Computer Graphics (Fall 2008)RadianceRadiance propertiesSlide 4Slide 5Slide 6Irradiance, RadiosityBuilding up the BRDFSlide 9BRDFSlide 11Slide 12Isotropic vs AnisotropicRadiometryReflection EquationSlide 16Slide 17Slide 18Brdf Viewer plotsDemoSlide 21Slide 22Slide 23Slide 24Analytical BRDF: TS exampleTorrance-SparrowTorrance-Sparrow ResultOther BRDF modelsComplex LightingEnvironment MapsSlide 31ConclusionWhat’s NextComputer Graphics (Fall 2008)Computer Graphics (Fall 2008)COMS 4160, Lecture 19: Illumination and Shading 2http://www.cs.columbia.edu/~cs4160Radiance•Power per unit projected area perpendicular to the ray per unit solid angle in the direction of the ray •Symbol: L(x,ω) (W/m2 sr)•Flux given by dΦ = L(x,ω) cos θ dω dARadiance properties•Radiance is constant as it propagates along ray–Derived from conservation of flux –Fundamental in Light Transport. 1 21 1 1 2 2 2d L d dA L d dA dw wF = = = F2 21 2 2 1d dA r d dA rw w= =1 21 1 2 22dA dAd dA d dArw w= =1 2L L\ =Radiance properties•Sensor response proportional to radiance (constant of proportionality is throughput)–Far away surface: See more, but subtends smaller angle–Wall equally bright across viewing distancesConsequences–Radiance associated with rays in a ray tracer–Other radiometric quants derived from radianceIrradiance, Radiosity•Irradiance E is radiant power per unit area•Integrate incoming radiance over hemisphere–Projected solid angle (cos θ dω)–Uniform illumination: Irradiance = π [CW 24,25]–Units: W/m2•Radiosity –Power per unit area leaving surface (like irradiance)Building up the BRDF•Bi-Directional Reflectance Distribution Function [Nicodemus 77]•Function based on incident, view direction•Relates incoming light energy to outgoing light energy•We have already seen special cases: Lambertian, Phong•In this lecture, we study all this abstractlyBRDF•Reflected Radiance proportional to Irradiance•Constant proportionality: BRDF [CW pp 28,29]–Ratio of outgoing light (radiance) to incoming light (irradiance)–Bidirectional Reflection Distribution Function –(4 Vars) units 1/sr( )( , )( ) cosr ri ri i i iLfL dww ww q w=( ) ( ) ( , ) cosr r i i i r i iL L f dw w w w q w=Isotropic vs AnisotropicIsotropic vs AnisotropicIsotropic: Most materials (you can rotate about normal without changing reflections)Anisotropic: brushed metal etc. preferred tangential directionIsotropicAnisotropicRadiometryRadiometryPhysical measurement of electromagnetic energyWe consider light fieldRadiance, IrradianceReflection functions: Bi-Directional Reflectance Distribution Function or BRDFReflection EquationSimple BRDF modelsReflection Equationiwrw( ) ( ) ( , )( )r r i i i r iL L f nw w w w w= gReflected Radiance(Output Image)Incident radiance (fromlight source)BRDFCosine of Incident angleReflection EquationiwrwSum over all light sources( ) ( ) ( , )( )r r i i i r iiL L f nw w w w w=�gReflected Radiance(Output Image)Incident radiance (fromlight source)BRDFCosine of Incident angleReflection EquationiwrwReplace sum with integralidw( ) ( ) ( , )( )r r i i i r i iL L f n dw w w w w wW=�gReflected Radiance(Output Image)Incident radiance (fromlight source)BRDFCosine of Incident angleRadiometryRadiometryPhysical measurement of electromagnetic energyWe consider light fieldRadiance, IrradianceReflection functions: Bi-Directional Reflectance Distribution Function or BRDFReflection EquationSimple BRDF modelsBrdf Viewer plots Brdf Viewer plots Diffusebv written by Szymon RusinkiewiczTorrance-SparrowAnisotropicDemoDemoAnalytical BRDF: TS exampleAnalytical BRDF: TS exampleOne famous analytically derived BRDF is the Torrance-Sparrow model.T-S is used to model specular surface, like the Phong model.more accurate than Phonghas more parameters that can be set to match different materialsderived based on assumptions of underlying geometry. (instead of ‘because it works well’)Torrance-SparrowTorrance-SparrowAssume the surface is made up grooves at the microscopic level.Assume the faces of these grooves (called microfacets) are perfect reflectors.Take into account 3 phenomenaShadowingMasking InterreflectionTorrance-Sparrow ResultTorrance-Sparrow Result( ) ( , ) ( )4cos( ) cos( )i i r hi rF G Df    Fresnel term:allows for wavelength dependencyGeometric Attenuation:reduces the output based on the amount of shadowing or masking that occurs.Distribution:distribution function determines what percentage of microfacets are oriented to reflect in the viewer direction.How much of the macroscopic surface is visible to the light sourceHow much of the macroscopic surface is visible to the viewerOther BRDF modelsOther BRDF modelsEmpirical: Measure and build a 4D tableAnisotropic models for hair, brushed steelCartoon shaders, funky BRDFsCapturing spatial variationVery active area of researchComplex LightingComplex LightingSo far we’ve looked at simple, discrete light sources.Real environments contribute many colors of light from many directions.The complex lighting of a scene can be captured in an Environment map.Just paint the environment on a sphere.Environment MapsEnvironment MapsInstead of determining the lighting direction by knowing what lights exist, determine what light exists by knowing the lighting direction.Blinn and Newell 1976, Miller and Hoffman, 1984Later, Greene 86, Cabral et al. 87DemoDemoConclusionConclusionAll this (OpenGL, physically based) are local illumination and shading modelsGood lighting, BRDFs produce convincing resultsMatrix movies, modern realistic computer graphicsDo not consider global effects like shadows, interreflections (from one surface on another)Subject of next unit (global illumination)What’s NextWhat’s NextHave finished basic material for the classTexture mapping lecture later todayReview of illumination and ShadingRemaining topics are global illumination (written assignment 2): Lectures on rendering eq, radiosityHistorical movie: Story of Computer Graphics Likely to finish these by Dec 1: No class Dec 8, Work instead on HW 4, written assignments Dec 10? will be demo session for HW