A signal-processing framework for reflection

We present a signal-processing framework for analyzing the reflected light field from a homogeneous convex curved surface under distant illumination. This analysis is of theoretical interest in both graphics and vision and is also of practical importance in many computer graphics problems---for instance, in determining lighting distributions and bidirectional reflectance distribution functions (BRDFs), in rendering with environment maps, and in image-based rendering. It is well known that under our assumptions, the reflection operator behaves qualitatively like a convolution. In this paper, we formalize these notions, showing that the reflected light field can be thought of in a precise quantitative way as obtained by convolving the lighting and BRDF, i.e. by filtering the incident illumination using the BRDF. Mathematically, we are able to express the frequency-space coefficients of the reflected light field as a product of the spherical harmonic coefficients of the illumination and the BRDF. These results are of practical importance in determining the well-posedness and conditioning of problems in inverse rendering---estimation of BRDF and lighting parameters from real photographs. Furthermore, we are able to derive analytic formulae for the spherical harmonic coefficients of many common BRDF and lighting models. From this formal analysis, we are able to determine precise conditions under which estimation of BRDFs and lighting distributions are well posed and well-conditioned. Our mathematical analysis also has implications for forward rendering---especially the efficient rendering of objects under complex lighting conditions specified by environment maps. The results, especially the analytic formulae derived for Lambertian surfaces, are also relevant in computer vision in the areas of recognition, photometric stereo and structure from motion.