In this work we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis In particular we consider a physical model that describes the vibration modes and frequencies of a surfac ID: 24015
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2K.Hildebrandt,C.Schulz,C.vonTycowicz,K.PolthiereigenmodesoftheHessianofquadraticenergiesthataredenedonaspaceoffunctionsonasurface.Onaplanardomain,theeigenfunctionsoftheLaplacianserveasamodelforthevibrationmodesofa atplate(Chladniplates).Forcurvedsurfacesmoreelaboratemodelsarerequiredtodescribethevibrationmodesofasurface.WeconsideraphysicalmodelthatdescribesvibrationmodesofasurfacemeshthroughtheeigenfunctionsoftheHessianofadeformationenergy.Ingeneral,computingtheHessianofadeformationenergyisadelicateandlaborioustask.But,tocomputethevibrationmodeswedonotneedtocomputetheHessianatallpointsinthespaceofpossiblesurfaces,butonlyatthepointthatrepresentsthereferencesurface.Wederiveasimpleformula,thatcanbeusedtocomputetheHessianatthereferencesurfaceforageneralclassofdeformationenergies.Wehopethatthisframeworkwillstimulatefurtherexplorationoftheeigenmodesandeigenfrequenciesofdeformationenergies.TheDirichletenergyofasurfaceisaquadraticfunctionalonanappropri-atespaceoffunctionsonasurface.TheHessianofthisenergyistheLaplace-Beltramioperatorofthesurface.WeproposeaquadraticfunctionalthatcanbederivedfromtheDirichletenergy,butisnotintrinsic.Theeigenfunctionsofthisenergyaresensitivetotheextrinsiccurvatureofthesurface.Wediscussthreeapplicationsthatusetheproposedeigenmodesandspectra.Wedenetwo(multi-scale)signatures,thevibrationsignature,basedonthevibrationmodes,andthefeaturesignature,basedontheeigenmodesofthemodiedDirichletenergy.Toeachofthetwosignaturesweassociatea(multi-scale)pseudo-metriconthesurface.Theresultingvibrationdistancecanbeusedasasimilaritymeasureonthesurfaceandthefeaturedistancecanidentifyfeaturesofamesh.Furthermore,wetestthespectralsurfacequadrangulationmethodofDongetal.[6]withspecicvibrationmodes,insteadofeigenfunctionsoftheLaplacian.Theresultingquadrangulation,inouropinion,alignsbetterwiththeextrinsiccurvatureofthesurface.Relatedwork.Recently,wehaveseenaboomofpapersthatusetheeigen-valuesandeigenfunctionsoftheLaplace-Beltramioperatorasaningredienttoalgorithmsingeometryprocessingandshapeanalysis.Anoverviewofthisde-velopmentcanbefoundintherecentsurveybyZhangetal.[29]andinthecoursenotesofaSiggraphAsia2009courseheldbyLevyandZhang[16].Here,wecanonlybrie youtlinetheworkthathasbeenmostrelevantforthispaper.ThespectrumoftheLaplace-BeltramioperatorofaRiemannianmanifoldcontainsasignicantamountofinformationaboutthemanifoldandthemetric.ThoughitdoesnotfullydeterminetheRiemannianmanifold,itcanbeusedasapowerfulshapedescriptorofaclassofisometricRiemannianmanifolds.Reuteretal.[21,22]usethespectrumoftheLaplace-Beltramioperatortoconstructangerprintofsurfaces,whichtheycalltheShape-DNA.Byconstructionthisngerprintisinvariantunderisometricdeformationsofasurface.AmongotherapplicationstheShape-DNAcanbeusedforshapematching,copyrightprotec-tion,anddatabaseretrieval.Rustamov[23]developedtheGlobalPointSignature(GPS),asignaturethatcanbeusedtoclassifyshapesuptoisometry.Based Eigenmodesofsurfaceenergiesforshapeanalysis5 Fig.2.Twoeigenmodesofthelowerspectrumonthedoubletoruswithsharpfeatures,left:Laplacian,andright:modiedDirichletenergy.whereeiisthedihedralangleattheedgeei,Aeiisthecombinedareaofthetwotrianglesincidenttoeiandkeikisthelengthoftheedge.Thequantitieskeik;Aei;andeiaremeasuredonthereferencemesh.Towritethis