3 Aiwij1 2cot ijcot ij7where ijand ijarethetwoanglesoppositetotheedgevivjandAiistheVoronoiareaofvertexviThelatterisdenedin45astheareaofthesurfaceregionbuiltbyconnectingincidentedgesm ID: 336221
Download Pdf The PPT/PDF document "On liner Variational surface deformation..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
3 (a)(b)(c)(d)Fig.1.TheoriginalsurfaceS(a)iseditedbyminimizingitsdeformationenergy,subjecttouser-denedconstraintsthatxthegraypartFofthesurfaceandprescribethetransformationoftheyellowhandleregionH.Thelinearizedenergy(2)consistsofstretchingandbendingterms,andtheexamplesshowpurestretchingwithks=1,kb=0(b),purebendingwithks=0,kb=1(c),andaweightedcombinationwithks=1,kb=10(d).Asaconsequence,theLaplaceoperatorw.r.t.theparametrizationpturnsintotheLaplace-BeltramioperatorS=divSrSw.r.t.themanifoldS[19]:�ksSd+kb2Sd=0:(4)Noticethatthisvariationalminimizationiscloselyrelatedtothedesignoffairsurfaces[47],[68],wheresurfaceareaandcurvatureareminimizedinsteadoftheirchanges,i.e.,stretchingandbending.Thelinearizedmembraneandthin-plateenergiescorrespondingto(2)aredenedas~Ememb(p)=Z kpuk2+kpvk2dudv;~Eplate(p)=Z kpuuk2+2kpuvk2+kpvvk2dudv:(5)Analogouslyto(4)theircorrespondingEuler-Lagrangeequa-tionsare�Sp=0and2Sp=0,respectively.SincetheLaplaciansorbi-Laplaciansvanishontheresultingsurfaces,thosearestationarysurfacesofLaplacianandbi-Laplacianowstypicallyusedinsurfacesmoothing[18],[63]:pt=Spandpt=�2Sp:TheorderkofpartialderivativesintheenergyorinthecorrespondingEuler-Lagrangeequations(�1)kkSd=0denesthemaximumcontinuityCk�1forinterpolatingdis-placementconstraints[10].Hence,minimizing(2)bysolving(4)providesC1continuoussurfacedeformations,ascanalsobeobservedinFig.1.B.DiscretizationTheenergiesandPDEspresentedsofarwereformulatedforcontinuoustwo-manifoldparametricsurfacesS=p( ).However,ournalgoalistorepresentthesurfaceSbyatrianglemesh,sincethisallowsforhighertopologicalexibilityandcomputationalefciency[13].InthefollowingwedenotebySatrianglemesh,whosetopologyisdeterminedbynvertices(v1;:::;vn)andmtriangles(t1;:::;tm),ti2f1;:::;ng3,andwhosepiecewiselineargeometricembeddingisdenedbythevertexpositionspi=p(vi)2IR3.Inthediscretemeshsetting,theuserselectscertainverticesasthexedpartFandthehandleregionH,andtypicallyprescribeseitherpositionsp0i=ci2S0orcorrespondingdisplacementsdi=ci�piforthem.Fortherestofthepaper,letusassumew.l.o.g.thatfromthenvertices(v1;:::;vn)therstn0verticesarefree,whilethelastk=n�n0vertices(vn0+1;:::;vn)areconstrained,i.e.,theyconstitutethexedpartFandhandleregionH.Inordertodiscretizetheaboveequationsfortrianglemeshesonecaneitheremploynitedifferencesorniteelements.TheFiniteElementMethod(FEM)leadstomoreaccurateapproximationsingeneral,butforthinshellproblemslike(1)or(2)ittheoreticallyrequiresC1continuousshapefunctions[5].Inparticularontriangulatedmanifoldsthosearerathercomplicatedtodesign[15].MeshsubdivisionprovidesanelegantformulationforC1basisfunctions,asproposedin[16],[17]and[65]forstaticanddynamicdeformations,respectively.Asanalternative,so-callednon-conformingC0elementsarefrequentlyandsuccessfullyemployedinprac-tice[32],althoughlackingsometheoreticalguarantees.IncomparisontoFEM,adiscretizationbasedonnitedifferencesisconsiderablyeasiertouse,inparticularsincetheEuler-Lagrangeequations(4)onlyrequireadiscretizationoftheLaplace-Beltramioperator.Givenapiecewiselinearscalarfunctionf:S!IRdenedonthemeshS,itsdiscreteLaplace-BeltramiatavertexvihastheformSf(vi)=wiXvj2N1(vi)wij(f(vj)�f(vi));(6)wherevj2N1(vi)aretheincidentone-ringneighborsofvi(cf.Fig.2).Thediscretizationdependsontheper-vertexnormalizationweightswiandtheedgeweightswij=wji.Whilethereareseveralvariationsoftheseweights(seealsoacomparisoninSectionV-A),thede-factostandardisthecotangentdiscretization[18],[45],[51]:wi=1 Ai;wij=1 2(cot ij+cotij);(7)where ijandijarethetwoanglesoppositetotheedge(vi;vj),andAiistheVoronoiareaofvertexvi.Thelatterisdenedin[45]astheareaofthesurfaceregionbuiltbycon-nectingincidentedges'midpointswithtrianglecircumcenters(foracutetriangles)ormidpointsofoppositeedges(forobtusetriangles),asshowninFig.2. Fig.2.Theangles ijandijandthe(darkgrey)VoronoiareaAiusedtodiscretizetheLaplace-BeltramiSatavertexviinequations(6)and(7). 