A Bayesian Framework for Image segmentation_10

Copyright(c)2010IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail]edforpublicationinafutureissueofthisjournal,buthasnotbeenfullyedited.Contentmaychangepriortofinalpublication.IEEETRANSACTIONSONIMAGEPROCESSING,VOL.??,NO.??,??1ABayesianFrameworkforImageSegmentationwithSpatiallyVaryingMixturesChristophorosNikouMember,IEEE,AristidisLikasSeniorMember,IEEEandNikolaosGalatsanosSeniorMember,IEEEAbstractAnewBayesianmodelisproposedforimagesegmentationbasedonGaussianmixturemodels(GMM)withspatialsmoothnessconstraints.ThismodelexploitstheDirichletcompoundmultinomial(DCM)probabilitydensitytomodelthemixingproportions(i.e.theprobabilitiesofclasslabels)andaGauss-Markovrandomfield(MRF)ontheDirichletparameterstoimposesmoothness.Themainadvantagesofthismodelaretwo.First,itexplicitlymodelsthemixingproportionsasprobabilityvectorsandsimultaneouslyimposesspatialsmoothness.Second,itresultsinclosedformparameterupdatesusingamaximumaposteriori(MAP)expectation-maximization(EM)algorithm.PreviouseffortsonthisproblemusedmodelsthatdidnotmodelthemixingproportionsexplicitlyasprobabilityvectorsorcouldnotbesolvedexactlyrequiringeithertimeconsumingMarkovChainMonteCarlo(MCMC)orinexactvariationalapproximationmethods.NumericalexperimentsarepresentedthatdemonstratethesuperiorityoftheproposedmodelforimagesegmentationcomparedtootherGMM-basedapproaches.Themodelisalsosuccessfullycomparedtostateoftheartimagesegmentationmethodsinclusteringbothnaturalimagesandimagesdegradedbynoise.IndexTermsImagesegmentation,Bayesianmodel,Gaussianmixture,spatiallyvaryingfinitemixturemodel,Gauss-Markovrandomfieldprior,Dirichletcompoundmultinomialdistribution.C.NikouandA.LikasarewiththeDepartmentofComputerScience,UniversityofIoannina,POBox1186,45110Ioannina,Greece,(email:[e-mail],[e-mail]).N.GalatsanosiswiththeDepartmentofElectricalandComputerEngineering,UniversityofPatras,26500Rio,Greece,(email:[e-mail])Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineat:JILINUNIVERSITY.DownloadedonAugust02,2010at10:00:37UTCfromIEEEXplore.Restrictionsapply.Copyright(c)2010IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail]edforpublicationinafutureissueofthisjournal,buthasnotbeenfullyedited.Contentmaychangepriortofinalpublication.I.INTRODUCTIONManyapproacheshavebeenproposedtosolvetheimagesegmentationproblem[1],[2].Amongthem,clusteringbasedmethodsrelyonarrangingdataintogroupshavingcommoncharacteristics[3],[4].Duringthelastdecade,themainresearchdirectionsintherelevantliteraturearefocusedongraphtheoreticapproaches[5],[6],[7],[8],methodsbasedonthemeanshiftalgorithm[9],[10]andratedistortiontheorytechniques[11],[12].Modelingtheprobabilitydensityfunction(pdf)ofpixelattributes(e.g.intensity,texture)withfinitemixturemodels(FMM)[13],[14],[15]isanaturalwaytoclusterdatabecauseitautomaticallyprovidesagroupingbasedonthecomponentsofthemixturethatgeneratedthem.Furthermore,thelikelihoodofaFMMisarigorousmetricforclusteringperformance[14].FMMbasedpdfmodelinghasbeenusedsuccessfullyinanumberofapplicationsrangingfrombioinformatics[16]toimageretrieval[17].TheparametersoftheFMMmodelwithGaussiancomponentscanbeestimatedthroughmaximumlikelihood(ML)estimationusingtheExpectation-Maximization(EM)algorithm[13],[18],[14].However,itiswell-knownthattheEMalgorithmfinds,ingeneral,alocalmaximumofthelikelihood.Furthermore,itcanbeshownthatGaussiancomponentsallowefficientrepresentationofalargevarietyofpdf.Thus,Gaussianmixturemodels(GMM),arecommonlyemployedinimagesegmentationtasks[14].AdrawbackofthestandardMLapproachforimagesegmentationisthatcommonalityoflocationisnottakenintoaccountwhengroupingthedata.Inotherwords,thepriorknowledgethatadjacentpixelsmostlikelybelongtothesameclusterisnotused.Toovercomethisshortcoming,spatialsmoothnessconstraintshavebeenimposed.Imposingspatialsmoothnessiskeytocertainimageprocessingapplicationssinceitisanimportantaprioriknownpropertyofimages[19].Examplesofsuchapplicationsincludedenoising,restoration,inpaintingandsegmentationproblems.Inaprobabilisticframework,smoothnessisexpressedthroughapriorimposedonimagefeatures.AcommonapproachistheuseofanMRF.ManyMRFvariantshavebeenproposed,seeforexample[20].However,determinationoftheamountoftheimposedsmoothnessautomaticallyrequiresknowledgeofthenormalizationconstantoftheMRF.Sincethisisnotknownanalytically,learningstrategieswereproposed[21],[22],[23].Researcheffortsinimposingspatialsmoothnessforimagesegmentationcanbegroupedintotwocategories.Inthemethodsofthefirstcategory,spatialsmoothnessisimposedonthediscretehiddenvariablesoftheFMMthatrepresentclasslabels,seeforexample[24],[25],[7],[26].Theseapproachesmaybecategorizedinamoregeneralareainvolvingsimultaneousimagerecoveryandsegmentationwhichisbetterknownasimagemodeling[27],[28],[29],[30].Morespecifically,spa