Multiple-scattering microfacet BSDFs with the Smith model

Abstract

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.

A Survey of BSDF Measurements and Representations

“…Some BSDF representations [19] are only validated on simulated data in the absence of such a well-known BSDF database. A similar situation is valid for existing data fitting and visualization tools.…”

“…However, Gu et al's BSDF model cannot be used to represent spatial variation of blurring and anisotropy, which can be seen in Figure 11. Dai Heitz et al [19] proposed an analytical microfacet-based BSDF model, which accounts for microsurface single and multiple scattering terms. Heitz [25] as this coordinate system helps to represent specular highlights more accurately.…”

Efficient rendering of layered materials using an atomic decomposition with statistical operators

“…e classical microfacet theory only accounts for single sca ering in the microgeometry and thus omits the energy due to multiple bounces in the microgeometry. According to Heitz et al [2016], a scaling factor of the directional albedo approximates multiple sca ering well. us, it should have no effect on the mean and variance and only a ect the energy term.…”

A Physically‐based Appearance Model for Special Effect Pigments

“…Microfacet theory based BSDF models are widely used in physically‐based energy‐conserving shading and have consistently proven instrumental and effective in reproducing the appearance of real‐world materials [CT82, NDM05, PJH16]. Recently, microfacet theory has been extended to handle multiple scattering [HHdD16, GQGP17], retroreflection [GP17] and wave‐optical effects [BB17, HP17]. Since these models usually adopt a smooth NDF to describe the microfacet orientations, they are only valid for very distant views, assuming that the micro‐scale details are indistinguishable [YHJ*14].…”