ShapeMatchingandAnisotropyMichaelKazhdanPrincetonUniversityThomasFunk - PDF document

ShapeMatchingandAnisotropyMichaelKazhdanPrincetonUniversityThomasFunkhouser†PrincetonUniversitySzymonRusinkiewicz‡PrincetonUniversityAbstractithrecentimprovementsinmethodsfortheacquisitionandren-deringof3Dmodels,theneedforretrievalofmodelshasgainedprominenceinthegraphicsandvisioncommunities.Avarietyofmethodshavebeenproposedthatenabletheefcientqueryingofmodelrepositoriesforadesired3Dshape.Manyofthesemethodsusea3Dmodelasaqueryandattempttoretrievemodelsfromthedatabasethathaveasimilarshape.Inthispaperweconsidertheimplicationsofanisotropyontheshapematchingparadigm.Inparticular,weproposeanovelmethodformatching3Dmodelsthatfactorstheshapematchingequationasthedisjointouterproductofanisotropyandgeometriccomparisons.Weprovideageneralmethodforcomputingthefactoredsimilaritymetricandshowhowthisapproachcanbeappliedtoimprovethematchingperformanceofmanyexistingshapematchingmethods.CRCategories:I.5.3[ComputingMethodologies]:PatternRecognition—SimilarityMeasures;I.5.4[ComputingMethodolo-gies]:Applications—ComputerVisionKeywords:shapematching,anisotropy1IntroductionWithrecentimprovementsinmethodsfortheacquisitionandren-deringof3Dmodels,theneedforeffectiveretrievalofmodelshasgainedprominenceinthegraphicsandvisioncommunities.Theabilitytoretrieveexistingmodelsfacilitatesthetasksofprofession-alsineldsrangingfromentertainmenttoscienticresearch,byallowingthemtoobtaindesiredmodelsquicklywithoutrequiringtheexpenditureoflargeamountsoftimemodelingthe3Dshape.Toaddressthisneed,avarietyofretrievalmethodshavebeenpro-posedthatenabletheefcientqueryingofmodelrepositoriesforadesired3Dshape[Princeton3DModelSearchEngine;ProteinDataBank;CCCC;ShapeSifter].Manyofthesemethodsusea3Dmodelasaqueryandattempttoretrievemodelswithmatchingshapefromthedatabase.Inthispaperweconsidertheimplicationsofanisotropyonshapematching.Inparticular,weproposeanovelmethodformatching3Dmodelsthatfactorstheshapematchingequationasthedisjointouterproductofanisotropyandgeometriccomparisons.Wepro-videageneralmethodforcomputingthefactoredsimilaritymetricandshowhowthisapproachcanbeappliedtoimprovethematch-ingperformanceofmanyexistingshapematchingmethods.e-mail:mkazhdan@cs.princeton.edu†e-mail:funk@cs.princeton.edu‡e-mail:smr@cs.princeton.eduFigure1:Whentwomodelshavedifferentanisotropicscales(left),itishardertoes-tablishcorrectcorrespondencesbetweenthetwo.Thus,matchingmethodsthatdependoncorrespondencesforevaluatingsimilaritywillbeinaccurateinthiscase.Incontrast,whenthemodelsaretransformedsothateachisisotropic(right),thecorrespondencesaremoreaccurateandthemeasureofshapesimilarityismorediscriminating.Thekeyideaofourapproachisbasedontheobservationthatmuchofthechallengeofshapematchingistheestablishingofcorrespon-dences,andthatitiseasiertoestablishcorrespondencesbetweentwomodelsiftheyareisotropic—havingconstantvarianceineachdirection.Figure1demonstratesthisformodelsofanarmchairandasofa.Whenthemodelsareattheirinitialanisotropicscales(left),itisdifculttoestablishcorrespondencesbetweensimilarregions.Methodssuchasassociatingtoapointononemodelthenearestpointontheother(commonlyusedinICP-typeapproaches[BeslandMcKay1992])willmappointsonthecornersofthearmchairtopointsinthemiddleofthesofa,pointsonthebottomofthearm-restofthearmchairtopointsonthetopofthearm-restofthesofa,etc.Thus,manypoorcorrespondenceswillbeestablished,result-inginaninaccuratemeasureofsimilarity.Ifinsteadbothmodelsarerescaledtobeisotropic(right),thenthecorrespondencesestab-lishedmoreaccuratelyreectcorrespondingregionsintheshape.Whilemanyexistingmatchingmethodscomparemodelswithoutexplicitlyestablishingcorrespondences,theunderlyingsimilaritymetricisoftendesignedtorepresentthedistancebetweenpointsonthesurfacesofthetwomodels.Theseobservationsmotivateustodesignashapematchingparadigmthatcomparestwomodelsby(1)transformingeachofthemintoisotropicmodels,(2)comparingthegeometricsimilarityoftheisotropicmodels,and(3)deningthemeasureofmodelsimilarityasafunctionofboththesimilarityoftheisotropicmodels,andthedifferenceintheirinitialanisotropicscales.Figure2demonstratesthisprocessfortwodifferentmodelsofatable.Eachtableisrepresentedbyitsisotropicversionanditsinitialanisotropicscale,representedbythecovarianceellipsoidoftheoriginalmodel.Thedistancebetweenthetwotablesisthendenedastheouterproductofthedistancebetweentheisotropictablesandthedistancebetweentheinitialanisotropicscales.Theremainderofthepaperisstructuredasfollows.Section2pro-videsageneraloverviewofexistingshapematchingapproaches,highlightingsomeofthecentralchallengesinthisarea.Section3describesourproposedmethod,andSection4providesempiricalresultsdemonstratingtheefcacyofourapproachinimprovingthematchingperformanceofmanyexistingshapemetrics.Wecon-cludeinSection5bysummarizingourresults. Figure2:Ourapproachistocomparetwomodelsbyrescalingeachmodelsothatitisisotropicandthende®ningthedistancebetweentwomodelsastheouterproductofthedifferencesbetweentheisotropicmodelsandtheirinitialanisotropicscales.2RelatedWorkTraditionalmethodsforretrievalofmodelsfromlargerepositoriesfocusondesigningamethodfordeningameasureofsimilaritybetweenaquerymodelandeverytargetmodelinthedatabase.Themodelsinthedatabasearethensortedbythismeasureofsimilarity,andthenearestmodelsarereturnedasmatches.Inthecontextofmatching3Dshapes,themostcommonapproachistoestablishcorrespondencesbetweenthequerymodelandthetargetmodel,andthentodenethemeasureofsimilarityintermsofthedistancesbetweencorrespondingpoints.Twogeneralclassesofmethodshavebeenproposedthatcomputeameasureofshapesimilaritybyexplicitlyestablishingsuchcorrespondences.Therstapproachisalocalone,seekingtoestablishcorrespondencesbe-tweenpairsofpointsonthetwomodels,andthendeningthemea-sureofshapesimilarityasthesumofthesquareddistancesbetweenpairsofpointsincorrespondence[BeslandMcKay1992].Thesec-ondmethodismoregeneral,decomposingamodelintoconstituentparts,andthenrepresentingthemodelasagraphcharacterizingtherelationshipbetweenthedifferentsegments[Siddiqietal.1998;Hi-lagaetal.2001].Correspondencesbetweentwomodelscanthenbeestablishedusingsubgraphisomorphismtechniques,whichsimul-taneouslydenethecorrespondencesbetweenthenodesofthetwographrepresentations,andgivethequalityofthecorrespondences.Forbothoftheseapproaches,theestablishingofcorrespondencesisadifcultandtimeconsumingtaskthatneedstobeperformedonaper-pair-of-modelsbasis.Thus,muchofthenecessarycomputationcanonlybeperformedatruntime,onceaqueryisspecied.Thismakesthesemethodsimpracticalfortheretrievalofmodelsfromlargedatabases,whereefcientcomparisonisessential.Thecomputationalcomplexityofestablishingcorrespondencesbe-tweenmodelshasmotivatedalargebodyofresearchintheareaofshapedescriptors.Thegeneralapproachofthesemethodsistode-neamappingfromthespaceofmodelsintoaxed-dimensionalvectorspace,andthentodenethemeasureofsimilaritybetweentwomodelsasthedistancebetweentheircorrespondingdescriptors[Horn1984;KangandIkeuchi1991;Ankerstetal.1999;Osadaetal.2001;VranicandSaupe2001;Funkhouseretal.2003].Themappingisoftenchosensothatthedistancebetweentwodescrip-torsmeasurestheproximityofpointsonthesurfacesofthetwomodels,sothatacorrespondence-basedmeasureofsimilaritycanbeobtainedwithouttheoverheadofexplicitlyestablishingthecor-respondences.Thisapproachhastheadvantageofaddressingthematchingproblemonaper-modelbasis,allowingforthecompu-tationofdescriptorsinanofineprocess.Then,atruntime,thedescriptorofthequeryiscomputedandcomparedagainstthe(pre-computed)descriptorsofallthemodelsinthedatabase,givingrisetomethodsthatcansatisfytheefciencyrequirementsofinterac-tivesearch.Foramoregeneralsurveyofshapedescriptors,wereferthereadersto[AltandGuibas1996;Loncaric1998;Pope1994;TangelderandVeltkamp2004].Aspecicchallengethatshapedescriptorapproachesneedtoad-dressisthatinthecontextof3Dshapematching,amodelanditsimageunderasimilaritytransformationareconsideredtobethesame.Ingeneral,thisissueisaddressedinoneoftwomanners:(1)Themappingischosentobeinvarianttosimilaritytransforma-tion,sothatthesameshapedescriptorisdenedforeveryorien-tationofasinglemodel.(2)Eachmodelisnormalizedbyplacingitintoitsowncanonicalcoordinatesystem,andthentheshapede-scriptorofthealignment-normalizedmodeliscomputed.Methodsfornormalizingamodel'stranslationandscalearebasedon[Horn1987;Hornetal.1988].Inthiswork,theauthorsde-scribeamethodforsolvingforthealignmentminimizingthesumofsquaredifferencesbetweentwoorderedpointsets.Whiletheso-lutionfortheoptimalrotationdependsonthecorrespondencebe-tweenthetwopointsets,theoptimaltranslationandscalecanbecomputedonaper-modelbasis,withtheoptimaltranslationbeingtheonethattransformsamodel'scenterofmasstotheorigin,andtheoptimalscalegivingrisetoamodelwhosemeanvariancefromtheoriginisequaltoone.Methodsforaddressingrotationalsimilarityhaveeithertakenthenormalizationapproach,aligningamodelsothatitsprincipalaxestransformtothex-,y-,andz-axes,orhaveobtainedrotationinvari-antrepresentationsbydiscardingsphericalphaseandobtainingacollectionofamplitudesthatareindependentofamodel'salign-ment[BurelandHenocq1995;Kazhdanetal.2003].3MethodologyInordertoseparateanisotropyfromtheshapematchingequation,weproposeamethodformatchingtwo3Dmodelsthatrstre-movestheanisotropyfromeachofthemodels,comparesthege-ometryoftheisotropicmodels,andthenexpressesthemeasureofsimilarityofthetwomodelsasafunctionofbothgeometricandanisotropicsimilarity.Thisapproachismotivatedbyearlierworkintheareaofisotropicscalenormalization,whichwereviewinthenextsubsection.Wethenshowhowtheseresultscanbegeneral-izedtoanisotropicscaleanddescribeamethodforremovingtheanisotropyfrommodel.Weconcludebydescribingamethodforcomparingtwomodels,providingafamilyofshapemetricsparam-eterizedbytheimportanceassignedtoanisotropy.3.1IsotropicScaleIn[Hornetal.1988;Horn1987],theauthorsaddresstheissueofsolvingfortheoptimalscalethatminimizesthesumofsquaredis-tancesbetweentwoorderedpointssets.Theyposetheproblemasfollows:Giventwoorderedpointsets,P=fp1;:::;pngandQ=fq1;:::;qng,ndthevalueofathatminimizesthesumofsquareddistances:nåi=1\rapiqia\r2Thisformulationoftheoptimalscaleproblemhasthepropertythatthemeasureofsimilarityattheoptimalscaleisindependentofthe