 5On average the FMQABC method shows more than 44of the time improvement for the eight figures when119872 = 4and more than 41 when 119872 = 5 From the perspectiveof FMQABCA the maximum and minimum running timeincrease by 40 and 14 for the Aeroplane and Lena imagerespectively when119872 = 4 And when119872 = 5 the maximumand minimum running time increase by 38 and 21 forthe Lena and Aeroplane image respectively On averagethe FMQABCA method shows more than 28 of the timeimprovement for the eight figureswhen119872 = 4 andmore than26 when119872 = 5 Therefore the FMQABC and FMQABCAmethods ensure the optimal threshold so as to obtain the besttime efficiency

From Tables 4ndash6 and Figures 4ndash9 together with visualeffect analysis it can be seen that FMQABCA deliversbetter segmentation results and enjoys obvious advantagesin running time More importantly MQABC is far superiorto ABC over convergence performance Therefore it canbe safely concluded that FMQABCA is a desirable multi-level segmentation method with high speed and of highquality


M = 2 M = 3 M = 4 M = 5

Figure 8 The segmentation results of (e)ndash(h) in Figure 1

7 Conclusion

In this paper MQABC has been employed to optimize fuzzyKapurrsquos entropy in order to obtain a set of thresholds Basedon the set values a set of fuzzy degrees is assigned toeach pixel through Pseudo Trapezoid-Shaped membershipfunction and then aggregation operation is conducted in theway that fuzzy degrees of neighborhood are aggregated toachieve the final segmentation When compared with ABCMQABC by improving on the distance strategies in theonlooker bee phase not only achieves the higher objectivefunction and betters PSNR value but also shows obviousimprovement in convergence performance and running timeIn the process of fuzzy aggregation FMQABCA gets higherPSNR values obtained by median average and iterativemethods than EMOABCMQABC FDE and FMQABCandshow more efficiency in running time than EMO and FDEAlthough compared with ABC MQABC and FMQABCFMQABCA takes a slightly longer time but such a littleprice is completely acceptable if we take the high quality ofimage segmentation into account In conclusion FMQABCand FMQABCA are able to obtain better image segmentation

quality and also show obvious superiority in terms of runningtime and convergence speed

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (nos 61300239 71301081and 61572261) The Postdoctoral Science Foundation(nos 2014M551635 and 1302085B) the Innovation Projectof graduate students Foundation of Jiangsu Province(KYLX15 0841) Higher Education Revitalization PlanFoundation of Anhui Province (nos 2013SQRL102ZDKJ2016A554 KJ2016A556 2015xdjy196 and 2015jyxm728)Natural Science Fund for Colleges andUniversities in JiangsuProvince (no 16KJB520034) and a Project Funded by thePriority Academic Program Development of Jiangsu Higher


Table 5 PSNR metrics of the test images segmented with different threshold

Image M PSNREMO ABC MQABC FDE FMQABC FMQABCA

Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096

Table 6 The running time to convergence for EMO ABC MQABC FDE FMQABC and FMQABCA

Image M Running time (second)EMO ABC MQABC FDE FMQABC FMQABCA

Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan

Image FDE FMQABC FMQABCA

Figure 9 The comparison of grayscale segmentation results about (a)ndash(d) in Figure 1

Education Institutions (PAPD) Jiangsu Collaborative Inno-vation Center on Atmospheric Environment and EquipmentTechnology (CICAEET)

References

[1] S Aja-Fernandez AH Curiale andG Vegas-Sanchez-FerreroldquoA local fuzzy thresholdingmethodology for multiregion imagesegmentationrdquoKnowledge-Based Systems vol 83 no 1 pp 1ndash122015

[2] N Shi and J Pan ldquoAn improved active contoursmodel for imagesegmentation by level set methodrdquo Optik vol 127 no 3 pp1037ndash1042 2016

[3] S Masood M Sharif A Masood M Yasmin and M RazaldquoA survey on medical image segmentationrdquo Current MedicalImaging Reviews vol 11 no 1 pp 3ndash14 2015

[4] J Torres-Sanchez F Lopez-Granados and J M Pena ldquoAnautomatic object-based method for optimal thresholding inUAV images application for vegetation detection in herbaceouscropsrdquoComputers and Electronics inAgriculture vol 114 pp 43ndash52 2015

[5] M Sonka V Hlavac and R Boyle Image Processing Analysisand Machine Vision Springer US Boston Mass USA 1993

