We present methods for synthesizing 3D shape features on subdivision surfaces using multiscale procedural techniques Multiscale synthesis is a powerful approach for creating surfaces with different levels of detail Our methods can also blend multipl ID: 26750
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ProceduralShapeSynthesisonSubdivisionSurfacesELHOERLINEXINGENNINGIERMANNIMPA-InstitutodeMatematicaPuraeAplicadaMediaResearchLab/NYUAbstract.Wepresentmethodsforsynthesizing3Dshapefeaturesonsubdivisionsurfacesusingmultiscale analysissynthesis eeve Figure1:Synthesisandanalysisdiagramsformultiresolutionsurfaces.2MultiresolutionSurfacesHerewebrieyreviewsubdivision-basedmultiresolutionsurfaces;detailscanbefoundin[9,14,18].Subdivisionsurfacescanbeviewedasgeneralizationofsplinestoarbitrarycontrolmeshes.Subdivisiondeasmoothsurfacerecursively,asalimitofasequenceofmeshes.nermeshisobtainedfromthecoarsermeshbyusingasetofxedrenementrulese.g.Loop[8]orCatmull-Clark[4]subdivisionrules.Inourwork,weuseCatmull-Clarksubdivision.Multiresolutionsurfacesextendsubdivisionsurfacesbyintroducingdetailsateachlevel.Eachtimeanermeshiscomputed,itisobtainedbyaddingdetailoffsetstothesubdividedcoarsermesh.Asdetailscanbespeciedonlyatanitenumberoflevels,theprocessreducestostan-dardsubdivisiononcewerunoutofdetails.Theprocessofreconstructingasurfacefromthecoarsemeshandde-tailsiscalledsynthesis(Figure1).Theinverseprocessofconvertingthedataspeciedonaneresolutionleveltothesequenceofdetailsetsandthecoarsestlevelmeshisanalysis.Foranalysis,weneedawayofobtainingthecoarsemeshfromthenemesh.Thiscanbedoneinanumberofways:simpleLaplaciansmoothingorTaubinsmoothing[15],quasi-interpolationortting.Forourpur-poses,quasi-interpolationappearstobethemostsuitableAnaspectofmultiresolutionsurfacesimportantformod-cationoperationsisthatdetailsarerepresentedinlocalframes,whicharecomputedfromthecoarserlevel;thisisanalogoustorepresentingdetailsurfaceintheframecom-putedfromthebasesurface.Notealso,thatwhenwein-cludeproceduralshapesynthesisintothismodel,itispos-sibletogenerateanarbitraryamountofdetailonasmoothsurface. Moreprecisely,thelimitsurfaceisthepointwiselimitofasequenceofpiecewiselinearfunctionsdenedontheinitialcontrolmesh.3ComputationalFrameworkforMultiscaleSynthesisInthissectionwedescribethecomputationalframeworkbehindmultiscaleproceduralshapesynthesis.Weexploitthefactthatsubdivisionsurfacesmakeshapeinformationavailablefordisplayandforeditingasasequenceofsep-aratedifferentlyscaledlevel-of-detailcomponents.Thisstructuregivesustheopportunitytomixdatawithproce-durallygeneratedsyntheticdeformationtextures.Thebasicparadigmistoexpressaproceduraldisplacementasasumofscale-limitedcomponents.Theneachcomponentcanbeusedtomodifytheequivalentlevelofdetailofasubdivisionsurface.Therearetwospatialdomainsinwhichtheprocedu-raldeformationdatacanbedened:(i)Intheunderlying3DEuclideanvolume(asin[10])and(ii)overaparamet-riccoordinatesystemimposedwithinthesurfacemanifold.Wewilldemonstratetheabilitytomixthesetwotogetherinusefulways.Thecomputationalframework(Figure2)startswithasetofshapedenitions.Eachoftheseiseitheranacquiredandstoredshapedescription(e.g.adigitizedskullmesh),orasynthesizedshapesignal(e.g.atorus).Eachshapedenitiontakesasitsdomaineitheranx;y;zcoordinatelocation,orau;vparametricloca-tiononabasesurface.Theoutputofeachshapedenitionisasetofdisplacementcontrolpointsateachscale