It is one of the most challenging elements too d ue to the large number of hairs on a human head their length and th eir complex interactions Hair appearance in rendering and si mula tion is dominated by collective properties yet most of the current ID: 10682
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VolumetricMethodsforSimulationandRenderingofHairLenaPetrovicMarkHenneJohnAndersonPixarTechnicalMemo#06-08PixarAnimationStudiosAbstractHairisoneofthecrucialelementsinrepresentingbelievabledigi-talhumans.Itisoneofthemostchallengingelements,too,duetothelargenumberofhairsonahumanhead,theirlength,andtheircomplexinteractions.Hairappearance,inrenderingandsimula-tion,isdominatedbycollectiveproperties,yetmostofthecurrentapproachesmodelindividualhairs.Inthispaperwebuildontheexistingapproachestoilluminationandsimulationbyintroducingavolumetricrepresentationofhairwhichallowsustoefcientlymodelcollectivepropertiesofhair.Weusethisvolumetricrepre-sentationofhairtodescribehairresponsetoillumination,hairtohaircollisions,andtosubtlydirectsimulationsofhair.Ourmethodproducesrealisticresultsfordifferenttypesofhaircolorsandstylesandhasbeenusedinaproductionenvironment.1IntroductionLonghairisoneofthemostinterestinganddifcultmaterialstorepresentincomputergraphics.Itisintriguingthatalthoughtheopticalanddynamicalpropertiesofsinglehairsareactuallyverysimpleandwellunderstood,thecompositepropertiesofahairvol-umecanbeextremelycomplicated.Thesecollectivepropertiesgiverisetochallengingcomputationalproblemssincethedynamicsoflonghairisalmostcompletelydominatedbymillionsofhairtohaircollisionswhiletheopticalpropertiesarestronglyinuencedbymillionsofmicroscaleshadows.Inordertoefcientlyemulatethesecomplexinteractionswerepresentthehairasavolume,andcalculateitsbulkproperties.Mostcurrentapproachestosimulatingandrenderinghairrelyonmethodsthattakeindividualhairsandapplyparameterizedrep-resentationsfortheinteractionswiththeenvironment.Inrender-ing,examplesofsuchapproachesincludetheilluminationmodelofKajiyaandKay[1989]inconjunctionwithLokovicandVeach's[2000]deepshadowmapapproximationstohairshadowing.InsimulationAnjyoetal.[1992],andRosenblumetal.[1991]havedevelopedsinglehairmass-springdynamicmodels.Changetal.[2002]simulateonlyasmallnumberofhairstrands,calledguidehairsandinterpolatethemotionoftherestofthehairs.Inthesemodelsilluminationandsimulationpropertiesofasinglehairarewellrepresented,butthecomplexhair-to-hairinteractionsaretooexpensivetocompute.HadapandMagnenat-Thalmann[2001]haveaddeduiddynamicalforcestoaugmenttheirkeyhairdy-namicmodelwithtermsrepresentinghair-to-hairinteractions. Figure1:Simulatedandshadedvolumetrichair.Ourworkbuildsuponthesemodelsthroughtheadditionofvol-umetricapproachestodynamicsandrendering.Ourapproachisbasedontheobservationthathairbehavesandisperceivedtobeabulkmaterialinitsinteractionswiththeenvironment.Forexample,weoftenperceivetheappearancehairasasurfacewithilluminationeffectscorrespondingtosurface"normals"eventhoughthatsurfacedoesnotproperlyexist.InclassicalPhongillumination,complexscatteringoflightoffsurfacemicrofacetshasbeensimpliedtoanobservableeffectoflightreectingaboutasurfacenormal.InthispaperweutilizethelevelsetsurfacesofOsherandFedkiw[2002]andSethian[1993]toautomaticallyconstructhairsurfaces.Fromthesehairsurfaceswederivenormalstobeusedinillumination.Similarlytotheopticalbehavior,wecanoftenviewthedynami-calbehaviorofahairbodyasacollectivematerial.Thismaterialischaracterizedbyresiliencetoshear,resiliencetocompression,andviscousdamping.Representingtheinteractionofeachindividualhairwithitsenvironmentisprohibitivelyexpensive.Wedevelopavolumetricmodelthatallowsustocheaplyaugmentsinglehairdynamicstoincludehairtohairinteractions.Sincehaircanoccludeoraccentuatefacialfeatures,itisacru-cialperformanceelement,andwewouldliketheabilitytocontrolitsmotion.Hairsimulation,likeallphysicalsimulation,isdifculttodirectwhilepreservingtheplausibilityofthemotion.Treuilleetal.[2003]haveintroducedavolumetricmethodfordirectingsim-ulationsofunconnectedparticlessuchassmoke.Inspiredbytheirapproach,weintroduceasimulationforcebasedonvolumetrichairdensity,whichdirectsagroupofconnectedhairparticlestowardsthedesiredshape.1 Inthispaperwealsobuildonanalogousphysicalmodelsofvol-umessuchasthedynamicalbehaviorofviscoelasticmaterialsandtheopticalbehaviorofcloudstoaugmentthepropertiesofthedy-namicandrenderedindividualhairs.2RelatedWorkKajiyaandKay[1989]considereachhairasaninnitelythincylin-der,andcalculatethediffuseandspecularlightresponsebasedonthe3Dvectorrepresentinghairdirection.Banks[1994]general-izesKajiya'sapproachtoarbitrarydimensions.Lengyel[2000]andLengyeletal.[2001]bringthemethodofKajiyaandKay[1989]andBanks[1994]totherealmofrealtimebyrenderingthevolu-metricdataasasetoftexturemappedslicesthroughthevolume.BeingaKajiyabasedmodel,itsuffersfromthesamedrawbacks,asexplainedinsection4.LokovicandVeach[2000]introducedeepshadowmapstoap-proximatehairselfshadowing.Mertensetal.[2004]implementarealtimeGPUtechniquesimilartodeepshadows.Deepshad-owsaremapsinwhicheachpixelencodesapiecewiselinearfunc-tiondescribinghowtheintensityoflightchangesalongthelightraygoingthroughthatpixel.Sincedeepshadowsaredependentonthelocationofthelight,theyneedtoberecomputedforeachlightinthescene,makingthemexpensiveforuseinrendering.Forprofessionallightersusedtoworkingwithsurfacesembeddedin3Dspace,deepshadows,duetotheirlackofsurfacenormals,donotprovideintuitiveartisticcontrols.Aphysicallybasedmodelofhair-lightinteractionhasbeenrecentlydescribedbyMarschneretal.[2003].Intheirworktheyconsiderlightscatteringinsideasin-glehair,andtraceuptotwoscatteredrays.Thismethodproduceshighlyrealisticpicturesofdarkhair,wheremostofthelightisab-sorbedaftertwoscatterevents.Forblondhair,however,wherelightscattersmultipletimesoffneighboringhairsbeforeitisabsorbed,thisapproachdoesnotachievethesamelevelofrealismasfordarkhair.InsimulationAnjyoetal.[1992],andRosenblumetal.[1991]havedevelopedsinglehairmass-springdynamicmodels.Inthisapproachthedynamicsofasinglehairarewellrepresented,butthecomplexhair-to-hairinteractionistooexpensivecomputationally.Toaddressthisproblem,Planteetal.[2001]introducealayeredwispmodeltorepresenthairclusters.Insimulationthesewispscol-lideorslideagainsteachotherdependingontheirorientation.Wardetal.[2003]extendPlante's[2001]workbyintroducingahierar-chicalmodelofselectivelysubdividedgeneralizedcylinders.EachcylinderrepresentsalockofhairwhichissubdividedbasedonanLODmetric.SimilarlyBartailsetal.