a The character and cage shown in black at bindtime b and c are two poses from an animated clip All images DisneyPixar Abstract In this paper we consider the problem of creating and controlling volume deformations used to articulate characters for u ID: 31182
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BindMeanValueHarmonic (a)(b)(c) (d)(e)(f)Figure2:Acomparisonbetweendeformationsbasedonmeanvalueandharmoniccoordinates.Therstrowshowsatorsobendandthesecondrowshowsalegbendforatypicalcharacter.Theleftcolumnshowsthecage(showninblack)andobject(showninbeige).Themiddlecolumnshowsmodiedcagesandthecorrespondingdeformedobjectsusingmeanvaluecoordinates.Therightcolumnshowsmodiedcagesanddeformedobjectsusingharmoniccoordinates.Noticethatthetwomethodsperformsimilarlyforthetorsobend.However,instronglyconcavesituationssuchasthelegs,harmoniccoordinatesproducedsignicantlymorepleasingresults.areaformofgeneralizedbarycentriccoordinatesthathaveanum-berofuses,buttheyareparticularlyinterestinginthecontextofcharacterarticulationbecause:Thecagethatcontrolsthedeformationcanbeanyclosedtri-angularsurfacemesh,sothereisagreatdealoftopologicalandgeometricexibilitywhendesigningthecage.Thecoordinatesaresmooth,sothedeformationissmooth.Thecoordinatesreproducelinearfunctions,sotheobjectdoesn't“pop”whenitisbound.Thatis,thecoordinatesaresuchthatsettingC0itoCiinEquation1resultsinp0reducingtop.However,meanvaluecoordinateshaveadrawbackforuseinchar-acterarticulation,asillustratedbythebipedalcharactershowninFigure2.NoticehowthemodiedcageverticesforthelegontheleftinFigure2(e)signicantlyinuencethepositionofobjectpointsinthelegontheright.Thisoccursbecausemeanvalueco-ordinatesarebasedonEuclidean(straight-line)distancesbetweencageverticesandobjectpoints.Sincethedistancebetweenthemodiedcageverticesandtheobjectpointsinthelegontherightarerelativelysmallinthebindpose,theinuenceisrelativelylarge.Noticetoothatthedisplacementofthoseobjectpointsisinadirec-tionoppositetothedisplacementofthecagevertices.Thisoccursbecausethemeanvaluecoordinatesarenegative,asshowninFig-ure4(b).Thisundesirablemovementisparticularlystrikinginin-teractiveuse,asdemonstratedinaccompanyingvideo(see[Joshietal.2007]).Suchbehaviorisunacceptableforthearticulationofcharactersinfeaturelmproduction.Theundesirablebehaviorillustratedaboveoccursbecausemeanvaluecoordinateslacktwopropertiesthatareessentialforhigh-endcharacterarticulation;namely:Interiorlocality:Informally,thecoordinatesshouldfalloffasafunctionofthedistancebetweencageverticesandobjectpoints,wheredistanceismeasuredwithinthecage.Non-negativity:AsillustratedinFigures2(e)and4(b),ifanobjectpointwhosecoordinaterelativetoacagevertexisneg-ative,theobjectpointandcagevertexwillmoveinoppositedirections.Topreventthisunintuitivebehavior,weseekcoor-dinatesthatareguaranteedtobenon-negativeontheinteriorofthecage,eveninstronglyconcavesituations.Inthispaper,weshowthatcoordinatespossessingthethesetwocriticalpropertiescanbeproducedassolutionstoLaplace'sequa-tion.SincesolutionstoLaplace'sequationaregenericallyreferredtoasharmonicfunctions,wethereforecallthesecoordinateshar-moniccoordinates,andthedeformationstheygenerateharmonicdeformations.1Unlikemeanvaluecoordinates,harmoniccoordinatesdonot,in 