Enforcing Energy Preservation in Microfacet Models

Microfacet models suffer from a significant limitation: they only simulate a single interaction between light and surface, ignoring the subsequent scattering across the microfacets. As a consequence, the BSDF is not energy preserving, resulting in an unexpected darkening of rough specular surfaces. Energy compensation methods face this limitation by adding to the BSDF a secondary component accounting for multiple scattering contributions. While these methods are fast, robust and can be added to a renderer with relatively minor modifications, they involve the computation of the directional albedo. This quantity is expressed as an integral that does not have a closed-form solution, but it needs to be precomputed and stored in tables. These look-up tables are notoriously cumbersome to use, in particular on GPUs. This work obviates the need of look-up tables by fitting an analytic approximation of the directional albedo, which is a more practical solution. We propose a 2D rational polynomial of degree three to fit conductors and a 3D rational polynomial of degree three to fit dielectrics and materials composed of a specular layer on top of a diffuse one, such as plastics. We enforce energy preservation by rescaling the specular albedo, thus maintaining the same lobe shape. We validated our results via the furnace test, highlighting that materials rendered using our analytic approximations match almost exactly the behaviour of the ones rendered with the use of look-up tables, resulting in an energy-preserving model even at maximum roughness. The software we use to fit coefficients is open-source and can be used to fit other BSDF models as well.