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To appear at the Eurographics Symposium on Geometry Processing Alexander Belyae

We argue that de64257ning a modeling operation by asking for rigidity of the local transformations is useful in various settings Such formulation leads to a nonlinear yet conceptually simple energy formulation which is to be minimized by the deforme

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To appear at the Eurographics Symposium on Geometry Processing Alexander Belyae






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O.Sorkine&M.Alexa/As-Rigid-As-PossibleSurfaceModeling originalmeshuniformweightscotanweightsuniformweightscotanweightsFigure2:Demonstrationoftheimportanceofproperedgeweightingintheenergyformulation( 3 ).Deformationusinguniformweighting(wij=1)leadstoasymmetricalresults,whereascotangentweightingenablestoeliminatetheinuenceofthemeshingbias. LetusdenotebySithecovariancematrixSi=åj2N(i)wijeije0Tij=PiDiP0Ti;(5)whereDiisadiagonalmatrixcontainingtheweightswij,Piisthe3jN(vi)jcontainingeij'sasitscolumns,andsimilarlyforP0i.ItiswellknownthattherotationmatrixRimaximizingTr(RiSi)isobtainedwhenRiSiissymmetricpositivesemi-denite(ifMisapsdmatrixthenforanyor-thogonalR,Tr(M)�Tr(RM)).OnecanderiveRifromthesingularvaluedecompositionofSi=UiSiVTi:Ri=ViUTi;(6)uptochangingthesignofthecolumnofUicorrespondingtothesmallestsingularvalue,suchthatdet(Ri)�0.2.2.ThelocalrigidityenergyOursimpleideaformeasuringtherigidityofadeformationofthewholemeshistosumupoverthedeviationsfromrigiditypercell,asexpressedby( 3 ).Thus,weobtainthefollowingenergyfunctional:E�S0=nåi=1wiE(Ci;C0i)= (7) =nåi=1wiåj2N(i)wij �p0i�p0j�Ri(pi�pj) 2;wherewi,wijaresomexedcellandedgeweights.NotethatE�S0dependssolelyonthegeometriesofS,S0,i.e.,onthevertexpositionsp;p0.Inparticular,sincethereferencemesh(ourinputshape)isxed,theonlyvariablesinE�S0arethedeformedvertexpositionsp0i.ThisisbecausetheoptimalrotationsRiarewell-denedfunctionsofp0,aswasshownintheprevioussection.Thechoiceofper-edgeweightswijandper-cellweightswiisimportantformakingourdeformationenergyasmesh-independentaspossible,asdemonstratedinFigure 2 .Theweightsshouldcompensatefornon-uniformlyshapedcells andpreventdiscretizationbias.Wethereforeusethecotan-gentweightformulaforwij[ PP93 , MDSB03 ]:wij=1 2�cotaij+cotbij;whereaij;bijaretheanglesoppositeofthemeshedge(i;j)(foraboundaryedge,onlyonesuchangleexists).Wefur-thernotethatthedeviationfromrigidity,asdenedby( 3 ),isanintegratedquantity,sothatthecellenergyispropor-tionaltothecellarea,andwecansetwi=1.Analternativeexplanationforthiswouldbeusingthearea-correctededgeweightsw0ij=(1=Ai)wij,whereAiistheVoronoiareaofcellCi[ MDSB03 ],andthenalsosettingthecellweightstobetheVoronoiarea:w0i=Ai.Theareatermsimplycancelsout,andweareleftwiththesymmetriccotangentweightswij.3.ModelingframeworkInamodelingframeworkweneedtosolveforpositionsp0ofS0thatminimizeE(S0),undersomeuser-denedmodel-ingconstraints.Thismeanswedonotknowtherigidtrans-formationsfRigaprioriandthusneedtosolveforthemaswell.Therefore,werstinterpretE(S0)asafunctionofp0andfRigandinanymodelingsituationweseekthemini-mumenergyunderthevariationinbothsets.Tosolveforthenextlocalminimumenergystate(start-ingfromagiveninitialvectorofpositionsandrotations),weproposetouseasimplealternatingminimizationstrat-egy.Thismeans,foragivenxedsetofrigidtransforma-tions,wendpositionsp0thatminimizeE(S0).Then,wendtherigidtransformationsfRigthatminimizeE(S0)forthegivensetofpositionsp0.Wecontinuetheseinterleavediterationsuntilthelocalenergyminimumisreached.LetusrstlookhowtondoptimalrigidtransformationsfRigforagivensetofmodiedpositionsp0.Eachterminthesum( 