1Sinceeachcomponentofaharmonicdeformationisaharmonicfunc-tion,manytextsrefertosuchdeformationsasharmonicmaps.Wepreferthetermharmonicdeformationbecauseofthecontextinwhichthey'reusedinthispaper. Figure3:Meanvaluevsharmonicinterpolation.(a)Thestraight-linepathscorrespondingtomeanvalueinterpolation.(b)TheBrownianpathscorrespondingtoharmonicinterpolation.designedtointerpolateatanitesetofpoints,whereasourmethodisdesignedtointerpolateovertheentirecontinuousboundaryofthecage.2TheoryInthissectionweformalizethediscussionoftheprevioussectionandprovethatharmoniccoordinatespossessallthepropertiesnec-essaryforuseinhigh-endcharacterarticulation.Webeginbyconsideringtheconstructionofmeanvaluecoordi-natesasdescribedin[Floateretal.2005]and[Juetal.2005].Theyderivecoordinatesstartingwitha“meanvalueinterpolant”ofafunctionfdenedonaclosedboundary.Tocomputeaninterpolantvalueforeachinteriorpointp,theyconsidereachpointxontheboundary,multiplyf(x)bythereciprocaldistancefromxtop,thenaverageoverallx(seeFigure3(a)).Thisdenitionmakesitclearthatmeanvaluecoordinatesinvolvestraight-linedistancesirrespec-tiveofthevisibilityofxfromp.Forcharacterarticulation,thecageoftenhaslargeconcavities,andamoreusefulinterpolantwouldre-spectvisibilityofancagevertexfromanobjectpoint.Toconstructsuchaninterpolant,wecanaveragenotoverallstraight-linepaths,butratheroverallBrownianpathsleavingp,wherethevalueas-signedtoeachpathisthevalueoffatthepointthepathrsthitsthecageboundary(seeFigure3(b)).Atrst,thisinterpolantseemsintractabletocompute.However,afamousresultfromstochas-ticprocesses(c.f.[PortandStone1978],[Bass1995])statesthattheinterpolantthusproduced(inanydimension)infactsatisesLaplace'sequationsubjecttotheboundaryconditionsgivenbyf2.Therefore,wecanobtainimprovedgeneralizedbarycentriccoordi-natesfromanumericalsolutionofLaplace'sequationinthecageinterior.Moreformally,letacageCbeapolyhedroninddimensions–thatis,aclosed(notnecessarilyconvex)volumewithapiecewiselinearboundary.Intwodimensions,acageisaregionoftheplaneboundedbyaclosedpolygon(suchastheonesshowninFigure4),andinthreedimensionsacageisaclosedregionofspaceboundedbyplanar(thoughnotnecessarilytriangular)faces.ForeachoftheverticesCiofthecage,weseekafunctionhi(p)denedonCsubjecttothefollowingconditions(listedintheorderthattheyareprovedlater):1.Interpolation:hi(Cj)=di;j.2.Smoothness:Thefunctionshi(p)areatleastC1smoothintheinteriorofthecage.3.Non-negativity:hi(p)0,forallp2C. 2WethankMichaelKassforpointingoutthisconnectiontous.4.Interiorlocality:WequantifythenotionofinteriorlocalityintroducedinSection1asfollows:interiorlocalityholds,if,inadditiontonon-negativity,thecoordinatefunctionshavenointeriorextrema.5.Linearreproduction:Givenanarbitraryfunctionf(p),thecoordinatefunctionscanbeusedtodeneaninterpolantH[f](p)accordingto:H[f](p)=åihi(p)f(Ci)(2)FollowingJuetal.