Animating Pictures with Stoc hastic Motion xtures ungY - PDF document

AnimatingPictureswithStochasticMotionTexturesYung-YuChuang1;3DanBGoldman1KeColinZheng1BrianCurless1DavidH.Salesin1;2RichardSzeliski21UniversityofWashington2MicrosoftResearch3NationalTaiwanUniversity(a)JapaneseTemple(b)Harbor(c)BoatStudio(d)Argenteuil(e)SunowersFigure1Sampleinputimagesweanimateusingourtechnique.ThersttwopicturesarephotographsofaJapaneseTemple(a)andaharbor(b).Thepaintingsshownin(c)and(d)areClaudeMonet'sTheBoatStudioandTheBridgeatArgenteuil.WealsoapplyourmethodtoVanGogh'sSunower(e)toanimatetheowers.(ThelastthreepaintingsarecourtesyofWebMuseum,http://www.ibiblio.org/wm/.)Abstractthispaper,weexploretheproblemofenhancingstillpictureswithsubtlyanimatedmotions.Welimitourdomaintoscenescon-tainingpassiveelementsthatrespondtonaturalforcesinsomefash-ion.Weuseasemi-automaticapproach,inwhichahumanuserseg-mentsthesceneintoaseriesoflayerstobeindividuallyanimated.Then,a“stochasticmotiontexture”isautomaticallysynthesizedus-ingaspectralmethod,i.e.,theinverseFouriertransformofalterednoisespectrum.Themotiontextureisatime-varying2Ddisplace-mentmap,whichisappliedtoeachlayer.Theresultingwarpedlayersarethenrecompositedtoformtheanimatedframes.There-sultisaloopingvideotexturecreatedfromasinglestillimage,whichhastheadvantagesofbeingmorecontrollableandofgener-allyhigherimagequalityandresolutionthanavideotexturecreatedfromavideosource.Wedemonstratethetechniqueonavarietyofphotographsandpaintings.CRCategories:I.3.3[ComputerGraphics]:Picture/ImageGeneration—Displayalgorithms;I.4.9[ImageProcessingandComputerVision]:ApplicationsKeywords:Animation,image-basedanimation,image-basedren-dering,naturalphenomena,physicalsimulation,videotexture1IntroductionWhenweviewaphotographorpainting,weperceivemuchmorethanthestaticpicturebeforeus.Wesupplementthatimagewithourlifeexperiences:givenapictureofatree,weimagineitsway-ing;givenapictureofapond,weimagineitrippling.Ineffect,webringtobearastrongsetof“priors,”andthesepriorsenrichourperception.http://grail.cs.washington.edu/projects/StochasticMotionTextures/Inthispaper,weexplorehowasetofexplicitlyencodedpriorsmightbeusedtoanimatestillimagesonacomputer.Thefullyau-tomaticanimationofarbitraryscenesis,ofcourse,amonumentalchallenge.Inordertomakeprogress,wemaketheproblemeasierintwoways.First,weuseasemi-automatic,user-assistedapproach.Inparticu-lar,ausersegmentsthesceneintoasetofanimatablelayersandassignscertainparameterstoeachone.Second,welimitourscopetoscenescontainingpassiveelementsthatrespondtonaturalforcesinsomefashion.Weexplorearangeofpassiveelementsincludingplantsandtrees,water,oatingobjectssuchasboats,andclouds.Themotionofeachoftheseobjectsisdrivenbyasinglenaturalforce,namely,thewind.Althoughthissetofobjectsandmotionsmayseemlimited,theyoccurinalargevarietyofpicturesandpaintings,asshowninFigure1.Wehavefoundthatalloftheseelementscanbeanimatedusingauniedapproach.First,wesegmentthepictureintoasetofuser-speciedlayersusingBayesianmatting[Chuangetal.2001].Aseachlayerisremovedfromthepicture,“inpainting”isusedtollintheresultinghole.Next,theuserannotatesoneormorelayerswithamotionarmature,alinesegmentwhichapproximatesthestruc-tureofalayer.Usingtheseconstraints,wesynthesizeastochasticmotiontextureusingspectralmethods[Stam1995].Spectralmeth-odsworkbygeneratingarandomnoisespectruminthefrequencydomain,applyingaphysicallybasedspectrumltertothatnoise,andcomputinganinverseFouriertransformtocreatethestochasticmotiontexture.Thismotiontextureisatime-varying2Ddisplace-mentmap,whichisappliedtothepixelsinthelayer.Finally,thewarpedlayersarerecompositedtoformtheanimatedpictureforeachframe.Theresultingmovingpicturecanbethoughtofasakindofvideotexture[Sch¨odletal.2000]—although,inthiscase,avideotexturecreatedfromasinglestaticimageratherthanfromavideosource.Thus,theseresultshavepotentialapplicationwherevervideotex-turesdo,i.e.,inplaceofstillimagesonWebsites,asscreensaversordesktop“wallpapers,”orinpresentationsandvacationslideshows.Inaddition,thereareseveraladvantagestocreatingvideotexturesfromastaticimageratherthanfromavideosource.First,becausetheyarecreatedsynthetically,theyallowgreatercreativecontrolintheirappearance.Forexample,thewinddirectionandamplitude