[2003]useadaptivewisptreestomodeldynamicsplittingandmergingofhairclusters.Themetricforsubdividingthewisps,unlikeWard's[2003]renderingLOD,isbasedonlocalcomplexityofsimulatedmotion.Thismethodisbestsuitedfordrasticmovementofhair.Alloftheabovemethodsrelyonrepresentinghairbystaticordynamicclusters.Insimulationresultsproducedbythesemethods,theclusterconstructionofhairisoftenapparent.Bandoetal.[2003]haveintroducedavolumetricapproachtosimulatinghair-to-hairinteractions.Theirsimulationmodelisbasedonlooselyconnectedparticlesandtheconnectivityofhairchangesduringsimulation.Bycontrastwemodelindividualhairsasstrands,anduseaxedgridvolumetricapproachforhair-to-hairinteractions.Thisallowsustosimulateanddirecthair-dosstylizedwithhairgel,wheretherestshapeofhairsiscurved(gure6(a)andgure1).TheworkbyHadapandMagnenat-Thalmann[2001]isinmanywaysthemostrelevantwithrespecttoourdiscussionofsimulationtechniques.Intheirpapertheyfollowasimilarapproachtooursinthattheymixapointchainhairmodelwithuidlikeforcesrep-resentinghairdynamics.Theimplementationofthetwosystemsishoweverquitedifferent.TheirapproachfollowsthetheSmoothParticleHydrodynamics(SPH)formalismwhichrepresentshair-to-hairforcesasparticle-to-particleforcesusingoverlappingkernelswhileourapproachisbuiltonagriddedapproachsimilartoanEu-lerianuidmodel.Wehavechosenthegridapproachbothforthecommonalityofimplementationwiththelevelsetrenderingmeth-odsaswellastheextensibilityofthisapproachtoallowustoaddadditionalfunctionalitysuchasdensitytargetingandorientationdependenthair-to-haircoupling.Trueilleetal.[2003]havedevelopedavolumetricmethodfordirectingsimulationsofsmoke.Theyintroducesimulationforcesbasedondensitydifferencesbetweenthegivenanddesiredshape.Sinceweconsiderhairtobeaparticlevolumewithhighdegreeofconnectivity,weutilizeasimilarmethodforcreatingvolumetricforces,inthesimulationofconnectedhairparticles.3VolumetricRepresentationofHairAswehavementionedaboveoneofthechallengesofdealingwithcomputergraphicshairsistheirsheernumber.WeutilizetheideasofChangetal.[2002]andWardetal.[2003]andconsideronlyasubsetofthehairs,whichwecallkeyhairs.Therestofthehairsareinterpolatedfromtheserepresentativekeyhairs. Figure2:(a)Hairmodel(b)SigneddistancefunctionOurkeyhairrepresentationissufcientlynesothatthelinearinterpolationofkeyhairsisnotvisibleinrendering,andwehaveenoughinformationtocomputecollectivepropertiesofhair;yetitissufcientlycoarsethatthecomputationofthebulkpropertiescanbeperformed.WeuseaCartesianvoxelrepresentationtorepresentbothilluminationandsimulationpropertiesofhair.Weconstructavolumetricrepresentationofthehairmass,bycalculatingthekey-hairdensityateachvertexofthevoxelgrid.Eachkeyhairisorigi-nallyrepresentedasacontrolhullofaB-spline.Tocreatethevol-ume,wesumupspatialinuencesofkeyhaircontrolvertices.Theinuenceofakeyhairvertexisa3Dtentfunction,withvalueoneatthekeyhairvertexlocationandlinearlydecreasingalongeachofthecoordinateaxes,becomingzeroatadistanceequaltothegridunit.Thus,thehairdensityateachvoxelvertexis:Dxyz=Xi(1 jPix xj)(1 jPiy yj)(1 jPiz zj);(1)whereDxyzisthedensityatthevoxelgridvertexlocatedat(x;y;z),(Pix,Piy,Piz)aretheith'spointscoordinatesinworldspace,andthesumisonlyoverpointsthatlieinthegridcellsadja-centtothevertex.2 