[6] J Li X Li B Yang and X Sun ldquoSegmentation-based imagecopy-move forgery detection schemerdquo IEEE Transactions onInformation Forensics and Security vol 10 no 3 pp 507ndash5182015

[7] D Oliva E Cuevas G Pajares D Zaldivar and V OsunaldquoA multilevel thresholding algorithm using electromagnetismoptimizationrdquo Neurocomputing vol 139 pp 357ndash381 2014

[8] B Akay ldquoA study on particle swarm optimization and artificialbee colony algorithms for multilevel thresholdingrdquo Applied SoftComputing Journal vol 13 no 6 pp 3066ndash3091 2013


[9] V Osuna-Enciso E Cuevas and H Sossa ldquoA comparison ofnature inspired algorithms for multi-threshold image segmen-tationrdquo Expert Systems with Applications vol 40 no 4 pp 1213ndash1219 2013

[10] T Kurban P Civicioglu R Kurban and E Besdok ldquoCompari-son of evolutionary and swarmbased computational techniquesfor multilevel color image thresholdingrdquo Applied Soft Comput-ing Journal vol 23 pp 128ndash143 2014

[11] S Sarkar S Das and S S Chaudhuri ldquoA multilevel color imagethresholding scheme based on minimum cross entropy anddifferential evolutionrdquo Pattern Recognition Letters vol 54 pp27ndash35 2015

[12] J N Kapur P K Sahoo and A K C Wong ldquoA new methodfor gray-level picture thresholding using the entropy of thehistogramrdquo Computer Vision Graphics and Image Processingvol 29 no 3 pp 273ndash285 1985

[13] N Otsu ldquoThreshold selection method from gray-level his-togramsrdquo IEEE Trans Syst Man Cybern vol 9 no 1 pp 62ndash661979

[14] X Li Z Zhao and H D Cheng ldquoFuzzy entropy thresholdapproach to breast cancer detectionrdquo Information SciencesmdashApplications vol 4 no 1 pp 49ndash56 1995

[15] J Kittler and J Illingworth ldquoMinimum error thresholdingrdquoPattern Recognition vol 19 no 1 pp 41ndash47 1986

[16] M Sezgin and B Sankur ldquoSurvey over image thresholdingtechniques and quantitative performance evaluationrdquo Journal ofElectronic Imaging vol 13 no 1 pp 146ndash168 2004

[17] I Bloch ldquoFuzzy sets for image processing and understandingrdquoFuzzy Sets and Systems vol 281 pp 280ndash291 2015

[18] J Anitha and J D Peter ldquoA spatial fuzzy based level setmethod for mammogram mass segmentationrdquo in Proceedingsof the IEEE 2nd International Conference on Electronics andCommunication Systems (ICECS rsquo15) pp 1ndash6 2015

[19] C I Gonzalez P Melin J R Castro et al ldquoColor imageedge detection method based on interval type-2 fuzzy systemsrdquoin Design of Intelligent Systems Based on Fuzzy Logic NeuralNetworks and Nature-Inspired Optimization pp 3ndash11 SpringerInternational Publishing 2015

[20] F Perez-Ornelas O Mendoza P Melin J R Castro ARodriguez-Diaz and O Castillo ldquoFuzzy index to evaluate edgedetection in digital imagesrdquo PLoS ONE vol 10 no 6 Article IDe0131161 2015

[21] Y Zheng B Jeon D Xu Q M J Wu and H Zhang ldquoImagesegmentation by generalized hierarchical fuzzy C-means algo-rithmrdquo Journal of Intelligent and Fuzzy Systems vol 28 no 2pp 961ndash973 2015

[22] H-P Chen X-J Shen and J-W Long ldquoHistogram-basedcolour image fuzzy clustering algorithmrdquoMultimedia Tools andApplications vol 57 no 18 pp 11417ndash11432 2015

[23] F Zhao H Liu and J Fan ldquoA multiobjective spatial fuzzyclustering algorithm for image segmentationrdquo Applied SoftComputing vol 30 pp 48ndash57 2015

[24] C I Gonzalez P Melin J R Castro O Mendoza and OCastillo ldquoAn improved sobel edge detection method based ongeneralized type-2 fuzzy logicrdquo Soft Computing vol 20 no 2pp 773ndash784 2016

