Localized Components Analysis(2)

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

520D.Alcantaraetal.

Fig.1.ShapecharacteristicsofcorporacallosacapturedbybasisvectorsgeneratedwithPCAandLoCA.Arrowsstartatpointstracingtheaveragecorpuscallosum;theirmagnitudesindicatethedegreethatpointsmovewhenthecorrespondingshapeparameterisvaried.ThePCAvectorrepresentsacomplex,globalpatternofshapecharacteristicswhiletheLoCAvectorfocusesonthegenu.

bepresentedintermsofasmallnumberofparameters,eachofwhichrepresentsaneasily-graspedaspectofregionshape.Thiscouldpromoteinterpretationsoftheshapedi erenceintermsofdiseasecausesore ects.

Ourgoalistoencourageinterpretabilityofresultsbygeneratingshapepara-meterizationsthatarebothconcise–capturingsalientshapecharacteristicsinasmallnumberofparameters–andspatiallylocalized–accountingfortheshapeofaspatiallyrestrictedsub-regionineachparameter.Thehypothesisunderlyingthispaperisthatspatially-localizedandconciseshapeparameterizationsaremoreintuitiveforendusersbecausetheyallowthemtoconceptualizeobjectshapeintermsasmallnumberofobjectparts,whichareoftena ecteddi eren-tiallybyphysicalphenomena.Intheaboveexample,shapechangeduetodiseaseprocessesisknowntooccurinspatially-localizedbrainsub-regionsinavarietyofdisorders[1].Inaddition,conciseparameterizationsareattractivebecausethestatisticalpoweroftestsonthoseparametersisreducedaslittleaspossiblebycorrectionsformultiplecomparisons[2].

Wefollowthelinearsubspaceparadigmofexpressingeachshapeasalinearcombinationofprototypical,orbasisshapes.Thatis,ifeachshapeisrepresentedasavectorvjofthe2mor3mcoordinatesofmpointssampledfromitsboundary(i.e.,vj=[vj,1,vj,2,···vj,m],vj,k=[xk,yk]for2Dshapes),vjisapproximatedasalinearcombinationofkbasisvectors{e1,e2,···ek}:

vkj=k

i=1αj,i ei

Theshapeparametersarethecoe cientsαj,i.Linearsubspacemethodsareattractivebecausetheirlinearityineiallowsthemtobemanipulatedusingstandardtoolsfromlinearalgebra.

However,linearsubspacemethodsdonotinherentlyencouragelocality.Fig-ure1(left)depictsatypicaleigeneratedbytheclassicallinearsubspacemethod,principalcomponentsanalysis(PCA),appliedtotracingsofthecorpuscallosum(CC),ahumanbrainregion.ThebasisshapesummarizesacomplexpatternofshapecharacteristicsacrosstheentiretyoftheCC.Therefore,ifthecorrespond-ingαidi ersbetweengroups,theexplanationofthegroupdi erenceinphysical