Multiple-scattering microfacet BSDFs with the Smith model

Modeling multiple scattering in microfacet theory is considered an important open problem because a non-negligible portion of the energy leaving rough surfaces is due to paths that bounce multiple times. In this paper we derive the missing multiple-scattering components of the popular family of BSDFs based on the Smith microsurface model. Our derivations are based solely on the original assumptions of the Smith model. We validate our BSDFs using raytracing simulations of explicit random Beckmann surfaces. Our main insight is that the microfacet theory for surfaces with the Smith model can be derived as a special case of the microflake theory for volumes, with additional constraints to enforce the presence of a sharp interface, i.e. to transform the volume into a surface. We derive new free-path distributions and phase functions such that plane-parallel scattering from a microvolume with these distributions exactly produces the BSDF based on the Smith microsurface model, but with the addition of higher-order scattering. With this new formulation, we derive multiple-scattering micro-facet BSDFs made of either diffuse, conductive, or dielectric material. Our resulting BSDFs are reciprocal, energy conserving, and support popular anisotropic parametric normal distribution functions such as Beckmann and GGX. While we do not provide closed-form expressions for the BSDFs, they are mathematically well-defined and can be evaluated at arbitrary precision. We show how to practically use them with Monte Carlo physically based rendering algorithms by providing analytic importance sampling and unbiased stochastic evaluation. Our implementation is analytic and does not use per-BSDF precomputed data, which makes our BSDFs usable with textured albedos, roughness, and anisotropy.