exuralenergyinthegeneralform(1)wesetfi=eiand!i=3keik2 Aei:Themembraneenergyconsistsoftwoterms:onemeasuringthestretchingoftheedges,EL=1 2Xi1 keik(keik�keik)2;(3)andonemeasuringthechangeofthetriangleareasAiEA=1 2Xi1 Ai(Ai�Ai)2:(4)Herethesecondsumrunsoverthetrianglesofthemesh.WecandescribeELinthegeneralform(1)bysettingfi=keikand!i=1 keik;andtodescribeEAwesetfi=Aiand!i=1 Ai:3ModesofDeformationEnergiesModalanalysisprovideswaystocomputethemodesofasurfacewithrespecttoadeformationenergy.Toinspectthemodesofamesh,givenbya3n-vectorx,weconsideradeformationenergyE(x)thathasxasareferencesurface.Then,weareinterestedintheeigenvaluesandeigenmodesoftheHessianofthedefor-mationenergyEatthemeshx2X. 6K.Hildebrandt,C.Schulz,C.vonTycowicz,K.PolthierTheHessianofadeformationenergy(ormoregeneralofafunction)doesnotdependsolelyonthedi erentiablestructureofX,butalsoonthemetriconX,hencebelongstoRiemanniangeometry.Therefore,beforeconsideringtheHessianofEweequipXwithametric.SinceXequalsR3n;thetangentspaceTxXatameshxcanbeidentiedwithR3n.WecaninterpretanelementofTxXasavectoreldonx,thatassignsavectorinR3toeveryvertexofx.Then,anaturalchoiceofascalarproductonTxXisadiscreteL2-product,e.g.,themassmatrixusedinFEM[28]orthediscreteL2-productusedinDEC[5,27].Wedenotethematrix,thatdescribesthescalarproductonTxXbyMx.Forcompleteness,wewouldliketomentionthatifxisameshthathasdegeneratetriangles,thediscreteL2-productonTxXmaybeonlypositivesemi-denite.However,awayfromtheclosedsetofmeshesthathaveatleastonedegeneratetriangle,XequippedwiththediscreteL2-productisaRiemannianmanifold.Wedenoteby@Exthe3n-vectorcontainingtherstpartialderivativesofEatxandby@2Exthematrixcontainingthesecondpartialderivativesatx.Wewouldliketoemphasizethat@Exand@2ExdonotdependonthemetriconX,whereasthegradientandtheHessianofEdo.ThegradientofEatxisgivenbygradxE=M�1x@Ex:(5)TheHessianofEatameshxistheself-adjointoperatorthatmapsanytangentvectorv2TxXtothetangentvectorhessxE(v)2TxXgivenbyhessxE(v)=rvgradxE;(6)whereristhecovariantderivativeofX.Hessiancomputation.Ingeneral,itisadelicatetasktoderiveanexplicitrepresentationoftheHessianofadeformationenergyandoftenonlyapproxi-mationsoftheHessianareavailable.Here,wederiveasimpleexplicitformulafortheHessianofadeformationinthegeneralform(1)atthepointx,whichinvolvesonlyrstderivativesofthefi's.SincethegradientofEvanishesatx,onecanshowthatatxtheHessianofEtakesthefollowingformhessxE=M�1x@2Ex:Hence,atxwedonotneedderivativesofthemetrictocomputehessxE.Fur-thermore,tocomputethesecondpartialderivativesofEatxwedonotneedtocalculatesecondderivatives,butweonlyneedtherstderivativesofthefi's.Wepresentanexplicitformulafor@2ExinthefollowingLemma.Lemma1(ExplicitHessian).LetEbeadeformationenergyoftheform(1).Then,thematrix@2ExcontainingthesecondderivativesofEatxhastheform@2Ex=Xi!i(x)@fix@fixT;(7)where@fixTdenotesthetransposeofthevector@fix. Eigenmodesofsurfaceenergiesforshapeanalysis9 Fig.4.VisualizationofmodesoftheLaplacian(threeimagesontheleft)andmodesofthethinshellenergyrestrictedtonormalvariations(threeimagesontheright).where'isafunctiononthemesh.Thecorrespondingeigenvalueproblemis@2ENx'=Mx';whereMxisthemassmatrixthatrepresentsthediscreteL2-productoffunctionsonthemeshx.4QuadraticEnergiesInadditiontoenergiesdenedonthespaceofmeshesX,weconsiderenergiesthataredenedonanappropriatespaceoffunctionsonasurface.Inparticular,weconsidertheDirichletenergy,thatonacompactsmoothsurfaceisdenedforweaklydi