4Higher-orderLaplaciansarethensimplydenedrecur-sively:kSf(vi)=wiXvj2N1(vi)wij�k�1Sf(vj)�k�1Sf(vi);0Sf(vi)=f(vi):C.NumericalSolutionUsingthediscretization(6),theLaplace-Beltramioperatorforthewholemeshcanbewritteninmatrixnotation0B@Sf(v1)...Sf(vn)1CA=M�1Ls| {z }L0B@f(v1)...f(vn)1CAwhereMisadiagonal“mass”matrixofthenormalizationweightsMii=1=wi=Ai,andLsisasymmetricmatrixcontainingtheedgeweightswij:(Ls)ij=8�&#x]TJ ;� -1;.93; Td;&#x[000;:�Pvk2N1(vi)wik;i=j;wij;vj2N1(vi);0;otherwise:TheEuler-Lagrangeequation(4)thenleadstoasparsennlinearsystem��ksL+kbL2d=0:Theboundaryconstraintsareincorporatedintothissystembymovingeachcolumncorrespondingtoaconstrainedvertexvi2F[Htotheright-handside,andremovingtherespectiverowfromthesystem(seealsoSectionV-C).Thisyieldsanon-zeroright-handsideb2IRn03andleadstoan0n0systemthatissolvedforthex-,y-,andz-componentsofthedisplacementsd=(d1;:::;dn0).Noticethatfornotationalconveniencewestilldenotethen0n0sub-matricesbyLandL2.Pre-multiplyingtheabovesystembyMnallyyieldsthesymmetricsystem��ksLs+kbLsM�1Lsd=Mb;(8)whichinadditioncanbeshowntobepositivedenite[51].Inaninteractiveapplicationtheabovelinearsystemhastobesolvedforthedeformedsurfaceeachtimetheuserchangestheboundaryconstraints,e.g.,bymovingtheconstrainedpoints,sincethatchangestheright-handsideb.Sincethesystemissparse,symmetric,andpositivedenite,aniterativemethodlikeconjugategradients[25]couldbeemployed,buttheresultingO(n2)computationalcomplexityisprohibitiveforlargemeshes.Solvingthesystemonamultigridhierarchyofsuccessivelycoarsenedmeshes,asproposedin[10],[34],yieldslinearO(n)complexityandhencealsoworksforcomplexmeshes.However,theimplementationofanefcientmultigridsolvercanbequitecomplex,sinceitrequiresseveralproblem-dependentdesigndecisions[1],[54].Whilemultigridsolversareanefcienttool,theydonotexploitthefactthatthesamelinearsystem(8)issolvedmanytimes(threetimeseachframe),onlyfordifferentright-handsidesMb.Incontrast,sparsedirectCholeskysolversrstfactorthematrix,suchthatforeachnewright-handsideonlyanefcientback-substitutionhastobeperformed. Fig.3.Amultiresolutioneditingframeworkconsistsofthreemainoperators:thedecompositionoperator,whichseparatesthelowandhighfrequencies,theeditingoperator,whichdeformsthelowfrequencies,andthereconstructionoperator,whichaddsthedetailsbackontothemodiedbasesurface.Thankstoamatrixpre-orderingtheresultingCholeskyfactorisalsosparse,leadingtobasicallylinearcomplexityofboththefactorizationandtheback-substitution.Incomparisonwithiterativemultigridsolversthedirectsolversarenotonlyeasiertouse,butalsoprovidebetterperformanceforso-calledmultiple-right-hand-sideproblems[7],[54].D.MultiresolutionHierarchiesThedeformationtechniquesdescribedaboveapproximatethenon-linearshellenergy(1)bythequadraticenergy(2)inordertoreducetheper-framecoststothesolutionofthelinearsystem(8).AlthoughtheglobalenergyminimizationguaranteessmoothandC1continuoussurfacedeformations,thelinearizationcausesgeometricdetailsandprotrudingfea-turestobedistorted.AscanbeseeninFig.4,evenapuretranslationofthehandleHisintuitivelyexpectedtolocallyrotatethegeometricdetails.Unfortunately,determiningtherequiredlocalrotationsfrompositionconstraintsaloneisanon-linearproblem,andthereforecannotbesolvedbyalinearizedtechnique(cf.Fig.4b).Inordertostillbeabletoachieveintuitivedetailpreservationwhileusingalineardeformationtechnique,onecancomplementthelineardeformationmodelbyaso-calledmultiresolutionormulti-scalehierarchy.ThemainideaofmultiresolutiondeformationsistoconsiderthesurfaceSasa“geometricsignal”,andtoseparatethelowfrequenciesfromthehighfrequencies.ThelowfrequenciesconstitutetheglobalshapeofthemodelandarerepresentedbyasmoothbasesurfaceB.ThehighfrequenciesarethedifferencebetweenSandB,i.e.,thegeometricdetailsD=S B.TheoriginalsurfaceScanbereconstructedbyaddingthegeometricdetailstothebasesurface,S=BD.Amul-tiresolutiondeformationcannowbecomputedbydeformingBtoB0andreconstructingS0=B0D.ThismodiestheglobalshapeB,butpreservesthene-scaledetailsD.ThewholeprocessisschematicallydepictedinFig.3. 8where(u;v)areparametersoverthedomainmeshand i()arethepiecewiselinear“hat”basisfunctionsassociatedwiththedomainmeshvertices,i.e., i(vk)=ik.Thegradientoffisthenrf(u;v)=nXi=1fir i(u;v):(14)Thegradientsr i(u;v)areconstantwithineachdomainmeshface;if(pi;pj;pk)aretheverticesofadomainmeshtrianglethenthegradientsofthecorrespondinghatfunctions i; j; kare(r i;r j;r