Computer Graphics (Fall 2005)RadiositySlide 3Advantages and DisadvantagesGeneral ApproachEarliest Radiosity picturesOutlineRendering EquationChange of VariablesSlide 10Rendering Equation: Standard FormRadiosity EquationSlide 13Discretization and Form FactorsForm FactorsMatrix EquationSlide 17Nusselt’s AnalogHemicubeHemicubesMonte Carlo Ray TracingSlide 22Computer Graphics (Fall 2005)Computer Graphics (Fall 2005)COMS 4160, Lecture 23: Radiosityhttp://www.cs.columbia.edu/~cs4160RadiosityRadiosityCornell box with color bleeding [Goral et al 84]Photograph of a sculpture.The front faces are all diffuse whiteThe color is because of reflectionfrom rear-facing colored facesRaytracing makes all faces white.It can handle specular reflection andshadows, but not diffuse-diffuse interreflection or color bleedingRadiosity correctly captures the color bleeding from the back of the boards to the front.Advantages and DisadvantagesAdvantages and DisadvantagesRadiosity methods track rate at which energy (radiosity) leaves [diffuse] surfacesDetermine equilibrium of light energy in a view-independent wayAllows for diffuse interreflection, color bleeding, and walkthroughsDifficult to handle specular objects, mirrorsGeneral ApproachGeneral ApproachAssume diffuse surfaces discretized into a finite set of patches or finite elementsRadiosity equation is a matrix equation or set of simultaneous linear equations derived by approximations to the rendering equationSolve iteratively using numerical methodsEarliest Radiosity picturesEarliest Radiosity picturesRadiosity was first developed in other fieldsHeat transport, Lighting DesignIn graphics: Goral et al. 84 Parry Moon and Domina Spencer (MIT), Lighting Design, 1948OutlineOutlineRendering equation reviewRadiosity equationForm factorsMethods to compute form factorsHigh-level overview only. Best textual reference is probably Sections 16.3.1 and 16.3.2 in FvDFH. This will be handed out. If curious, read the rest of 16.3 and parts of Cohen and Wallace.Rendering Equationiwrwx( , ) ( , , ) c( , ) ( , ) ose r i rr r i ir iL x L xL x f x dw w ww q wwW= +�-�Reflected Light(Output Image)EmissionReflectedLightBRDFCosine of Incident angleidwSurfaces (interreflection)dAx�UNKNOWNUNKNOWNKNOWNKNOWN KNOWNix xw�-:Change of Variables Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables)( , ) ( , ) ( , ) ( , , ) cosr r e r r i i r i iL x L x L x df xw w w w w wqW�= + -�xx�dA�iwiw-iqoqidw2cos| |oidAdx xqw�=�-Change of Variables Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables) ( , ) ( , ) ( , ) ( , , ) cosr r e r r i i r i iL x L x L x df xw w w w w wqW�= + -�2cos| |oidAdx xqw�=�-all visible2 to cos cos( , ) ( , ) ( , ) ( , , )| |i or r e r r i i rx xL x L x L x f xxdxAq qw w w w w��= + -�-��2cos cos( , ) ( , )| |i oG x x G x xx xq q� �= =�-Rendering Equation: Standard Form Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables) Domain integral awkward. Introduce binary visibility fn V( , ) ( , ) ( , ) ( , , ) cosr r e r r i i r i iL x L x L x df xw w w w w wqW�= + -�2cos| |oidAdx xqw�=�-all visible2 to cos cos( , ) ( , ) ( , ) ( , , )| |i or r e r r i i rx xL x L x L x f xxdxAq qw w w w w��= + -�-��2cos cos( , ) ( , )| |i oG x x G x xx xq q� �= =�-all surfaces ( , ) ( , ) ( , ) ( , , ) ( , ) ( , )r r e r rxi i rL x L x L x f x G x dAx x V xw w w w w�� � �= + -��Same as equation 2.52 Cohen Wallace. It swaps primedAnd unprimed, omits angular args of BRDF, - sign.Same as equation above 19.3 in Shirley, except he has no emission, slightly diff. notationRadiosity Equationall surfaces ( , ) ( , ) ( , ) ( , , ) ( , ) ( , )r r e r rxi i rL x L x L x f x G x dAx x V xw w w w w�� � �= + -��Drop angular dependence (diffuse Lambertian surfaces)( ) ( ) ( ) ( ) ( , ) ( , )Sr e rL x L x f x L x G dAx x V x x� � �= +��Change variables to radiosity (B) and albedo (ρ)( , ) ( , )( ) ( ) ( ) ( )SG x x V x xB x E x x B x dArp� ��= +��Same as equation 2.54 in Cohen Wallace handout (read sec 2.6.3)Ignore factors of π which can be absorbed. Expresses conservation of light energy at all points in spaceOutlineOutlineRendering equation reviewRadiosity equationForm factorsMethods to compute form factorsSection 16.3.1,2 (eqs 16.63-65) in FvDFHDiscretization and Form FactorsDiscretization and Form Factors( , ) ( , )( ) ( ) ( ) ( )SG x x V x xB x E x x B x dArp� ��= +��ji i i j j ijiAB E B FAr�= +�F is the form factor. It is dimensionless and is the fraction of energy leaving the entirety of patch j (multiply by area of j to get total energy) that arrives anywhere in the entirety of patch i (divide by area of i to get energy per unit area or radiosity).Form FactorsForm FactorsjdAiqjqidArjAiA( , ) ( , )i i j j j i i jG x x V x xA F A F dA dAp� �� �= =��2cos cos( , ) ( , )| |i oG x x G x xx xq q� �= =�-Matrix EquationMatrix Equationji i i j j ijiAB E B FAr�= +�( , ) ( , )i i j j j i i jG x x V x xA F A F dA dAp� �� �= =��i i i j i jjB E B Fr�= +�i i j i j ijB B F Er�- =�ij j i ij ij i i jjM B E MB E M I Fr�= = = -�OutlineOutlineRendering equation reviewRadiosity equationForm factorsMethods to compute form factorsSection 16.3.2 in FvDFHNusselt’s AnalogNusselt’s AnalogAnalytically projectinto hemisphere abovepoint. Then project onto hemisphere baseForm factor is ratio of area on base to area of entire base This computes differentialpoint to patch form factorWhy does it work?HemicubeHemicubeHemicubesHemicubesEach small hemicube cell has a precomputed delta form factor: add up to get final valueWe can render the scene using normal Z-buffer scan conversion onto the faces of the hemicube!ArFpip2coscosMonte Carlo Ray TracingMonte Carlo Ray TracingCan be used to find form factors (slow)Can be used directly to shoot