canbetunedforaparticulardesiredeffect.Second,consumer-gradedigitalstillcamerasgenerallyprovidemuchhigherimagequalityandgreaterresolutionthantheirvideocameracounterparts.Theseadvantagesallowanimatedstillstobeusedinnewsituationssuchasanimatedmattepaintingsforspecialeffects.Furthermore,theycanbeappliedtosourcesthatexistonlyinastaticformsuchaspaintingsandhistoricphotographs.Forthemostpart,thealgorithmswedescribeinthispaperareap-plicationsoftechniquesfromavarietyofdisparatesourcessuchasimagemattingandinpainting,andphysicallybasedanimationofnaturalphenomena.Weshowhowthesetechniquescanbecom-bined,seamlesslyandsynergisticallyintoaneasy-to-usesystemforanimatingstillimages.Thus,ourmajorcontributionsareinthefor-mulationoftheoverallproblem,includingtherecognitionthataninterestingclassofphenomenacanallbeanimatedattractivelyviaasinglewindsourceusingsimplecontrols;themarshallingofavari-etyoftechniques,mostnotablystochasticmotiontextures,tosolv-ingthisproblem;thedesignofauserinterfacethatallowsnoviceuserstoanimatepictureswithlittleornotraining;andlastly,aproofoftheviabilityandqualityofapplyingimagewarpingapproachestosynthesizingappealinganimatedpictures.1.1RelatedworkOurgoalistosynthesizeastochasticvideofromasingleimage.Hence,ourworkissimilarinspirittotheworkonvideotexturesanddynamictextures[SzummerandPicard1996;Sch¨odletal.2000;WeiandLevoy2000;Soattoetal.2001;WangandZhu2003].Likeourwork,videotexturesfocuson“quasi-periodic”scenes.However,theinputstovideotexturealgorithmsareshortvideosthatcanbeanalyzedtomimictheappearanceanddynam-icsofthescene.Incontrast,theinputtoourworkisonlyasingleimage.workis,inspirit,similartothe“TourIntothePicture”sys-temdevelopedbyHorryetal.[1997].Theirsystemallowsuserstomapa2Dimageontoasimple3Dboxscenebasedonsomeinter-activelyselectedperspectiveviewingparameterssuchasvanishingpoints.Thisapproachallowsuserstointeractivelynavigateintoapicture.Criminisietal.[2000]proposeanautomatedtechniquethatcanproducesimilareffectsinageometricallycorrectway.Morerecently,Ohetal.[2001]developedanimage-baseddeptheditingsystemcapableofaugmentingaphotographwithamorecompli-cateddeptheldtosynthesizemorerealisticeffects.Inourwork,insteadofsynthesizingadeptheldtochangetheviewpoint,weaddmotioneldstomakethescenechangeovertime.Forcertainclassesofmotions,oursystemrequirestheusertospecifyamotionarmatureforalayer,andthenperformsphysically-basedsimulationonthearmaturetosynthesizeamo-tioneld.ItisthereforesimilartothemethodofLitwinowiczandWilliams[1994],whichuseskeyframelinedrawingstodeformim-agestocreate2Danimations.Theirsystemisquiteusefulfortra-ditional2Danimation.However,theirtechniqueisnotsuitableformodelingthenaturalphenomenawetargetbecausesuchmotionsaredifculttokeyframe.Also,theyuseasmoothscattereddatainterpolationtosynthesizeamotioneldwithoutanyphysicaldy-namics.workisalsorelatedtotheobject-basedimageeditingsystemproposedbyBarrettandCheney[2002],namely,objectselection,matteextraction,andholelling.Indeed,Barrettetal.havealsodemonstratedhowtogenerateavideofromasingleimagebyedit-ingandinterpolatingkeyframes.LikeLitwinowicz'ssystem,thefo-cusisonkey-framedratherthanstochastic(temporaltexture-like)motions.etal.[1991]previouslyattemptedtocreatetheillusionofmotioninastaticimageintheir“Motionwithoutmovement”paper.Theyapplyquadraturepairsoforientedlterstovarythelocalphaseinanimagetogivetheillusionofmotion.Whilethemotionisquitecompelling,theband-passlteredimagesdonotlookphotorealistic.Evenearlier,attheturnofthe20thcentury,peoplepaintedout-doorscenesonpiecesofmaskedvellumpaperandusedseriesofsequentiallytimedlightstocreatetheillusionofdescendingwa-terfalls[Hathawayetal.2003].Peoplestillmakethiskindofde-vice,whichisoftencalledakineticwaterfall.AnotherexampleofasimpleanimatedpictureisthepopularJavaprogram,Lakeap-plet,whichtakesasingleimageandperturbstheimagewithasetofsimpleripples[Grifths1997].Thoughvisuallypleasing,theseresultsoftendonotlookrealisticbecauseoftheirlackofphysicalproperties.orkingonaninverseproblemtoours,Sunetal.[2003]proposeavideo-inputdrivenanimation(VIDA)systemtoextractphysi-calparameterssuchaswindspeedfromrealvideofootage.Theythenusetheseparameterstodrivethephysicalsimulationofsyn-theticobjectstointegratethemconsistentlywiththesourcevideo.Theyestimatephysicalparametersfromobserveddisplacements;wesynthesizedisplacementsusingaphysicalsimulationbasedonuser-speciedparameters.Theytargetasimilarsetofnaturalphe-nomenatothosewestudy:plants,waves,andboats,whichcanallbeexplainedasharmonicoscillations.Tosimulatedynamics,weusephysically-basedsimulationtech-niquespreviouslydevelopedincomputergraphicsformodelingnaturalphenomena.Forwaves,weusetheFourierwavemodeltosynthesizeatime-varyingheighteld.Mastinetal.[1987]werethersttointroducestatisticalfrequency-domainwavemodelsfromoceanographyintocomputergraphics.Inasimilarway,wesynthe-sizestochasticwindelds[ShinyaandFournier1992;StamandFiume1993]byapplyingadifferentspectrumlter.Whenapply-ingthewindeldtotrees,sincetheforceisoscillatoryinnature,thecorrespondingmotionsarealsoperiodicandcanbesolvedmorero-bustlyandefcientlyinthefrequencydomain[Stam1997;Shinyaetal.1998].Aokietal.