k)=0B@(pi�pk)T�pj�pkTnT1CA�10@10�101�10001A;wherenistheunitnormalofthetriangle.Thisformulationensuresthatthegradientslieinthetriangle'splane(fordetailsonthederivationsee[14]).Onecanformulate(14)usingaglobaloperatorG,expressedasa3mnmatrixthatmultipliesthen-vectorfofthediscretevaluesfitoobtainavectorofmstackedgradients,eachgradienthaving3spatialcoordinates(mbeingthenumberoftriangles).Thus,onecanwritedownthefollowingformulafortheinputmesh:Gx0=gx;andthesamefortheothertwocoordinatefunctions.Whenthegradientsofthesurfaceareknown(asfunctionsoverthedomainmesh)andthecoordinatefunctionsareunknown,wecanndthembyminimizing(11)withGbeingthedifferentialoperator.Thuswesolve(12),wherethe3m3mweightmatrixMcontainstheareasofthetriangles:GTMGx0=GTMgx;ThematrixGTMcorrespondstothediscretedivergenceoperatorassociatedwiththedomainmesh,andGTMGisnonotherthanthecotangentdiscretizationoftheLaplace-Beltramioperator[14],discussedinSectionII-B,sowecansimplywriteLsx0=GTMgx;(15)whichisthediscretizedversionof(13).Todeformasurfaceusingthisgradientrepresentation,adi-rectadaptationofPoissonImageEditing[50]wouldbesimplytoaddDirichletboundaryconditionsto(15),correspondingtouser-denedmodelingconstraintsp0i=ci(16)forthexedverticesFandhandleverticesH.However,theresultofsucheditingapproachisnotsatisfac-tory,becauseittriestopreservetheoriginalmeshgradients,withtheirorientationintheglobalcoordinatesystem.Thisignoresthefactthatinthedeformedsurfacethegradientsshouldrotate,sincetheyalwayslieinthetriangles'planes,whichtransformasaresultofthesurfacedeformation.TheeffectisdemonstratedinFig.6b,clearlyshowingthattheresultingdeformationisnotintuitive.Thislocaltransformationsproblemiscentralinalldif-ferentialeditingapproaches;itstemsfromthefactthatthe (a)(b)(c)(d)Fig.6.Usinggradient-basededitingtobendthecylinder(a)by90.Reconstructingthemeshfromnewhandlepositions,butoriginalgradientsdistortstheobject(b).Applyingdampedlocalrotationsderivedfrom(25)totheindividualtrianglesbreaksupthemesh(c),butsolvingthePoissonsystem(15)re-connectsitandyieldsthedesiredresult(d).representationisdependentontheparticularplacementofthesurfaceinspace,i.e.,itisnotrigid-invariant,andthuswhenthesurfacedeforms,therepresentationmustbeupdated.Unfortunately,itisachicken-and-eggprobleminitsessence,becausethedeformedsurfaceisunknown.WereviewthedifferentapproachestoobtainthelocaltransformationsinSectionIII-B.2)Laplacian-basedrepresentation:Laplacian-basedap-proachesrepresentthesurfacebytheso-calleddifferentialco-ordinatesorLaplaciancoordinates[3],[59].ThesecoordinatesareobtainedbyapplyingtheLaplacianoperatortothemeshvertices,i.e.,takingfpin(6);theresultingvectoristhemeancurvaturenormal:i=S(pi)=�Hini;(17)whereHiisthemeancurvatureH=1+2atvi.Modelingdirectlywiththesecoordinatesismeanttocircumventtheneedtodecomposethesurfaceintoalow-frequencybasesurfaceandhigh-frequencydetails,asinthemultiresolutionapproachesdiscussedinSectionII-D.Laplacianeditingwasdevelopedconcurrentlywithgradient-basedediting,andsimilarlytothelatter,therstnaiveattemptwouldbetoformulatethedeformationbydirectlyminimizingthedifferencefromtheinputsurfacecoordinatesi.Inthecontinuoussettingtheenergyminimizationisformulatedasminp0Z kp0�k2dudv:(18)TheEuler-Lagrangeequationderivedfortheaboveminimiza-tionis2p0=:Whenweconsiderthisequationtakingtheinputsurfaceastheparameterdomain,theLaplaceoperatorturnsintoLaplace-BeltramiSandwearriveatthediscretizedequationL2p0=L;(19)whichcanbeseparatedintothreecoordinatecomponents;theequationisconstrainedbythemodelingconstraintsoftheform(16).Itisalsopossibletoarriveatthisequationbydiscretizingthecontinuousenergy(18):minp0XiAikS(p0i)�ik2:(20) 10 Fig.7.Anon-uniformtwistusingthematerial-awaredeformationtech-nique[52],withstiffnessweightscolor-codedinthebottomimage.smoothtransformationpropagation,withtheaddedadvantageofeconomiccomputation,whenthematrixin(26)isthesamematrixusedforediting(15),thusthesamefactorizationcanbeused.3)Material-awarepropagation:Itispossibletocontrolthesurfacematerialproperties,namelylocalstiffness,bycarefullydesigningtheinterpolationweightssi,aswasdonein[52].Theusermaydenestiffnessbyapaintinginterface,whichprovidesascalareld'ijoverthemeshedges;theinterpo-lationweightsarethendeterminedbysolvingthefollowingweightedquadraticminimization:minsXj2N1(vi)'ijksi�sjk2:Thus,wherethestiffnessparameter'ijishigher,theinter-polationparameterstendtobecloser,whichassignssimilarlocaltransformationsandmakesthesurfacelocallystiffer(cf.Fig.7).Theconstraintsoftheminimizationabovearethesameasinharmonicpropagation,i.e.,0forxedverticesFand1forthehandleH.Curiously,theminimizationofthequadraticenergyaboveleadstoaLaplaceequation(see[20]fordetails),weightedbythestiffnessparameters.Notethatanytransformationpropagationtechniquewouldonlyworkifthetransformationofthehandle(e.g.rotation)isactuallyprovided.Ifthehandleisonlytranslated,allpropagatedlocaltransformationswillequaltheidentity.Thisphenomenoniscalledtranslation-insensitivity[12],becausethemethodmightnotgenerateintuitivelocalrotationswhenthemodelingconstraintcontainstranslation.4)Explicitoptimization:Thismethodproducesthelocaltransformationsbysolvingforthelocalframesofthede-formedsurfaceintheleast-squaressense[42],using(22).Thecoefcientsarederivedfromtheoriginalmesh.Theideaistooptimizethelocaltransformationssoastopreservetherelationshipsbetweenthelocalframes.Thehandleframesneedtobeconstrained,similarlytothetransformationprop-agationmethods,suchthatthismethodisalsotranslation-insensitive.Notethattheleast-squaressolutionmightproducenon-orthonormalframes,whichmayleadtoareaandvolumeshrinkage.Ifthemodelingconstraintsontheframesinvolvesolelyorthogonaltransformations,itisthereforeadvisabletonormalizeandorthogonalizetheframes.5)Estimationfromaninitialguesssolution:Localtrans-formationsmaybeestimatedfromanaivesolutionofthedeformation(computedwithouttransformingthedifferentialrepresentation).Thisapproachisakintothemultiresolutiontechniquesinthesensethattheinitialguessofthedeformedsurfacelacksdetails;thedetailsarethentransformedbytheestimatedlocaltransformations.Thelocaltransformationsareestimatedfromtheinitialguessbycomparingk-ringvectorsVioftheoriginalsurfacewithcorrespondingk-ringsV0kinthedeformedsurface[41](thecolumnsofViarevectorsfromthecentervertexpitoitsk-orderneighbors,andinadditionthenormalatpi),oralternativelyfrompairsofcorrespondingtrianglesandtheirnormals[14].Aleast-squarestofthelocaltransformationis~Ti=V0kV+k;where()+denotesthepseudo-inverseofamatrix.ThelocaltransformationsarethenorthogonalizedtoobtainrigidTi's(isotropicscalesmayalsobeallowed,dependingontheuser'srequirements).Theassumptionisthatthedeformedsurfaceismostlysmooth,andthusthenaivesolutionprovidesagoodguessforthedeformedunderlyingbasesurfaceandonlysmall-scaledetailsneedtoberotatedtocorrecttheirorientation.Thelargerkthesmoothertheestimationis,naturallyatlargercomputationalcost.Notethatthismethodissensitivetotranslationsofthehandle.6)Implicitoptimization:Implicitoptimizationoftransforma-tionstriestotacklethe“chickenandegg”problemoflocaltransformationsbyexpressingtheseunknowntransformationsintermsoftheunknowndeformedgeometry:Ti=Ti(p0).Thelocaltransformationsarethenfoundtogetherwiththedeformedsurfaceintheglobaloptimizationprocess(24).Noticethatthethreespatialmeshcoordinatefunctionsarenolongerdecoupledintheglobaloptimization.Thelocaltrans-formationsshouldbealsoconstrainedtorigidorsimilaritytransformationsonly.ThecoefcientsofTiarefunctionsofp0;intheoptimalcaseTiwouldbeconstrainedtorotationsalone,butthiswouldrequiretousenon-linearcombinationsofp0,turning(24)intoaglobalnon-linearoptimization.Itispos-sible,however,tolinearizethesimilaritytransformations[60]:Ti=0@s�h3h2h3s�h1�h2h1s1A:(27)Theparameterss;haredeterminedbywritingdownthedesiredtransformationconstraints,i.e.,Ti(pi�pj)=p0i�p0jforeachvj2Nk(vi)andthusextractings;haslinearcom-binationsofp0.Theprecisederivationcanbefoundin[40].PluggingthelinearexpressionforTibackinto(24)resultsinalinearleast-squaresproblem.ItshouldbenotedthattheexpressionforTiaccommodatesisotropicscalesinadditiontorotations,thereforewhenthehandleis“pulled”,forexample,thedeformedsurfacewillscaleandinate.Whenthiseffectisundesired,itcanbeeliminatedbyscalingthedifferentialrepresentations(gradients/Laplacians)ofthedeformedsurfacebacktotheirlengthintheoriginalsurfaceandsolving(11)withthiscorrectedrepresentation.Anothersolution,proposedin[33]istoscalethetrianglesofthedeformedsurfacebackto 11 (a)(b)(c)Fig.8.A100ktriangleversionofthemesh(a)from[27]wasbenttocomparetheoriginalLaplacianeditingformulationLTLp=LT(b),andthecorrectoneL2p=L(c).Both(b)and(c)usethecotangentLaplacian.theiroriginalsizeandre-stitchingthemeshusingthePoissonsetup(15).Theimplicitoptimizationoflocaltransformationsisalsosensitivetotranslationalmodelingconstraints.C.RelatedapproachesInthissectionwelistanddiscussseveraldeformationapproachesthatemploydifferentialsurfacerepresentationsandcategorizethemaccordingtotheparticularrepresentationandlocaltransformationhandling.TherstuseofdifferentialcoordinatesformesheditingwassketchedbyAlexain[2];hesuggestedusingtheoriginalsurfaceLaplacianswithsoftmodelingconstraintsin(21).Sincenoappropriatelocaltransformationswerecomputed,thisapproachwassuitableforeditingsmoothsurfaceswithnofeaturesorforperformingdeformationsthatalmostdonotinvolverotations.In2004,Lipmanetal.