Figure3:(a)Hairisosurface(b)Normalsillumination(c)KajiyailluminationWeapproximatehairtohaircouplingbydiffusiononthevoxelgrid.Intuitively,thiscorrespondstocorrelatingthebehaviorofneighboringkeyhairs.Fromthisvolumetricdensityeld,wecon-structahairisosurface,fromwhichweobtainashadingnormal.Adensityweightedvelocityeldiscalculatedonthegridandisusedtoimplementmomentumdiffusion,mimickinghairtohairfrictionandcollisions.Weusethevolumetrichairdensityrepresentationtodirecthairsimulation,bycreatingsimulationforcesbasedonthedifferencesbetweengivenanddesireddensityelds.4IlluminationWerenderhairgeometrybydrawingeachindividualhairasaB-spline.Toshadethehairs,webaseourilluminationalgorithmonKajiyaandKay's[1989]illuminationmodel,treatingeachhairasaninnitelythincylinder.Kajiyailluminationconsiderseachhairindividually,andcalculatesanilluminationresponsebasedonthehair'stangentvector,independentlyofthesurroundinghairs.Withtheadditionofdeepshadows[LokovicandVeach2000]weachievetheeffectofhairselfshadowing.Inreality,lightscatteringfromneighboringhairsaccountsformuchoftheilluminationonanindi-vidualhair.Webuilduponthesemodelsbyaddinganapproxima-tionoflightscatteringoffneighboringhairs.Weapproximatelightscatteringbyconstructingasurfacenormalaboutwhichthelightrayreects,similarlyinspirittoPhongillumination.4.1IlluminationNormalsfromDensityVolumesHairilluminationmodelshavethecommonproblemofhavingtolighta1Dobjectembeddedina3Dworld.However,unlikestan-dardsurfaceilluminationmodels,hairilluminationdoesnothaveanavailablenormalfromwhichtocomputethelightingresponse,sincehairisaninnitelythincylinderhavingnowelldenednor-malatanygivenpoint.Giventhevolumetrichairdensityrepresen-tationinequation(1)wecanconstructthisnormal.Inordertocalculatethenormal,weconstructanisosurfaceatadesiredhairdensity(gure3(a)).Wecallthisisosurfacethehairshell.FollowingthelevelsetapproachofSethian[1993],andOs-herandFedkiw[2002],wesolveforthesigneddistanceeldSfrom:jrSj=1withS=0attheisosurface.Asigneddistancefunctionsisascalarvolumetricfunctionwhichrepresentsdistancetothenearestpointontheisosurface.Negativesigneddistancevalues,showninblueingure2(b),representtheinteriorofthehair.Unlikethedensityeld,thesigneddistanceeldissmoothlydifferentiableacrossthehairshellboundary(blackingure2(b)).Figure2(b)showstwocuttingplanes(orange)throughthehairvolume(showninblue),andgure2(a)showsthecharacterandherhairfromwhichwebuildthelevelset.Derivativediscon-tinuitiesinthesigneddistanceeldoccurfarfromourregionofinterest,inthemiddleofthehairshell,wheretherearemultipleminimaldistancedirections(whiteregionsingure2(b)).WecanobtainhairnormalsbyprojectingeachhairB-splinecon-trolvertexontothehairshellandreadingthenormalofthesur-face.However,doingtheprojectionisslow.Insteadweusethegradientofthesigneddistanceeldasnormalsinthehairvol-ume:N=rS:(2)Wenowhaveawelldenedhairnormalforagivenvolumeofhair,whichwecanuseinshading.Ingure3theimageontheleftistheisosurfacerepresentinghair.Figure3(b)isourhairilluminationusingnormalsfromthehairisosurface,andgure3(c)istheKajiyailluminatedhair.Noticeingure3howtheshadowonthehairshellingure3(a)andonthehairingure3(b)lookthesame.Also,notethattheshadowterminatorsontheneckandhairlineupingure3(a)and(b),whileingure3(c)theshadowonthehairhasreceded.Thehairvolumechangesfromframetoframewithanimationandsimulationofhair.Thenormalsdescribedaboveareonlycom-putedoncefortherestpositionofthehair.Insteadofrepeatingtheprocedureforeveryframe,wetransformtherestposenormalstofollowthehairdeformation.Totransformthenormalsfromframetoframe,weexpresstheminlocalcoordinateframesalongthelengthofthehair.Asthehairmovesthesecoordinateframeswillchangesmoothly,andwecanusethemtoreposethenormals(usingthesamelocalcoordinates).Tocomputethecoordinateframeswestartattheroothairvertex,withacoordinateframexedwithrespecttothehead,withthexaxisalongthehairdirection.Therstsegmentofthehairisrigid.Wethenpropagatethisframealongthehairusingparalleltransportframes[Bishop1975]betweensuccessivecontrolvertices.Theseframescomputetheminimumrotationtoaccountforthechangeinhairdirectionbetweentwoconsecutivevertices.5SimulationDuetoitslargenumberofdegreesoffreedomhairhasalwaysbeendifculttosimulate.Weresorttosimulatingonlythekeyhairs,andinterpolatingthemotionoftherestofthehairs.Werepresentthedynamicsofanindividualkeyhairbyspringsconnectingpointmasses.Thesespringsresiststretchingandtwistinginthehair,andproduceplausiblemotionforasinglehair.Incollisionswithnon-hairgeometry,suchasscalp,ahairstrandistreatedasasetofparticlesatthepointmasspositions.Treatingeachhairindividu-ally,however,doesnotaccountforhairtohaircollisionandfrictionduringsimulation.Wehavedevelopedaninexpensivevolumetricmethodtomodeltheseinteractions.3 Sincehaircanoccludeoraccentuatefacialfeatures,itiscrucialinenhancingacting,andwewouldliketheabilitytocontrolitsmotion.Hairsimulation,likeallphysicalsimulation,isdifculttodirect,whilepreservingplausibilityofthemotion.Insteadofover-constrainingthesimulationandintroducingaforcewhichpushesagivenhairparticletoadesiredposition,weintroduceaweakerforcewhichpushesagroupofhairparticlestowardsadesireddensity. abFigure4:(a)Hairself-intersecting.(b)Velocitysmoothedhair.5.1HairtoHairInteractionWecreatecomputationallycheapandrealistichairtohairinterac-tionbyspatiallydiffusinghairparticlevelocities.Insteadofmod-elingthecollisionofeachhairparticle,wegrouphairparticlesandtheircorrespondingvelocitiesintoa3Dvoxelgrid.Eachgridvertexrepresentsanaveragevelocityofallthehairparticlesintheadjacentvoxels.Inequation(1)wepopulatedthegridwithhairdensities.Thistimewepopulatethegridwithhairvelocities:Vxyz=Pi(1 jPix xj)(1 jPiy yj)(1 jPiz zj)vi Dxyz;whereVxyzistheaveragevelocityatthevoxelvertex(x,y,z),(Pix,Piy,Piz)aretheith'sparticlecoordinatesinworldspace,viistheparticlevelocity,Dxyzisthevoxeldensityfromequation(1),andthesumlikeinequation(1)isoverpointsthatlieinthegridcellsadjacenttothevertex.Theprocessofaveraginghasalreadycorre-latedvelocitiesofnearbyhairparticles.Wesmooththeaverageve-locitiesfurtherbyusingalterkernelontheCartesiangrid,whichconservesthetotalenergy.Adragtermforeachhairisthencom-putedbasedonthevelocitydifferencebetweenthehairparticleandthegridvelocity.ThistreatmentisequivalenttothesecondorderLaplaciandiffusionwhichoccursinNewtonianuidsandwillal-waysleadtothedissipationofenergy.Forexample,ifaparticleismovingfasterthanitsneighbors,withvelocitysmoothingitwillslowdown.Usingthistechniqueinterpenetratinghairlockscanbeavoidedbyconstrainingnearbyhairstomoveatsimilarvelocities.Fig-ure4showsstillsfromahairsimulationwithandwithoutvelocitysmoothing.Ingure4(a)thelockoflighterhairinthehighlightedregioncomesfrominterpenetratinglocks,whileingure4(b),thisregionissmooth.Seethevideoforthesimulatedloop.5.2DirectingHairAswementionedabove,inordertoachieveanatural,believablemotionwecreateaforcewhichdirectsthehairtowardthedesired(target)shape,whileintroducingminimalamountofenergyintothesimulation.Usingequation(1)wecreatedensitygridsforboththestartingshapeandthetargetshape.Tomatchthedensityofthestartingshapetothedensityofthetargetshapewecreateagradientforcebetweenthetwogrids.Theenergyinthegridis:E=1 