Bidirectional reflection distribution function expressed in terms of surface scattering modes

This work indicates how to construct a set of orthonormal polynomials on the Cartesian product of the hemisphere with itself with the required symmetry and invariance properties.

Abstract

In many applications one needs a concise description of the Bidirectional Reflection Distribution Function (BRDF) of real materials. Because the BRDF depends on two independent directions (thus has four degrees of freedom) one typically has only a relatively sparse set of observations. In order to be able to interpolate these sparse data in a convenient and principled manner a series development in terms of an orthonormal basis is required. The elements of the basis should be ordered with respect to angular resolution. Moreover, the basis should automatically respect the inherent symmetries of the physics, i.e., Helmholtz's reciprocity and (most often) surface isotropy. We indicate how to construct a set of orthonormal polynomials on the Cartesian product of the hemisphere with itself with the required symmetry and invariance properties. These "surface scattering modes" form a convenient basis for the description of BRDF's.