[1999]coupledphysically-basedanimationsofplantswithimagemorphingtechniquesasanefcientalternativetoex-pensivephysically-basedplantsimulationandsynthesis.However,theyonlydemonstratetheirconceptonsyntheticimages.Inourwork,wetargetrealpicturesanduseourapproachasawaytosyn-thesizevideotexturesforstochasticscenes.Oursystemrequiresuserstosegmentanimageintolayers.Tosup-portseamlesscomposites,asoftalphamatteforeachlayerisre-quired.Weuserecentlyproposedinteractiveimagemattingalgo-rithmstoextractalphamattesfromtheinputimage[RuzonandTomasi2000;Chuangetal.2001].Tollinholesleftbehindaf-terremovingeachlayer,weuseaninpaintingalgorithm[Bertalmioetal.2000;Criminisietal.2003;JiaandTang2003;Drorietal.2003].OverviewWebeginwithasystemoverviewthatdescribesthebasicowofoursystem(Section2).Wethenaddressourmostimportantsub-problem,namelysynthesizingastochasticmotiontexture(Sec-tion3).Finally,wediscussourresults(Section4)andendwithconclusionsandideasforfuturework.2SystemoverviewGivenasingleimage,howcanwegenerateacontinuouslymovinganimationquicklyandeasily?Onepossibilityistouseakeyframe-basedapproach,asdidLitwinowiczandWilliams[1994].However,suchanapproachisproblematicforna¨veusers:specifyingthemo-tionsiscomplex,andachievinganykindofmovementresemblingphysicalrealismisquitedifcult.Anotherstraightforwardapproachistousecompositionsofsinusoidstocreateoscillatorymotions displacement map...(a)(b)(c)(d)(e)......= L1L2Ll-2Ll-1LlL (t)1L (t)2L (t)l-2L (t)l-1L (t)l displacement map displacement map displacement map displacement mapd (t) l-1d (t) ld (t) l-2d (t) 2d (t) 1type=“boat”type=“still”type=“tree”type=“cloud”type=“water”Figure2Overviewofoursystem.Theinputstillimage(a)ismanuallysegmentedintoseverallayers(b).EachlayerLiisthenanimatedwithadifferentstochasticmotiontexturedi(t)(c).Finally,theanimatedlayersLi(t)(d)arecompositedbacktogethertoproducethenalanimationI(t)(e).[Grifths1997],buttheresultingeffectmaynotmaintainaviewer'sinterestovermorethanashortperiodoftime,onaccountofitspe-riodicityandpredictability.Theapproachweultimatelysettledupon—whichhastheadvan-tagesofbeingquitesimpleforuserstospecify,andofcreatinginteresting,complex,andplausiblyrealisticmotion—istobreaktheimageupintoseverallayersandtothensynthesizeadiffer-entmotiontexture1foreachlayer.Amotiontextureisessentiallyatime-varyingdisplacementmapdenedbyamotiontype,asetofmotionparameters,andinsomecasesamotionarmature.Thisdisplacementmapd(p;t)isafunctionofpixelcoordinatespandtimet.ApplyingitdirectlytoanimagelayerLresultsinaforwardwarpedimagelayerL0suchthatL0(p+d(p;t))=L(p)(1)However,sinceforwardmappingisfraughtwithproblemssuchasaliasingandholes,weactuallyuseinversewarping,denedasL0(p)=L(p+d0(p;t))(2)WedenotethisoperationasL0=L\nd0.Wecouldcomputetheinversedisplacementmapd0fromdusingthetwo-passmethodsuggestedbyShadeetal.[1998].Instead,sinceourmotioneldsareallverysmooth,wesimplydilatethembytheextentofthelargestpossiblemotionandreversetheirsign.Withthisnotationinplace,wecannowdescribethebasicworkowofoursystem(Figure2),whichconsistsofthreesteps:layeringandmatting,motionspecicationandediting,andnallyrendering.Layeringandmatting.Therststep,layering,istosegmenttheinputimageIintolayerssothat,withineachlayer,thesamemotiontexturecanbeapplied.Forexample,forthepaintinginFig-ure2(a),wehavethefollowinglayers:oneforeachofthewater,sky,bridgeandshore;oneforeachofthethreeboats;andoneforeachoftheeleventreesinthebackground(Figure2(b)).Toaccom-plishthis,weuseaninteractiveobjectselectiontoolsuchasapaint-ingtoolorintelligentscissors[MortensenandBarrett1995].Thetoolisusedtospecifyatrimapforalayer;wethenapplyBayesian1Weusethetermsmotiontextureandstochasticmotiontextureinter-changeablyinthispaper.ThetermmotiontexturewasalsousedbyLiet.al[2002]torefertoalineardynamicsystemlearnedfrommotioncapturedata.mattingtoextractthecolorimageandasoftalphamatteforthatlayer[Chuangetal.2001].Becausesomelayerswillbemoving,occludedpartsoftheback-groundmightbecomevisible.Hence,afterextractingalayer,weuseanenhancedinpaintingalgorithmtolltheholeintheback-groundbehindtheforegroundlayer.Weuseanexample-basedin-paintingalgorithmbasedontheworkofCriminisietal.[2003]be-causeofitssimplicityanditscapacitytohandlebothlinearstruc-turesandtexturedregions.Notethattheinpaintingalgorithmdoesnothavetobeperfectsinceonlypixelsneartheboundaryoftheholearelikelytobecomevis-ible.Wecanthereforeacceleratetheinpaintingalgorithmbycon-sideringonlynearbypixelsinthesearchforsimilarpatches.Thisshortcutmaysacricesomequality,soincaseswheretheautomaticinpaintingalgorithmproducespoorresults,weprovideatouch-upinterfacewithwhichausercanselectregionstoberepainted.Theautomaticalgorithmisthenreappliedtothesesmallerregionsus-ingalargersearchradius.Wehavefoundthatmostsignicantin-paintingartifactscanberemovedafteronlyoneortwosuchbrush-strokes.Althoughthismayseemlessefcientthanafullyautomaticalgorithm,wehavefoundthatexploitingthehumaneyeinthissim-plefashioncanproducesuperiorresultsinlessthanhalfthetimeofthefullyautomaticalgorithm.Notethatifalayerexhibitslargemotions(suchasawildlyswingingbranch),artifactsdeepinsidetheinpaintedregionsbehindthatlayermayberevealed.Inprac-tice,theseartifactsmaynotbeobjectionable,asthemotiontendstodrawattentionawayfromthem.Whentheyareobjectionable,theuserhastheoptionofimprovingtheinpaintingresults.Afterthebackgroundimagehasbeeninpainted,weworkonthisimagetoextractthenextlayer.Werepeatthisprocessfromtheclosestlayertothefurthestlayertogeneratethedesirednumberoflayers.EachlayerLicontainsacolorimageCi,amattei,andacompositingorderzi.Thecompositingorderispresentlyspeciedbyhand,butcouldinprinciplebeautomaticallyassignedwiththeorderinwhichthelayersareextracted.Motionspecicationandediting.Thesecondcomponentofoursystemletsusspecifyandeditthemotiontextureforeachlayer.Currently,weprovidethefollowingmotiontypes:trees(swaying),water(rippling),boats(bobbing),clouds(translation),andstill(nomotion).Foreachmotiontype,theusercantunethemotionparam-etersandspecifyamotionarmature,whereapplicable.WedescribethemotionparametersandarmaturesinmoredetailforeachmotiontypeinSection3. Sinceallofthemotionswecurrentlysupportaredrivenbythewind,theusercontrolsasinglewindspeedanddirection,whichissharedbyallthelayers.Thisallowsallthelayerstorespondtothewindconsistently.Ourmotionsynthesisalgorithmisfastenoughtoanimateahalf-dozenlayersinreal-time.Hence,thesystemcanprovideinstantvisualfeedbacktochangesinmotionparameters,whichmakesmotioneditingeasier.EachlayerLihasitsownmo-tiontexture,di,asshowninFigure2(c).Rendering.Duringtherenderingprocess,foreachtimein-stancetandlayerLi,adisplacementmapdi(t)issynthesized.(Here,wehavedroppedthedependenciesofLianddionpfornotationalconciseness.)ThisdisplacementmapisthenappliedtoCianditoobtainLi(t)=Li(0)\ndi(t)(Figure2(d)).Noticethatthedisplacementisevaluatedasanabsolutedisplacementofthein-putimageI(0)ratherthanarelativedisplacementofthepreviousimageI(t1).Inthisway,repeatedresamplingandnumericalerroraccumulationareavoided.Finally,allthewarpedlayersarecompositedtogetherfrombacktofronttosynthesizetheframeattimet,I(t)=L1(t)L2(t):::Ll(t),wherez1z2


zlandisthestandardoveroperator[PorterandDuff1984](Figure2(e)).3StochasticmotiontexturesInthissection,wedescribeourapproachtosynthesizingthestochasticmotiontexturesthatdrivetheanimatedimage.Werstdescribethebasicprinciplesonwhichoursystemisbased(Sec-tion3.1).Wethendescribethedetailsofeachmotiontype,i.e.,trees(Section3.2),water(Section3.3),bobbingboats(Section3.4),andclouds(Section3.5).3.1StochasticmodelingofnaturalphenomenaManynaturalmotionscanbeviewedasharmonicoscillations[Sunetal.2003],and,indeed,hand-craftedsuperpositionsofasmallnumberofsinusoidshaveoftenbeenusedtoapproximatenaturalphenomenaforcomputergraphics.However,thissimpleapproachhassomelimitations,aswediscoveredafterexperimentingwiththisidea.Firstofall,itistedioustotunetheparameterstoproducethedesiredeffects.Second,itishardtocreatemotionsforeachlayerthatareconsistentwithoneanothersincetheylackaphysicalbasis.Lastly,theresultingmotionsdonotlooknaturalsincetheyarestrictlyperiodic—irregularityactuallyplaysacentralroleinmodelingnaturalphenomena.Onewaytoaddrandomnessistointroduceanoiseeld.Intro-ducingthisnoisedirectlyintothetemporalorspatialdomainoftenleadstoerraticandunrealisticsimulationsofnaturalphenomena.Instead,wesimulatenoiseinthefrequencydomain,andthensculptthespectralcharacteristicstomatchthebehaviorsofrealsystemsthathaveintrinsicperiodicitiesandfrequencyresponses.Specicspectrumltersneedtobeappliedtomodelspecicphenomena,leadingtoso-calledspectralmethods[Stam1995].Thespectralmethodforsynthesizingastochasticeldhasthreesteps:(1)generateacomplexGaussianrandomeldinthefre-quencydomain,(2)applyadomain-specicspectrumlter,and(3)computetheinverseFouriertransformtosynthesizeastochas-ticeldinthetimeorfrequencydomain.Anicepropertyofthismethodisthatthesynthesizedstochasticeldcanbetiledseam-lessly.Hence,weonlyneedtosynthesizeapatchofreasonablesizeandtileittoproduceamuchlargerstochasticsignal.Thistilingap-proachworksreasonablywellifthesizeofthepatchislargeenoughtoavoidobjectionablerepetition.Furthermore,eachlayercanuseapatchofadifferentsize,whichobscuresanyrepetitivemotionthatmayremaininindividuallayers.Torealisticallymodelnaturalphenomena,theltershouldbelearnedfromreal-worlddata.Forthephenomenawesimulate,plantsandwaves,suchexperimentaldataandstatisticsareavail-ablefromotherelds,e.g.,structuralengineeringandoceanogra-phy,andhavealreadybeenusedbythegraphicscommunitytocre-atesyntheticimagery[ShinyaandFournier1992;StamandFiume1993;Mastinetal.1987].Afterexperimentingwithseveraldiffer-entvariantspublishedinboththecomputergraphicsandsimulationliterature,weselectedthefollowingsetoftechniquestosynthesizestochasticmotiontexturesthatarebothrealisticandeasytocontrol.3.2PlantsandtreesThebranchesandtrunksoftreesandplantscanbemodeledasphys-icalsystemswithmass,damping,andstiffnessproperties.Thedriv-ingfunctionthatcausesbranchestoswayistypicallywind[Stam1997].Ourgoalistomodelthespectrallteringduetothedy-namicsofthebranchesappliedtothespectrumofthedrivingwindforce.omodelthephysicsofbranches,wetakethesimpliedviewintro-ducedbySunetal.