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

翻译：史密斯微面元模型被广泛应用于计算机图形学中的材质表示。传统微面元模型未考虑微观几何结构上的多重弹射，导致可见能量缺失，尤其在粗糙表面上更为显著。随后，随着微面元与体积等效关系的揭示，随机游走解被提出以引入多重弹射，但代价是高方差。近年来，位置无关性被引入多重弹射模型，显著降低了噪声，但也引入了偏差或复杂的推导过程。本文提出了一种利用不变性原理推导多重弹射史密斯微面元双向反射分布函数（BRDFs）的简洁方法。该模型的核心是面向由方向集合（而非分离的弹射）构成的路径间阴影-遮蔽函数。得益于简明的公式，该模型确保无偏性，并在同等计算时间内相较先前工作产生更低的噪声。此外，我们还提出了一种针对BRDF多重重要性采样的新概率密度函数（PDF），该函数与多重弹射BRDF的匹配度更高，相较于先前朴素近似方法能进一步降低噪声。