An Analysis of the In‐Out BRDF Factorization for View‐Dependent Relighting

Interactive rendering with dynamic natural lighting and changing view is a long‐standing goal in computer graphics. Recently, precomputation‐based methods for all‐frequency relighting have made substantial progress in this direction. Many of the most successful algorithms are based on a factorization of the BRDF into incident and outgoing directions, enabling each term to be precomputed independent of viewing direction, and re‐combined at run‐time. However, there has so far been no theoretical understanding of the accuracy of this factorization, nor the number of terms needed. In this paper, we conduct a theoretical and empirical analysis of the BRDF in‐out factorization. For Phong BRDFs, we obtain analytic results, showing that the number of terms needed grows linearly with the Phong exponent, while the factors correspond closely to spherical harmonic basis functions. More generally, the number of terms is quadratic in the frequency content of the BRDF along the reflected or half‐angle direction. This analysis gives clear practical guidance on the number of factors needed for a given material. Different objects in a scene can each be represented with the correct number of terms needed for that particular BRDF, enabling both accuracy and interactivity.