Gaussian Process for Radiance Functions on the $\mathbb{s}^2$ Sphere

Abstract

Efficient approximation of incident radiance functions from a set of samples is still an open problem in physically based rendering. Indeed, most of the computing power required to synthesize a photo-realistic image is devoted to collecting samples of the incident radiance function, which are necessary to provide an estimate of the rendering equation solution. Due to the large number of samples required to reach a high-quality estimate, this process is usually tedious and can take up to several days. In this paper, we focus on the problem of approximation of incident radiance functions on the $\mathbb{S}^2$ sphere. To this end, we resort to a Gaussian Process (GP), a highly flexible function modelling tool, which has received little attention in rendering. We make an extensive analysis of the application of GPs to incident radiance functions, addressing crucial issues such as robust hyperparameter learning, or selecting the covariance function which better suits incident radiance functions. Our analysis is both theoretical and experimental. Furthermore, it provides a seamless connection between the original spherical domain and the spectral domain, on which we build to derive a method for fast computation and rotation of spherical harmonics coefficients.