Last TimeTodayForm Factors (recall)Form Factor ComputationsDirect IntegrationDirect Integration – Point-RectDirect Integration – Rect-RectContour IntegralForm Factor AlgebraProjection MethodsNusselt’s AnalogySame Projection – Same Form FactorMonte-Carlo Form FactorsThe HemicubeHemicube, cont.Problems with Gauss-SeidelRadiosity Eqn to Energy EqnRelaxation and ResidualsSouthwell RelaxationUpdating ResidualsSouthwell SummaryPhysical InterpretationGathering and ShootingProgressive RefinementTypical EquationsMore EquationsAmbient Correction01/29/03 © 2003 University of WisconsinLast Time•Radiosity01/29/03 © 2003 University of WisconsinToday•Form Factor Computations•Progressive Radiosity01/29/03 © 2003 University of WisconsinForm Factors (recall)•The proportion of patch i’s power received by patch j•Also a point-patch form: the proportion of the power from a differential area about point x received by jdydxyxVrAFi jPx Pyiji),(coscos12,  jPyjxdyyxVrF ),(coscos2,01/29/03 © 2003 University of WisconsinForm Factor Computations•Unoccluded patches:–Direct integration–Conversion to contour integration–Form factor algebra•Occluded patches:–Monte Carlo integration–Projection methods•Hemisphere•Hemicube01/29/03 © 2003 University of WisconsinDirect Integration•Only works for simplest cases•For example: Point to a Disc–A point to something form factor is that from a small area, dx, to the patch–Integrating over the points on a patch gets you the patch-patch integral–Note that the point is on the line perpendicular to the disc through the center222,vuuFdiscx01/29/03 © 2003 University of WisconsinDirect Integration – Point-Rect•Note that you could get any point-rectangle configuration using the additive property212212,1tan11tan121YXYYXYXXFrectxwvYwuX  ,01/29/03 © 2003 University of WisconsinDirect Integration – Rect-Rect•Note that we can do this only under the constant radiosity over patch assumption•There is a formula for 2 isolated polygons, but it assumes they can see each other fully!         222222222222222222221221111ln11ln111ln41 1tan1tan1tan1YXYYXYYYXXYXXXYXYXXYXYXYYXXXF01/29/03 © 2003 University of WisconsinContour Integral•Use Stokes’ theorem to convert the area integrals into contour integrals•For point to polygon form factors, the contour integral is not too hard•Care must be taken when r0i jC CiijydxrdAFln2101/29/03 © 2003 University of WisconsinForm Factor Algebra•Use the properties of form factors and direct integration to compute new form factors–Additivity–Reciprocity–Point to disc form factors–Rectangle to rectangle form factors•Difficult to apply automatically–Useful only in tiny scenes•Unoccluded case only01/29/03 © 2003 University of WisconsinProjection Methods•For patches that are far apart compared to their areas, the inner integral in the form factor doesn’t vary much–That is, the form factor is similar from most points on a surface i•So, compute point to patch form factors and weigh by areadyyxVrFPyPx),(coscos2,01/29/03 © 2003 University of WisconsinNusselt’s Analogy•Integrate over visible solid angle instead of visible patch area:PdFPxcos1,Fx,P is the fraction of the area of the unit disc in the base plane obtained by projecting the surface patch P onto the unit sphere centered at x and then orthogonally down onto the base plane.01/29/03 © 2003 University of WisconsinSame Projection – Same Form Factor•Any patches with the same projection onto the hemisphere have the same form factor–Makes sense: put yourself at the point and look out – if you see equal amounts, they get equal power•It doesn’t matter what you project onto: two patches that project the same have the same form factor01/29/03 © 2003 University of WisconsinMonte-Carlo Form Factors•We can use Monte-Carlo methods to estimate the integral over visible solid angle•Simplest method:–Uniformly (in area) sample the disc about the point–Project up onto the hemisphere, then cast a ray out from the point in that direction–Form factor for each patch is the weighted sum of the number of rays that hit the patch•There are even better Monte-Carlo methods that we will see later01/29/03 © 2003 University of WisconsinThe Hemicube•We have algorithms, and even hardware, for projecting onto planar surfaces•The hemicube consists of 5 such faces•A “pixel” on the cube has a certain projection, and hence a certain form factor•Something that projects onto the pixel has the same form factor222222)1()1(yzAzFyxAFfacesidefacetop01/29/03 © 2003 University of WisconsinHemicube, cont.•Pretend each face of the hemicube is a screen, and project the world onto it•Code each polygon with a color, and count the pixels of each color to determine C(j)•Quality depends on hemicube resolution and z-buffer depth)(,jCqqPxFFj01/29/03 © 2003 University of WisconsinProblems with Gauss-Seidel•All the form factors are required before any image can be generated–So you wait a long time to see anything•Reducing the number of form factors requires reducing the number of patches, which severely impacts quality•We desire a progressive solution, that starts with a rough approximation and refines it–This also opens the possibility of computing some pieces (the bits you can see) before others01/29/03 © 2003 University of WisconsinRadiosity Eqn to Energy Eqn•The radiosity equation is in terms of power per unit area•Rewrite this equation in terms of energy values per patch (instead of per unit area)NiFBEBijijjiii1for jiiijijjiijjiiiijijjjiiiiiiiiiiiFFFAAEABAEAεBAβKεKβ with ,Let iNote that the form factor is now from j to i01/29/03 © 2003 University of WisconsinRelaxation and Residuals•Relaxation methods start with an initial guess, (0), and