orderofPandQ,andisminimizedwhena=4såkqik2åkpik2:Thus,ifeachofthepointsetsisindependentlyscaledsothatitsmeanvarianceisequalto1,thentheoptimalscalevalueforaligningthetwonormalizedpointsetsisa=1andthepointsetsareinfactoptimallypairwisescale-aligned.Theimplicationofthisresultforshapematchingisthatscalenor-malizationcanbedoneonaper-modelbasis,independentofcor-respondence.Aswithusingthecenterofmassfortranslationnor-malization,thismethodisonlyprovablycorrectwhenmodelsarecomparedbysummingdistancesbetweencorrespondingpoints.Inpracticehowever,manyexistingshapedescriptorsimplicitlydeneshapesimilarityintermsofthedistancebetweensurfacesandwendthattranslatingmodelssothattheircenterofmassisattheoriginandscalingthemsothattheirmeanvarianceisequaltooneprovidesarobustmethodfortranslationandscalenormalization.3.2AnisotropicScaleWenowshowhowtheresultsforoptimalscalecanbegeneralizedtosolvefortheoptimalanisotropicscale.GiventwopointsetsP=fp1;:::;png,withpi=(pi;pi;pi),andQ=fq1;:::;qng,withqi=(qi;qi;qi),thesumofsquareddifferencesbetweenthetwopointsetsisgivenbytheequation:nåi=1(piqi)2+(piqi)2+(piqi)2:Itfollowsfromtheworkdescribedabovethatifwesearchfortheoptimalanisotropicscaleinanysingledirectionv,thenthisoccurswhenthepointsetsPandQarenormalizedsothattheirvarianceinthedirectionvisequalto1.Consider,forexample,thecaseofsolvingfortheoptimalanisotropicscaleinthexdirection.Inthiscase,wewouldliketosolveforthevalueofathatminimizesnåi=1(apiqi=a)2+(piqi)2+(piqi)2:Asinthecaseoftheisotropicscale,thevalueofathatminimizestheerroris:a=4så(qi)2å(pi)2andthemodelsareoptimallyscale-alignedinthexdirectionifthevarianceofeachmodel,inthex-direction,isequalto1.Moregenerally,ifbothpointsetssatisfythepropertythatthevarianceinanydirectionisequalto1,thenitfollowsthatanyanisotropicscalingofoneofthetwopointssetswillonlyincreasethesumofsquareddifferences,andthemodelsareinfactoptimallyanisotropicallyaligned.Inordertotransformanarbitrarypointsetintoonethathasunitvarianceinanydirection,itsufcestocomputeitscovariancema-trixCandthenapplythetransformationC1=2tothepointset.(Sinceweassumethatthepointsarenotallcoplanar,thematrixCispositivedeniteandhencecanbeinverted,andhasarealsquareroot.)Toseethis,notethatthecovariancematrixofapointsetP=fp1;:::;pngcanbedenedbytheequation:CP=nåi;j=1(pipj)
(pipj)t;Figure3:Uniformpointsamplesfromthesurfaceofanirismodelareshownontheleft.Thesamepointsafteranisotropicrescaling,areshownontheright.Thoughthepointsetontherighthasconstantvarianceineverydirection,itnolongerrepresentsauniformsamplingfromthesurfaceoftheanisotropicallyrescalediris.wherethedoublesummationistakeninordertoaccountforthevariancewithrespecttocenterofmass.IfwesetQtobethetransformedpointsetQ=C1=2Pp1;:::;C1=2Ppnthenthecovari-ancematrixofQisgivenby:CQ=nåi;j=1C1=2P(pipj)
(pipj)tC1=2P=C1=2P nåi;j=1(pipj)
(pipj)t!C1=2P=1Thus,thecovariancematrixofthetransformedpointsetisequaltotheidentity,andthevarianceinanydirectionisequalto1.Aswithisotropicrescaling,thisapproachhastheadvantagethatitcannormalizeforanisotropicrescaleonaper-modelbasis,allowingamodeltobetransformedinapre-processingstage,independentofthemodelthatitwillbematchedagainst.ThedifcultywithapplyingthismethoddirectlytotriangulatedmodelsisillustratedinFigure3whichshowspointsuniformlysam-pledfromamodelofaniris(left).Afterananisotropictransforma-tionisappliedtothepointset(right),thepositionsofthepointsaretransformedandtheynolongerrepresentauniformsamplingofthesurface.Notethatpointsonthestemaretightlyclustered,whilepointsonthepetalbecomemorespreadout.Thispropertyof3Dmeshesresultsintheundesiredpropertythatoftenthetransformedmodelisstillnotisotropic.Inordertoaddressthisissue,weproposeaniterativeapproachtotransformingthemodel.Ateachstepoftheiteration,themodelisrsttranslatedsothatitscenterofmassisattheorigin,thecovari-ancematrixiscomputed,andnallythemodelisrescaledbytheinversesquarerootofthecovariancematrix.Inourexperiments,wendthatthisapproachconvergesefcientlytoanisotropicmodeland,inpractice,nomorethanveiterationsofthisprocessarenec-essarytoobtainanearlyisotropicshape.Figure4showsamodelofapenandthetransformedmodelafterseveralstepsoftheitera-tionprocess.Thegurealsodrawstheassociatedcovarianceellip-soids,whichconvergetoasphereasthemodelbecomesisotropic.Notethataftertherstiteration,thetransformedmodelisstillnotisotropic,though,asthegureindicates,theiterativeprocesscon-vergesquicklytoanisotropicmodel.Weprovideaproofoftheconvergenceofthisapproachintheappendix. Figure4:Avisualizationofapenmodelanditscovarianceellipsoidisshownontheleft.Thetransformedmodelanditsassociatedcovarianceellipsoid,afterone,two,andthreeiterationsareshownontheright.Notethatthoughthemodelisveryanisotropic,afterthethirditerationofanisotropicrescalingweobtainamodelthatisnearlyisotropic,withthecovarianceellipsoidconvergingtoasphere.3.3AnisotropyFactoringThemethodthatweproposeforanisotropyfactoringisageneralonethatcanbeappliedtoanyofthemanymethods[Horn1984;KangandIkeuchi1991;Ankerstetal.1999;VranicandSaupe2001;Osadaetal.2001;Funkhouseretal.2003]thatmatchestwomodelsbyindependentlyrepresentingeachonebyafeaturevec-tor,andthendeningthemeasureofmodelsimilarityastheL2-differencebetweenthecorrespondingfeaturevectors.Inparticular,weanisotropicallyrescaleamodelMtoobtainanisotropicmodel˜M,storingthesortedtripletofeigenvalueslM=(lM1;lM2;lM3)ofthematrixtransformingMinto˜M.ThetripletlMisarotationin-variantrepresentationoftheanisotropyofMand,forsimplicity,wenormalizethetripletsothat\rlM\r=1.WecomputethefeaturevectorvÄMoftheisotropicmodeland,usingthefactthattheinfor-mationcontainedinvÄMisorthogonaltotheinformationcontainedinlM,werepresenttheinitialmodelMbythenewfeaturevectorvÄMlM,asshowninFigure5.Atruntime,whenaquerymodelQispresentedtothedatabase,wecomputetheanisotropyfactorizationofQanddenethemeasureofsimilaritybetweenQandadatabasemodelMtobethevalue:Dg(M;Q)=kvÄMk2+kvÄQk22hvÄM;vÄQihlM;lQig:Ifg=1thenDg(M;Q)istheL2-differencebetweenthevectorsvÄQlQandvÄMlM.Moregenerally,gcanbetreatedasaxedconstantrepresentingtheimportanceofanisotropyinforma-tioninthecontextofshapematching.Thus,inthecasethatg=0,anisotropyinformationplaysnoroleinthematchingandthematch-ingmethodisinvarianttoanisotropicscale.Ifadditionallythefea-turevectorisitselfrotationinvariant[Ankerstetal.1999;Osadaetal.2001;Funkhouseretal.2003],thenweobtainamatchingmethodthatisinvarianttoallafnetransformations.Figure5:Wecreateanewfeaturevectorforamodelbycomputingtheouterproductoftheanisotropicscaleswiththefeaturevectoroftheisotropicmodel.Theadvantageofthismatchingapproachisthattheshapemet-ricdenessimilarityastheouterproductofthesimilarityofthefeaturevectorsandthesimilarityoftheanisotropyvectors.Thus,thenewfeaturevectorsonlyneedtostorethreeadditionalval-ues,correspondingtothenormalizedeigenvaluesofthesymmet-ricmatrixtransformingananisotropicmodelintoanisotropicone.Thismeansthatneitherthestoragenorthecomparisontimeoftheanisotropyfactorizedfeaturevectorissignicantlylargerthanthecorrespondingstorageandcomparisontimefortheoriginalone.4ResultsTomeasuretheefcacyoftheanisotropicrescalingscaleapproachintasksofshaperetrieval,wecomputedanumberofshapede-scriptorsandcomparedmatchingresultswhenthedescriptoroftheoriginalanisotropicmodelwasusedwiththeresultsobtainedwithanisotropyfactoring.Thedescriptorsweusedwere:ShapeHistogram(Shells)[Ankerstetal.1999]:Arepresen-tationofa3Dmodelasahistogramofthedistancesofsurfacepointsfromthecenterofmass.D2[Osadaetal.2001]:Arepresentationofa3Dmodelasahistogramofthedistancesbetweenpairsofsurfacepoints.ExtendedGaussianImage[Horn1984]:Arepresentationofa3Dmodelasasphericalhistogramofthedistributionofnormaldirectionsoverthesurfaceofamodel.ShapeHistogram(Sectors)[Ankerstetal.1999]:Arepre-sentationofa3Dmodelasasphericalfunctionassociatingtoeachdirectionfromtheoriginthemeasureofthesurfaceareainthatdirection.SphericalExtentFunction[VranicandSaupe2001]:Arep-resentationofa3Dmodelasasphericalfunctionassociatingtoeachdirectionfromtheoriginthedistancetothelastpointofintersectionofthemodelwiththeray.GaussianEuclideanDistanceTransform[Funkhouseretal.2003]:Arepresentationofa3Dmodelasavoxelgrid,wherethevalueateachpointisgivenbythecompositionofaGaus-sianwiththeEuclideanDistanceTransformofthesurface.Thersttwoshapedescriptorsarerotation-invariantbydesign,whiletheotherfourcaneitherbenormalizedforrotationbyalign-ingthemodelwithPCAorcanbemaderotation-invariantusingthesphericalpowerspectrum[BurelandHenocq1995;Kazhdanetal.2003];wepresentmatchingresultsforbothtypesofapproaches.(Notethatsinceanisotropyfactorizationmakesthecovariancema-trixofamodelamultipleoftheidentity,PCAalignmentneedstobeperformedpriortothefactorization.)Weevaluatedtheperformanceofeachmethodbymeasuringhowwellitclassiedmodelswithinatestdatabase.ThedatabasewasprovidedbythePrincetonShapeBenchmark[PrincetonShapeBenchmark],andconsistsof1814modelsdecomposedintotwogroupsofroughly900models,correspondingtotrainingandtestdatasets.Eachgroupisprovidedwithaclassication,associatingeachofthemodelstooneofroughly90distinctclasses.Clas-sicationperformancewasmeasuredusingprecision/recallplots,whichgivethepercentageofretrievedinformationthatisrelevantasafunctionofthepercentageofrelevantinformationretrieved.Thatis,foreachtargetmodelinclassCandanynumberKoftopmatches,“recall”representstheratioofmodelsinclassCreturnedwithinthetopKmatches,while“precision”indicatestheratioofthetopKmatchesthatareinclassC.Thus,plotsthatappearshiftedupindicatesuperiorretrievalresults. Figure6:TheimprovementinprecisionofanisotropyfactorizationforfourPCA-alignedand®verotation-invariantrepresentations.Theplotsindicatethatwhentheimportanceofanisotropydifferencesisampli®ed(a=3)retrievalperformanceisimproved,andwhenanisotropydifferencesareignored(a=0)retrievalperformanceishampered.Foreachshapedescriptor,wecomparedtheprecisionversusre-callresultsobtainedusingthedescriptorappliedtotheoriginalmodelwiththoseobtainedwithanisotropyfactorization.FortheanisotropyfactorizationweusedthemetricsDgwithg=0todampentheimportanceofanisotropyinretrievalandg=3toam-plifytheimportanceofanisotropyinretrieval.Figure6showstheresultsforthedifferentPCA-alignedandrotation-invariantrepre-sentations,withprecisionversusrecallplotsaveragedoverthedif-ferentmodelsinthedatabase.Sinceweareprimarilyevaluatingtheeffectofanisotropyfactorizationonthematchingperformanceofagivendescriptor,theplotsshowtheimprovementinprecisionofanisotropyfactorizationovertheresultsobtainedwithoutfactoriza-tion.Theresultsindicatethattheanisotropicscaleofamodelisanessentialclassierofshapeandwhenitisignored(g=0),retrievalperformancedeteriorates.Ontheotherhand,theresultsalsoindi-catethatitiseasiertomatchtwomodelswhentheyareanisotropi-callyaligned,sothatmethodsthatcomparethetwomodelsintheiranisotropynormalizedframesandthenpenalizefordifferencesintheinitialanisotropicscales(g=3)giverisetomatchingresultswithimprovedprecision.Finally,wenotethatwhiletheresultsindicatethatanisotropyfac-torizationisatechniquethatworkswellonaverage,therearespe-cictypesofmodelsforwhichthismethodcanfail.Inparticular,wehavefoundthatwhentheinitialmodelisveryanisotropic(thedifferencebetweenthelargestandsmallestprincipaleigenvaluesislarge)anisotropyfactorizationdoesnotalwaysworkwell.Webe-lievethatthereasonforthisisduethemannerinwhichanisotropicrescalingactsonashape.Whenoneoftheprincipaleigenvaluesismarkedlysmallerthantheothers,anisotropynormalizationwillscalethemodeldisproportionatelyinoneprincipaldirection.Asaresult,featuresthatmayhavebeenunimportantintheinitialmodelwillplayamoredominantroleindeningmodelssimilarity.Asanexample,Figure7demonstratesthisforachesssetmodel.Whenthemodelisrescaledtohaveconstantvarianceineverydirection,thesideoftheboardandthechesspiecesbecomemoreprominentfeaturesinthemodelandplayamoredominantroleinmatching.5ConclusionInthispaperwehavedescribedamethodforfactoringtheshapematchingequationintotheproductofananisotropycomparisonandageometriccomparison.Thisfactorizationprovidesafam-ilyofshapemetricsthatallowsuserstospecifytheimportanceofanisotropywithinthegeneralcontextofshapesimilarity.Wehaveshownthatthismethodissufcientlygeneraltobeappliedtoawidecollectionofexistingshapematchingalgorithms,improvingFigure7:Whenamodelisnormalizedforanisotropy,partsthatmayhavebeenunimportantintheinitialmodelmaybecomemorepronouncedfeaturesofthemodel,adverselyaffectingshapematching.thematchingperformanceofmostwithoutinducingalargeover-headincomputationtime.Thus,thedescribedfactorizationiswellsuitedformanyofthenascentapplicationsthatstrivetoprovideamethodforefcientlyandeffectivelyretrievingmodelsfromlargerepositoriesof3Dshapes.AcknowledgementsWewouldliketothankViewpoint,Cacheforce,andJoseMariaDeEspona,whodonatedcommercialdatabasesofpolygonalmod-elsforexperiments.TheNationalScienceFoundationfundedthisprojectundergrantsCCR-0093343,IIS-0121446,andDGE-9972930.ALT,H.,ANDGUIBAS,L.J.1996.Discretegeometricshapes:Matching,interpolation,andapproximation:Asurvey.Tech.Rep.B96-11,EVL-1996-142,InstituteofComputerScience,FreieUniversit¨atBerlin.ANKERST,M.,KASTENM¨ULLER,G.,KRIEGEL,H.,ANDSEIDL,T.1999.3dshapehistogramsforsimilaritysearchandclassi-cationinspatialdatabases.InAdvancesinSpatialDatabases,6thInternationalSymposium,207–226.BESL,P.,ANDMCKAY,N.1992.Amethodforregistrationof3-dshapes.IEEEPAMI14,239–256. 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(pq)tdpdqsothatthevarianceofMinadirectionvisgivenby:Var(M;v)=vtCMv=ZMZMhpq;vi2dpdq:Wewillrstshowthatamodelwithnon-zerovarianceinanydi-rectioncanalwaysberescaledsothattheminimumandmaximumeigenvaluesarereciprocals.Next,weprovetwolemmasdescribingthedecompositionMintoanevenpartitionandthecorrespondingdecompositionofthevarianceofMacrosssuchapartition.Finally,weusethelemmastoshowthattheextremaleigenvaluesmustcon-vergeto1.IsotropicRescaling:GivenamodelMandscalefactors,theco-variancematrixofsMisdenedas:CsM=ZsMZsM(pq)
(pq)tdpdq=s6CM:Thus,givenamodelMwhosecovariancematrixCMhaseigenval-ues0l1l2l3,wecanrescalethemodelby(pl1