[25] PGhamisiM S Couceiro FM LMartins and J A Benedikts-son ldquoMultilevel image segmentation based on fractional-orderDarwinian particle swarm optimizationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 52 no 5 pp 2382ndash23942014

[26] A K Bhandari A Kumar and G K Singh ldquoModified artificialbee colony based computationally efficient multilevel thresh-olding for satellite image segmentation using Kapurrsquos Otsu andTsallis functionsrdquo Expert Systems with Applications vol 42 no3 pp 1573ndash1601 2015

[27] S Sarkar S Paul R Burman et al ldquoA fuzzy entropy basedmulti-level image thresholding using differential evolutionrdquo in SwarmEvolutionary and Memetic Computing pp 386ndash395 SpringerInternational Publishing 2014

[28] M Zhao A M N Fu and H Yan ldquoA technique of three-level thresholding based on probability partition and fuzzy 3-partitionrdquo IEEE Transactions on Fuzzy Systems vol 9 no 3 pp469ndash479 2001

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

[30] G Q O Pratamasunu Z Hu A Z Arifin et al ldquoImagethresholding based on index of fuzziness and fuzzy similaritymeasurerdquo in Proceedings of the IEEE 8th InternationalWorkshopon Computational Intelligence and Applications (IWCIA rsquo15) pp161ndash166 IEEE Hiroshima Japan November 2015

[31] M Muppidi P Rad S S Agaian and M Jamshidi ldquoImage seg-mentation by multi-level thresholding based on fuzzy entropyand genetic algorithm in cloudrdquo inProceedings of the 10th Systemof Systems Engineering Conference (SoSE rsquo15) pp 143ndash148 IEEESan Antonio Tex USA May 2015

[32] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[33] D Karaboga and B Gorkemli ldquoA quick artificial bee colony-qABC- algorithm for optimization problemsrdquo in Proceedingsof the International Symposium on INnovations in IntelligentSysTems and Applications (INISTA rsquo12) 5 1 pages July 2012

[34] S-L Jui C LinW XuW Lin DWang andK Xiao ldquoDynamicincorporation of wavelet filter in fuzzy C-means for efficientand noise-insensitive MR image segmentationrdquo InternationalJournal of Computational Intelligence Systems vol 8 no 5 pp796ndash807 2015

[35] E Cuevas F Sencion D Zaldivar M Perez-Cisneros andH Sossa ldquoA multi-threshold segmentation approach based onartificial bee colony optimizationrdquo Applied Intelligence vol 37no 3 pp 321ndash336 2012

[36] P ArbelaezMMaire C Fowlkes and JMalik ldquoContour detec-tion and hierarchical image segmentationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 33 no 5 pp898ndash916 2011

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

Journal of


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Complex AnalysisJournal of


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

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Of nonparametric methods gray level global thresh-olding adopting information entropy is the most popularone Such method takes the intelligent optimization as thetool and by optimizing some information entropy to getthe maximum (or minimum) objective function to achievethe final segmentation The most commonly used objectivefunctions include the maximization of the entropy (egKapurrsquos entropy) [12] maximization of the between-classvariance (eg Otsursquos method) [13] the use of the fuzzysimilarity measure [14] and minimization of the Bayesianerror [15] All of these techniques have been extended into thearea ofmultilevel image segmentation But the computationalcomplexity for the application of such methods in MTgrows dramatically As a consequence scholars in recentyears have extensively used evolutionary and swarm-basedintelligent computational techniques to promote multilevelcomputational efficiency for multilevel image segmentationand precision of threshold [11] But the global multilevelthresholds generally get the final segmentation results bymaximizing the separation between classes Yet it neglectsthe complexity in uneven illumination and soft transitionsbetween gray levels and fails to consider the spatial rela-tionships between pixels which lead to the imperfectionsof segmentation results [1] The biggest problem with theglobal thresholding lies in the fact that it is pixel orientedrather than region oriented Therefore pixels that have thesame gray levels are segmented into the same regions if suchfactors as connectivity of pixels and closeness of objects arenot considered As a result such methods are prone to createisolated pixels