[2005],werequireH[f](p)tobeexactforlinearfunctions.AsshownbyJuet.al,takingf(p)=pmeansthatp=åihi(p)Ci(3)whichisthe“non-popping”conditionmentionedinSection1.6.Afne-invariance:åihi(p)=1forallp2C.7.Strictgeneralizationofbarycentriccoordinates:whenCisasimplex,hi(p)isthebarycentriccoordinateofpwithrespecttoCi.Meanvaluecoordinatespossessallbuttwooftheseproperties,namely,non-negativityandinteriorlocality.Weclaimthatcoor-dinatefunctionssatisfyingallsevenpropertiescanbeobtainedassolutionstoLaplace'sequation52hi(p)=0;p2Int(C)(4)iftheboundaryconditionsareappropriatelychosen.Togainsomeinsightintohowtheboundaryconditionsaredeter-mined,weconsiderrsttheconstructionofharmoniccoordinatesintwodimensions.Itwillthenbeclearhowtheconstructiongener-alizestoddimensions.Forreasonsthatwillsoonbecomeapparent,theappropriateboundaryconditionsforhi(p)intwodimensionsareasfollows.Let¶pdenoteapointontheboundary¶CofC,thenhi(¶p)=fi(¶p);forall¶p2¶C(5)wherefi(¶p)isthe(univariate)piecewiselinearfunctionsuchthatfi(Cj)=di;j.Forexample,ifCisthecageshowninFigure4(a),thenfi(¶p)isthepiecewiselinearfunctiondenedontheedgese1;:::;e19suchthatfi(Cj)=di;j,fori;j=1;:::;19.WenowshowthatfunctionssatisfyingEquation4subjecttoEqua-tion5possessthepropertiesenumeratedabove.1.Interpolation:byconstructionhi(Cj)=fi(Cj)=di;j.2.Smoothness:AwayfromtheboundaryharmoniccoordinatesaresolutionstoLaplace'sequation,andhencetheyareC¥inthecageinterior.Ontheboundarytheyareonlyassmoothastheboundaryconditions,andhenceareonlyguaranteedtobeC0ontheboundary.3.Non-negativity:harmonicfunctionsachievetheirextremaattheirboundaries.Sinceboundaryvaluesarerestrictedto[0;1],interiorvaluesarealsorestrictedto[0;1],andarethereforenon-negative.AnexampleisshowninFigure4(c).4.Interiorlocality:followsfromnon-negativityandthefactthatharmonicfunctionspossessnointeriorextrema.5.Linearreproduction:Letf(p)beanarbitrarylinearfunc-tion.WeneedtoshowthatH[f](p)=f(p),whereH[f](p)isdenedasinEquation2.WebeginbyestablishingthatH[f](p)=f(p)everywhereontheboundaryofC.If¶pisapointontheboundaryofC,thenbyconstructionH[f](¶p)=åihi(¶p)f(Ci)=åifi(¶p)f(Ci)(6) (a)(b)(c)Figure7:Dynamicbinding.(a)Anobjectandcageintheirrestposes.(b)theobjectafterapplyingthearmtwist.Thisistheposeinwhichtheobjectisboundtothecage.(c)Thedeformedobjectafterapplyingtheharmonicdeformation.Dynamicbindingallowsthearmtwistaswellastheharmonicdeformationtobeperformedinreal-time.shouldinsteadsolveforonlythosecoordinatesthatareinu-encedbythemodiedregion.Thatis,weshouldnotre-solveforcoordinateswhose(sparsied)valueiszerointheregionofchange.Happily,thebenetsofthisoptimizationincreaseasthecagebecomesmorecomplicated.Sincecagesarespecicallydesignedtobefairlycoarse,weexpectcagecomplexitiestorangefromtensofvertices(especiallyforlocalusecasessuchastheoneshowninFigure5),tonomorethanathousandverticesforafullbodycageofacomplicatedcharacter.Usingtheimprovementslistedabove,weexpecttoreducefullsolvetimesintothesubminuterangeevenforthemostcomplicatedcageswearelikelytoencounter.Harmoniccoordinatesareaformofgeneralizedbarycentriccoordi-natesandcanbedenedforanydimension.Generalizedbarycen-triccoordinatesarefundamentalbuildingblocksinanumberofotherareassuchastheconstructionofN-sidedsurfacepatches[LoopandDeRose1989]andniteelementanalysis[Wachpress1975].Asasecondareaoffutureresearchitwouldbeinterest-ingtoinvestigatetheuseofddimensionalharmoniccoordinatesinapplicationareasotherthancharacterarticulation.ReferencesBASS,R.1995.ProbabilisticTechniquesinAnalysis.Springer-Verlag.CAPELL,S.,GREEN,S.,CURLESS,B.,DUCHAMP,T.,ANDPOPOVIC,Z.2002.Amultiresolutionframeworkfordynamicdeformations.InACMSIGGRAPHSymposiumonComputerAnimation,ACMSIGGRAPH,41–48.CARR,J.C.,BEATSON,R.K.,CHERRIE,J.B.,MITCHELL,T.J.,FRIGHT,W.R.,MCCALLUM,B.C.,ANDEVANS,T.R.2001.Reconstructionandrepresentationof3Dobjectswithra-dialbasisfunctions.InSIGGRAPH2001,ComputerGraphicsProceedings,ACMPress/ACMSIGGRAPH,E.Fiume,Ed.,67–76.CHOE,B.,LEE,H.,ANDKO,H.-S.2001.Performance-drivenmuscle-basedfacialanimation.TheJournalofVisualizationandComputerAnimation12,2,67–79.DEROSE,T.,ANDMEYER,M.2006.Harmonicco-ordinates.PixarTechnicalMemo06-02,PixarAnima-tionStudios,January.http://graphics.pixar.com/HarmonicCoordinates/.DUCHON,J.1977.Splinesminimizingrotationinvariantsemi-normsinsobolevspaces.InLectureNotesinMathematics,Springer-Verlag,vol.571.FLOATER,M.S.,KOS,G.,ANDREIMERS,M.2005.Meanvaluecoordinatesin3d.ComputerAidedGeometricDesign22,623–631.FLOATER,M.S.,HORMANN,K.,ANDKOS,G.2006.Ageneralconstructionofbarycentriccoordinatesoverconvexpolygons.AdvancesinComp.Math.24,311–331.FLOATER,M.2003.Meanvaluecoordinates.ComputerAidedGeometricDesign20,1,19–27.IGARASHI,T.,MOSCOVICH,T.,ANDHUGHES,J.F.2005.As-rigid-as-possibleshapemanipulation.InSIGGRAPH'05:ACMSIGGRAPH2005Papers,ACMPress,NewYork,NY,USA,1134–1141.JOSHI,P.,TIEN,W.C.,DESBRUN,M.,ANDPIGHIN,F.2006.Learningcontrolsforblendshapebasedrealisticfacialanima-tion.InSIGGRAPH'06:ACMSIGGRAPH2006Courses,ACMPress,NewYork,NY,USA,17.JOSHI,P.,MEYER,M.,DEROSE,T.,GREEN,B.,ANDSANOCKI,T.2007.Harmoniccoordinatesforchar-acterarticulation.PixarTechnicalMemo06-02b,PixarAnimationStudios.http://graphics.pixar.com/HarmonicCoordinatesB/.JU,T.,SCHAEFER,S.,ANDWARREN,J.2005.Meanvaluecoor-dinatesforclosedtriangularmeshes.ACMTrans.Graph.24,3,561–566.LEWIS,J.P.,CORDNER,M.,ANDFONG,N.2000.Posespacedeformation:auniedapproachtoshapeinterpolationandskeleton-drivendeformation.InProceedingsofthe27thannualconferenceonComputergraphicsandinteractivetech-niques,165–172.LOOP,C.T.,ANDDEROSE,T.D.1989.Amultisidedgeneraliza-tionofb´eziersurfaces.ACMTrans.Graph.8,3,204–234.MACCRACKEN,R.,ANDJOY,K.I.1996.Free-formdeformationswithlatticesofarbitrarytopology.InProceedingsofSIGGRAPH'96,AnnualConferenceSeries,181–199.MEYER,M.,LEE,H.,BARR,A.,ANDDESBRUN,M.2002.Gen-eralizedbarycentriccoordinatesforirregularpolygons.JournalofGraphicsTools7,1,13–22. 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