Rustamov Purdue University West Lafayette IN Abstract A deformation invariant representation of surfaces the GPS embedding is introduced using the eigenvalues and eigenfunctions of the LaplaceBeltrami differential operator Notably since the de64257n ID: 24011
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R.M.Rustamov/Laplace-BeltramiShapeRepresentation AscalarlforwhichtheequationhasanontrivialsolutioniscalledaneigenvalueofD;thesolutionfiscalledaneigen-functioncorrespondingtol.Notethatl=0isalwaysaneigenvalue–thecorrespondingeigenfunctionsareconstantfunctions.TheeigenvaluesoftheLaplace-Beltramioperatorarenon-negativeandconstituteadiscreteset.Wewillassumethattheeigenvaluesaredistinct,sowecanputthemintoas-cendingorderl0=0l1l2:::li:::Theappropriatelynormalizedeigenfunctioncorrespondingtoliwillbedenotedbyfi.ThenormalizationisachievedusingtheL2innerproduct.Giventwofunctionsfandgonthesurface,theirinnerproductisdenotedbyhf;gi,andisdenedasthesurfaceintegralhf;gi=ZSfg:Thus,werequirethathfi;fii=1.SincetheLaplace-BeltramioperatorisHermitian,theeigenfunctionscorrespondingtoitsdifferenteigenvaluesareorthogonal:hfi;fji=ZSfifj=0;wheneveri6=j.Givenafunctionfonthesurface,onecanexpanditintermsoftheeigenfunctionsf=c0f0+c1f1+c2f2+;wherethecoefcientsareci=hf;fii=ZSffi:Thus,eigenfunctionsofthecontinuousLaplace-Beltramioperatorgiveanorthogonalbasisforthespaceoffunctionsdenedonthesurface.Lévy'smainpointin[ Lév06 ]isthatthisbasisistheone;theexpansioncoefcientsprovideacanonicalparametrizationoffunctionsdenedonthesur-face,andallgeometryprocessingshouldbecarriedoutinthiscoefcientdomain.Forexample,tosmoothafunctiononeshouldsimplydiscardthecoefcientscorrespondingtothelargereigenvalues,i.e.truncatetheinniteexpansionabove.WhethertheLaplace-Beltramieigenbasisistheoneornot,itstillwouldberelevantfordeformableshapematching–thisisbecauseofLaplace-Beltrami'sisometryinvariance.Perhaps,anotherfactortosingleouttheLaplace-Beltramioperatoramongtheinnitudeofdifferentialoperatorswouldbeits“simplicity”andwell-studiedness.4.GlobalPointSignaturesCanweintrinsicallycharacterizeapointonasurface–de-scribeitslocationwithoutreferringtoanexternalcoordinate system?LetusremindthereaderthattheLaplace-Beltramioperatoranditseigenfunctionsareintrinsicinthatsense:thevaluesofeigenfunctionscanbethoughtasnumbersattachedtothepointsonthesurface,thesenumbersdonotdependonhowthesurfaceislocatedinCartesiancoordinates.Thus,itisnaturaltotrytocharacterizethepointsbythevaluesoftheeigenfunctions.Givenapointponthesurface,wedeneitsGlobalPointSignature,GPS(p),astheinnite-dimensionalvectorGPS(p)= 1 p l1f1(p);1 p l2f2(p);1 p l3f3(p);:::!;wherefi(p)isthevalueoftheeigenfunctionfiatthepointp.Noticethatf0isleftoutbecauseitisvoidofinforma-tion.Thenamecomesfromtheintuitionthatthisinnite-dimensionalvectorisasignature,acharacterizationofthepointwithintheglobal“context”ofthesurface.Ourmotiva-tionfornormalizingbytheinverserootofeigenvalueswillbeprovidedshortly.GPScanbefurtherconsideredasamappingofthesurfaceintoinnitedimensionalspace.TheimageofthismapwillbecalledtheGPSembeddingofthesurface.WewillrefertotheinnitedimensionalambientspaceofthisembeddingastheGPSdomain.LetuslistsomeofthepropertiesoftheGPSembedding.First,theGPSembeddingofasurfacewithoutself-intersectionshasnoself-intersectionseither.WeneedtoprovethatdistinctpointshavedistinctimagesundertheGPS.Tothisend,supposethatfortwosurfacepointsp6=qwehaveGPS(p)=GPS(q).Thismeansthattheeigenfunc-tionsoftheLaplace-Beltramioperatorsatisfytheequalityfi(p)=fi(q).Givenanyfunctionfonthesurface,consideritsexpansionintermsoftheeigenfunctions.Undersomemildconditions,theexpansionwillconvergetofpointwise;consequently,theequalityf(p)=f(q)willhold.However,onecaneasilyimaginea“nice”functionthattakesdistinctvaluesatthosetwopoints–acontradictionmeaningthattheGPSsmusthavebeendifferent.Second,GPSembeddingisanisometryinvariant.Thismeansthattwoisometricsurfaceswillhavethesameim-ageundertheGPSmapping.Indeed,theLaplace-Beltramioperatorisdenedcompletelyintermsofthemetricten-sor,whichisitselfanisometryinvariant.Consequently,theeigenvaluesandeigenfunctionsofisometricsurfacescoin-cide,i.e.theirGPSembeddingsalsocoincide.Third,giventheGPSembeddingandtheeigenvalues,onecanrecoverthesurfaceuptoisometry.Infact,eigenvaluesandeigenvectorsoftheLaplace-Beltramioperatoruniquelydeterminethemetrictensor.Thisstemsfromcompletenessofeigenfunctions,whichimpliestheknowledgeofLaplace-Beltrami,fromwhichoneimmediatelyrecoversthemetrictensor[ Ber03 ],andso,theisometryclassofthesurface.Fourth,theGPSembeddingisabsolute,itisnotsubjectto c TheEurographicsAssociation2007. R.M.Rustamov/Laplace-BeltramiShapeRepresentation 7.SummaryandfutureworkWehavedescribedanewframeworktorepresentnon-rigidshapes.OurmaincontributionistointroducetheGPSem-beddingasameansofrippingasurfacefromits“transient”,Euclideanembeddingrelatedproperties,tokeepitsessence–featuresthatareisometryinvariant.TodemonstratethepracticalrelevanceoftheGPSembed-dingweintroducedG2-distributionsasshapedescriptors,andhaveconductedinitialstudiesoftheirdiscriminativepower,robustnesstolocalshapechanges,includingtopol-ogymodications,anddeformationindependence.Weplantoperformlarge-scaleexperimentstofurtherunderstandthepropertiesoftheGPSembeddingbasedsignatures.ThemaindrawbackoftheLaplace-Beltramiframeworkisitsinabilitytodealwithdegeneratemeshes.Wedidnotmen-tionsurfaceswithboundariesneither.However,wethinkthatoneshouldbeabletohandlethembyimposingappro-priateboundaryconditions.Weshouldalsomentionthatinpracticetherearetwoproblemswhileworkingwitheigenvaluesandeigenvectorsingeneral[ JZ06 ]:thesignsofeigenvectorsareundened,andtwoeigenvectorsmaybeswapped.Usingd2distribu-tionsindirectlyaddressesbothoftheseissues.Furtheranal-ysisisneededtoclarifytheconsequencesofthesefactorsforshapeprocessingwhentheGPSembeddingisuseddirectly.Apartfromshapeclassication,weexpecttheGPSem-beddingtoberelevantinthecontextofshapecorrespon-denceandsegmentation.Figure 1 showsapreliminaryresultfromoursegmentationexperimentsusingtheGPS.Simplek-meansclusteringbasedondistancesintheGPSdomainwasperformedtosegmenttheArmadillosintosixpatches,nofurtheroptimizationhasbeendone.Theguredemon-stratespose-obliviousnatureofsuchsegmentation.Thesere-sultsandapplicationsdescribedin[ Lév06 , VL07 ]raisehopethatLaplace-Beltramieigenfunctionswillprovidetogeome-tryprocessingwhatFourierbasishasprovidedtosignalpro-cessing.8.AcknowledgementsAllofthemodelsexcepttheDinopetandthesphereweredownloadedfromAIM@SHAPEShapeRepository.SevendeformationsofArmadilloobtainedbymethodsof[ YBS06 ]arecourtesyofShinYoshizawa.TherestofthemodelsarecourtesyofINRIA.Iamdeeplygratefultotheanonymousreviewersfortheirdetailedandusefulcommentsthathelpedimprovethearticleimmensely.References [BBK07] BRONSTEINA.M.,BRONSTEINM.M.,KIM-MELR.:Jointintrinsicandextrinsicsimilarityforrecog- nitionofnon-rigidshapes.Tech.Rep.CIS-2007-01-2007,ComputerScienceDepartment,Technion,March2007. 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