[41]proposedtoaddlocalrotationestimationtothesimpleLaplacianeditingparadigm,com-putingitfromthenaivesolutionof[2].Theyhaveshownthatsmoothingtheestimatedtransformationsbyusinglargerneighborhoodswithspecialweightedaveragingmaysigni-cantlyimprovetheresults,althoughatlargercomputationalcost.Still,notethatthistwo-stepdeformationprocessonlyrequirestwosolvesbyback-substitution(plusintermediatetransformationcomputations),sincethesystemmatrix(21)remainsthesameandcanbepre-factored.Theapproachworkswellforrelativelysmoothsurfaceswithnolargelyprotrudingfeatures;otherwisetheunderlyingassumptionthattheinitialguessbynaiveLaplacianeditingprovidesagoodrotationguessnolongerholds,andtherotationestimationfails.Inparticular,theapproachmayhavedifcultywithfeaturesthatcannotbedescribedasaheighteldoverthebasesmoothsurface.Botschetal.[14]proposedaconceptuallysimilartechnique:theyestimatethelocalrotationsfromabasesurfaceBanditsdeformedversionB0(seeFig.3andSectionV-E);theythenapplytheserotationstothegradientsoftheinputmeshtoreconstructthenalresultusingthePoissonframework(15).Sorkineetal.[60]proposedtousetheLaplacianrepre-sentationcoupledwithimplicittransformationoptimization,derivedfromone-rings.Toeliminateisotropicscaling,theyrescaletheLaplaciansofthedeformedsurfacebacktotheiroriginallengthandsolve(21).Thistechniquecanhandlemorecomplexsurfaceswithlargefeatures;itislimited,however,intheallowedrotationrange,becausethelinearizedapproximationoflocalrotationsisonlyvalidforsmallangles.Inpractice,rotationsofupto=2canbewell-performed[57];forlargerrotationsseveralstepsofthetechniqueshouldbeappliedtobreakthelargerotationintosmallerones.ThemethodofSorkineetal.[60]wasappliedtomanipu-lationoftriangulated2DshapesbyIgarashietal.[33].Theyusedimplicittransformationoptimization;notethatin2Dsim-ilaritytransformationscanbeexactlylinearlyparameterized:S=ab�ba:Inasecondstep,Igarashietal.removetheunwanteduniformscalingfromthelocaltransformationsandre-solveforthever-texpositionsusingedgeequations,asin(23).Thetechniqueisveryeffectivefor2Dshapeediting,thankstotheexactrotationformulationandthemeshingoftheinterioroftheshape.TheideaofcombiningtheLaplacianrepresentationwithimplicittransformationoptimizationwasfurtherdevelopedbyFuetal.[23].Theyproposeahybridapproachthatcombinesimplicitoptimizationwithtwo-steplocaltransformationesti-mation.Intherststage,Laplacianeditingisperformedwithimplicittransformations,thatarenotconstrainedtolinearizedsimilaritytransformedbutinsteadareallowedtobeanyafnetransformationsTi=U0iU+i,whereUiaretheone-ringvectorsofvertexvi.Inaddition,thelocaltransformationsareaskedtobelocallysmooth,whichisexpressedbyadditionalquadratictermsinthedeformationenergy:kTi�Tjk2forneighboringverticesvi;vj.Theresultingdeformedsurfaceisthenusedasaninitialguesstoestimatetheactuallocaltransformations;thoseareorthogonalizedandLaplacianedit-ing(21)isapplied.Thisapproachenableslargerrotationsthan[60],butitrequirestweakingtherelativeweightingoflocaltransformationsmoothnessterms;moreovertheformu-lationofimplicittransformationsmaybeill-denedforatone-rings,inwhichcaseaperturbationisrequired.Itisworthnotingthattheaboveapproachesusedaslightlyerroneousversionofthediscreteenergy(20)sincetheyomit-tedtheareaweightsAi,whichcorrespondtodiscretizationoftheL2productonthemesh[67].ThisleadtonormalequationsoftheformLTLp0=LT;whichdifferfromthecorrectlydiscretized(21).Suchformu-lationmayleadtoproblemsonirregularmeshes,asdemon-stratedinFig.8.Laplacianeditingwasfurtherusedforasketch-basededit-ingsystem[49]andvolumegraphdeformations[73].Nealenatal.[49]employedimplicittransformationpropagationandproposedusingsketchedcurvesonthesurfaceashandlesanddeformationconstraints,whichleadstoanintuitivesilhouetteandfeatureeditingtool.Intheclassicalhandlemetaphorthepositionofthehandleisdirectlymanipulatedbytheuserandthushardpositionalconstraintsarepreferred;inasketch-basedsystemsoftconstraintsareactuallyadvantageous,sincetheyallowtheusertoplaceimprecisestrokesthatareonlymeanttohintatthedesiredshapebutnotspecifyitexactly.Thus,Nealenetal.allowedvaryingtheweightonthesketchedpo-sitionalconstraintstoachieveroughsketching(smallweights) 