[2003].Inparticular,themotionofeachbranchisconstrainedbyamotionarmature;a2Dlinesegmentparameter-izedbyu,whichrangesfrom0to1.Thislinesegmentisdrawnbytheuserforeachlayer.Notethat,tomodelacorrectmechanicalstructure,thelinesegmentmayneedtoextendoutsidetheimage.Displacementsofthetipofthebranchdtip(t)aretakentobeper-pendiculartothelinesegment.Modalanalysisindicatesthatthedisplacementperpendiculartothelineforotherpointsalongthebranchcanbesimpliedtotheform:d(u;t)=h13u443u3+2u2idtip(t)(3)Weapproximatethe(scalar)displacementofthetipinthedirectionoftheprojectedwindforceasadampedharmonicoscillator:dtip(t)+\r_dtip(t)+42f2odtip(t)=w(t)=m(4)wheremisthemassofthebranch,fo=k=misthenaturalfrequencyofthesystem,and\r=c=misthevelocitydampingterm[Sunetal.2003].Theseparametershaveamoreintuitivemeaningthanthedamping(c)andstiffness(k)termsfoundinmoretraditionalformulations.Thedrivingforcew(t)isderivedfromthewindforceincidentonthebranch,asdetailedbelow.TakingthetemporalFouriertransformFfgofequation(4)andnot-ingthatFf_dtip(t)g=i2fFfdtip(t)g,wearriveat42f2Dtip(f)+i2\rfDtip(f)+42f2oDtip(f)=W(f)m(5)wherei=p1andDtip(f)andW(f)aretheFouriertransformsofdtip(t)andw(t),respectively.SolvingforDtip(f)andexpress-ingtheresultincomplexexponentialnotationgivesDtip(f)=W(f)ei22m[2(f2f2o)]2+\r2f2 1=2(6)whereW(f)istheFouriertransformofthedrivingwindforce,afunctionoffrequencyf,asdenedinequations(8)and(9)below.Thephaseshiftisgivenbytan=\rf2(f2f2o)(7)Next,wemodeltheforcingspectrumforwind.Anempiricalmodelmadefromexperimentalmeasurements[SimiuandScanlan1986,p.55]indicatesthatthetemporalpowerspectrumofthewindve-locityatapointtakesthefollowingform:PV(f)vmean(1+f=vmean)5=3(8) wherevmeanisthemeanwindspeedandisgenerallyafunctionofaltitude,whichwetaketobeaconstant.Thevelocityspectrumisgivenbythesquarerootofthepowerspectrum.WethereforemodulatearandomGaussiannoiseeldG(f)withthevelocityspectrumtocomputethespectrumofaparticular(random)windvelocityeld:V(f)=G(f)pPV(f)(9)Theforceduetothewindiscomplicatedbythepresenceofturbu-lence[Feynmanetal.1964,Fig.41-4],butcanbegenerallymod-eledasadragforceproportionaltothesquaredwindvelocity.How-ever,inourexperiments,wehavefoundthatmakingthewindforcedirectlyproportionaltowindvelocityproducesmorepleasingre-sults.,weassembleEquations(6)-(9)toconstructthespectrumofthetipdisplacementDtip(f),taketheinverseFouriertransformtogeneratethetipdisplacementdtip(t),anddistributethedisplace-mentoverthebranchaccordingtoequation(3).Weapplythedis-placementasarotationofeachpointabouttherootpositionofthebranch.Thedisplacementsofpointsinthelayerawayfromthemo-tionarmaturearegivenbythedisplacementofthepointonthear-maturethatisthesamedistancefromtheroot.Theusercancontroltheresultingmotionappearancebyindepen-dentlychangingthemeanwindspeedvmeanandthenatural(oscil-latory)frequencyfo,massm,andvelocitydampingterm\rofeachbranch.WaterWatersurfacesbelongtoanotherclassofnaturalphenomenathatexhibitoscillatoryresponsestonaturalforceslikewind.Inthissec-tionwedescribehowonecanspecifya3Dwaterplaneinaphoto-graphandthendenethemappingofwaterheightoutofthatplanetodisplacementsinimagespace.Wethendescribehowtosynthe-sizewaterheightvariations,againusingaspectralmethod.Themotionarmatureforwaterissimplyaplane;weassumethattheimageplaneisthexyplaneandthewatersurfaceisthexzplane.Tocorrectlymodeltheperspectiveeffect,theuserroughlyspecieswheretheplaneis.ThisperspectivetransformationMcanbefullyspeciedbythefocallengthandthetiltofthecamera,whichcanbevisualizedbydrawingthehorizon[Criminisietal.2000].Afterspecifyingthe3Dwaterplane,thewaterisanimatedusingatime-varyingheighteldh(q;t),whereq=(xq;y0;zq)Tisapointonthewaterplane,andy0=0istheelevationofthewa-terplane.Toconverttheheighteldhtothedisplacementmapd(p;t),foreachpixelpwerstndthatpixel'scorrespondingpointq=Mponthewaterplane.Wethenaddthesynthesizedheighth(q;t)asaverticaldisplacement,whichgivesusapointq0=(xq;h(q;t);zq)T.Wethenprojectq0backtotheimageplanetogetp0=M1q0.Thedisplacementvectorford(p;t)=p0pistherefored(p;t)=M1[Mp+(0;h(Mp;t);0)T]p(10)Notethatpandp0areafnepoints,disavector,andMisa33matrix.abovemodelistechnicallycorrectifwewanttodisplaceob-jectsonthesurfaceofthewater.Inreality,theshimmerinthewateriscausedbylocalchangesinsurfacenormals.Therefore,amorephysicallyrealisticapproachwouldbetousenormalmapping,i.e.,toconvertthesurfacenormalscomputedfromthespatialgradi-entsofh(q;t)intotwo-dimensionaldisplacementsofthereectedrays.However,wehavefoundthatapplyingthisnormalmappingapproachwithouta3-dimensionalmodelofthesurroundingenvi-ronmentproducesconfusingdistortionscomparedtoourcurrentapproach,whichgenerallyproducespleasing,realistic-lookingre-ectionsaslongasthewaveamplitudeisrelativelysmall.Tosynthesizeatime-varyingheighteldforthewater,weusetheuser-speciedwindvelocitytosynthesizeaheighteldmatchingthestatisticsofrealoceanwaves,asdescribedbyMastinetal.