l3)1=6toobtainanewmodelwhosecovariancematrixhasasitssmallestandlargesteigenvaluesthereciprocalspl1=l3andpl3=l1.Lemma1:GivenacontinuousfunctionfdenedonM,thereexistsanevenpartitionofMintosubsetsM+andMandavaluemsuchthatjM+j=jMjandf(p+)mf(p)forallp+2M+andallp2M.Proof:Toprovethatsuchadecompositionmustexist,wedenethefunctionF(t)thatgivestheareaofthesubsetofMwithvaluelessthanorequaltot:F(t)=f1(¥;t]ThenF(t)isanon-decreasing,right-continuousfunctionthatstartsat0andgrowstojMj,andisdiscontinuousatpointst0suchthatjf1(t0)j�0.SetFtobetheclosureofthesetofvaluestforwhichF(t)jMj=2.SinceF(t)ismonotonicweknowthatF=(¥;m],forsomevaluem.ThenforalltmwehaveF(t)jMj=2andforalltmwehaveF(t)jMj=2.IfF(m)=jM=2jwecansetMequaltotheinverseimageoffontherange(¥;m],andwecansetM+=MM.OtherwisethefunctionF(t)isdiscontinuousatmandwemusthavef1(m)F(m)jMj=2.ThuswecansetM+tobetheunionoftheinverseimageoffontherange(m;¥)andanysubsetoff1(m)thathasareaF(m)jMj=2Lemma2:GivenapartitionofMintoequalsizedsubsetsM+andM,thevarianceacrossM+andMisatleastaslargeashalfthe variancewithinM+andhalfthevariancewithinM.Thatis,if:I++M(v)=ZM+ZM+hpq;vi2dpdqI+M(v)=ZM+ZMhpq;vi2dpdqIM(v)=ZMZMhpq;vi2dpdqthenwemusthave:I++M(v)2I+M(v)andIM(v)2I+M(v):Proof:WeshowthatIM(v)2I+M(v),byintegratingI+M(v)overMandusingthetriangleinequality.2I+M(v)=2ZM+ZMhpq;vi2dpdq=1jMjZMZMZM+hpp+;vi2++hqp+;vi2dp+dpdqBythetriangleinequality,weknowthat:hpp+;vi2+hqp+;vi2hpq;vi2sothat:2I+M(v)1jMjZMZMZM+hpq;vi2dp+dpdq=jM+jjMjZMZMhpq;vi2dpdq=IM(v)asdesired.TheproofforI++M(v)2I+M(v)isanalogous.AnisotropicRescaling:GivenamodelM,wesetCMtobethecovariancematrixofM,0l1l2l3=1=l1tobetheeigen-valuesofCM,andBM=C1=2MtobetheinversesquarerootofCM.ApplyingBMtothemodelM,weobtainamodelwhosevarianceindirectionvisgivenby:Var(BM(M);v)=ZBM(M)ZBM(M)hpq;vi2dpdq(1)=ZMZMhpq;BM(v)i2b(p)b(q)dpdqwhereb(p)isthedifferentialchangeofareaatthepointMandmustsatisfy:pl1pl2b(p)1pl1
l2:UsingthefactthateachsummandinEquation1ispositive,wecanapplytheaboveinequalitiestoget:ZMZMhpq;BM(v)i2dpdql1l2Var(BM(M);v)Var(BM(M);v)ZMZMhpq;BM(v)i2dpdq1l1
l2sothatl1l2Var(BM(M);v)1l1
l2:Weobservethatwhenwerescalethemodelsothatminimalandmaximaleigenvaluesofthecovariancematrixarereciprocals,theminimaleigenvalueisnosmallerthanl1sothattransformingMbyBMcannotmaketheminimalvariancesmaller,norcanitmakethemaximalvariancelarger.Toshowthattheminimalandmaximaleigenvaluesmustactuallyconvergeto1,weusethelemmasabove.Todothis,weusethefunctionb(p)andLemma1toevenlypartitionMintoM+andMandobtainavaluemsatisfying:sl1l2b(p)mb(p+)s1l1
l2forallp2Mandp+2M+.(Thoughb(p)isnotcontinuousonM,itisonlydiscontinuousonaclosedsubsetwith0area,soLemma1stillholds.)ExpressingthevarianceofBM(M)inthedirectionvintermsofthispartitionweget:Var(BM(M);v)=ZMZMhpq;BM(v)i2b(p)b(q)dpdq+2ZMZM+hpp+;BM(v)i2b(p)b(q+)dpdp++ZM+ZM+hp+q+;BM(v)i2b(p+)b(q+)dpdp+Thisallowsustoboundtheminimalvarianceby:Var(BM(M);v)IM(BM(v))l1l2+2I+M(BM(v))msl1l2+I++M(BM(v))m2:SinceIM(BM(v))+2I+M(BM(v))+I++M(BM(v))=1,sincepl1=l2m,andsince2I+M(BM(v))IM(BM(v)),itfollowsthattheminimumvarianceisboundedby:l1=l2+mpl1=l22Var(BM(M);v):Inasimilarmannerwecangetanupperboundforthevariance:l1l2+mrl1l22Var(BM(M);v)1l1
l2+mq1l1
l22IsotropicallyrescalingBM(M)togetamodel˜Mwithminimalandmaximalvariancesthatarereciprocals,weget:l1pf(m)Var(˜M;v)withf(t)=1+tpl2=l11+tpl2
l1:Inordertondtheminimumofthevariance,wecomputethederivative:f0(t)=pl2=l1pl2
l1(1+tpl2
l1)2:Sincel11,thederivativeisnevernegative,andhencethevari-anceof˜Misminimizedwhenmisassmallascanbe,whichistosaym=pl1=l2.Inthiscaseweget:l1s21+l1Var(˜M;v)1l1r1+l12:Thus,theminimalandmaximalvariancesofthemodelareguar-anteedtoconvergeto1,andtheiterativemethoddescribedinSec-tion3isguaranteedtoconvergetoamodelwithvariance1ineverydirection.