Localized Components Analysis(6)

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

524D.Alcantaraetal.

itsrangeto[0,1].Wechoseasinusoidalfthatisnon-zerooverahalf-period:f(x)=0.5(cos(πx

ρ)+1).Largerρselectforgroupsofpointswhichco-varyover

largerspatialextents.Itwassetto0.25inalloftheexperimentsbelow.

Optimization.Ouroptimizationprocedureissimilartothatusedin[7].PCAprovidesaninitialorthonormalbasise,andeverypossiblepairei,ejarero-tatedtogetherinthetwo-dimensionalplanetheyspan.Becausetherotatingpairiskeptorthogonaltoeachotherandstayintheir2Dplane,thebasisre-mainsorthonormalthroughoutoptimization.EachpairisrotatedbytheangleθthatminimizesEvar+λEloc.TheoptimalθisfoundnumericallyusingBrent’smethod[16].NoticethatsinceEvarandElocarebothsummationsoftermsthateachdependsolelyonanindividualei,onlythetermscorrespondingtothecurrentei,ejpairneedtobeupdatedduringoptimization.

Thepairsarerotatedindecreasingorderofshapevariationaccountedfor.Thesetofallei,ejpairsareadjustedrepeatedly,andoptimizationceaseswhenad-justingthemchangestheobjectivefunctionlessthana xedthreshold.Between50and150iterationswererequiredforeachexperimentbelow.

DataPreparation.Weassumethatwearegivenanensembleofnobjects,eachrepresentedbympointsonitsboundary,andthecompatibilitymatrixB.Overalldi erencesinobjectscale,rotationandtranslationovertheensembleareremovedthroughgeneralizedProcrustesalignment[5].Theresultingscaledandaligneddatasetsareusedasinputtotheaboveoptimization.

4Results

Below,wecompareLoCAtoPCA,ICA,andS-PCAonthreedatasets:CCs,colobinemonkeyskulls,andhumerifromvariousprimates2.Foreachbasis,lo-calityisevaluatedvisuallyusingrenderingsoftheentriesineachbasisvector,andthroughlocalitygraphs (seeFigure2).Concisenessofeachbasisisassessed

kquantitativelybychartingn

j=1||vj vj||L2overallk,andmorespeci cally

byrecordingthenumberofeirequiredtocapture90%ofshapevariation,i.e.reducethisreconstructionerrorto10%.

LoCAbehaviordependsstronglyonλ,theparameterthatmodulatesthetradeo betweenconcisenessandlocality.Forλ=0,LoCAreducestoPCA.Forsmallλ,LoCAbasisvectorsaccountingforthehighestamountsofshapevaria-tionresemblePCAbasisvectors,whiletherestofthebasisisclearlylocalized(Figure2).Forlargerλ,allLoCAbasisvectorsarelocal,andthebasesrequiremorebasisvectorstoaccountforshapevariationinthedata.InFigures3,5,and6,LoCAandS-PCAbasisvectorsaredepictedforthesmallestvalueofλforwhichthebaseslackedglobalbasisvectors.S-PCAperformssimilarlytoLoCAforsmallvaluesofλ,inagreementwithearlierS-PCAresults[7].However,S-PCArequiredamuchlargerbasis–morebasisvectorsfor10%reconstructionerror–beforetheglobalbasisvectorsdisappeared;thisislikelyduetothevery2Moviesandlargerimagesareat:http://idav.ucdavis.edu/～dfalcant/loca.html