Approximating the Location of Integrand Discontinuities for Penumbral Illumination with Area Light Sources

The problem of computing soft shadows with area light sources has received considerable attention in computer graphics. In part, this is a difficult problem because the integral that defines the radiance at a point must take into account the visibility function. Most of the solutions proposed have been limited to polygonal environments, and require a full visibility determination preprocessing step. The result is typically a partitioning of the environment into regions that have a similar view of the light source. We propose a new approach that can be successfully applied to arbitrary environments. The approach is based on the observation that, in the presence of occluders, the primary difficulty in computing the integral that defines the contribution of an area light source, is that of determining the visible domain of the integrand. We extend a recent shadow algorithm for linear light sources in order to calculate a polygonal approximation to this visible domain. We demonstrate for an important class of shadowing problems, and in particular, for convex occluders, that the shape of the visible domain only needs to be roughly approximated by a polygonal boundary. We then use this boundary to subdivide an area light source into a small number of triangles that can be integrated efficiently using either a deterministic solution, or a low degree numerical cubature.