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1SequentialKernelDensityApproximationandItsApplicationtoReal-TimeVisualTrackingBohyungHan,Member,IEEEDorinComaniciu,SeniorMember,IEEEYingZhu,andLarryS.Davis,Fellow,IEEEJune22,2007DRAFT2AbstractVisualfeaturesarecommonlymodeledwithprobabilitydensityfunctionsincomputervisionprob-lems,butcurrentmethodssuchasamixtureofGaussiansandkerneldensityestimationsufferfromeitherthelackofexibility,byxingorlimitingthenumberofGaussiancomponentsinthemixture,orlargememoryrequirement,bymaintaininganon-parametricrepresentationofthedensity.Theseproblemsareaggravatedinreal-timecomputervisionapplicationssincedensityfunctionsarerequiredtobeupdatedasnewdatabecomesavailable.Wepresentanovelkerneldensityapproximationtechniquebasedonthemean-shiftmodendingalgorithm,anddescribeanefcientmethodtosequentiallypropagatethedensitymodesovertime.Whiletheproposeddensityrepresentationismemoryefcient,whichistypicalformixturedensities,itinheritstheexibilityofnon-parametricmethodsbyallowingthenumberofcomponentstobevariable.Theaccuracyandcompactnessofthesequentialkerneldensityapproximationtechniqueisillustratedbybothsimulationsandexperiments.Sequentialkerneldensityapproximationisappliedtoon-linetargetappearancemodelingforvisualtracking,anditsperformanceisdemonstratedonavarietyofvideos.IndexTermskerneldensityapproximation,mean-shift,modepropagation,on-linetargetappearancemodeling,objecttracking,real-timecomputervisionI.INTRODUCTIONDensityestimationisbroadlyusedtostatisticallymodelvisualfeaturesincomputervisionapplications.Theunderlyingprobabilitydensityofthefeaturescanbedescribedbyaparametric(e.g.,GaussianormixtureofGaussians)ornon-parametric(e.g.,histogramorkerneldensity-based)representation.However,theserepresentationscreateatrade-offbetweentheexibilityofthemodelanditsdatasummarizationproperty.Inotherwords,parametricmethodsaresimpleandefcient,buthavedifcultyrepresentingmulti-modaldensityfunctionseffectively.Also,theyusuallyrequireapre-denedparameterforthenumberofcomponents,soitishardtouseparametricdensityfunctionsinreal-timeapplications,especiallywhentherearealargenumberofmodesintheunderlyingdensityorthenumberofmodesisfrequentlychanging.Ontheotherhand,non-parametricmodelsareveryexibleandcanaccommodatecomplexdensities,butrequirealargeamountofmemoryfortheirimplementation.Wepresentanewmethodtoapproximateamulti-modaldensityfunctionwithamixtureofJune22,2007DRAFT3GaussiansKernelDensityApproximation(KDA),whichwasoriginallyintroducedin[15],[16].Kerneldensityapproximationisaexiblemulti-modaldensityrepresentationmethodsinceeveryparameterfortheGaussianmixtureisdeterminedautomatically.Thistechniqueisappliedtoareal-timecomputervisionproblemon-linetargetappearancemodelingforobjecttracking.A.RelatedWorkAGaussiandistribution,whichisthesimplestdensity-basedmodelingmethod,isfrequentlyusedforvariouscomputervisionproblems,suchasbackgroundsubtractionandobjecttracking[14],[17],[24],[32].However,thisrepresentationcannothandlemulti-modaldensityfunctions,sotheaccuracyofmodelingisseverelylimited.Mixturemodelsbasedonmultiplecomponentshavebeenutilizedinnumerousapplications.In[22],atargetappearancemodelbasedoncolorisconstructedbyamixtureofGaussianswhichisreplacedineachframe,andthesamedensityrepresentationisemployedforopticalowestimationbyJepsonandBlack[18].ArecursiveupdateofaGaussianmixturemodelisproposedin[20],[29]forbackgroundmodeling,butthesemethodsarenotexibleenoughtomodelcomplexdensityfunctionssincetheytypicallyrequirethemaximumnumberofcomponentsinthemixtureinadvance.Forobjecttracking,adaptivetargetappearancemodelingbya3-componentmixtureisdescribedin[19],wherethemixturedensityfunctionisupdatedovertimebyanon-lineEMalgorithm.However,itisgenerallydifculttoaddorremovecomponentsintheexistingadaptivemixturemodelsinaprincipledway.Therefore,mostreal-timeapplicationsrelyonmodelswithaxednumberofmixtures[18],[19],[22]orapplyad-hocstrategiestoadaptthenumberofmixturesintime[20],[27],[29],wheretheadditionanddeletionofaGaussiancomponentishighlydependentonthepre-denedthresholdvalues.Thereisamoreelaboratedmethodtodeterminethenumberofcomponentsusingalayermodel,butitalsorequiresthemaximumnumberoflayersasaparameterandthedecisionregardingthenumberoflayersisbasedonanadditionalcomplexprocess[30].Kerneldensityestimation[12]isoneofthemostpopularnon-parametrictechniquestomodeldensities,becauseitprovidesaexibleframeworktorepresentmulti-modaldensities.However,itsveryhighmemoryrequirementsandcomputationalcomplexityinhibittheuseofthismethodinreal-timeapplications,eventhoughtherehavebeenseveralattemptstoreducethecomputationalcost[11],[33].June22,2007DRAFT4B.OurApproachIncontrasttopreviousapproaches,wepresentanewstrategyformulti-modaldensityap-proximationanditson-linelearningthatreliesonmodelingandpropagationofdensitymodes.Themodes(localmaxima)ofadensityfunctionrepresentregionswithhigherlocalprobability;hence,theirpreservationisimportanttomaintainalowdensityapproximationerror.WerepresentthedensityasaweightedsumofGaussians,whosenumber,weights,meansandcovariancesareautomaticallydetermined.
本文标题:Sequential Kernel Density Approximation and Its Ap
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