Localized Components Analysis(4)

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

522D.Alcantaraetal.

Fig.2.E ectofλonthe rst(top)andthird(bottom)basisvectors,wherevectorsareorderedbytheamountofshapevariationcaptured.Asλincreases,thenumberofvectorsrequiredtocapture90%ofthevariation(inparentheses)increases.Forsmallvaluesofλ,vectorscapturingsubstantialvariationrepresentaglobaldeformationoftheentireshape.Asλisincreased,moreoftheLoCAvectorsbecomelocaldeformations,untiltheentirebasisconsistsoflocalvectors.S-PCAbecomessparsemoreslowly,sothatthe rstvectorisstillaglobaldeformationontheright.Thethirdvectorissparse,butthereissomeperturbationacrosstheentireshape.Eachvectorisaccompaniedbyagraphshowingitslocality,whereeverypointinthegraphrepresentsapointontheout-line.Thecenterpointisde nedasthepointminimizingEloc,asdescribedinSection3.Analternativeapproachfordeterminingspatially-localizeddi erencesbe-tweenshapeensemblesistoperformstatisticalteststhatcomparecorrespondingvj,kbetweengroups;spatialmapsthencolor-codeeachvj,kbythee ectsizeorpvalueofthetest.Visualinspectionoftherenderingshasrevealedspatially-localizedshapedi erencesinavarietyofmedicalconditions(see,e.g.,[1]);how-ever,misgenerallysolargethatthesigni cancethresholdofthestatisticaltestsmustbereduceddramaticallytoguardagainstdetectionofspuriousgroupdi erences[2].Thisreducesthesensitivityofspatialmappingtechniquestode-tectsubtleshapedi erences.LoCAusesalinearsubspacetoreducethenumberofvariablesrequiredforlocalizedshapecomparisons,andthereforeboostthepowerofstatisticaltests.

3Methods

PCAproducesthemostconcisebasispossibleundertheL2norm;thatis,foreach nk,j=1||vj vkj||L2isminimizedwhene1···ekarethe rstkeigenvectorsofthecovariancematrixofthevj.WeuseaformulationofPCAastheminimizationofanenergyfunctionEvar,andmodifyitbyminimizingEvar+λEloc,whereElocisanewenergytermthatsummarizesthespatiallocalityoftheei.Theλbalancesthetradeo betweenthecompetinginterestsofconcisenessandlocality(Figure2).