14withprotrudingfeatureslikeinthecactusexample.However,sincethislinearsystemdoesnotconsiderpositionalcon-straints,thismethodisalsotranslationinsensitive.Inaddition,thelinearsystemforreconstructingthepositionsfromlocalframescorrespondstoauniformLaplacian,whichcausestheasymmetriesfortheregulartessellationofthebumpyplane.Fromthoseexamplesonecanderivethefollowingguide-linesforpickingthe“right”deformationtechniqueforaspecicapplicationscenario:Intechnical,CAD-likeengineeringapplicationstherequiredshapedeformationsaretypicallyrathersmall,sinceinmanycasesanexistingprototypeonlyhastobeadjustedslightly,buttheyhavehighrequirementsonsurfacefairness,boundarycontinuity,andtheprecisecontrolthereof.Forsuchkindofproblemsalinearizedshellmodellike[10]wasshowntobewellsuited.Incontrast,applicationslikecharacteranimationmostlyinvolve(possiblylarge)rotationsoflimpsaroundbendsandjoints.Here,methodsbasedondifferentialcoordinatesclearlyarethebetterchoice.Moreover,therequiredrotationsmightbeavailablefrom,e.g.,asketchinginterface[49],[73]oramotioncapturesystem[54].Applicationsthatrequirebothlarge-scaletranslationsandrotationsareproblematicforalllinearapproaches.Inthiscaseonecaneitheremployaremorecomplexnon-lineartechnique,orsplituplargedeformationsintoasequenceofsmallerones.Whilethenon-lineartechniquesarecomputationallyandimplementation-wisemoreinvolved,splittingupdeformationsorprovidingadensersetofconstraintscomplicatestheuserinteraction.Withtherapidlyincreasingcomputationalpoweroftoday'scomputers,non-linearmethodsbecomemoreandmoretractable,whichalreadyleadtoarstsetofnon-linear,yetinteractive,meshdeformationapproaches[4],[12],[31],[36],[53],[62],[66].V.DEFORMATIONFAQAfterdescribing,comparing,anddiscussingthevariousshapeeditingtechniquesofSectionsIIandIII,wenallywanttoanswerasetofquestionsmostfrequentlyaskedinthecontextofmesh-basedsurfacedeformations.A.“WhatistheinuenceoftheLaplaciandiscretization?”MostoftheapproachesdescribedinSectionIIandSec-tionIIIderivethedeformedsurfacebysolvingaLaplacianorbi-Laplacianlinearsystem.Hence,theyallrequireadis-cretizationoftheLaplacianoperator,andtheirresultsstronglydependonthischoice.ThereexistseveralvariationsoftheweightsusedinthetypicallyemployedLaplaciandiscretiza-tion(6).TheuniformLaplacian,employedforinstancein[34],[41],[60],[63],usestheweightswij=1;wi=1 Pjwij:Sincethisdiscretizationtakesneitheredgelengthsnoranglesintoaccount,itcannotprovideagoodapproximationforirregularmeshes.Betterresultscanbeachievedbywij=1 2(cot ij+cotij);wi=1;whichnowconsidersangles,butnotvaryingvertexdensities[69],[71].Thebestresultsareobtainedbyincludingtheper-vertexnormalizationweights(seeSectionII-B)wij=1 2(cot ij+cotij);wi=1 Ai;asproposedby[18],[45],[51]andemployedforinstancein[10],[11].AqualitativecomparisonofthethreediscretizationsisgiveninFig.11;inthisexamplecurvatureenergiesareminimizedbysolving2Sp=0,sincesmoothsurfacesarevisuallyeasiertoevaluatethansmoothdeformations.Amoredetailedanalysisofdifferentdiscretizations,withafocusontheirconvergenceproperties,canbefoundin[27],[70].Whilethecotangentdiscretizationclearlygivesthebestresults,itcanalsoleadtonumericalproblemsinthepresenceofnear-degeneratetriangles,sincethenthecotangentvaluesdegenerateandtheresultingmatricesbecomesingular.Inthiscasethedegeneratetriangleswouldhavetobeeliminated[8]inapreprocess.Alternatively,thewholebasesurfaceBcouldbere-meshedisotropically,asproposedin[11].B.“Whatisthedifferencebetweenthethin-plateenergyR21+22andthemeancurvatureenergyRH2?”WiththemeancurvatureH=1+2andGaussiancurvatureK=12,wehaveZ H2dudv=Z �21+22dudv+2Z Kdudv;i.e.,thetwoenergiesbasicallydifferintheintegralRK,whichbytheGauss-Bonnettheoremonlydependsonthe(xed)Dirichletboundaryconstraintson@ andthereforestaysconstant[19].Hence,theminimizersofthetwoenergiesareequivalentforidenticalDirichletboundaryconstraints.Thisalsoholdsforthelinearizedenergies,whichareZ 21+22dudvZ kpuuk2+2kpuvk2+kpvvk2dudvZ H2dudvZ kpuuk2+2pTuupvv+kpvvk2dudv:VariationalcalculusyieldstheidenticalEuler-Lagrangeequa-tion2p=0forbothlinearizedenergies,anditsdiscretiza-tion(andsymmetrization)leadstoLsM�1Lsp=0tobesolvedforitsminimizersurface(seeSectionII-C).EvenwhendiscretizingthemeancurvatureenergyinsteadoftheaboveEuler-Lagrangeequationsonearrivesatthesamelinearsystem[67].C.“Whatisthedifferencebetweenhardconstraintsandsoftleast-squaresconstraints?”AsintroducedinSectionII,therstn0vertices(v1;:::;vn0)areconsideredfree,andthelastk=n�n0vertices(vn0+1;:::;vn)areconstrainedbyprescribingpositionsciordisplacementsdi=ci�pi. 16preservene-scaledetails,i.e.,ndinglocalrotationsofthegeometricdetails.Whileallmultiresolutionapproachesderivethoserotationsfromthedeformationofthelow-frequencybasesurface(SectionII-D),thereareseveralapproachestorotatethedifferentialcoordinates(SectionIII-B).E.