7 )involvesonlytheper-cellrigidtransformationRi,i.e.,wecancomputeanoptimaltransformationforeachcellwithoutregardfortheothercellsandtheirrigidtrans-formations.Thus,weseekanRithatminimizesthepercellenergyin( 3 ).Thesolutiontothis,however,isdetailedinSection 2.1 ,namely,Equation( 6 ). c TheEurographicsAssociation2007. O.Sorkine&M.Alexa/As-Rigid-As-PossibleSurfaceModeling initialguess1iteration2iterationsinitialguess1iteration4iterationsFigure3:Successiveiterationsoftheas-rigid-as-possibleeditingmethod.TheinitialguessisthenaiveLaplacianeditingresult(asin[ LSCO04 ]butwithoutanylocalrotationestimation).TheoriginalstraightbarmodelisshowninFigure 6 . Inordertocomputeoptimalvertexpositionsfromgivenrotations,wecomputethegradientofE(S0)withrespecttothepositionsp0.Letuscomputethepartialderivativesw.r.t.p0i.NotethattheonlytermsinE(S0)whosederivativedoesnotvanisharethoseinvolvingverticesiandj2N(i):¶E(S0) ¶p0i=¶ ¶p0i åj2N(i)wij �p0i�p0j�Ri(pi�pj) 2++åj2N(i)wji �p0j�p0i�Rj(pj�pi) 2!==åj2N(i)2wij��p0i�p0j�Ri�pi�pj++åj2N(i)�2wji��p0j�p0i�Rj�pj�pi:Usingthefactthatwij=wji;wearriveat¶E(S0) ¶p0i=åj2N(i)4wij�p0i�p0j�1 2(Ri+Rj)�pi�pj:Settingthepartialderivativestozerow.r.t.eachp0iwearriveatthefollowingsparselinearsystemofequations:åj2N(i)wij�p0i�p0j=åj2N(i)wij 2�Ri+Rj�pi�pj:(8)Thelinearcombinationontheleft-handsideisnon-otherthanthediscreteLaplace-Beltramioperatorappliedtop0;thesystemofequationscanbecompactlywrittenasLp0=b;(9)wherebisann-vectorwhoseithrowcontainstheright-handsideexpressionfrom( 8 ).Wealsoneedtoincorporatethemodelingconstraintsintothissystem.Inthesimplestform,thosecanbeexpressedbysomexedpositionsp0j=ck;k2F;(10)whereFisthesetofindicesoftheconstrainedvertices.Theseconsistofstaticandhandlevertices,interactivelyma-nipulatedbytheuser.Incorporatingsuchconstraintsinto( 9 )simplymeanssubstitutingthecorrespondingvariables,ef-fectivelyerasingrespectiverowsandcolumnsfromLandupdatingtheright-handsidewiththevaluesck.NotethattherigidtransformationsfRigonlyinuencetheright-handsideofthesystem,whereasthesystemma-trixonlydependsontheinitialmesh.Thus,wecanemploy adirectsolver,andthesystemmatrixhastobefactoredonlyonceforminimizingE(S0).Moreover,sincep0consistsofthreecolumns(forthethreecoordinatefunctions),weonlyneedtoperformthreetimesback-substitutiontosolveforeachcoordinate,usingthesamennfactorization.SinceLissymmetricpositivedenite,thesparseCholeskyfac-torizationwithll-reducingreorderingisanefcientsolverchoice[ Tol03 ].Tosummarize,theoverallminimizationofE(S0)pro-ceedsasfollows.Firstlythecoefcientswijareprecomputedandthesystemmatrixof( 9 )ispre-factored.Givenaninitialguessp00,thelocalrotationsRiareestimated,asdescribedinSection 2.1 .Newpositionsp01areobtainedbysolving( 9 ),pluggingRiintotheright-handside.Thenfurtherminimiza-tionisperformedbyre-computinglocalrotationsandusingthemtodeneanewrighthand-sideforthelinearsystem,andsoon.Thisleadstoanefcientsolutionofthenon-linearproblemathand,sinceonlyback-substitutionsarenecessary.4.ResultsanddiscussionWehaveimplementedtheas-rigid-as-possibledeformationtechniqueusingC++onaPentium42.16GHzlaptopwith2GBRAM.WeusedthesparseCholeskysolverprovidedwiththeTAUCSlibrary[ Tol03 ]andstandardSVDimple-mentation(usedforpolardecompositionof33matrices)from[ PTVF92 ].WepresentsometypicaldeformationresultsobtainedwithourtechniqueinFigures 1 , 4 – 8 .Notethatnaturalde-formationsareobtained,evenwhenthemanipulationhandleisonlybeingtranslated,becausetheoptimizationautomati-callyproducesthecorrectlocalrotations.TheCactus(Fig-ure 7 )isaparticularlychallengingexample,especiallyforlinearvariationaldeformationmethods,duetoitslongpro-trudingfeatures[ BS07 ].TheresultsofourmethodcanbecomparedwithPRIMO[ BPGK06 ],astate-of-the-artnon-lineartechnique,aswellasvariouslinearvariationaltechniques,byobserv-ingthecanonicalexamplesinFigures 5 , 6 , 7 .Suchde-formationsappearinthecomparisontablein[ BS07 ];itisevidentthatourmethodperformsequallywelltoPRIMOandisgenerallysuperiortolinearmethods,especiallywhenhandletranslationisinvolved.Toemphasizethispoint,wecomparetheresultsofourmethodwithPoissonmeshedit-ing[ YZX04 , ZRKS05 ]inFigure 5 .Notethatsincethe c TheEurographicsAssociation2007. O.Sorkine&M.Alexa/As-Rigid-As-PossibleSurfaceModeling Figure4:Largedeformationofthespikyplane(seeFigure 2 fortheoriginalmesh).Notethatthehandle(inyellow)wasonlytranslated,withoutspecifyinganyrotation. PoissonourmethodPoissonourmethodFigure5:ComparisonwithPoissonmeshediting.Theorig-inalmodelsappearinFigures 2 and 7 .Theyellowhan-dlewasonlytranslated;thisposesaproblemforrotation-propagationmethodssuchas[ YZX04 , ZRKS05 , LSLCO05 ]. handlewasonlytranslated,Poissoneditingcannotgener-ateaproperrotationeld(sincethereisnohandlerota-tiontopropagate),whichresultsindetaildistortionandlackofsmoothnessneartheconstraints.Thesametranslation-insensitivitywouldbeobservedinthemethodofLipmanetal.[ LCOGL07 ];ourtechniquehandlestranslationwellbyoptimizingforthelocalrotations,atthepriceofaglobalnon-linearoptimization.Itisworthnotingthoughthattherequirednumericalmachineryandthesetupofthelinearsys-temisalmostidenticaltothelinearvariationalmethods.Theaccompanyingvideoshowsseveralshorteditingses-sionscapturedlive.Ourunoptimizedcoderunsinterac-tively(at10-30fps)forregionsofinterest(ROI)ofupto10Kvertices,using2-3iterationsperedit.Anumberofimprovementsarepossibletospeedupconvergence:afasterpolardecompositionroutine(i.e.,onethatreusespre-viousframecomputationsratherthanstartingfromscratcheachtime)andamultiresolutiontechnique,suchastheonein[ BPGK06 ]or[ HSL06 ],toallowtheoptimizationtorunonacoarseversionofthemeshinordertoquicklypropagatethedeformationacrosstheROI.Animportantimplementationissueistheinitialguesswhichstartstheoptimization;sincetheenergyweminimizeisnon-linear,multiplelocalminimamayexist,andtheso-lutiondependsontheinitialguessinsuchcase.Itisimpor-tanttouseareasonable-qualityinitialguess(i.e.,nottoofarfromtheinitialshapeandtheintuitivelyexpectedresult)toallowquickconvergence,yetitisdesirabletocomputeitquickly.Weexperimentedwithseveralpossibilities,whichcanbeusedindifferentscenarios:Previousframe(forinteractivemanipulation):Iftheuserinteractivelymanipulatesthecontrolhandle(s),itisreason-abletousetheresultofthepreviousframeastheinitial Figure6:Twistandrotationdeformations. ModelFigureRelativeRMSerror DinoFig. 