A First-Order Analysis of Lighting, Shading, and S

AFirst-OrderAnalysisofLighting,Shading,andShadowsRAVIRAMAMOORTHI,DHRUVMAHAJAN,andPETERBELHUMEURColumbiaUniversityTheshadinginascenedependsonacombinationofmanyfactors—howthelightingvariesspatiallyacrossasurface,howitvariesalongdifferentdirections,thegeometriccurvatureandreflectancepropertiesofobjects,andthelocationsofsoftshadows.Inthisarticle,weconductacompletefirst-orderorgradientanalysisoflighting,shading,andshadows,showinghoweachfactorseparatelycontributestosceneappearance,andwhenitisimportant.Gradientsarewell-suitedtoanalyzingtheintricatecombinationofappearanceeffects,sinceeachgradienttermcorrespondsdirectlytovariationinaspecificfactor.First,weshowhowthespatialanddirectionalgradientsofthelightfieldchangeaslightinteractswithcurvedobjects.ThisextendstherecentfrequencyanalysisofDurandetal.[2005]togradients,andhasmanyadvantagesforoperations,likebumpmapping,thataredifficulttoanalyzeintheFourierdomain.Second,weconsidertheindividualtermsresponsibleforshadinggradients,suchaslightingvariation,convolutionwiththesurfaceBRDF,andtheobject’scurvature.Thisanalysisindicatestherelativeimportanceofvariousterms,andshowspreciselyhowtheycombineinshading.Third,weunderstandtheeffectsofsoftshadows,computingaccuratevisibilitygradients,andgeneralizingpreviousworktoarbitrarycurvedoccluders.Asonepracticalapplication,ourvisibilitygradientscanbedirectlyusedwithconventionalray-tracingmethodsinpracticalgradientinterpolationmethodsforefficientrendering.Moreover,ourtheoreticalframeworkcanbeusedtoadaptivelysampleimagesinhigh-gradientregionsforefficientrendering.CategoriesandSubjectDescriptors:I.3.0[ComputerGraphics]:General;[ImageProcessingandComputerVision]:GeneralGeneralTerms:TheoryAdditionalKeyWordsandPhrases:Gradients,reflectance,shadows,FourieranalysisACMReferenceFormat:Ramamoorthi,R.,Mahajan,D.,andBelhumeur,P.2007.Afirst-orderanalysisoflighting,shading,andshadows.ACMTrans.Graph.26,1,Article2(January2007),21pages.DOI=10.1145/1186644.1186646ficientrendering.Understandinghowshadingchangesinpenumbraregionscanleadtoefficientandrobustsoftshadowcomputations,aswellasadvancesininverselighting-from-shadowalgorithms.Inthisarticle,weseektoaddressthesetheoreticalquestionsthroughafirst-order,orgradient,analysisoflighting,shading,andvisibility.Theappearanceofasurface(anditsgradients)dependsonmanyfactors.Theshadingisaffectedbylighting—spatiallightingvari-ationoveraflatobjectsurfaceduetoclosesources,aswellasangularvariationinlightingatapointfromdifferentdirections.Shadingalsodependsongeometriceffects,suchastheobject’scurvature,whichdeterminehowmuchthesurfacenormalororien-tationchangesbetweenneighboringpoints.Materialpropertiesarealsoimportant,sinceshadingiseffectivelyaconvolutionwiththeobjectBRDF[BasriandJacobs2001,2003;RamamoorthiandHan-rahan2001,2004].Thesefactorscancombineincomplexwaysinanimage,andeachfactormayhavelessormoreimportance,depend-ingonthesituation.Forexample,thespatialvariationinlightingThisworkwassupportedinpartbyNSFGrants0085864,0325867,0305322,0430258,and0446916,andaSloanResearchfellowship.Authors’address:R.Ramamoorthi,D.Mahajan,andP.Belhumeur,ComputerScienceDepartment,ColumbiaUniversity,450ComputerScienceBldg.,500W120St.,NewYork,NY10027;email:{ravir,dhruv,belhumeur}@cs.columbia.edu.Permissiontomakedigitalorhardcopiesofpartorallofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitordirectcommercialadvantageandthatcopiesshowthisnoticeonthefirstpageorinitialscreenofadisplayalongwiththefullcitation.CopyrightsforcomponentsofthisworkownedbyothersthanACMmustbehonored.Abstractingwithcreditispermitted.Tocopyotherwise,torepublish,topostonservers,toredistributetolists,ortouseanycomponentofthisworkinotherworksrequirespriorspecificpermissionand/orafee.PermissionsmayberequestedfromPublicationsDept.,ACM,Inc.,2PennPlaza,Suite701,NewYork,NY10121-0701USA,fax[phone],[e-mail].c2007ACM0730-0301/2007/01-ART2$5.00DOI10.1145/1186644.1186646flectionsfromanearsourceonaglossyflattable.Ontheotherhand,theangularvariationinlightingismostimportantforahighly-curvedbumpyobjectonthetable—theeffectofspatialvariationhereisoftensmallenoughthatthelightingcaneffectivelybetreatedasdistant(seeFigure5).Foranalysisoflightingandshading,thegradientisusuallyasumofterms,eachcorrespondingtovariationinaspecificfactor.Hence,afirst-orderanalysisisabletoisolatetheimpactofvariousshadingeffects.Ourcomputationofgradientsalsoenablesnewpracticalren-deringalgorithms,suchasefficientgradient-basedimagesampling,aswellasfastandaccurategradient-basedinterpolationofvisibility(Figure1).Specifically,wemakethefollowingcontributions:—Analysisoflightreflection:First,weanalyzethebasicconceptualstepsintherefle