Localized Components Analysis(7)

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

LocalizedComponentsAnalysis

525

Fig.3.Corporacallosabasiscomparison.Outof54basisvectors,the rstfewareshown.LoCAsuccessfullycapturesthemajorshapedeformationsofthegenuandspleniuminthe rstfourvectors,whilebothICAandS-PCAspreadthisvariationoverseveralvectors.

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

526D.Alcantaraetal.

Corpora callosa reconstruction error

PCA (7)

LoCA (10)

LoCA (15)

LoCA (26)

S PCA (11)

S PCA (15)

S PCA (26)

ICA (38)0.30

0.00

00.100.20Reconstruction error1020304050

Basis vectors included

Fig.4.Reconstructionerrorwhenusingonlythe rstkvectorsofthebasis.Thenum-bersinparenthesesdenotethenumberofvectorsrequiredtocapture90%ofvariationinthedataset.LoCAbasesarecomparedwithS-PCAbaseswhichhaveessentiallythesamereconstructionerrorfor10,15,or26vectorsrespectively.Thesechoicescorre-spondtothedi erentλingfewervectors,S-PCAhaslowerreconstructionerrorbecausethe rstfewS-PCAvectorsrepresentglobaldeformations(asseeninFigure2).

highspatiallocalityofS-PCAbasisvectors.Also,moreofthemwererequiredtodescribethedeformationofanyextendedsurfaceregion.

CorporaCallosa.55healthysubjectsandHIV/AIDSpatientsreceivedhigh-resolutionmagneticresonancebrainscansaspartofapreviously-describedstudy[3].TheCCwasmanuallytracedonallscansusingareliable,repeatableprotocol,andsparselandmarkswereplacedonalltracesusingtheWitelsoncriteria[17].103pointcorrespondenceswereestablishedbetweenallCCtracesbasedontheWitelsonlandmarksusingasparse-to-densecorrespondencealgorithm[18].

Figure3comparesthebasisvectorsfromeachmethodthatcapturedthemostshapevariation.PCArequired7basisvectorsfor90%ofshapevariation,whileICArequiredthemostat38.Notetheglobale ectsofPCAvectors,theextremelocalityofICAandS-PCA,andthespatiallybroadere ectsofLoCA.MajordeformationsofmeaningfulCCsub-regions,thegenuandsplenium,arerepre-sentedbythe rstfourLoCAvectors,whilethenextsixrepresentdeformationsofthecorpuscallosum’slongcentralbody.

ReconstructionerrorforallmethodsisgraphedinFigure4.ICArequiresalargenumberofcomponentsforaccurateshapereconstruction,andPCArequiresthefewest;S-PCAandLoCArequiremorebasisvectorsforbasesthataremorelocalorsparse(i.e.,higherλ).Note,however,thatforcomparablereconstruc-tionerrorcurves,S-PCAbasestendtocontainglobalshapecomponentswhileLoCAdoesnot;forexample,compareLoCA(26)andS-PCA(26)inFigures4and2.