2Xx;y;z(Ds Dt)2(3)whereDsisthestartingdensityandDtisthetargetdensity.Usingequation(3)theforceonparticlePiisthen:F=(@E @Px;@E @Py;@E @Pz)=Xx;y;z(Ds Dt)(@Ds @Px;@Ds @Py;@Ds @Pz)Let:D=Ds Dtax=(1 jPx xj)ay=(1 jPy yj)az=(1 jPz zj)TakingthederivativeofDfromequation(1)withrespecttoPixweget:Fix=(ay)(az)(D(x;y;z) D(x+1;y;z))+(1 ay)(az)(D(x;y+1;z) D(x+1;y+1;z))+(ay)(1 az)(D(x;y;z+1) D(x+1;y;z+1))+(1 ay)(1 az)(Dd(x;y+1;z+1) Dd(x+1;y+1;z+1))togetarounddiscontinuitiesinthederivative,ifPx;Py;orPzequalx;y;orz,respectively,wetakethederivativetobe0.Similarly,weobtainFyandFz.Weintroducetheseforcesintothesimulationofourspring-masssystem.Figure5showsthetargetshape,thestartingshapeandtheshapeachievedonlybyapplyingdensityforces.Thetargetshapeisasin-gleframefromasimulationofwindblowinginthehair.Todrivethestartingshapetowardsthetargetshape,weonlyuseindivid-ualspring-masshairdynamics,andthedensityforcesdescribedabove.Thereisnogravity,inertia,orcollisionsinthesimula-tion.Thesimulationisdamped,andafterafewoscillations,hairsettlesintotheshapeingure5(c).Forthewholesimulationseethevideo.Noticethatthetargetshapeandtheachievedshapearenotexactlymatched.Theshapedifferencesareduetotheindivid-ualhairspringsresistingdeformation,andtothecoarsenessofthevoxelgrid.Byforcingdensitiesbetweenthetargetandthesimulatedshapetobethesame,insteadofforcingspecichairparticlestobeinthesameposition,weareimposinglessofaconstraintonthesystem,thusdecreasingtheamountofexternalenergyintroducedintothesimulation.Ifthetargetshapeissignicantlydifferentfromthestartingshape,itmaynotbeclearinwhatdirectiontoproceedtoobtainthetargetshape.Inthiscasewestartoffthesimulationwithalowresolutiondensitygrid.Asthesimulationprogressesandtheshapesbecomecloserweincreasethegridresolution.6DiscussionandFutureWorkInconclusion,wehavepresentedvolumetricmethodsforhairillu-mination,hairtohaircollision,anddirectinghair.Bycreatingasurfacerepresentationofthehairvolumewehavecreatedsmoothlyvaryingcoherentnormalstoapproximatelightscattering.Usingthisapproachinilluminationwecanproducecompellingimagesforawiderangeofhaircolor,length,thickness,andstyleasdemon-stratedingure6(a).Usingvolumetricrepresentationofhairwecancreateinexpensivehairinteractioninsimulation,preventingthe4 Figure5:(a)Startingshape(b)Targetshape(c)Achievedshapehairsfrompassingthrougheachother(gure4).Byaddinginden-sitygradientforceswecandirectthehairsimulationtoachieveadesiredshape(gure5).Inconjunctionwiththemass-springsystemrepresentingindividualhairswesimulatehighlystylizedorveryloosehairstyles,asshowningure6(b).Inthisworkwehavepresentedavolumetricrepresentationofhairforsimulationandillumination,whichbuildsontheprevi-ouslyexistingsinglehairmodels.InilluminationwehaveappliedaBRDFmodeltohair.Anaturalextensionistoapplyasubsur-face,BSSRDreectionmodel[WannJensenetal.2001]toen-hancetherealism.Insimulationwehavedevelopedaspatialve-locitycorrelationltertorepresenthairtohairinteraction.Fordirectingsimulationwehaveusedaforwarddifferencing,densitytargetingscheme.Amorerobustapproachwouldbetouseavaria-tionalscheme[Treuilleetal.2003]usingshootingoradjointmeth-odswhichwouldallowtheuseofsmaller,moresubtletargetingforcestoachievethedesiredshape.7VideoCaptionsIncludedwiththepaperaresixvideoclipsdemonstratingourtech-niques:1.SuppVideo_0391.mp4showstheconstructedlevelsetsurface.2.AnonSupp1_0391.mp4showshairshadedusingthenormalsderivedfromthehairsurface.3.AnonSupp2_0391.mp4showsakeyhairsimulationwithoutanyvelocitysmoothing.4.AnonSupp3_0391.mp4showsakeyhairsimulationwithve-locitysmoothing.5.AnonSupp4_0391.mp4showsafullhairrenderingoftheaboveloop,VelocitySmoothing.mp4.6.AnonSupp5_0391.mp4showsasimulationofhairdirectedtowardatargethairstyleusingdensityforces.Notethatinthissim-ulationwehaveturnedoffhair-bandcollisionsinordertodemon-strateonlydensityforces.ReferencesANJYOI.K.,USAMIY.,ANDKURIHARAT.1992.Asamplemethodforextractingthenaturalbeautyofhair.ComputerGraphics(Proc.SIG-GRAPH),111120.BANDOY.,CHENB.Y.,ANDNISHITAT.2003.AnimatingHairwithLooselyConnectedParticlesComputerGraphics(EurographicsCom-puterGraphicsForum),411-418.BANKSD.1994.IlluminationinDiverseCodimensions.ComputerGraph-ics(Proc.SIGGRAPH),327-334.BERTAILSF.,KIMT-Y.,CANIM-P.,NEUMANNU.2003.AdaptiveWispTree-amultiresolutioncontrolstructureforsimulatingdynamicclusteringinhairmotionComputerGrahpics(EurographicsSymposiumonComputerAnimation).BISHOP,R.L.March1975.Thereismorethanonewaytoframeacurve.(AmericanMathematicsMonthly),246-251.CHANGJ.T.,JINJ.,ANDYUY.2002.ApracticalmodelforhairmutualinteractionsComputerGraphics(Proc.ACMSIGGRAPHSymposiumonComputerAnimation),7380.HADAPS.,ANDMAGNENAT-THALMANNN.2001.ModelingDynamicHairasaContinuumEurographics2001KAJIYA,J.,ANDKAY,T.1989.RenderingFurwithThreeDimensionalTexturesComputerGraphics(Proc.SIGGRAPH),271280.LENGYELJ.2000.Real-timefur.ComputerScience(EuropgraphicsRen-deringWorkshop2000),243-256.LENGYELJ.,PRAUNE.,FINKELSTEINA.,HOPPEH.2001.Real-timefuroverarbitrarysurfaces.ComputerScience(SymposiumonInteractive3DGraphics),227-232.LOKOVIC,T.,ANDVEACH,E.2000.DeepShadowMapsComputerGraphics(Proc.SIGGRAPH),385392.MAGNENAT-THALMANNN.,HADAPS.,KALRAP.2000.StateoftheArtinHairSimulationInternationalWorkshoponHumanModelingandAnimation,39.MARSCHNERS.,WANNJENSENH.,CAMMARANOM.,WORLEYS.,ANDHANRAHANP.,780-7902003.LightScatteringfromHumanHairFibersComputerGraphics(Proc.SIGGRAPH).MERTENST.,KAUTZJ.,BEKAERTP.,VANREETHF.2004.ASelf-ShadowAlgorithmforDynamicHairUsingDensityClusteringCom-puterGraphics(EurographicsSymposiumonRendering2004).OSHERS.J.,FEDKIWR.P.LevelSetMethodsandDynamicImplicitSur-facesSpringerVerlag,2002.PLANTEE.,CANIM.P.,POULINP.2001.ALayeredWispModelforSimulatingInteractionsInsideLongHairComputerScience(ComputerAnimationandSimulationProceedings).ROSENBLUMR.,CARLSONW.,ANDTRIPPE.1991.Simulationthestruc-tureanddynamicsofhumanhair:Modeling,renderingandanimation.JournalofVisualizationandComputerAnimation2,141148.SETHIANJ.A.LevelSetMethodsandFastMarchingMethods:Evolv-ingInterfacesinComputationalGeometry,FluidMechanics,ComputerVision,andMaterialsScienceCambridgeUniversityPress,1993.TREUILLEA.,MCNAMARAA.,POPOVI´CZ.,ANDSTAMJ.,716-7232003.KeyframeControlofSmokeSimulationsComputerGraphics(Proc.SIGGRAPH).5 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