Of all the research findings for the various thresholdingmethods each one of them is designed specifically to dealwith one particular kind of images Sezgin and Sankur [16]were the first to classify the image thresholding methods intosix categories Aja-Fernandez et al [1] further grouped theminto three kinds based on the six classifications (a) Methodsthat calculate a global threshold for the whole image (b)Methods that use an adaptive local threshold (c) Methodsthat use spatial local information for classifying the pixelsThe first kind was the earliest to be employed But due tothe drawbacks inherent in it currently scholars have directedtheir focus on the latter two The second kind is an improve-ment on and complement to the first kind which selects thethreshold locally through adaptability so as to avoid localnonadaptability but the computational complexity will growaccordingly The third kind is mainly employed to make upfor the deficiency in spatial relationship and introduces thelocal spatial information to heighten the effects of segmentedresults In this paper we introduce local spatial informationto solve segmentation problems and at the same time explorethe applications of fuzzy logic and fuzzy theory in imagethresholding The starting point of introducing local spatialinformation comes from the definition of one class or targetwhere the membership of a pixel is highly correlated withthe membership of the pixels of surrounding regions [1]In order to promote the quality of image segmentationfuzzy membership function is used to assign each pixel withfuzzy degrees And according to fuzzy theory the traditionalhard assigning regions or classes will be replaced with soft

assigning patternDetailedly the fuzzyKapurrsquos entropywhichis modified by Kapurrsquos entropy is adopted in this methodas the objective function It combines with the MQABCalgorithm to generate the optimal thresholds which are usedas the centroids for fuzzy membership assignment and localinformation aggregation

The rest of the paper is organized as follows Section 2introduces related work of fuzzy entropy intelligent opti-mization and local information aggregation in the field ofMT In Section 3 the nonfuzzy hard thresholding and fuzzysoft thresholding are formulated and then the correspondingKapurrsquos entropy for image segmentation and the ultimateobjective function are presented In Section 4 the standardbee colony algorithm is briefly described and the proposedMQABC is dealt with in detail Section 5 analyzes themultiplethresholds-based fuzzy membership initialization and fuzzyaggregationmethodsThe experiment results and discussionsare presented in Section 6 The last section presents theconclusions

2 Related Works

Fuzzy theory and fuzzy logic are well-known to be flexibletools where imprecise knowledge or not-well defined featureshave to be used [17] It has been extensively used in systemcontrol domain and applied in image processing to someextent [18ndash21] In the past 10 years thresholding segmenta-tion approaches have been proposed based on fuzzy logicand fuzzy calculations In such approaches fuzzy logic iscombined with clustering method [22] fuzzy compactness[23] the interpretation of thresholds as type 2 fuzzy logic[24] and the soft calculations (such as ant colonies beecolonies and bacteria migration heuristics) [11] respectivelymainly used to search for the optimal thresholds but theyfailed to take the pixel spatial relationship segmentationmethods into full account In these fuzzy based thresholdingmethods clustering methods are among the most popu-lar but they are highly complicated where computationaltime is concerned With the progress and upgradation ofoptimization methods fuzzy optimization in thresholdingsegmentation gets more and more attention and it has beenproved feasible to employ intelligent optimization to solvethresholding problems [10] A large number of literature showthat intelligent optimizations are far superior to traditionalmethods in processing speed precision and robustness [510] Oliva et al [7] introduced Electromagnetism Optimiza-tion into thresholding segmentation and compared it withKapurrsquos entropy and Otsursquos method The experiments showthat Kapurrsquos entropy is more efficient Ghamisi et al [25]have fully analyzed the performance in thresholding of suchmethods as Particle swarm optimization (PSO) DarwinianParticle Swarm Optimization (DPSO) and Fractional-orderDarwinianParticle SwarmOptimization (FODPSO) In com-parison with Bacteria Foraging algorithm (BF) and geneticalgorithms (GA) FOPDSO delivers better performance inovercoming local optimum trap and computational speedKurban et al [10] have conducted a comparative study ofthe applications of the computational techniques involvedin MT exhaustively According to the statistical analysis of














1198670 = minus1199051minus1sum119894=0

1199011198941205960 ln1199011198941205960 1205960 =

1199051minus1sum119894=0

119901119894

1198671 = minus1199052minus1sum119894=1199051

1199011198941205961 ln1199011198941205961 1205961 =

1199052minus1sum119894=1199051

119901119894

119867119895 = minus119905119895+1minus1sum119894=119905119895

119901119894120596119895 ln119901119894120596119895 120596119895 =

119905119895+1minus1sum119894=119905119895

119901119894

119867119898 = minus119871minus1sum119894=119905119898

119901119894120596119898 ln119901119894120596119898 120596119898 =

119871minus1sum119894=119905119898

119901119894

(2)