“Whatistherelationofgradient-baseddeformationanddeformationtransfer?”In[61]SumnerandPopovi´ctransferthedeformationS7!S0,foragivensourcemeshSanditsdeformedversionS0,ontoatargetmeshT,whichyieldsadeformedmeshT0suchthatthetwodeformationsS7!S0andT7!T0areassimilaraspossible.Theyaddtoeachtriangletiafourthpoint,turningthetriangleintoatetrahedron,suchthatthesefourpointsinSandS0uniquelydeterminetheafnetransformationx7!Six+ti.Theythenconsiderthegradientofthisafnemapping(so-calleddeformationgradient),whichisthe33matrixSicontainingtherotationandscale/shearpart.Finally,newvertexpositionsp0i2T0arefoundsuchthattheresultingdeformationgradientsTiforTareclosetothegivenSi,whichleadstothearea-weightedleastsquaressystem~GTM~G0B@p01T...p0~nT1CA=~GTM0B@ST1...STm1CA;where~n=n+m3nisthenumberofverticesincludingtheadditionalfourthpoints,and~Gisthe3m~nmatrixthatcomputesthedeformationgradientsfromthevertexpositions.Inthiscontext,thegradient-baseddeformation[71]issimilar,butheretheuserdirectlyprescribesthelocalrotationsSi,whicharethenappliedtothegradientsGi2IR33oftheoriginalmeshT,resultinginG0i.InordertondthemeshT0thathasthedesiredgradientsG0ithePoissonsystemGTMG0B@p01T...p0nT1CA=GTM0B@G01...G0m1CAissolved,asdescribedinSectionIII.Itwasshownin[14]thatonecansafelyremovethefourthpointsfromtherstsystem,whichreducesthesizeofthesystemfrom~n~ntonn.AfterthatreductionthematricesGand~G—andhencethewholelinearsystems—canbeshowntobeequal.Hence,deformationtransfercanalsobeconsideredasaspecialcaseofPoissonediting,wherethelocalper-triangletransformationsaredeterminedfromSandS0.VI.CONCLUSIONSInthissurveyweattemptedtogiveasystematicdescriptionandclassicationoftheplethoraofsurfaceeditingmethodsthatcanbegenerallyseenaslinearvariationaltechniques.Ourgoalwastorstofallexplaintheoriginalmotivationbehindthesetechniques,thatcomesfromcontinuousformulationsandiscloselyrelatedtophysically-basedenergiesandclassicaldifferentialgeometry.Thenweshowedhowthedifferentmeth-odssimplifyanddiscretizethesesettingsinordertoachieveinteractiveandrobustmeshdeformationmethods.Finally,weperformedpracticalcomparisonofseveralrepresentativemethodstorevealthecharacteristicstrengthsanweaknessesofeachapproachinextremedeformationcases.Wehopethatourqualitativedescriptionandpracticalillustrationswillhelpthereaderstounderstandtheideasbehindthesemethodsandalsotochoosetherightmethodforeachparticulareditingscenario.Wefocusedonlinearvariationalmethods,sincetheycom-prisealargebodyofworkovertherecentyears,yettheyhavenotbeensurveyedinanelaboratedandcomparativemanner.Inaddition,thisgroupofmethodshasgainedhighvisibilityincomputergraphicsresearch,asisevidentbythenumberofcitations.Thispopularityisowedtotherobustnessandeaseofimplementationoftheseapproaches,especiallythankstotheavailabilityofadvancedsparselinearsolvers.Oneobviousconclusionofthissurvey,however,isthatthereisnoperfecttechniquethatwouldworksatisfactoryineverycase.Apartfromthefactthata“perfect”resultmaybeasubjectiveandapplication-dependentnotion,allthereviewedmethodssharethesameproperty:forthesakeofspeedandrobustnesstheylinearizetheinherentlynon-lineardeformationproblem.Thevariousmachinerythatismeanttomaskthelinearizationerrorsworksinsomescenarios,butfailsinothers,aswedemonstrated.Ascomputingresourcesbecomefasterandfaster,andpreviouslyinfeasiblenumericalmethodsbecometractable,thereisnowroomfornon-linearmethodsandoptimizationstobeexploredininteractiveapplications.Inlightoftheabove,wefeltthatthisisanappropriatepointintimetosummarizethelinearvariationaldeformationmethods.Wearecondentthatthesetechniquesareyettoconquerthecommercialmodelingapplications,andthatresearch-wisethereareyetmanyareaswheretheycanbeincorporatedandexploredfurther.Moreover,weanticipatefurtherdevelopmentofnon-lineardeformationtechniques,exploitingtheknowledgeandexperiencegainedfromthelinearmethods.Afterall,eachgeneralproblemissolvedbyiteratingandreninglinearapproximations.ACKNOWLEDGMENTSWewishtothankLeifKobbeltandDanielCohen-Orforencouragingustopreparethissurveyandforco-authoringnumerouspapersrecitedhere.Wearealsogratefultothem,andtoMarcAlexa,MarkusGross,DenisZorin,MaxWardet-zky,KlausHildebrandtforthevariousdiscussionsthathelpedtoimprovethismanuscript.Wealsothanktheanonymousreviewersfortheirvaluablecommentsandsuggestions.TheresultsinFig.7arecourtesyofTiberiuPopaandAllaSheffer.REFERENCES[1]BurakAksoylu,AndreiKhodakovsky,andPeterSchr¨oder.Multilevelsolversforunstructuredsurfacemeshes.SISC,26(4):1146–1165,2005.