erentiablefunctions':!R(thatvanishattheboundaryof)byE(')=1 2Zkgrad'k2dA:(13)Thisisanintrinsicenergy,therefore,theenergyanditseigenmodesdonotchangeunderisometricdeformationsofthesurface.Forsomeapplicationsthisisadesiredfeature,forotherapplicationsitisnot.BymodifyingtheDirichletenergy,weconstructanewenergythatisextrinsic.AssumethatisanorientablesurfaceinR3andletdenotethenormalof.Then,allthreecoordinateskofaresmoothfunctionsandforaweaklydi erentiablefunction'theproduct'kisweaklydi erentiable.WedeneEN(')=3Xk=1E('k):(14) 10K.Hildebrandt,C.Schulz,C.vonTycowicz,K.PolthierThisenergysatisestheequationEN(')=E(')+1 2Z'2(21+22)dA;(15)where1and2aretheprincipalcurvaturesof.ThismeansthatEND(')isthesumoftheDirichletenergyof'andthe'2-weightedtotalcurvatureof.Discreteenergies.Inthediscretesetting,weconsiderameshx2XandthespaceFxofcontinuousfunctionsu:x!Rthatarelinearineverytriangle.Suchafunctionisdi erentiableintheinteriorofeverytriangle,andhenceonecandirectlyevaluatetheintegral(13).ForarigoroustreatmentofthisdiscreteDirichletenergyandaconvergenceanalysissee[7,13].AfunctioninFxisalreadydeterminedbyitsfunctionvaluesattheverticesofx,henceitcanberepresentedbyann-vector.Then,thediscreteDirichletenergyEDisaquadraticfunctionalonFx,andforafunctionu2Fx(representedbyann-vector)ED(u)isexplicitlygivenbyED(u)=1 2uTSu;(16)whereSistheusualcotan-matrix,see[19,28].TodiscretizetheenergyENwexanormaldirectionateveryvertexofthemesh,andwedenotetheorientedunitnormalvectoratavertexvibyN(vi).Then,wesayacontinuousandpiecewiselinearvectoreldVonxisanormalvectoreldifforeveryvertexviofxthevectorV(vi)isparalleltoN(vi).Thespaceofnormalvectoreldsonxisann-dimensionalvectorspaceandthemapthatmapsafunctionu2FxtothenormalvariationVu,givenbyVu(vi)=u(vi)N(vi)forallvi2x,isalinearisomorphism.ThethreecoordinatefunctionsVkuofVuarefunctionsinFxandwedenethediscreteenergyENDanalogtoeq.(14)byEND(u)=3Xk=1ED(Vku):AsimplecalculationshowsthattheenergyENDsatisesEND(u)=1 2uTAu;(17)wheretheformulaAij=hN(vi);N(vj)iSijrelatestheentriesAijofthematrixAtotheentriesSijofthecotan-matrixS.ThecomputationoftheHessianoftheDirichletenergyEDandtheenergyENDissimple.Bothenergiesarequadratic,thereforetheHessianisconstantandthematrices@2EDand@2ENDarethematricesSandAgiveninequations(16)and(17).ThemetricweconsideronFxisthediscreteL2-productgivenbythemassmatrixM.TheeigenvaluesandeigenfunctionsoftheHessianofED(resp.END)satisfythegeneralizedeigenvalueproblemS =M (resp. 12K.Hildebrandt,C.Schulz,C.vonTycowicz,K.PolthierThesignaturesweconsideraremulti-scalesignatures,whichtakeapositivescaleparametertasinput.Foreverytsuchasignatureisafunctiononthemesh,i.e.,itassociatesarealvaluetoeveryvertexofthemesh.Letvbeavertexofameshxandlettbeapositivevalue.Then,wedenethevibrationsignatureofxatvertexvandscaletbySVibt(v)=Xje�jtkj(v)k2;(18)wherejandjdenotetheeigenvaluesandtheL2-normalizedvector-valuedvibrationmodesofameshx.Thevaluekj(v)kdescribesthedisplacementofthevertexvundertheL2-normalizedvibrationmodej.Foraxedtthevi-brationsignatureofvmeasuresaweightedaveragedisplacementofthevertexoverallvibrationmodes,wheretheweightofthejtheigenmodeise�jt.Theweightsdependontheeigenvaluesandonthescaleparameter.Forincreasing;thefunctione�trapidlydecreases,e.g.,themodeswithsmallereigenvaluereceivehigherweightsthanthemodeswithlargeeigenvalues.Furthermore,forincreasingtallweightsdecrease,and,moreimportantly,theweightsofthevi-brationmodeswithsmallereigenvaluesincreasesrelativetotheweightsofthemodeswithlargereigenvalues.Thefeaturesignatureisconstructedinasimilarmanner,butitusestheeigenmodesandeigenvaluesofthemodiedDirichletenergyEND.WedeneSFeatt(v)=Xje�jt