1 0.024 SpikyplaneFig. 4 left0.034 SpikyplaneFig. 4 right0.016 TwistedbarFig. 6 left0.095 ArmadilloFig. 8 (b)0.037 ArmadilloFig. 8 (c)0.013 ArmadilloFig. 8 (e)0.051 Table1:RelativeRMSerrorofedgelengthsforvariousde-formations.Whenthemodelingconstraintsdonotnecessi-tatestretching,theerrorisverylow.Thetwistexampledoesinvolvesomeslightstretchingbecausethetopofthebarisconstrainedtoremainatthesameheight,hencethehigherrelativeerrorinthiscase. guess,sincethehandlemovementand/ordeformationisex-pectedtobecontinuous.Therefore,inthiscasewesimplytakethepreviousframeandassigntheuser-denedpositionstotheconstrainedvertices.Thisapproachwasusedforalltheguresinthispaper,unlessexplicitlymentionedother-wise,andisalsodemonstratedintheaccompanyingvideo.Theuserexperienceremindsalotofinteractingwithphysi-calmaterial.NaiveLaplacianediting:ThestartingguessisobtainedbysimplelinearminimizationofkLp0�dk2undertheposi-tionalmodelingconstrains( 10 ),whered=Lparethedif-ferentialcoordinatesoftheinputmesh.Althoughthisguessproducesdistortedresultsforlargedeformations,thesubse-quentiterationsmanagetorecover,asdemonstratedinFig-ure 3 .Forsignicantlydistortedinitialguesstheconver-gencemaybeslow,however.Rotation-propagation:Ifthemanipulationofthehandlein-volvesexplicitrotation(alongwithtranslation),onecanuseanyofthetechniquesthatexplicitlypropagatethespeciedrotationtotheunconstrainedregions,suchas[ LSLCO05 , ZRKS05 , LCOGL07 ].Subsequentoptimizationofouren-ergyallowstoconsolidatetheotherwisedecoupledrotationandtranslationandimprovestheresults;convergenceistyp-icallyveryfastsincethestartingrotationalcomponentofthedeformationisalreadygood.Aninterestingpropertyofouras-rigid-as-possiblesur-facedeformationisedgelengthpreservation,totheextentallowedbythemodelingconstraints.Ifthemodelingcon-straintsdonotimposestretchingonthesurface,theopti-mizationalwaysstrivestoconvergetoastatewheretheedgelengtherrorissmall.Thisisclearlyvisibleinthedeforming c TheEurographicsAssociation2007. O.Sorkine&M.Alexa/As-Rigid-As-PossibleSurfaceModeling (a)(b)(c)(d)(e)(f)Figure7:BendingtheCactus.(a)istheoriginalmodel;yellowhandlesaretranslatedtoyieldtheresults(b-f).(d)and(e)showsideandfrontviewsofforwardbending,respectively.Notethatin(b-e)asinglevertexatthetipoftheCactusservesasthehandle,andthebendingistheresultoftranslatingthatvertex,norotationconstraintsaregiven. (a)(b)(c)(d)(e)(f)Figure8:EditingtheArmadillo.(a)and(d)showviewsoftheoriginalmodel;therestoftheimagesdisplayeditingresults,withthestaticandhandleanchorsdenotedinredandyellow,respectively. planeexample(Figure 4 ),forinstance,whichbehavessim-ilartorubber-likematerial.Table 1 