119891 (119879) = 119898sum119894=0




119877 = (119894 120583119903 (119894)) | 119894 isin 119868 0 le 120583119903 (119894) le 1 119903 isin [1119873] (4)




1205830 (119896) =


1205831 (119896) =


120583119898 (119896) =


(5)


1198670 = minus119871minus1sum119894=0

1205830 (119894) lowast 1199011198941205960 ln(1205830 (119894) lowast 1199011198941205960 )

1205960 =119871minus1sum119894=0

(1205830 (119894) lowast 119901119894)

1198671 = minus119871minus1sum119894=0

1205831 (119894) lowast 1199011198941205961 ln(1205831 (119894) lowast 1199011198941205961 )

1205961 =119871minus1sum119894=0

(1205831 (119894) lowast 119901119894)

119867119895 = minus119871minus1sum119894=0

120583119895 (119894) lowast 119901119894120596119895 ln(120583119895 (119894) lowast 119901119894120596119895 )

120596119895 =119871minus1sum119894=0

(120583119895 (119894) lowast 119901119894)

119867119898 = minus119871minus1sum119894=0

120583119898 (119894) lowast 119901119894120596119898 ln(120583119898 (119894) lowast 119901119894120596119898 )

120596119898 =119871minus1sum119894=0

(120583119898 (119894) lowast 119901119894)

(6)



120601 (1198861 1198862 119886119899) = 119891 (119879) =119898sum119894minus0


(7)


1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(8)









V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (10)



fit (119905119895) = 1

(1 + 119891 (119905119895)) 119891 (119905119895) ge 01 + abs (119891 (119905119895)) 119891 (119905119895) lt 0

(11)












119889119895119894 = sumSN119909=1 119905119909119894SN

119894 = 1 2 SL (14)










V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (17)





119889119895119894 = sumSN119909=1 119905119909119894SN


(19)






Step 11 End for

Step 12 End while





1

09

08

07

06

05

04

03

02

01

0


Mem

bers

hip

prob

abili

ty

250200150100500


1

09

08

07

06

05

04

03

02

01

0

Mem

bers

hip

prob

abili

ty

0 50 100 150 200 250





12058310158400 (119896) =


12058310158401 (119896) =


1205831015840119898 (119896) =


(21)



120583 (119868 (119894)) = [1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894))] (22)





120583aggr (119868 (119894))= median119868(119894)isin119873(119868(119894))

1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894)) (23)




120583 (119868 (119894)) (24)


119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)







1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)











(a) (b)

(c) (d)

(e) (f)

(g) (h)




Swam size 10 20 30 40 50PSNR 203650 204675 207193 204243 204071

304

302

30

298

296

294

292


Obj

ectiv

e fun

ctio

n va

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Obj

ectiv

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ctio

n va

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50

50454035302520151050


1009080706050403020100

304

306

302

30

298

296

294

292


100 120 140 160 180 200806040200


100 150 200 250 300 350 400 500450500


306

304

302

30

298

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294

292

29

288

Obj

ectiv

e fun

ctio

n va

lue

308

31

306

304

302

30

298

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294

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Obj

ectiv

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ctio

n va

lue






PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




Obj

ectiv

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lue


304

306

308

31

302

30

298

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ectiv

e fun

ctio

n va

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ABCMQABC



ABCMQABC


30

298

296

294

292

29

288

286

284

282

Obj

ectiv

e fun

ctio

n va

lue

(b) Lena with 5 th


ABCMQABC


316

315

314

313

312

311

31

309

308

Obj

ectiv

e fun

ctio

n va

lue









M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















1198670 = minus1199051minus1sum119894=0

1199011198941205960 ln1199011198941205960 1205960 =

1199051minus1sum119894=0

119901119894

1198671 = minus1199052minus1sum119894=1199051

1199011198941205961 ln1199011198941205961 1205961 =

1199052minus1sum119894=1199051

119901119894

119867119895 = minus119905119895+1minus1sum119894=119905119895

119901119894120596119895 ln119901119894120596119895 120596119895 =

119905119895+1minus1sum119894=119905119895

119901119894

119867119898 = minus119871minus1sum119894=119905119898

119901119894120596119898 ln119901119894120596119898 120596119898 =

119871minus1sum119894=119905119898

119901119894

(2)


