Hierarchical algorithms for illumination

This dissertation is a discussion and development of hierarchical algorithms for illumination. These algorithms operate through recursive, adaptive refinement of the environment into hierarchical meshes--rather than computing light transport only between elements at the finest level of refinement, the algorithms allow computation of transport between higher level subpatches, as controlled by user specified error bounds. As discussed in this dissertation, employment of hierarchical methods yields significant savings in computation. The initial work in hierarchical methods was that of Hanrahan and Salzman for the computation of radiosity over unoccluded environments. In this dissertation, we discuss extension of the algorithm to occluded environments, incorporating visibility heuristics and acceleration via radiosity weighting. Given an environment consisting of k polygonal patches and n elements at the finest level of refinement, the algorithm requires at most $O(n+k\sp2)$ transport interactions; traditional methods require $O(n\sp2).$ Application of hierarchical transport to nondiffuse reflection is developed through the derivation of a radiance formulation for discrete three point transport, incorporating a new measure and description of reflectance: area reflectance. This formulation and associated reflectance allow an estimate of error in the computation of radiance across triples of surface elements, and lead directly to a hierarchical refinement algorithm for global illumination. We have implemented and analyzed this algorithm over surfaces exhibiting glossy and diffuse reflection. Theoretical growth in transport computation is shown to be $O(n+k\sp3)$--this growth is exhibited in experimental trials. Naive application of three point transport would require computation over $O(n\sp3)$ element triples. Global illumination within nondiffuse environments is ideally suited for computation under importance and radiance driven refinement: a transport interaction is of significance only if it lies within paths of directional reflection of both radiance originating at a light source, and importance originating at the eye. We have thus derived the adjoint to the radiance transport formulation, and present preliminary results of application of this adjoint in the form of an importance driven version of our implementation. These results show significant reduction in computation, and indicate that importance and radiance driven hierarchical techniques possess great potential for efficient evaluation of global illumination over general reflection.