Return to Assumptions Recall the two fundamental assumptions

Return to Assumptions • Recall the two fundamental assumptions of the standard radiosity approach: – Constant radiosity over each element (today) – Perfectly diffuse surfaces (next week) • We wish to more accurately represent the radiosity at every point: – Regardless of the quality of the solution (form factors, subdivision, etc. ), if the correct radiosity cannot be represented, it won’t be computed

Better Radiosity Representations • Two approaches: – Work around the constant radiosity assumption by constructing meshes where elements really should have constant (or bilinear) radiosity – Use more expressive representations for radiosity (work under different assumptions)

Meshing Artifacts • Where does poor meshing really show? • Find the (many) artifacts in the following image

Typical Meshing • Lays down an orthogonal grid in some orientation • May subdivide, but the principle directions remain unchanged • Require subdivision down to pixel resolution to remove artifacts

Staircase Effect • Most evident at shadow boundaries: – Elements are dark if they have more than half shadow, light otherwise • Solution: – Subdivide the mesh to sub-pixel resolution (costly) – Blur the boundaries (introduces other problems) – Align element boundaries with the shadow boundaries (discontinuity meshing, later)

Light and Shadow Leaks • Occur when the mesh does not respect geometric boundaries – Interior walls meeting the floor • The wall appears to float – Shadows leaking out from under objects • No reasonable perceptual interpretation • Solutions: – Ensure that there are element boundaries at geometric intersections (hard to do without user help) – Discontinuity meshing

Interpolation Artifacts • Have to “fill in” radiosity between vertices, or blur element boundaries • Interpolation schemes are frequently poorly implemented, particularly those in hardware • Solutions: – Meshes that make interpolation easier – More expensive interpolation

Preprocessing the Mesh • Seek to remove inconsistencies before starting the solution • Ensure invariants are met: – Objects are solid – Objects don’t intersect • This is very hard to do automatically, it should be part of the modeling process • But, commercial modelers are generally poor in this regard

Fixing Interpolation • Pre-process to remove concave polygons – Replace them with triangles - not too hard • Enforce subdivision constraints – Neighboring patches cannot differ by more than one subdivision level • Post-process the remove T-vertices – Subdivide mismatched neighbors and interpolate new vertices correctly

Smoother Interpolants • Bi-linear (hardware accelerated) interpolation uses only vertices of given polygon • Build higher order interpolants – Must make sure not to use a high-order interpolant across a low order discontinuity • Gather to a texture or cloud of points • Gather to pixels in the image (slow)

Discontinuity Meshing • Identify expected discontinuities and mesh around them – Sharp boundaries due to point light sources or object contact – Derivative discontinuities due to area sources and multi-object shadows • Related to aspect graphs in computer vision – Places where the set of visible things changes

Two Types of Discontinuities • Assume polygonal environment • Vertex-Edge events – Discontinuities where the plane defined by a vertex and an edge intersects other objects • Edge-Edge – Higher order discontinuities at places where three edges appear to meet at a point – Generally ignored

Meshing With Discontinuities • Construct VE planes • Intersect them with surfaces • Mesh the resulting edges – Constrained triangulation is a difficult problem • Mesh must be able to store different radiosity values at one point

Using Discontinuity Meshes • Very high number of possible discontinuities: O(n 6) for n vertices • Only find discontinuities due to bright light sources • Try to only find visible discontinuities • Research topic: Integrate into hierarchical scheme

Better Radiosity Representations • Standard approach: Each point takes on the value of the patch on which it lies: • Finite Element Approach: The radiosity at each point is given by a linear combination of basis functions evaluated at that point:

Finite Element Formulation • Note the similarity to splines: a set of weights multiply a set of basis functions to give a value • Choose a set of basis functions that can capture the desired behavior • Find the coefficients, Bj, that give the best solution

Galerkin Method • Find the set of weights that minimize the variation of the found solution from the true solution • In other words: Find the closest expressible solution to the true one • The standard radiosity equation, with accurate form factors, is a Galerkin method with constant basis functions of finite support (supported by each patch)

Point-Collocation Method • Find the set of weights that zero the error at a fixed set of points • The hemicube algorithm implements a point -collocation method for the radiosity equation • Not as accurate as the Galerkin method: – Only locally accurate, as opposed to globally optimal

Alternate Bases • Linear basis functions • Wavelets: – Multi-resolution representation – Behaves like hierarchical radiosity, but without redundant information