Localized Components Analysis(5)

Abstract. We introduce Localized Components Analysis (LoCA) for describing surface shape variation in an ensemble of biomedical objects using a linear subspace of spatially localized shape components. In contrast to earlier methods, LoCA optimizes explicit

LocalizedComponentsAnalysis523

EnergyFunction.EachsuccessivePCAcomponentaccountsforasmuchoftheshapevariationaspossible;thatis,thedistributionofshapevariationoverthePCAbasisvectorsisasconcentratedaspossibleontheleadingei.Moreformally,onecande netherelativevarianceβiofeachbasisvectoreias

n2j=1 (vj μ),ei βi= n2j=1||vj μ||

whereμrepresentsthemeanofthedatavectorsvj.Theentropyofthedistribu- ktion i=1βilogβiisminimized,overallorthogonalbases,bythePCAbasis,sowede nethistobeEvar,asin[7].TheS-PCAconstructioninthatpaperbalancesEvaragainstanotherenergyfunctionthatrewardssparseei–thatis,asmanyentriesaspossibleineacheiareencouragedtohavezeromagnitude.Weinsteadoptimizeforlocality,de ningElocasfollows.

Weencourageeacheitohavesimultaneousnonzeroentriescorrespondingtopointspiandpjifandonlyifpiandpjareclosetoeachother.Todoso,weintroduceapairwisecompatibilitymatrixBwhoseentriesB[i,j]tendtoward1whenpiandpjareneareachother,andtendtowards0whentheyaredistant;wede neBbelow.TheBmatrixde nesacostfunctionC:

C(ei,pc)=m

j=1(B[c,j] ||ei,j||L2)κ

Theeihaveunitlength,sobothB[c,j]and||ei,j||varybetween0and1.Intuitively,pointspcandpjcontributesigni cantlytoCif:1.pcandpjareincompatible,butei,jhashighmagnitude;or2.pcandpjarecompatible,but||ei,j||iscloseto0.Theexponentκcantakeonanyvaluebetween1and2todealwithoutliere ects.Forourexperiments,κwas1.5.

Foreachbasisvectorei,eachpcyieldsadi erentC.Wede nethelocalityofeiusingthebestpossiblepc,thatis,theonethatminimizesthiscostfunctionC.Eachpcdi ersinthedistributionofitsdistancestoallotherpj–forexample,pointsatoneendofahumerusboneinFigure5areextremelydistantfrommanypointsattheoppositeendofthebone,whilepointsinthemiddlearenot.SowenormalizeCasfollows:

C(ei,pc)

pcmaxebadC(ebad,pc)i Thedenominatorforagivenpcissimplyjmax(|B[c,j] 1|,|B[c,j] 0|)κ.It

needstobecomputedonlyonce.

ThecompatibilityB[i,j]canbecomputedinwhateverwayisappropriateforthedataset;here,B[i,j]isbasedonthedistanceD(pi,pj)betweenpiandpj.FortheCCdatasetconsideredbelow,Disthegeodesicdistancecomputedfromdensesurfacemeshes.Forthe3Dhumeriandskulldatasets,Discomputedfromanadjacencygraphconstructedbetweenthelandmarks.ThecompatibilityisB[i,j]=f(D(pi,pj)),wherefisafunctionthatmodulatesDtoadjustEloc=min