,Theconstructedshapeisdenedasasmoothreconstructionofthedisplacementcontrolpointsateveryscale,followedbyasumoverallscalesofthereconstructedsignals.Theseshapedenitionsareblendedtogetherviaanal-phasignal.Thealphasignalmayeitherbeinteractivelypaintedbyauser,ordenedprocedurally.Theresultoftheblendingisadetailedsurfacedenition.Thiscanbepost-processedviaashader,toproduceanaldetailedsur-facedenition,whichisthenrendered.Aanimationtimeparametercanfeedinto:(i)anysyn-thesizedshapedenition,(ii)thesynthesizedalpha,and(iii)thepost-processingshader. Input Shape InfoAcquired DataSynthesized CombinerDisplacer PostProcesses Synthesis UserInteraction(Painting) componentsDetailDisplacedSurface FinalDisplacedSurface Figure2:Diagramofcomputationalframework.4DeÞningMultiscaleShapeDetailInthissectionwedescribetheprinciplesofproceduralgen-erationofshapefeaturesonsurfacesandgiveexamplesofproceduralshapemodels.4.1BasicPrinciplesOurmultiscaleproceduralshapesynthesisisaccomplishedthroughtheadditionofgeometricdetailsatthevariouslev-elsofthemultiscaleshapemodel.Forthis,wedesignaproceduraldenitionofthebasicshapefeaturethatwewanttopasteonasurfaceatsomescalelevel.Thisprocedureisafunctionthatsynthesizesthedifferencebetweenthefeatureattwosuccessivescalelevels.Theinputofthefunctionisapointonthesur-faceandascalelevel.Theoutputisadisplacementtobeaddedasadetailatthatlevel,p;lwhereisapointgiveneitherinlocalintrinsicsurfacecoordinatesu;voringlobalextrinsiccoordinatesx;y;z;andisadisplacementrelativetothesurfaceatlevelWehavedesignedandexperimentedwithafewshapedetailproceduralmodels.Thesemodelsexploitthetwoba-siccharacteristicsoftheproceduraldenition:(i)thetypeofcoordinatesand;(ii)themagnitudeofdisplacementsrel-ativetothelevel.ModelsbasedonglobalcoordinatesleadtovolumetricshapedeÞnitions,i.e.featuresaretakenfromthethreedi-mensionalspaceinwhichthesurfaceisembeded.ModelsbasedonlocalcoordinatesleadtosurfaceshapedeÞnitionsi.e.featuresaregrownonthesurface.Someofourmod-elsarebasedonintrinsiccoordinatesandsomeonextrinsiccoordinates.Themagnitudeofthedisplacementsisusuallyrelatedtothescalelevel.Modelsinwhichdisplacementisin-verselyproportionaltoscaleleadtofractal-likefeaturesModelsinwhichdisplacementisdirectlyproportionaltoscaleleadtomorphogenic-likefeatures.Whenthemagni-tudeofthedisplacementisindependentofscale,thefea-turesareessentiallyarbitrary.Thisisappropriateforman-madeshapesorevenphysicalphenomena,suchassmallwaves.Weexperimentedwithallthesekindsofdisplace-ment.Belowwepresentachartrelatingtheaboveclassitionwithsomeproceduralmodelsthatwecreated,andareillustratedbyexamplesinthenextsection. Morphogenic Global(3D) mushroomcloud Local(2D) Ò 4.2ExamplesHereweshowsomeresultsofusingourmultiscaleproce-duralshapesynthesis.Rockisanexampleofavolumetricfractalshapemodel.Thespatialcoordinatesofthereferencesurfaceareusedastheinputofanoisefunctiongenerator.Thedisplacementsaregiveninafashion,whereisrelatedtothescalelevel.TraditionallyaproceduralshaderisdenedasasumofPerlinNoisefunctions[10].However,ifoneworks withinamultiscaleframeworkthatcontainsaB-splinere-constructionlterateverysuccessivescalelevel,itwasdemonstratedin[13]thatitisonlynecessarytospecifyarandomvalueateverycontrolpoint.Thealgorithmisthendoneintwosuccessivepasses,intherstpass,thesurfacepointsateachlevelaregivenarandomperturbationvalue,basedeitheronthex;y;zlo-cationorontheu;v;levelcoordinatesofthebasesurface(whichactsareferenceshape). rock()for(level=0;levelnLevels;level++)forallu;vonthislevelu;v,level)=random() Inthesecondpassthestoreddisplacementvaluesarere-trievedfromtheparametricdomain: Vectorrock detail(u,v,level)value=0for(l=0;llevel;l++)value+=reconstructlevel,l)returnvalue Inpracticewesetthedetailsforasurfacepointatalllevelsinonepass.Thismakestheevaluationofthisprocedure,whereisthenumberofmeshvertices.TherocktextureisusedinsomeoftheexamplesinSection6,andshowninFigure3. Figure3:FractalRock.Berryisanexampleofahybridsurface/volumeshapemodel.TheinitialseedtothecellfeaturesisasetofpointsplacedonthesurfaceaccordingtoaPoisson-diskdistribution.Fromtheseinitialpointsatabaselevel,spheri-caldomesaregrownrecursivelyonthesurfaceateachlevelofdetail.Inthisexample,wedeneacoherentproceduraltex-turewithinasurface,byspreadingasetofequallyspacedseedpoints,asin[16].Thisallowsustodeneabaseleveltexture.Thenwecreateeachsuccessiverecursivedetaillevelbydeningavolumetexturearoundeachseedpointfromthepreviouslevel,todenethepositionsofaclusterofseedpoints.Weusethisstructuretomodifycontrolpointsonthesurface.Ateverylevel,eachcontrolpointonthesurfacewillbeclosesttooneseedpoint.WeusetheEuclideandistancefromthecontrolpointtothatseedpointtoweightaperturbationofthecontrolpointintothesurfacenormaldirection.Inordertoshapethedetailintoasectionofasphere(tocreatethebulgingfeature),weusetheseedsradiusofin.Giventhatthesurfacepointisadistancefromtheseedpoint,wedeneaperturbationintothesurfacenormaldirectionofmagnitude:Thesurfacenormaldirectionisredenedateachscalelevel,basedontheperturbationthathadbeenappliedonthepre-viousscalelevel.Forthisreason,theclusterfeaturesateachlevelgrowoutward,notfromtheoriginalsurface,butperpendicularlytotheevolvingdetailsurface.Figure4showstheconstructionprocessfortheberryshape. (a) (c)Figure4:Berry;(a)Basedomesfrominitialseedpointslevel1ofdetail;(b)rstrecursionaddedtodomeslevel2ofdetail;(c)nalberry3levelsofdetail.Tentaclesisanexampleofsurfacemorphogenicshapemodel.Frominitialseedpointsonthesurfacetentaclesaregrownoutward.Thedirectionandlenghtofthedisplacementsvaryateachlevel.Thedisplacementsaredirectlypropor-tionaltoscale.Belowwegivepseudo-codeoftheshapedetailproce- Vectortentacle detail(Point2seed,intlevel)Scalarmagnitude=reference lenght*levelScalarp=(PI/3)*levelVectordisplacement=(+p),+p),1)(is even(level))thendisplacement*=-1returndisplacement*magnitude (Notethattheintrinsiccoordinatesoftheseedpointmustcorrespondatalllevels.) Figure5illustratesthegrowthprocessofthetentaclefor4levelsofdetail,asthefeaturegrowsfromaseedpoint. Figure5:Growthprocessofthetentacle.Levels1to4.Theshapefeaturemodelhastwoparameters:refer-encelengthrotationangle.Thesetwoparameterscanbeusedformodelingpurposes.Theparameterscanbetime-varyingandbeusedforanimationWeremarkthatthebasicstructureofthetentacleshapedetailmodelcanbeusedasthebasistocreatemanytypesofmodelssuchasthesubmarineexplosivemineshowninFigure6. Figure6:Submarineexplosivemine.Othervariationsofthegrowthmodelarepossible.OneideiaistouseL-Systemstocreatebranchingstructures.Forthistypeofmodel,inadditiontothefeaturegrownfromtheinitalseedpoint,branchingfeaturesaregrownathigherlevelsofdetail.MushroomCloudisanexampleofhybridsurface/volumemorphogenicshapemodel.Thefeaturesgrowfromseedpointsonthesurface,butarebasedonthe3Dcoordinatesintheneighborhoodofeachseedpoint.Thedisplacementisdirectlyproportionaltoscale.Figure7showsanexampleofmushroomcloudfeaturesplacedonasphericalshape. Figure7:Mushroomplanet(inspiredontheplanetfromTheLittlePrinceofSaint-Exupery).5ApplyingMultiscaleDetailtotheSurfaceOncewehavedenedarepertoireofmultiscaleshapede-tailprocedures,wecanusethemtocreatenewshapesfrombaseshapes.Theseprocedurescanbeappliedgloballytoasurface,asshownintheexamplesoftheprevioussection.Wecanalsoapplytheshapedetailproceduresasalo-caloperationtoconstructasinglefeatureatagivenseedpointofthesurface.Thiscanbeaverypowerfulmodelingtoolifappliedinteractively.OursoftwareimplementationisfastenoughtoenableinteractivemodelingonaPentiumIII800Mhzclassmachinewith512MbytesofmemoryandanOpenGLgraphicscard.Inthissection,wedescribesomeresultsofinteractivemodelingusinglocalmultiscaledetailoperations.Wehaveexperimentedwithtwokindsofoperations:featureplace-mentandlocalshapemodication.5.1FeaturePlacementAlocalfeatureplacementoperationconsistsoftheapplica-tionoftheshapedetailprocedureatasingleseedpointofthesurface.Therearetwowaystoimplementthelocalfeatureplace-mentoperation:subordinateorindependentoftheparametriza-tionofthesubdivisionsurface.Inthisworkweadoptedtherstoption,wherefea-turesareplacedonlyatthecontrolpointsofthesubdivisionsurface.Thisoptionreliesonlyonthebasicsubdivisionsurfacestructure.Forthisreason,itissimplerandmoreef-cient,buthasthedisadvantagethatwhentheparametriza-tionisnotuniformsomedistortionscouldhappen.Notethatfeaturescanbeplacedatanyarbitraryinter-mediatelevelofscaleofthesubdivisionsurface.Belowwegivesomeexamplesofapplyingthefeatureplacementatasinglelevelandatmultiplelevels. Notethat,insomecases,thisuniformglobalplacementreliesontheasetofseedpointsevenlydistributedonthesurface. PlacementattheSameLevelWhenfeaturesareplacedatthesamelevel,theyallhavethesamesizeandusuallytheydonotinterferewitheachother.Figure8showsthreeexamplesoffeatureplacementatthesamelevelofthemanequinheadandskullmodels. Figure8:Localfeatureplacementatthesamelevel.PlacementatDifferentLevelsWhenfeaturesareplacedatdifferentlevels,theyhavedifferentsizesandusuallyinterferewitheachother.Thisenablesaverypowerfulmodelingframework.Figure9showsoneexampleoffeatureplacementatdifferentlevels.InFigure9(a),weappliedaspurshapedetailproceduretoseveralpointsonlyatlevel1ofaspher-icalsurface.InFigure9(b),weappliedthesamelocalfeatureoperationsonlyatlevel3ofthesurface.InFig-ure9(c),weappliedthelocalfeatureoperationsatbothlevels.Notethatweobtainedacombinationoffeaturesatdifferentscales. (a) (c)Figure9:Combinationoffeaturesatdifferentlevels.5.2LocalDetailModiÞcationAlocaldetailmodicationoperationconsistsoftheappli-cationofasignalprocessingoperationtodetailsinasmallneighborhoodofapointofthesurface.Theoperationcanattenuateorenhancethedetailsassomelevels.Thisisverymuchinthespiritof[6],wherearangefrequencybandsofthesurfacefeaturesaremodied.Thelocalmethodhastheadvantagethat,ifappliedinteractivelygivesamuchnercontrolofthistechniquetotheuserasamodelingtool.Toimplementthisoperationisimportanttohavetwocomponents:adistancefunctionfromapointonthesur-facethatextendsovertheneighborhoodwherethemodi-cationisapplied;andasmoothdrop-offfunctionofdis-tance.Thesecomponentstogetherprovideawaytoapplythemodicationwithoutcreatingdiscontinuitiesonthesur-face.Inourimplementationwecurrentlyemployatopolog-icaldistancefunctionwithacubicdrop-offkernel.Figure10showsanexampleofsurfacelocalsignalprocessingappliedtotheskullmodel.InFigure10(a)wepresenttheoriginalskullmodelandinFigure10(b)wepresentthemodiedresult.Wesmoothedthenoseareaandenhancedthejawanddetailsontopoftheheadtocreateahornycarnivalmask.Theinteractiveeditsessiontooklessthan5minutes. (a) Figure10:Localsignalprocessingformeshes.6CombiningMultiscaleDetailsInthissection,wedescribemustiscaleshapeblending.Thisisapowerfulshapecombinationoperation,thatcanbeei-therappliedlocallyorglobally.Weremarkthatthereareothermultiscalecombinationoperationsforshapes,whichwedidntconsider,andremainasatopicforfurtherre-search.TheblendingoperationbetweentwomultiscaleshapesisdoneinthesamewaythatBurt[3]denedamulti-scaleblendingbetweentwoimages:ateachscalelevel,atransi-tionoccurswhichisproportionaltothesizeofonesampleatthatscalelevel.AsBurtdemonstratedforimages,theresultisasurfacedenitionthatdoesnothaveanexplicitvisualtransition.Instead,theeffectisthatonetypeofsur-faceisgraduallyandnaturallytransformedintotheother.Thisstepinvolvesonlyalinearcombinationofthetwosig-nals. 6.1BlendingofDifferentProceduralModelsTherustedfuseisanexampleofamultiscaleblendingoftwoshapesgeneratedbydifferentproceduralmodels.Onemodelisarockandtheotherisafuse.ThefuseshapeisasurfaceofrevolutionwhoseproisdenedbyFigure11showstheresultofblendingbetweenthesetwoshapes.Theblendingisspeciedbyaplaneorientedinthedirection.Inordertoavoidaliasingasofttransitionregionisusedtoblendbetweenthethedetailco-cientsofthetwomodels.Thisregionchangesfromleveltolevelaccordingto,whereisaconstantthatdependsonthesizeoftheobject.Notethewhilethetran-sitionissharpatthenestlevel,thecoarselevelfeaturesofoneshapeinuencestheotherbeyondthedividingblending Figure11:Blendingbetweentwoproceduralshapes.6.2CombiningInstancesofaProceduralModelTheplanetisanexampleofcombiningthesameproceduralmultiscaleshapemodelwithdifferentparameters.Theproceduralmodelusedfortheplanetistheonenedfortherock.Thedifferenceisthatinsteadofgen-eratingadisplacementofthesurfacewegenerateanscalarfunctionthatisfedintoapost-processingoperation.Theresultofshapeblendingisthattwovaluesarede-nedateverydetailpoint:(i)amulti-scaleblendedscalarfunctionvalue,and(ii)ablendparameter,between0.0and1.0,whichindicatedtherelativeinuenceofeachsub-planetonthenalscalarvalue.Post-processingOncetheblendingiscomplete,therewillbeasin-glescalarvaluedenedateverydetailpointonthesur-facemesh.Thisscalarvaluerepresentsinformationfromallscalesthatcanbeusedtogeneratefeaturesrequiringnon-linearshaderoperations,suchassnowcaps,mountains,lowlands,lakes,oceans.Thepost-blendingproceduralshaderusesthisscalarvalue,togetherwiththeblendvariable,inarbitrarywaystogeneratethevariousterrestrialfeatures.Theblendparameterisusedtoinuencethecolorproducedbythisproceduralshader.Figure12(c)showsasyntheticplanetwhichisthere-sultblendinginmultiscaleanearth-likeplanet,showninFigure12(a),withanplanet,showninFigure12(b).Notehowthedifferentcharacteristicsofthecoastlinesandtopographyblendseamlessly.Onecansee,scanningacrossindividualfeatureswhichstraddlethetransitionregion,thattheygraduallychangetheir(statisticallydened)appear-ance.Forexample,asinglelakethatappearsjagged,withhighfractaldimension,ononesideofthetransition,gradu-allyturnsintoasmoothlycontouredlake. (a) (c)Figure12:Planet. Importanceofpost-processing:Itisimportantthatsomeportionoftheproceduralshadingcanbedoneafterthemul-tiscaleblendinghasoccurred.Thisallowsthefeaturescre-atedbythatshader,whichmayinvolvenon-linearopera-tions,tobevisuallycoherentacrossthetransitioncreatedbythelinearmulti-scaleblendingoperation.7ConclusionsandFutureWorkWehavedemonstratedhowmulti-scalerepresentationscanenableacquiredshapedataandsyntheticproceduraltex-turegeneratorstobeusedtogetherasapowerfulandgen-eralshapemodelingparadigm.Thesetechniquescanbeap-pliedlocallyandinteractivelytopartsofamodel,andcanbeusedtoseamlesslyfusetogetherandreconcilemodelswhichhavedifferentshapeandtexturalcharacteristics.Theabilitytoworkwithindifferentlevelsofamulti-scalerep-resentationallowsadesignertointeractivelymakechangesatverydifferentlevelsofscale,aswellastorapidlyshiftbetweenlargescaleanddetailwork.Infuturework,weplantousethesetechniquestobuildafully-featuredproceduralshapepaintingandedit-ingsystem.Weplantoincorporateinnitelyzoomablesurfacerepresentations,inaextensionoftherepresentationschemesthatwerepresentedforzoomabletexturalpaintingin[13].Thiswillallowdesignerstocreateprocedurally-enhanceddetailsofarbitraryscale.Thesesurfacerepresen-tationscanrelyonlazyevaluation,sothatthenestvisi-bledetailsofprocedurallyenhancedmultiscaleshapeanddisplacementtexturesneedeverbeevaluatedonlywhencloselyviewed.References[1]AdobeSystems.AdobePhotoshop.Softwarepack-[2]DeborahF.Berman,JasonT.Bartell,andDavidH.Salesin.MultiresolutionPaintingandCompositing.ProceedingsofSIGGRAPHÕ94,pages8590,July[3]P.J.BurtandE.H.Adelson.Amultiresolutionsplinewithapplicationtoimagemosaics.2(4):217236,Oc-tober1983.[4]EdCatmullandJamesClark.RecursivelygeneratedB-splinesurfacesonarbitrarytopologicalmeshes.10(6):350355,1978.[5]DavidS.Ebert,F.KentonMusgrave,DarwynPeachey,StevenWorley,andKenPerlin.TexturingandModeling.MorganKaufmannPublishers,July[6]IgorGuskov,WimSweldens,andPeterSchroder.Multiresolutionsignalprocessingformeshes.Pro-ceedingsofSIGGRAPH99,pages325334,August[7]PaulE.Haeberli.Paintbynumbers:Abstractimagerepresentations.ComputerGraphics(ProceedingsofSIGGRAPH90),24(4):207214,August1990.[8]CharlesLoop.Smoothsubdivisionsurfacesbasedontriangles.Mastersthesis,UniversityofUtah,Depart-mentofMathematics,1987.[9]MichaelLounsbery,TonyDeRose,andJoeWarren.Multiresolutionanalysisforsurfacesofarbitrarytopo-logicaltype.TransactionsonGraphics,16(1):34January1997.[10]KenPerlin.Animagesynthesizer.ComputerGraph-ics(ProceedingsofSIGGRAPH85),19(3):287July1985.HeldinSanFrancisco,California.[11]KenPerlinandEricM.Hoffert.Hypertexture.ComputerGraphics(ProceedingsofSIGGRAPH89)23(3):253262,July1989.[12]KenPerlinandLuizVelho.Awaveletrepresentationforunboundedresolutionpainting.Technicalreport,NewYorkUniversity,NewYork,1992.[13]KenPerlinandLuizVelho.LivePaint:PaintingWithProceduralMultiscaleTextures.In95ConferenceProceedings,pages153160,August[14]K.PulliandM.Lounsbery.Hierarchicaleditingandrenderingofsubdivisionsurfaces.TechnicalReportUW-CSE-97-04-07,Dept.ofCS&E,UniversityofWashington,Seattle,WA,1997.[15]GabrielTaubin.Asignalprocessingapproachtofairsurfacedesign.InSIGGRAPH95ConferencePro-,pages351358,1995.[16]GregTurk.Generatingtexturesforarbitrarysur-facesusingreaction-diffusion.ComputerGraphics(ProceedingsofSIGGRAPH91),25(4):289298,July[17]StevenP.Worley.Acellulartexturebasisfunction.ProceedingsofSIGGRAPH96,pages291294,Au-gust1996.[18]DenisZorin,PeterSchroder,andWimSweldens.In-teractivemultiresolutionmeshediting.ProceedingsofSIGGRAPH97,pages259268,August1997.