119891 (119879) = 119898sum119894=0




119877 = (119894 120583119903 (119894)) | 119894 isin 119868 0 le 120583119903 (119894) le 1 119903 isin [1119873] (4)




1205830 (119896) =


1205831 (119896) =


120583119898 (119896) =


(5)


1198670 = minus119871minus1sum119894=0

1205830 (119894) lowast 1199011198941205960 ln(1205830 (119894) lowast 1199011198941205960 )

1205960 =119871minus1sum119894=0

(1205830 (119894) lowast 119901119894)

1198671 = minus119871minus1sum119894=0

1205831 (119894) lowast 1199011198941205961 ln(1205831 (119894) lowast 1199011198941205961 )

1205961 =119871minus1sum119894=0

(1205831 (119894) lowast 119901119894)

119867119895 = minus119871minus1sum119894=0

120583119895 (119894) lowast 119901119894120596119895 ln(120583119895 (119894) lowast 119901119894120596119895 )

120596119895 =119871minus1sum119894=0

(120583119895 (119894) lowast 119901119894)

119867119898 = minus119871minus1sum119894=0

120583119898 (119894) lowast 119901119894120596119898 ln(120583119898 (119894) lowast 119901119894120596119898 )

120596119898 =119871minus1sum119894=0

(120583119898 (119894) lowast 119901119894)

(6)



120601 (1198861 1198862 119886119899) = 119891 (119879) =119898sum119894minus0


(7)


1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(8)









V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (10)



fit (119905119895) = 1

(1 + 119891 (119905119895)) 119891 (119905119895) ge 01 + abs (119891 (119905119895)) 119891 (119905119895) lt 0

(11)












119889119895119894 = sumSN119909=1 119905119909119894SN

119894 = 1 2 SL (14)










V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (17)





119889119895119894 = sumSN119909=1 119905119909119894SN


(19)






Step 11 End for

Step 12 End while





1

09

08

07

06

05

04

03

02

01

0


Mem

bers

hip

prob

abili

ty

250200150100500


1

09

08

07

06

05

04

03

02

01

0

Mem

bers

hip

prob

abili

ty

0 50 100 150 200 250





12058310158400 (119896) =


12058310158401 (119896) =


1205831015840119898 (119896) =


(21)



120583 (119868 (119894)) = [1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894))] (22)





120583aggr (119868 (119894))= median119868(119894)isin119873(119868(119894))

1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894)) (23)




120583 (119868 (119894)) (24)


119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)







1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)











(a) (b)

(c) (d)

(e) (f)

(g) (h)




Swam size 10 20 30 40 50PSNR 203650 204675 207193 204243 204071

304

302

30

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292


Obj

ectiv

e fun

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50

50454035302520151050


1009080706050403020100

304

306

302

30

298

296

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292


100 120 140 160 180 200806040200


100 150 200 250 300 350 400 500450500


306

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31

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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




Obj

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ABCMQABC


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ABCMQABC


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M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120601 (1198861 1198862 119886119899) = 119891 (119879) =119898sum119894minus0


(7)


1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(8)









V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (10)



fit (119905119895) = 1

(1 + 119891 (119905119895)) 119891 (119905119895) ge 01 + abs (119891 (119905119895)) 119891 (119905119895) lt 0

(11)












119889119895119894 = sumSN119909=1 119905119909119894SN

119894 = 1 2 SL (14)










V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (17)





119889119895119894 = sumSN119909=1 119905119909119894SN


(19)






Step 11 End for

Step 12 End while





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1

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12058310158400 (119896) =


12058310158401 (119896) =


1205831015840119898 (119896) =


(21)



120583 (119868 (119894)) = [1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894))] (22)





120583aggr (119868 (119894))= median119868(119894)isin119873(119868(119894))

1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894)) (23)




120583 (119868 (119894)) (24)


119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)







1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)











(a) (b)

(c) (d)

(e) (f)

(g) (h)




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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




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ABCMQABC



ABCMQABC


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ABCMQABC


316

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M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119895119894 = sumSN119909=1 119905119909119894SN

119894 = 1 2 SL (14)










V119895119894 = 119905119895119894 + 120593119895119894 lowast (119905119895119894 minus 119905119903119894) (17)





119889119895119894 = sumSN119909=1 119905119909119894SN


(19)






Step 11 End for

Step 12 End while





1

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12058310158400 (119896) =


12058310158401 (119896) =


1205831015840119898 (119896) =


(21)