summarizesroot-mean-squareedgelengtherrormeasurementsforseveraldeforma-tionspresentedinthispaper;itcanbeseenthattherelativeRMSerrorisverylow.5.ConclusionsTheimportantfeaturesofourapproachare(1)robustness,resultingfromtheminimizationprocedurethatisguaranteedtonotincreaseenergyineachstep;(2)simplicity,aseachstepoftheminimizationisconceptuallysimilartoLapla-cianmodeling;and(3)efciency,becausetheLaplacesys-temmatrixisconstantthroughouttheiterationsandhastobefactoredonlyonce.Wehavelearnedduringourexperimentsthatthiscom-binationisnotevident,i.e.,simplyupdatingtheright-handsideofadiscreteLaplacesysteminaseeminglyreasonablewaywouldfailtoconvergeinalmostallcases.Convergenceinourapproachistheresultofderivinganenergythatcan-notincreaseineachstepoftheiterations.Notethattheo-retically,thelocalminimumfoundbydecreasingtheenergymightnotbeunique,i.e.,therecouldbeaconnectedsetofminimumenergystates.However,wehavenotexperiencedthisproblemandbelievethatifitexistsatallthenonlyforparticularlyderivedexamples.Thefactthateachstepintheiterationscanbeperformedbysolvingalinearsystemwithaconstantmatrixthroughout theminimizationprocedurereallyistheresultofacarefuldesignoftheenergyfunctional.Thenumberofiterationsrequiredtogetreasonablyclosetoaminimumdependsontheconditionnumberofthe(anchored)Laplacianmatrix,whichisgenerallyproportionaltothemeshsize.Speci-cally,ifwekeeptheboundaryconditionsthesameandre-nethemesh,theconditionnumberwillgrowproportion-ally,eveniftheshapeofthemeshelementsisperfect(fordetailedanalysisandboundsontheconditionnumberoftheuniformanchoredLaplacianmatrix,see[ CCOST05 ];theuniformLaplaciancoincideswiththecotangentLaplacianfortessellationswithequilateraltriangles,andinothercasestheboundsforthecotangentLaplacianareprobablymorepessimistic).Thismeansasthemeshesarerenedstabilitydeteriorates,andtypicallymoreiterationsareneededuntilconvergence(inadditiontothefactthateachiterationbe-comesmorecostly).Thispracticalefciencyproblemcouldbeeasilyalleviatedwithmulti-resolutiontechniques.Anotherinterestingqualityofourapproachisthatittriv-iallyextendstovolumetriccells,e.g.,tetrahedra.Astherigidityismeasuredbasedontheedgesineachcell,nothingwouldhavetobechangedinthesetupoftheenergy–onewouldonlyhavetoplug-intheconnectivityofavolumet-ricgrid.So,ifpreservationofvolumeisofconcernratherthanpreservationofsurface,thenthiscouldeasilybeac-complished.Ofcourse,aswithotherrecentapproaches,theoptimizationcouldbeappliedtoacoarsevolumetricgrid c TheEurographicsAssociation2007. O.Sorkine&M.Alexa/As-Rigid-As-PossibleSurfaceModeling whichcontrolstheshapeembeddedinit,ratherthandirectlytothediscretesurfaceorvolume.Infuturework,wewishtoexperimentwithseveraldegreesoffreedomthatourmodelingframeworkoffers:changingthesizeandrelativeweightspercell,soastocon-troltheoverallandrelativelocalrigidityofthesurface.AcknowledgementWewishtothankMarioBotschandLeifKobbeltforin-sightfuldiscussionsandtheanonymousreviewersfortheirvaluablecomments.ThisworkwassupportedinpartbytheAlexandervonHumboldtFoundation.References [ACOL00] ALEXAM.,COHEN-ORD.,LEVIND.:As-rigid-as-possibleshapeinterpolation.InProceedingsofACMSIGGRAPH(2000),pp.157–164. [Ale01] ALEXAM.:Localcontrolformeshmorphing.InPro-ceedingsofSMI(2001),pp.209–215. [ATLF06] AUO.K.-C.,TAIC.-L.,LIUL.,FUH.:DualLapla-cianeditingformeshes.IEEETVCG12,3(2006),386–395. [BK04] BOTSCHM.,KOBBELTL.:Anintuitiveframeworkforreal-timefreeformmodeling.ACMTOG23,3(2004),630–634. 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