Monte Carlo approximation of form factors with error bounded a priori

Abstract

The exchange of radiant energy (e.g. visible light, infra-red radiation) in simple macroscopic physical models is sometimes approximated by the solution of a system of linear equations (energy transport equations). A variable in such a system represents the total energy emitted by a discrete surface element. The coefficients of these equations depend on the form factors between pairs of surface elements. A form factor is the fraction of energy leaving a surface element which directly reaches an other surface element. Form factors depend only on the geometry of the physical model. Determining good approximations of form factors is the most time consuming step in these methods, when the geometry of the model is complex due to occlusions. In this paper we introduce a new characterization of form factors based on concepts from integral geometry. Using this characterization, we develop a new and asymptotically efficient Monte Carlo method for the simultaneous approximation of all form factors in an occluded polyhedral environment. This is the first algorithm for which an asymptotic time bound and a bound on the absolute approximation error has been proved. This algorithm is one order of magnitude faster than methods based on the hemi-sphere paradigm, for typical scenes. Let A be a set of convex non-intersecting polygons in R3 with a total of n edges and vertices, covering the facets of the input polyhedra. Let ϵ be the error parameter and 6 be the confidence parameter. We compute an approximation of each non-zero form factor such that with probability at least 1 - δ the absolute approximation error is less than ϵ. The expected running time of the algorithm is 0((ϵ-2 log δ-1)(n log-2 n + K log n)), where K is the expected number of regular intersections for a random projection of A. The number of regular intersections can range from 0 to quadratic in n, but for typical applications it is much smaller than quadratic. The expectation is with respect to the random choices of the algorithm and the result holds for any input.