Localized Components Analysis(3)

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

LocalizedComponentsAnalysis521

termsiscomplex.Figure1(right),bycontrast,showsatypicaleigeneratedbythemethodpresentedbelow;di erencesinthecorrespondingαibetweengroupsgivesrisetoasimplephysicalexplanationintermsofthegenu,theCCsubregionwhoseshapeiscapturedbytheei.

WepresentLocalizedComponentsAnalysis(LoCA),amethodthatopti-mizestheeiforspatiallocalityandconcisenesssimultaneously.Itimprovesonpreviouslinearsubspacemethodsbyexplicitlyoptimizingforlocalizedshapeparametersandbyallowingtheusertomodulatethetradeo betweenlocal-ityandconcisenesswithgreater exibilitythanpreviousmethods.Theresult-ingshapecomponentscouldprovidesuccinctsummariesofspatially-localizedchangestobiomedicalstructuresduetoavarietyofphysicalphenomena;forexample,LoCAcouldprovideaconcisesummaryofthespatially-localizedCCshapechangesthatarethoughttoaccompanydiseasessuchasHIV/AIDS[3].Inprimateevolution,LoCAcouldsummarizetheshapesimilaritiesbetweentheskullsofgeneticallyrelatedspeciesusingafewintuitiveparameters.

WesummarizerelatedtechniquesinSection2,andpresentLoCAinSection3.AthoroughsetofexperimentsinSection4showstheintuitivenessand exibil-itygainedbyLoCAoverestablishedlinearsubspacemethodswhenappliedtohumanCC,colobinemonkeyskulls,andprimatehumeri(upperarm)bones.2RelatedWork

PCAhasbeenusedto ndconcisebasesforshapespacesinmedicalimageanalysis[4],morphometrics[5],computergraphics[6],andmanyothercontexts.InPCA,eiistheitheigenvectorofthecovariancematrixoftheexamplevjvec-tors;therefore,theeiareorthogonalandvkjisthebestk-thorderapproximationofvjundertheL2norm.TwoalgorithmsindependentlynamedSparsePCA(S-PCA)encourageasmanyentriesineitobezeroaspossible,eitherbyiterativelyadjustingthePCAbasis[7]orbyiterativelyconstructingsparseorthogonalvectors[8][9]1.Empiricallytheeioftenrepresentshapeinasmallnumberofspatially-localizedsubregions[9][11].Similarly,whileindependentcomponentsanalysis(ICA)andprincipalfactoranalysis(PFA)donotdirectlyoptimizealocality-relatedobjectivefunctionwhenestimatingei,theyappeartogeneratespatially-localizedcomponentsanyway[12][13].Alternatively,pre-de nedspa-tiallylocatedregionsofinterestcanbeintegratedintoPCA[14].OurapproachisinspiredbyS-PCAandfollowsasimilarstrategyofadjustingtheeiprovidedbyPCA;butweexplicitlyoptimizeforspatially-localized,ratherthansparse,ei.Unlike[14]worksoflocalizedmedialgeometricprimitiveshavethepotentialtocap-turelocalshapeinaconcisesetofparameters[15].Wefeelthatmedialandsurface-basedrepresentationscouldcapturecomplementaryshapeinformation.Wenote,however,thatnetworksofmedialprimitivescanbechallengingtocon-structinanautomatedwayandmaythereforebemorelabor-intensivethantheapproachwepresent.

1Athird,unrelatedSparsePCAsparsi esthevjbeforeapplyingstandardPCA[10].