120583 (119868 (119894)) = [1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894))] (22)





120583aggr (119868 (119894))= median119868(119894)isin119873(119868(119894))

1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894)) (23)




120583 (119868 (119894)) (24)


119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)







1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)











(a) (b)

(c) (d)

(e) (f)

(g) (h)




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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




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M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

09

08

07

06

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03

02

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12058310158400 (119896) =


12058310158401 (119896) =


1205831015840119898 (119896) =


(21)



120583 (119868 (119894)) = [1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894))] (22)





120583aggr (119868 (119894))= median119868(119894)isin119873(119868(119894))

1205830 (119868 (119894)) 1205831 (119868 (119894)) sdot sdot sdot 120583119898 (119868 (119894)) (23)




120583 (119868 (119894)) (24)


119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)







1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)











(a) (b)

(c) (d)

(e) (f)

(g) (h)




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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




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M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120583 (119868 (119894)) (24)


119908 = 13 lowast [[[

0 05 005 1 050 05 0

]]] (25)

120583aggr (119868 (119894)119905+1) = 119908 lowast 120583aggr (119868 (119894)119905) (26)







1199050 = 1198861 + 11988622 1199051 = 1198863 + 11988642

119905119898 = 119886119899minus1 + 1198861198992

(27)











(a) (b)

(c) (d)

(e) (f)

(g) (h)




Swam size 10 20 30 40 50PSNR 203650 204675 207193 204243 204071

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100 120 140 160 180 200806040200


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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




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ABCMQABC


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ABCMQABC


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31

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308

Obj

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M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(a) (b)

(c) (d)

(e) (f)

(g) (h)




Swam size 10 20 30 40 50PSNR 203650 204675 207193 204243 204071

304

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100 120 140 160 180 200806040200


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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




Obj

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ABCMQABC


30

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ABCMQABC


316

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31

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308

Obj

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M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Swam size 10 20 30 40 50PSNR 203650 204675 207193 204243 204071

304

302

30

298

296

294

292


Obj

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50

50454035302520151050


1009080706050403020100

304

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100 120 140 160 180 200806040200


100 150 200 250 300 350 400 500450500


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308

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PSNR (119894 119895) = 20 log10 ( 255RMSE (119894 119895)) (28)




Obj

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304

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ABCMQABC



ABCMQABC


30

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Obj

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(b) Lena with 5 th


ABCMQABC


316

315

314

313

312

311

31

309

308

Obj

ectiv

e fun

ctio

n va

lue









M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Obj

ectiv

e fun

ctio

n va

lue


304

306

308

31

302

30

298

296

294

292

Obj

ectiv

e fun

ctio

n va

lue


ABCMQABC



ABCMQABC


30

298

296

294

292

29

288

286

284

282

Obj

ectiv

e fun

ctio

n va

lue

(b) Lena with 5 th


ABCMQABC


316

315

314

313

312

311

31

309

308

Obj

ectiv

e fun

ctio

n va

lue









M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












M 2 3 4 5

Median aggregation

Average aggregation

Iterative averaging










Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Baboon

2 78 143 80 144 80 144 176799 176802 1768023 46 100 153 46 101 153 46 100 153 221331 221331 2213314 46 100 153235 45 100 153 236 45 100 152 235 265254 265249 2652655 34 75 115 160 235 34 75 116 161 236 33 74 115 162 235 306432 304394 306509

Lena

2 97 164 97 164 97 164 178234 178234 1782343 81 126 177 83 127 178 82126 176 221102 221091 2212014 16 82 127 177 16 87 123 174 16 83 125 176 261107 261064 26 11015 16 64 97 137 179 16 65 96 133 178 16 64 97 137 178 300052 299870 300029

Aeroplane

2 54 161 54 161 54 161 178618 178618 1786183 54 157 229 54 134 186 54 132 184 223106 222964 2231124 54 127 167 206 47 109 165 227 48 110 165 228 263540 266389 2664275 45 77 109 155 202 41 96 109 168 229 45 109 149 189 229 308109 308083 309162

Swan

2 66 150 66 150 66 150 181274 181274 1812743 64 132 200 64 133 198 64 131 199 229607 229621 2298874 64 117 161 210 64 117 164 212 64 117 162 210 273410 273493 2735095 63 108 140 179 218 63 97 132 170 212 63 98 134 174 212 312520 312512 312654