[1987].Notethatthisapproachdealsonlywithoceanwaves,whicharegravitywaves.Althoughitdoesnotphys-icallydescribeshort-lengthwaves,non-wind-generatedwavesonrivers/brooks/streamsorlargewavesonshallowwater,itgivesplau-sibleresultsforourapplication.ThespectrumlterweuseforwavesisthePhillipsspec-trum[Tessendorf2001],whichisapowerspectrumdescribingtheexpectedsquareamplitudeofwavesacrossallspatialfrequenciessPH(s)e[1=(sL)2]s4j^s
^vmeanj2(11)wheres=jsj,andL=v2mean=g,andgisthegravitationalcon-stant,and^sand^vmeanarenormalizedspatialfrequencyandwinddirectionvectorsinthexzplane,respectively.(Wedenote2Dvec-torsinboldface.)Thesquarerootofthepowerspectrumdescribestheamplitudeofwaveheights,whichwecanusetolterarandomGaussiannoiseeldG(s):H0(s)=aG(s)pPH(s)(12)whereaisaconstantofproportionalityandH0isaninstanceoftheheighteldwhichwecannowanimatebyintroducingtime-varyingphase.However,wavesofdifferentspatialfrequenciesmoveatdif-ferentspeeds.Therelationshipbetweenthespatialfrequencyandthephasevelocityisdescribedbythewell-knowndispersionrela-tion,!(s)=pgs(13)ThetimevaryingheightspectrumcanthusbeexpressedasH(s;t)=H0(s)ei!(s)t+H0(s)ei!(s)t(14)whereH0isthecomplexconjugateofH0[Tessendorf2001].Wecannowcomputetheheighteldattimeh(q;t)asthetwo-dimensionalinverseFouriertransformofH(s;t)withrespecttospatialfrequenciess.Wetakethegeneratedheighteldandtilethewatersurfaceusingascaleparameter,,tocontrolthespatialfrequency.Torecaptheprocess,giventhewindspeedanddirection,wesyn-thesizeaspectrumlterusingequation(11)andapplyittoaspatialGaussiannoiseeldtoobtainaninitialheighteld(12).Thisheighteldisthenanimatedusingequation(14)tosynthesizetheFouriertransformH(s;t)oftheheighteldh(q;t)attimet.TakingtheinverseFouriertransform,werecovertheheighteld,useittotilethewaterplaneandsubstituteitintoequation(10)tosynthesizemotiontexturediattimet.Therearethusseveralmotionparametersrelatedtowater:windspeed,winddirection,thesizeofthetileN,theamplitudescalea,andthespatialfrequencyscale.Thewindspeedanddirec-tionarecontrolledgloballyforthewholeanimation.WendthatatileofsizeN=256usuallyproducesnicelookingresultsforthesizesofimagesweused.Userscanchangeatoscaletheheightofthewaves/ripples.Finally,scalingthefrequenciesbychangesthescaleatwhichthewavesimulationisbeingdone.Simulatingatalargerfrequencyscalegivesarougherlook,whileasmallerscalegivesasmootherlook.Hence,wecalltheroughnessinouruserinterface. 3.4BoatsWeapproximatethemotionofabobbingboatbya2Drigidtrans-formationcomposedofatranslationforheavingandarotationforrolling.Aboatmovingonthesurfaceofopenwaterisalmostal-waysinoscillatorymotion[Sunetal.2003].Hence,thesimplestmodelistoassignasinusoidaltranslationandasinusoidalrota-tion.However,thisoftenlooksfake.Inprinciple,wecouldbuildasimplemodelfortheboat,converttheheighteldofwaterintoaforceinteractingwiththehull,andsolvethedynamicsequationfortheboattoestimateitsdisplacement.However,sinceourgoalistosynthesizeaquicklycomputablesolution,wedirectlyusetheheighteldofthewavetomovetheboat,asfollows.Welettheuserselectalineclosetothebottomoftheboat.Then,wesampleseveralpointsqialongthelineandassumethesepointsareonthewaterplanesurroundingtheboat.Attimet,foreachpointqi,welookupitsdisplacementvectord(pi;t)(10)andcalculatethecorrespondingpositionp0ofpiattimetaspi+d(pi;t).Finally,weuselinearregressiontotalinethroughthedisplacedpositions.Thepositionandorientationofthettedlinethendeterminetheheavingandrollingoftheboat.3.5CloudsAnothercommonelementforscenicpicturesisclouds.Inprinciple,cloudscouldalsobemodeledasastochasticprocess.However,weneedthestochasticprocesstomatchthecloudsintheimageatsomepoint,whichisharder.Sincecloudsoftenmoveveryslowlyandtheirmotiondoesnotattracttoomuchattention,wesimplyassignatranslationalmotioneldtothem.Weextendthecloudsoutsidetheimageframetocreateacyclictextureusingourinpaintingalgo-rithm,sincetheirmotioninonedirectionwillcreateholesthatwehavetoll.4ResultsWehavedevelopedaninteractivesystemthatsupportsmatting,in-painting,motionediting,andpreviewingtheresults.Wehaveap-pliedoursystemtoseveralphotographsandfamouspaintings.Theaccompanyingvideoprovidesasenseoftheuserinterfaceforcre-atingtheanimatedpictures,aswellasademonstrationoftheani-matedresults.Table1summarizesthenumberoflayersofeachtypecreatedfortheveanimatedpicturesshowninFigure1,themotionspecica-tion,alongwiththetimethatittookausertoperformthemattingandin-paintingsteps(whichareinterleavedintheprocess,andthusdifculttoseparateintime),andtheplaybackspeeds.Generallythemattingandin-paintingstepstakethelargemajorityofthetime.Inallcases,theanimatedpaintingstakefromalittleunderanhourtoafewhourstocreate.Notethattwooftheanimatedpictureswhosetimingsarepresentedabove,“BoatStudio”and“Sunowers,”werecreatedbyacompletenoviceuserwhoonlyhadafewminutesofinstructionbeforebeginningworkonthepictures.Weprovideplay-backspeedsforourcurrentunoptimizedsoftwareimplementation:Ourcodepresentlytakesnospecialadvantageofgraphicshard-ware,butalloftheoperationscouldbereadilymappedtoGPUs,therebygreatlyincreasingframerates.FortheJapaneseTemple(Figure1(a)),wemodelatotalof10branchesontheleftandtheright.Weuseasmallwaveamplitude(a=1:0)andhighroughness(=200)togivetheripplesane-grainedlook.FortheharborpictureinFigure1(b),weanimatethewaterandhavenineboatsswingwiththewater.Thecloudandskyareanimatedusingatranslationalmotioneld.Figure1(c)-(e)showsthreepaintingswehaveanimated.Ourtech-niqueworksreasonablywellwithpaintings,probablybecauseinthissituationweareevenlesssensitivetoanythingthatdoesnotlookperfectlyrealistic.ForClaudeMonet'spaintinginFigure1(c),weanimatethewaterwithloweramplituderoughnesstokeepthestrokesintact.Wealsolettheboatswaywiththewater.AnotherofMonet'spaintings,showninFigure1(d),isamorecomplexexam-ple,withmorethantwentylayers.Weusethisexampletodemon-stratethatwecanchangetheappearanceofthewaterbycontrol-lingthephysicalparameters.InFigure3,weshowtheappearanceofthewaterunderdifferentwindspeeds,directions,andsimulationscales.orVanGogh'ssunowerpainting(Figure1(e)),weuseourstochasticwindmodeltoanimatethetwenty-veplantlayers.Withasimplesinusoidalmodel,theviewerusuallycanquicklygureoutthattheplantsswinginsynchrony,andthemotionlosesalotofitsinterest.Withthestochasticwindmodel,theowers'motionsde-correlateinphaseandtheresultedanimationismoreappealing.Wealsoexperimentedwithaverysmallamountofscalingalongthebrancharmatureinordertosimulateforeshorteningoftheowersastheymoveinandoutoftheimageplane.5ConclusionandfutureworkInthispaper,wehavedescribedanapproachforanimatingstillpicturesofoutdoorscenesthatcontaindynamicelementsthatre-spondtonaturalforcesinasimplequasi-periodicfashion.Weseeourworkasjustarststepinthelargerproblemofanimatingamuchmoregeneralclassofpictures.Beforewebeganthiswork,itwasnotatallclearwhetheritwouldbepossibletomakestillimagescometolifeasanimatedscenes.Webelieveourjudiciousselectionandenhancementofrecentlydevel-opedmatting,inpainting,stochasticmotionsynthesis,imagewarp-ing,andcompositingalgorithmsprovidesaneffectiveandeasy-to-usesystemforgeneratingrealisticanimationsfromstaticimages.Wepointoutthatourchoiceoftechniquesisespeciallywell-suitedtothisproblem,inthatarelativelyhigh-qualitycompositeanima-tioncanbeproducedevenwhentheresultsofeachautomatedstepareofobjectivelylowerquality.First,theuseofmattingproduceslayersthatarecolor-coherentalongtheirboundaries,evenifthere-sultingmattedoesnotfollowobjectboundaries.Wheninmotion,theselayersoftenseemperceptuallyplausibleevenwhentechni-callyincorrect.Second,thelimitedamountofdisplacementweseektointroduceimpliesthattheinpaintingprocesscanberelativelylow-qualityandstillproduceseamlesscomposites.Thisallowsustouseheuristicmeasurestoreducethesearchspaceandspeeduptheinpaintingprocess.Finally,wedonotaskenduserstokeyframeanimations,butratherinuencethesceneinphysical,easilyunder-stoodterms,suchaswindspeedanddirection.Weprovideauserinterfacethatisaccessibletousersatalllevels.Manyusersarealreadyfamiliarwithmattingandinpaintingprocessesfromcom-mercialproductssuchasPhotoshop,andtheadditionalburdenofassigning“canned”motiontypesisminimal.Oursystemcurrentlymakesanumberofassumptionsthatwewouldliketorelax.Forexample,weassumethattheelementsoftheinputimageareintheirequilibriumpositions.Thisisoftennotthecase,e.g.,forascenewithwaterthatalreadyhasripples.Indeed,aninterestingchallengewouldbetousetheseripplestoestimatethewatermotion,unwarpthereferenceimageandthenanimateitcorrectly.Inaddition,wecurrentlyignoretheeffectsofshadows,transparency,andreections.Forexample,thereectionsoftheboatmovewiththedeformationsofthewater,butdonotaccountforanyadditionalmotionduetotheboat'sbobbingupanddown.Whenthemotionislarge,theresultsarelessrealistic.Onesolutionwouldbetosegmentoutreections,transparentlayersandshadowssomehow,andletthemmovewiththecastingobjectsaccordingly.Manyofourapproximationslimittheplausibilityofverylarge-scalemotions,inwhichpixelsarewarpedmorethanafewdozenpixelsfromtheirsourceposition.Forexample,wesimulateboatsrollingasa2Drigidmotion.Itmightbepossibletofakeaslight3Drotationwithanon-rigiddistortion,toallowformoreplausible (a)composite(b)lowerwindspeed(c)windofdifferentdirection(d)rougherwatersurfaceFigure3Wecancontroltheappearanceofwatersurfacebyadjustingsomephysicalparameterssuchaswindspeed.Weshowoneofthecomposites(a)asthereference,inwhichthewindblowat5m/sinzdirection.Wedecreasethewindspeedto3m/s(b)andchangethewinddirectiontobealongzaxis(c).In(d),wechangethescaleofthesimulationtorenderwaterwithnerripples.TreesWaterBoatsCloudsStillLayeringAnimatingRenderingResolutionJapaneseTemple10100245m10m7fps900x675Harbor0291590m10m3.8fps900x600BoatStudio0110130m10m10fps600x692Argenteuil161313120m15m4.1fps800x598Sunowers250001210m20m5.1fps576x480Table1ThenumberoflayersofeachtypeforeachoftheveexamplesinFigure1,alongwithapproximatetimesinminutesforausertoperformthelayeringsteps(includingmattingandinpainting),animatingstep(includingmotionspecicationandediting),andplaybackspeeds.large-scalemotions.Verylargewarpsofthewatersurfacecanap-peardistortedduetowarpingfromoutsidetheimageboundaries,andwhenthewaterwavesbecomelargeenoughunderverywindyconditions,weexpecttoseeanumberofadditionalreal-worldef-fectssuchaswater“lappingup”againsttheshoreorboats,“white-caps,”splashes,orotherturbulentsurfaceeffects.Ourmethodcurrentlyworksbestfortreesatadistance.Fornearbytrees,itispresentlydifcultandtedioustosegmenttheleafandbranchstructureproperly.Itwouldalsobeinterestingtoaddthe“shimmering”effectofleavesblowinginthewindbyapplyingtur-bulentoweldswithinthetreelayers.Thereareotherclassesofmotionthatcouldbemodeledusingasimilarapproach.Weimaginethatwaterfalls,oceanwaves,yingbirdsandothersmallanimals,ame,andsmokemayallbepos-sible.Forexample,waterfallscouldperhapsbeanimatedusingatechniquesimilarto”motionwithoutmovement”[Freemanetal.1991].Oceanwavescouldbesimulatedusingstochasticmodels,althoughmatchingtheappearanceofthesourceimageposessomeinterestingchallenges.Flyingbirdsandothersmallanimalscouldbeanimatedusingideasfromvideosprites[Sch¨odletal.2000].Webelievethatitmightalsobepossibletoanimateuidslikeameorsmoke.However,thiswouldrequireaconstrainedstochasticsimu-lation,sincethestateofsimulationshouldresembletheappearanceoftheinputimage.Recentadvancesincontrollingsmokesimula-tionbykeyframescouldbeusedforthispurpose[Treuilleetal.2003].oursystem,allthelayersarehookeduptogethertoasyntheticwindforce.Currently,thesamemeanwindvelocityisappliedev-erywhereinthescene.Itwouldbestraightforwardtoextendthefor-mulationtohandlecompletevectoreldsofevolvingwindforcesinordertoprovideamorerealisticstyleofanimationsuchasmov-inggustsofwind.Inaddition,wecouldaddmorecontrollabilitysothattheuserscouldinteractwithtreesindividually.Currently,weusephysically-basedsimulationtosynthesizeapara-metricmotioneld,butthequalityofthemotioncouldpotentiallybeimprovedusinglearningalgorithmstotransfermotionfromsim-ilartypeofobjectsinvideos.Furthermore,ourmotionmodeladdressesonlyarestrictedrangeofmotions.Weimaginefuturesystemsmighthandletransitionsbe-tweendifferenttypesofmotion,animationtoorfromareststate,waterfeaturessuchasstreamsthatmovecontinuouslyinasingledi-rection,andtransitionsbetweendifferentscenestatesand/ortypesofmotion(e.g.weatherchangingfromcalmtostormy,skieschang-ingfromcleartocloudy,boatstravelingtoandfromthehorizon,etc.).systempresentlyrequiresafairamountofuserinteraction.Wehopetofurtherreducethetimeandefforttocreatetheseanima-tionsbyexploitingcontinuedadvancesinintelligentimageselec-tionandmattingalgorithmssuchasGrabCut[Rotheretal.2004]orLazySnapping[Lietal.2004].Furthermore,anautomatedorsemi-automatedregionclassicationtoidentifyfeaturessuchasforegroundtreebranchesandwaterwouldenableamuchmoreautomatedprocess.Forexample,onecouldimagineautomaticallyidentifyingthe“whitewater”ofawaterfall,andthenautomaticallyanimatingthewaterfall.Foralakewithasimpleboundary,suchasinFigure1(a),itmightalsobepossibletoautomaticallysegmentthethewaterregionbyidentifyingreections.Anotherpossibilitywouldbetousemultiplepicturesasinput.Mostmoderndigitalcamerashavea“motor-drive”modethatal-lowsuserstotakehigh-resolutionphotographsatarestrictedsam-plingrate,around1–3framespersecond.Fromsuchasetofpho-tographswemightbeabletoautomaticallysegmentapictureintoseveralcoherentlymovingregionsandgureoutthemotionparam-etersfromthesamplestillimages.Itwouldalsobeinterestingtocombinehigh-resolutionstillswithlower-resolutionvideotopro-duceattractiveanimations.Ourapproachcouldalsobecombinedwith“Tourintothepicture”toprovideanevenricherexperience,withtheabilitytomovethecameraandlessconstrainedperspectiveplanes.conclusion,wehaveshowntheeasewithwhichitispossibletobreathelifeintopictures,basedonrecentlydevelopedmatting,inpainting,andstochasticmodelingalgorithms.Wehopethatourworkwillinspireothertoexplorethecreativepossibilitiesinthisrichdomain.AcknowledgmentsTheauthorswishtothankWilLifornarratingourvideo,andMiraDontchevaforuser-testingoursegmentationandinpaintingsytem.Wewouldalsoliketothankthereviewersfortheirhelpfulcom-ments.ThisworkwassupportedbytheUniversityofWashingtonAnimationResearchLabs,WashingtonResearchFoundation,NSF 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