Localized Components Analysis(10)

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

LocalizedComponentsAnalysis529

Humeri.3Dsurfacemodelsofhuman,gorilla,andchimpanzeehumerusboneswereobtainedbyscanningthebonesusingalaserrangescanneraspartofalong-termprojectonofprimateevolutionarymorphology.The3Dmodelswereannotatedbyasinglehumanoperator(D.A.)inagraphicalinterfacebyplacingcurvesatanatomicallandmarksontheshaftandproximalanddistalextremitiesofthebone.Acurveontheproximalextremityfollowedthelengthofthearticularneck.Threecurvesweretracedlongitudinallyalongridgesthatspannedthelengthoftheshaft.Onthedistalextremity,curvesfollowedtheridgeoftheolecranonfossa,theridgealongthemediallimitofthetrochlea,theridgealongthelaterallimitofthecapitulum,andtheridgebetweenthetrochleaandcapitulum.Surfacepointssampledfromthesecurvesweretheinputtotheshapeparameterizationmethods.

The rstfewbasisvectorsoftheresultsareshowninFigure5.LoCAidenti edbasisvectorsthatintuitivelydescribedeformationsofthetwojointsurfacesattheextremities,aswellasdeformationsofsub-regionsofthelongbody.

ColobineMonkeyCrania.Theshapespacewasbuiltfromasetof235cra-niafromcolobinemonkeys(SubfamilyColobinae,FamilyCercopithecidae),fromsixAsianspecies.Eachcraniumwasmarkedwith45correspondinglandmarkpoints.Variouscomparativeprimatemorphologistsmanuallymarkedeachcra-niumwith45landmarkpointsusingaMicroscribe3Ddigitizer[19],aspartofdatacollectionforalong-termprojectonOldWorldmonkeycranialevolution.Resultsareshownin gure6.ThelandmarksareshownonaColobusguerezacranium,whichiscoloredasin5.

TheusefulnessofthecompatibilitymatrixBforcreatinglocalizedbaseswithhigher-ordercharacteristicsisalsoillustratedinFigure6.Toencouragespatiallysymmetriccomponents,landmarkpointswerere ectedacrossthemidsagittalplane;patibilityB[i,j]wascomputedbetweenthere ectedpoints,sothattwopointsatcorrespondinglocationsacrossthemidsagittalplane(e.g.,rightandleftcheek)werehighlycompatible.Theresulting“SymmetricLoCA”basisintuitivelycapturesshapevariabilityincorrespondingrightandleftskullfeaturesineachcomponent.

5Discussion

LoCAprovidedasuperiortrade-o ofconcisenessandlocalitythanICAorS-PCAforabroadrangeofdatasets,atacostofgreatercomputationtime.Futureworkwillfocusondesigningcompatibilitymatricesforawiderangeofshapeapplications,usingnon-geodesicdistancemetricsanduser-de nedobjectregionsofinterest.Wewillalsogeneratelocalizedbaseswhosevectorsvarysigni cantlyintheirspatialsuppport;currently,therangeofpossiblespatialsupportsislimitedtoadegreebythedistancemodulatingfunctionf.