01/27/03© 2002 University of Wisconsin Last Time Radiometry A lot of confusion about Irradiance and BRDFs –Clarrified (I hope) today Radiance

01/27/03© 2002 University of Wisconsin Today Radiosity

01/27/03© 2002 University of Wisconsin Irradiance Irradiance is the power arriving at a surface, per unit area on the surface Usually denoted I To get it, integrate the “per steradian” and “perpendicular to the direction of travel” parts out of radiance Radiance: Wm -2 sr -1 Irradiance: Wm -2 Total irradiance (the power received from all directions) is obtained by integrating radiance over the incoming directions

01/27/03© 2002 University of Wisconsin BRDF Bidirectional reflectance distribution function The ratio of the radiance in the outgoing direction to the incident irradiance The BRDF is like a probability distribution function –The use of irradiance in the form on the bottom means the BRDF has no use unless integrated over a set of directions The BRDF has units: sr -1

01/27/03© 2002 University of Wisconsin BRDF Properties Easy to compute radiance leaving due to irradiance from one or all directions –Integrate over incoming hemisphere to get all outgoing light in one direction Not an arbitrary function: –Must be symmetric:  bd (x,  o,  o,  i,  i )=  bd (x,  i,  i,  o,  o ) –Cannot be large in too many places: light cannot be created

01/27/03© 2002 University of Wisconsin Global Illumination Equation All the light in the world must balance out – energy that is reflected or emitted must arrive somewhere Outgoing Radiance Emitted Radiance BRDFIrradiance

01/27/03© 2002 University of Wisconsin Qualitative Reasoning Before we go into solution methods for global illumination, some qualitative reasoning can help with understanding A lot about the brightness of a point can be deduced by considering what the world looks like from the point Examples: –Overcast sky, plane –Overcast sky, plane, high wall –Overcast sky, plane, half wall –Point light source, linear light source –Area lights and shadows

01/27/03© 2002 University of Wisconsin Radiosity 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 Surfaces are diffuse, so radiance is equal in all directions out from a point on a surface –Surfaces can be broken into patches with constant radiosity These assumptions allow us to linearize the global illumination equation

01/27/03© 2002 University of Wisconsin Radiosity Radiosity is the total power leaving a point on a surface per unit area on the surface Diffuse surfaces, by definition, have outgoing radiance that does not depend on direction

01/27/03© 2002 University of Wisconsin Directional Hemispheric Reflectance Used for surfaces for which the light exiting does not depend on the exit direction –But it might depend on the incoming direction Directional Hemispheric Reflectance is the fraction of the incident irradiance in a given direction that is reflected by the surface

01/27/03© 2002 University of Wisconsin Ideal Diffuse Surfaces For ideal diffuse surfaces, the BRDF is a constant: Directional hemispheric reflectance is generally denoted  d

01/27/03© 2002 University of Wisconsin Exitance Measure of the power of light sources Note the similarity to radiosity For constant emitted radiance (diffuse light sources): Exitance is the total power leaving a point on a surface per unit area on the surface

01/27/03© 2002 University of Wisconsin Radiance to Radiosity Recall: Simplifying the global illumination equation gives:

01/27/03© 2002 University of Wisconsin Switching the Domain We still have annoying radiance terms inside the integral Radiance is constant along lines, the radiance arriving is coming from a diffuse surface, y:

01/27/03© 2002 University of Wisconsin Switching the Domain (cont) We can convert the integral over the hemisphere of solid angles into one over all the surfaces in a scene: A continuous equation the radiosity of diffuse surfaces

01/27/03© 2002 University of Wisconsin Discrete Formulation Assume world is broken into N disjoint patches, P j, j=1..N, each with area A j Define:

01/27/03© 2002 University of Wisconsin Discrete Formulation (cont) Change the integral over surfaces to a sum over patches:

01/27/03© 2002 University of Wisconsin The Form Factor Note that we use it the other way: the form factor F ij is used in computing the energy arriving at i F ij is the proportion of the total power leaving patch P i that is received by patch P j

01/27/03© 2002 University of Wisconsin Form Factor Properties Depends only on geometry Reciprocity: A i F ij =A j F ji Additivity: F i(j  k) =F ij +F ik Reverse additivity is not true Sum to unity (all the power leaving patch i must get somewhere):

01/27/03© 2002 University of Wisconsin The 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

01/27/03© 2002 University of Wisconsin Solving 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 radiosity

01/27/03© 2002 University of Wisconsin Solving the Linear System The matrix is very large – iterative methods are preferred Start by expressing each radiance in terms of the others:

01/27/03© 2002 University of Wisconsin Relaxation Methods Jacobi relaxation: Start with a guess for B i, then (at iteration m): Gauss-Siedel relaxation: Use values already computed in this iteration:

01/27/03© 2002 University of Wisconsin Gauss-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 factors

01/27/03© 2002 University of Wisconsin Displaying the Results Color is handled by discretizing wavelength and solving each channel separately Smooth shading: –Patch radiosities are mapped to vertex colors by averaging the radiosities of the patches incident upon the vertex –Per-vertex colors then used to Gourand shade

01/27/03© 2002 University of Wisconsin Value for Computation Most of the time is spent computing form factors - must solve N visibility problems However, same form factors for different illumination conditions Result is view independent - have radiosities for all patches. May be good or may be wasteful

01/27/03© 2002 University of Wisconsin Next Lecture Some methods for solving the equation