Last TimeTodayRadiosityExitanceRadiosity AlgorithmsDiffuse Surface RadiosityRadiosity Light TransportSwitch the DomainDiscretize RadiosityDiscrete FormulationThe Form FactorForm Factor PropertiesThe Discrete Radiosity EquationSolving for RadiositySolving the Linear SystemRelaxation MethodsGauss-Seidel RelaxationDisplaying the ResultsValue for ComputationForm FactorsForm Factor ComputationsDirect Integration – e.g. Rect-RectContour IntegralProjection MethodsNusselt’s AnalogySame Projection – Same Form FactorMonte-Carlo Form FactorsThe HemicubeHemicube, cont.Next Time02/16/05 © 2005 University of WisconsinLast Time•Re-using paths–Irradiance Caching–Photon Mapping02/16/05 © 2005 University of WisconsinToday•Radiosity–A very important method in practice, because it is so much more efficient than Monte Carlo for diffuse environments–Can also be used in conjunction with Monte Carlo, if you’re very careful about partitioning the LTE into different components02/16/05 © 2005 University of WisconsinRadiosity•Radiosity is the total power leaving a surface, per unit area on the surface–Usually denoted B–The outgoing version of irradiance•To get it, integrate radiance over the hemisphere of outgoing directions:  2cos),(nxxHoodLdxdB02/16/05 © 2005 University of WisconsinExitance•Light sources emit light, they are sources of radiance•Exitance is the equivalent of radiosity for emitters:•Distinguish exitance from radiosity to simplify equations•Different from Intensity, which is power per unit solid angle•Exitance is not ill-defined for point light sources  2cos),(nxxHooedLE02/16/05 © 2005 University of WisconsinRadiosity Algorithms•Radiosity algorithms solve the global illumination equation under a restrictive set of assumptions–All surfaces are perfectly diffuse–We only care about the radiosity at surfaces•Some form of rendering pass is required to transfer to the image plane–Surfaces can be broken into patches with constant radiosity•Some algorithms extend this to linear combinations of basis functions•These assumptions allow us to linearize the global illumination equation02/16/05 © 2005 University of WisconsinDiffuse Surface Radiosity•Diffuse surfaces, by definition, have outgoing radiance that does not depend on direction•Same can be said for diffuse emitters•And recall the definition of the diffuse BRDF in terms of directional hemispheric reflectance   xxxnoHooLdLB2cos),(   xxxneHooeLdLE2cos),( hdf x02/16/05 © 2005 University of WisconsinRadiosity Light Transport•Simplifying the global illumination equation gives:•We have removed almost all the angular dependence, but we still have an integral of directions computing irradiance                   dLEBdLLLdLfLLhdhdeooeocos,cos,cos,,,,,xxxxxxxxxxxx02/16/05 © 2005 University of WisconsinSwitch the Domain•We can convert the integral over the hemisphere of solid angles into one over all the surfaces in a scene: otherwiseisiblemutually v arey and x if 01,cos2yxVrdyd       ShddAVrBEByyxyxxx ,coscos)(202/16/05 © 2005 University of WisconsinDiscretize Radiosity•Assume world is broken into N disjoint patches, Pi, i=1..N, each with area Ai•Assume radiosity is constant over patches•Define:iiPiiPiidxEAEdxBABxxxx)(1)(102/16/05 © 2005 University of WisconsinDiscrete Formulation•Change the integral over surfaces to a sum over patches:                 NjP PijiiiPNjPhdPiNjPhdi jijijddVrABEBddVrBEdBAdVrBEB121212,coscos1,coscos)(1,coscos)(x yxyxyxyyxxyyxyxxxxyyxyxxx02/16/05 © 2005 University of WisconsinThe Form Factor•Note that we use it the other way: the form factor Fij is used in computing the energy arriving at I•Also called the configuration factordydxyxVrAFi jPx Pyiij),(coscos12  Fij is the proportion of the total power leaving patch Pi that is received by patch Pj02/16/05 © 2005 University of WisconsinForm Factor Properties•Depends only on geometry•Reciprocity: AiFij=AjFji•Additivity: Fi(jk)=Fij +Fik•Reverse additivity is not true•Sum to unity (all the power leaving patch i must get somewhere):1,1NjijFi02/16/05 © 2005 University of WisconsinThe Discrete Radiosity Equation•This is a linear equation!•Dimension of M is given by the number of patches in the scene: NxN–It’s a big system–But the matrix M has some special propertiesNjjijiiiBFEB1)( where FIMΜBEFBEB02/16/05 © 2005 University of WisconsinSolving for Radiosity•First compute all the form factors–These are view-independent, so for many views this need only be done once–Many ways to compute form factors•Compute the matrix M•Solve the linear system–A range of methods exist•Render the result using Gourand shading, or some other method – but no additional lighting, it’s baked in–Each patch’s diffuse intensity is given by its radiosity02/16/05 © 2005 University of WisconsinSolving the Linear System•The matrix is very large – iterative methods are preferred•Start by expressing each radiance in terms of the others:ijiijijijNjijFMNiEBM ,1 ,1NiMEBMMBiiijNijjiiiji1 ,102/16/05 © 2005 University of WisconsinRelaxation Methods•Jacobi relaxation: Start with a guess for Bi, then (at iteration m):•Gauss-Siedel relaxation: Use values already computed in this iteration:NiMEBMMBiiimjNijjiiijmi1 ,)1(1)(NiMEBMMBMMBiiimjNijiiijmjijiiijmi1 ,)1(1)(11)(02/16/05 © 2005 University of WisconsinGauss-Seidel Relaxation•Allows updating in place•Requires strictly diagonally dominant:•It can be shown that the matrix M is diagonally dominant–Follows from the properties of form factorsNiMMNijjijii1 |,|||102/16/05 © 2005 University of WisconsinDisplaying the Results•Color