One of the goals of computer graphics is the simulation of global illumination, the interreflection of light between diffuse and specular surfaces in three-dimensional scenes. Like thermal radiation, as studied in mechanical engineering and physics, global illumination is governed by an integral equation. A distinguishing feature of the integral equations of global illumination is that their solutions have numerous discontinuities, such as shadow edges, caused by occlusion. We show that the most common global illumination algorithms, radiosity and ray tracing, are simple finite element and Monte Carlo methods for solving integral equations, respectively. In the process of re-deriving these techniques, a number of alternative algorithms with higher accuracy are suggested. The principal alternatives explored in this thesis are adaptive meshing techniques that resolve discontinuities.

Radiosity algorithms can be made more accurate by a priori discontinuity meshing, placing mesh boundaries on significant discontinuities. Discontinuities are found using an object-space visible surface algorithm from the point of view of object vertices. The accuracy of radiosity simulations can also be improved using linear, quadratic, or higher degree elements instead of constant elements. Algorithms are developed first for two-dimensional radiosity in flatland problems, then extended to three dimensions.

Scenes containing diffuse and specular surfaces are most easily simulated using Monte Carlo ray tracing techniques. Traditional ray tracing algorithms trace rays from the eye into the scene. A bidirectional ray tracing algorithm is demonstrated that traces rays from both the lights and the eye, storing the diffuse intensity function in a radiosity texture on each diffuse surface in the scene. These textures are adaptively subdivided in order to resolve shadow edges and other discontinuities.




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