Approximating the Location of Integrand Discontinuities for Penumbral Illumination With Linear Light Sources

One of the benefits of shading with linear light sources is also one of its major challenges: generating soft shadows. The primary difficulty in this task is determining the discontinuities in the linear light source integrals that are caused by occluding objects. We demonstrate in this paper that the computed location of each discontinuity only needs to be moderately accurate, provided that the expected value of this location is a continuous function of the actual value of the location. We introduce Random Seed Bisection (RSB), an algorithm that has this property. We use this algorithm to efficiently find the approximate location of a discontinuity, in order to partition the domain of integration into subintervals (panels) over which the integrand is naturally smooth, and approximate the integral efficiently over each panel using low-degree numerical quadratures. We demonstrate the effectiveness of this solution for shadowing problems with at most 1 discontinuity in the domain of integration. We also provide efficient heuristics that take advantage of the coherence in a scene to handle shadowing problems with at most 2 discontinuities in the domain of integration. This work is a first step toward a comprehensive approach to efficiently solving numerical integration problems for extended light sources.