Multipole expansion of the light vector

Computing the light field due to an area light source remains an interesting problem in computer graphics. This paper presents a series approximation of the light field due to an unoccluded area source, by expanding the light field in spherical harmonics. The source can be nonuniform and need not be a planar polygon. The resulting formulas give expressions whose cost and accuracy can be chosen between the exact and expensive Lambertian solution for a diffuse polygon, and the fast but inexact method of replacing the area source by a point source of equal power. The formulas break the computation of the light vector into two phases: The first phase represents the light source’s shape and brightness with numerical coefficients, and the second uses these coefficients to compute the light field at arbitrary locations. We examine the accuracy of the formulas for spherical and rectangular Lambertian sources, and apply them to obtaining light gradients. We also show how to use the formulas to estimate light from uniform polygonal sources, sources with polynomially varying radiosity, and luminous textures.