02/07/03© 2003 University of Wisconsin Last Time Finite element approach Two-pass approaches

02/07/03© 2003 University of Wisconsin Today Participating media Assignment 2

02/07/03© 2003 University of Wisconsin Volumetric Effects Emission –Energy generated by the volume (flame, sun) Absorption –Energy lost to the volume Out-scattering –Energy scattered out of a volume In-scattering –Energy scattered into a volume from the neighborhood

02/07/03© 2003 University of Wisconsin Functions Describing a Volume Absorption coefficient,  a –Amount of energy absorbed per unit length Scattering coefficient,  s –Amount of energy scattered per unit length Emitted radiance, L e Phase function, f(  ) –Function describing how much energy comes from direction  into another direction –Note the “other direction” doesn’t matter because we assume that there is no orientation to the medium

02/07/03© 2003 University of Wisconsin General Transfer Equation With, extinction coefficient Describes how radiance changes along a line –For instance, integrate it to find total radiance after traveling some distance through the media from some point Once again, not easy to handle in its full form

02/07/03© 2003 University of Wisconsin Transmittance  : the fraction of energy that goes straight through:

02/07/03© 2003 University of Wisconsin No Scattering Can use with ray tracing Constant absorption and emittance (fog models):

02/07/03© 2003 University of Wisconsin Two-Pass Method Assume isotropic medium –Scatters equally in all directions –Volumetric equivalent of diffuse surface –R k =  s /  t Break volume into chunks Compute incoming radiance for all chunks and surfaces Render in a raytracing pass, accumulating contribution along each ray from the eye

02/07/03© 2003 University of Wisconsin Zonal Method Equations Need exchange factors (generalized from factors): Fraction of energy leaving one surface/volume that arrives at another surface/volume

02/07/03© 2003 University of Wisconsin Exchange Factors

02/07/03© 2003 University of Wisconsin Up next We switch to Monte-Carlo methods But first, assignment 2

02/07/03© 2003 University of Wisconsin Assignment 2 Modify your raytracer to compute a progressive radiosity solution Input: a scene that is highly triangulated and only contains diffuse surfaces and area light sources Output: A radiosity image of the scene

02/07/03© 2003 University of Wisconsin Sample Scene

02/07/03© 2003 University of Wisconsin Review of Progressive Radiosity Each triangle stores two quantities, with their initial values: On each step, find the highest residual: Compute all the form factors into patch i –With ray-tracing in this assignment Continued…

02/07/03© 2003 University of Wisconsin More Equations Keep iterating until: –A fixed number of iterations (good for debugging) –Until the max residual is small

02/07/03© 2003 University of Wisconsin Algorithm Details Each triangle stores: –  B i, B i : These are colors, compute residuals by averaging colors, do each equation 3 times – once for each color –A sample point (or points) for computing the form factor –Its area Compute form factors from sample point –Cast rays to one (or more) points on the source patch –Use disc approximation for form factor: Produce final image by raytracing scene using stored B i as surface colors

02/07/03© 2003 University of Wisconsin Better Form Factors To compute the basic form factor, just use a single ray to check whether patch j can see patch i, and if so use disc approximation –Use center points of triangles as source and target for ray For better estimates, use multiple points on the target –Subdivide triangle using standard subdivision scheme –Place sample points at the center of each leaf triangle –Use area of each leaf for disc form factors and add to get total form factor

02/07/03© 2003 University of Wisconsin Better Radiosity Estimates Subdivide receiving patches Shoot to each sub-patch Area-weight result to get parent radiosities and residuals