Ladyhand

2 86 171 86 171 86 171 184477 184477 1844773 63 116 182 64 118 180 61 115 180 229799 229883 2298904 52 95 137 180 53 96 136 180 51 96 138 182 271901 271956 2719585 52 97 136 168 202 56 98 137 168 202 55 98 137 174 204 311297 311302 311321

Starfish

2 90 170 90 170 90 170 187518 187518 1875183 73 131 185 73 127 182 72 127 185 233105 233192 2332414 68 117 166 208 63 112 159 207 67 118 167 206 275621 275509 2756755 57 96 135 173 211 55 94 132 170 210 53 90 130 170 210 315536 315338 315519

Mountain

2 117 196 117 196 117 196 176737 176737 1767373 62 119 196 62 119 196 62 119 196 221255 221255 2212554 20 62 119 196 20 62 120 198 20 62 119 194 264474 264338 2644275 20 61 96 137 197 20 60 96 139 203 20 61 100 159 204 305513 305448 305562

Building

2 90 156 90 156 90 156 178071 178071 1780713 84 139 191 84 139 190 83 141 194 223895 223893 2238764 20 84 141 193 20 84 141 192 20 83 139 192 267108 267105 2671065 20 81 131 170 213 20 83 136 177 218 20 81 134 173 214 307119 308631 308983







M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M = 2 M = 3 M = 4 M = 5







M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M = 2 M = 3 M = 4 M = 5


7 Conclusion



Competing Interests


Acknowledgments





Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Baboon

2 159947 160070 160070 159533 162043 1887933 185921 185921 185921 178855 191648 1992284 185921 185435 186718 170614 198360 2041955 205234 205058 207193 195717 210266 210405

Lena

2 145901 145901 145901 167293 166456 1863283 172122 171178 172328 178298 178884 1903664 185488 183798 186011 183482 193349 2099905 203250 202440 203360 208519 209714 222737

Aeroplane

2 127187 127187 127187 145518 153454 1696513 159198 161541 162681 184402 183086 2007824 184136 186846 187095 210749 206326 2124945 191549 186876 210436 212442 211950 215475

Swan

2 148785 148785 148785 162123 163309 1736573 162369 162373 162407 162334 173286 2307854 166445 166571 166953 167511 169194 2327715 169943 169482 170144 233818 237829 246911

Ladyhand

2 154928 154928 154928 157940 158844 1699983 175422 175446 175744 179525 179778 2043944 187518 187950 189135 190744 205095 2132415 206768 207171 207392 207710 217443 220850

Starfish

2 143952 143952 143952 170598 165631 1736413 169727 171199 171430 173605 179680 1944804 181811 184836 185054 185546 185492 1974075 200649 202304 203400 212142 214146 218224

Mountain

2 108782 108782 108782 132357 150018 1765303 174652 174652 174652 216381 216748 2185474 180535 180726 182056 201892 209634 2240855 206893 204889 217589 203026 204539 233328

Building

2 147242 147242 147242 122905 139163 1530273 162205 162261 162340 181586 178589 1868184 179914 182581 183348 174129 189629 2038995 188962 191442 208872 184646 190172 209096



Baboon 4 6700095 2176596 2131156 3665105 2234492 29837495 8389706 2269232 2021364 5288985 3032174 3766464

Lena 4 6598210 2202426 2000834 3847913 2179925 33070055 8202104 2275810 2001621 5351902 3809324 4252020

Aeroplane 4 7877475 2439630 2318803 4966922 2562252 29577715 9311095 2590907 2448975 6188636 3045369 3810145

Swan 4 7778865 2520282 2305300 4741355 2194570 29584915 9368673 3253198 3045478 6227627 3575132 4198971

Ladyhand 4 7637196 2582458 2580046 4742558 3066803 35153095 9254623 3219210 2962865 6399271 4435080 5078334

Starfish 4 7722705 3271955 3013681 4995923 2806032 31789645 9123489 3358983 3075762 6647341 4417617 4978053

Mountain 4 7732155 2408338 2340408 4723301 2517399 30963325 9190040 3063349 2940541 6345679 3073803 4989546

Building 4 7612425 2532897 2452882 4825658 2951447 39965615 9296582 3184240 2981036 6318346 3586839 4960509


Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Baboon

Lena

Aeroplane

Swan




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Fuzzy Multilevel Image Thresholding Based...

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