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Smith microfacet models are widely used in computer graphics to representmaterials. Traditional microfacet models do not consider the multiple bounceson microgeometries, leading to visible energy missing, especially on roughsurfaces. Later, as the equivalence between the microfacets and volume has beenrevealed, random walk solutions have been proposed to introduce multiplebounces, but at the cost of high variance. Recently, the position-free propertyhas been introduced into the multiple-bounce model, resulting in much lessnoise, but also bias or a complex derivation. In this paper, we propose asimple way to derive the multiple-bounce Smith microfacet bidirectionalreflectance distribution functions (BRDFs) using the invariance principle. Atthe core of our model is a shadowing-masking function for a path consisting ofdirection collections, rather than separated bounces. Our model ensuresunbiasedness and can produce less noise compared to the previous work withequal time, thanks to the simple formulation. Furthermore, we also propose anovel probability density function (PDF) for BRDF multiple importance sampling,which has a better match with the multiple-bounce BRDFs, producing less noisethan previous naive approximations.

Multiple-bounce Smith Microfacet BRDFs using the Invariance Principle