[2]MarcAlexa.Localcontrolformeshmorphing.InProceedingsofShapeModelingInternational,pages209–215.IEEEComputerSocietyPress,2001.[3]MarcAlexa.Differentialcoordinatesforlocalmeshmorphinganddeformation.TheVisualComputer,19(2):105–114,2003.[4]OscarKin-ChungAu,Chiew-LanTai,LigangLiu,andHongboFu.DualLaplacianeditingformeshes.IEEETransactionsonVisualizationandComputerGraphics,12(3):386–395,2006. 18[57]OlgaSorkine.LaplacianMeshProcessing.PhDthesis,SchoolofComputerScience,TelAvivUniversity,2006.[58]OlgaSorkineandDanielCohen-Or.Least-squaresmeshes.InProceed-ingsofShapeModelingInternational,pages191–199.IEEEComputerSocietyPress,2004.[59]OlgaSorkine,DanielCohen-Or,andSivanToledo.High-passquanti-zationformeshencoding.InProceedingsoftheEurographics/ACMSIGGRAPHSymposiumonGeometryProcessing,pages42–51.Euro-graphicsAssociation,2003.[60]OlgaSorkine,YaronLipman,DanielCohen-Or,MarcAlexa,ChristianR¨ossl,andHans-PeterSeidel.Laplaciansurfaceediting.InProceed-ingsoftheEurographics/ACMSIGGRAPHSymposiumonGeometryProcessing,pages179–188.ACMPress,2004.[61]RobertW.SumnerandJovanPopovi´c.Deformationtransferfortrianglemeshes.ACMTransactionsonGraphics(ProceedingsofACMSIGGRAPH),23(3):399–405,2004.[62]RobertW.Sumner,MatthiasZwicker,CraigGotsman,andJovanPopovi´c.Mesh-basedinversekinematics.ACMTransactionsonGraphics(ProceedingsofACMSIGGRAPH),24(3):488–495,2005.[63]GabrielTaubin.Asignalprocessingapproachtofairsurfacedesign.InProceedingsofACMSIGGRAPH,pages351–358.ACMPress,1995.[64]DemetriTerzopoulos,JohnPlatt,AlanBarr,andKurtFleischer.Elas-ticallydeformablemodels.InProceedingsofACMSIGGRAPH,pages205–214.ACMPress,1987.[65]BernhardThomaszewski,MarkusWacker,andWolfgangStrasser.Aconsistentbendingmodelforclothsimulationwithcorotationalsubdi-visionniteelements.InProceedingsofACMSIGGRAPH/EurographicsSymposiumonComputerAnimation,pages107–116,2006.[66]WolframvonFunck,HolgerTheisel,andHans-PeterSeidel.Vectoreldbasedshapedeformations.ACMTransactionsonGraphics(ProceedingsofACMSIGGRAPH),25(3):1118–1125,2006.[67]MaxWardetzky,Mikl´osBergou,DavidHarmon,DenisZorin,andEitanGrinspun.Discretequadraticcurvatureenergies.CAGD,2007.ToappearinthespecialissueonDiscreteDifferentialGeometry.[68]WilliamWelchandAndrewWitkin.Variationalsurfacemodeling.InProceedingsofACMSIGGRAPH,pages157–166,1992.[69]DongXu,HongxinZhang,QingWang,andHujunBao.Poissonshapeinterpolation.InProceedingsoftheACMSymposiumonSolidandPhysicalModeling,pages267–274.ACMPress,2005.[70]GuoliangXu.DiscreteLaplace-Beltramioperatorsandtheirconver-gence.ComputerAidedGeometricDesign,21(8):767–784,2004.[71]Y.Yu,K.Zhou,D.Xu,X.Shi,H.Bao,B.Guo,andH.-Y.Shum.MesheditingwithPoisson-basedgradienteldmanipulation.ACMTransac-tionsonGraphics(ProceedingsofACMSIGGRAPH),23(3):644–651,2004.[72]RhalebZayer,ChristianR¨ossl,ZachiKarni,andHans-PeterSeidel.Harmonicguidanceforsurfacedeformation.InComputerGraphicsForum(ProceedingsofEurographics),pages601–609,2005.[73]KunZhou,JinHuang,JohnSnyder,XinguoLiu,HujunBao,BainingGuo,andHeung-YeungShum.Largemeshdeformationusingthevol-umetricgraphLaplacian.ACMTransactionsonGraphics(ProceedingsofACMSIGGRAPH),24(3):496–503,2005.[74]DenisZorin,PeterSchr¨oder,andWimSweldens.Interactivemultiresolu-tionmeshediting.InProceedingsofACMSIGGRAPH,pages259–268.ACMPress/Addison-WesleyPublishingCo.,1997. MarioBotschisapost-doctoralresearchassociateandlecturerattheComputerGraphicsLaboratoryofETHZurich.HereceivedhisMSinMathematicsfromtheUniversityofErlangen,Germany,in1999.From1999to2000heworkedasresearchassociateattheMax-PlanckInstituteforComputerScienceinSaarbr¨ucken,Germany.From2001to2005heworkedasresearchassociateandPhDcandidateattheRWTHAachenUniversityofTechnology,fromwherehereceivedhisPhDin2005.Hisresearchinterestsincludegeometryprocessingingeneral,andmeshgeneration,meshoptimization,shapeediting,andpoint-basedrepresentationsinparticular. OlgaSorkineisapostdoctoralresearcherattheComputerGraphicsGroupofTechnischeUniversit¨atBerlin.ShereceivedtheBScdegreeinmathematicsandcomputersciencefromTelAvivUniversityin2000andcompletedherPhDincomputersciencein2006,alsoatTelAvivUniversity.Herresearchinterestsareincomputergraphicsandincludein-teractivegeometricmodeling,shapeapproximation,shapeandimagemanipulation,computeranimationandexpressivemodelingtechniques.