j(v)2(19)wherethejaretheeigenvaluesandthe j(v)aretheL2-normalizedeigenmodesoftheHessianofthemodieddiscreteDirichletenergyEND.Multi-scaledistances.Fromeachofthetwosignatureswecanconstructthefollowing(multi-scale)pseudo-metriconthemesh:letv,~vbeverticesofthemeshx,thenwedene[t1;t2](v;~v)= Zt2t1(St(v)�St(~v))2 Pke�ktdlogt!1 2:(20)Byconstruction,foranypairofscalevaluest1t2,[t1;t2]ispositivesemi-deniteandsymmetric,andonecanshowthatitsatisesthetriangleinequal-ity.Wecallthepseudo-metricsconstructedfromSVibtandSFeattthevibrationdistanceandthefeaturedistance.Theideabehindtheconstructionofthepseudo-metricistousetheintegralRt2t1(St(v)�St(~v))2dt.However,theactualdenitionadditionallyincludestwoheuristics.First,sinceforincreasingtthevaluesSt(v)decreasesforallv,wenormalizethevalue(St(v)�St(~v))2bydividingitbythediscreteL1-normofSt,kStkL1=Pke�kt: 14K.Hildebrandt,C.Schulz,C.vonTycowicz,K.Polthier vFig.7.Comparisonoftwosimilaritymeasures.Distancetovertexvinbinaryaswellascontinuouscoloringbasedonourvibrationsignature(leftmost)andtheheatkernelsignature(rightmost).Spectralzoo.WecomparetheeigenmodesoftheLaplaciantotheonesofthemodiedDirichletenergyENDandtothevibrationmodesofthethinshellenergyrestrictedtonormalvariations.Toconveyanimpressionofthecharacteristicsofthemodesofthedi erentenergies,weshowsomeexamplesinFigures1,2and4.Tovisualizethemodesweusebluecolorforpositivevalues,whiteforzerocrossings,andorangefornegativevalues.Additionally,wedrawisolinesasblacklines.Asarstexample,westudyhowtheeigenmodeschangewhenweisometri-callydeforma atplate,seeFig.1.Ontheundeformed atplate,theeigenmodesofENDequaltheeigenmodesoftheLaplacian.AsshowninFig.1,therearecer-taindi erencesbetweenthethreetypesofconsideredmodeswhencomputedonthedeformedplate.DuetoitsintrinsicnaturetheLaplacianeigenmodesignorethenewlyintroducedfeature,Fig.1left.Incontrast,theeigenmodesofENDandthevibrationmodesaresensitivetothefeature,Fig.1middleandright.TheeigenmodesofENDcorrespondingtolowereigenvaluesalmostvanishatthefeatureandthevibrationmodesplaceadditionalextremaonthefold.Investigatingthedi erencesbetweentheeigenmodesoftheLaplacianandENDfurther,wecomputethemonthedoubletoruswithsharpfeaturesshowninFig.2.ItcanbeseenthateachoftheshownLaplacianeigenmodescontainsamoreorlessequallydistributedsetofextremaaswellasacertainre ectionsymmetry,Fig.2left.Thecorrespondingisolinessuggestalowin uenceofthesharpfeaturestotheconsideredLaplacianeigenmodes.SimilartotheLaplacianmodesthetwoeigenmodesforENDalsopossesare ectionsymmetry,Fig.2right.Butherewendthattheeigenmodesofthelowerpartofthespectrumcorrespondtooscillationsof atareassurroundedbysharpedges,Fig.2right.ThismatchesourconsiderationsinSection4.Forathirdcomparison,wechooseamodelwithoutsharpedges,thedancer(25kvertices).WecomparetheeigenmodesoftheLaplaciantothemodesofthethinshellenergyrestrictedtonormalvariations,seeFig.4.AsinthecaseofthetoruswenoticethattheLaplacianeigenmodesoscillateequallyoverthewhole