We propose a set of techniques to efficiently importance sample the derivatives of several BRDF models. In differentiable rendering, BRDFs are replaced by their differential BRDF counterparts which are real-valued and can have negative values. This leads to a new source of variance arising from their change in sign. Real-valued functions cannot be perfectly importance sampled by a positive-valued PDF and the direct application of BRDF sampling leads to high variance. Previous attempts at antithetic sampling only addressed the derivative with the roughness parameter of isotropic microfacet BRDFs. Our work generalizes BRDF derivative sampling to anisotropic microfacet models, mixture BRDFs, Oren-Nayar, Hanrahan-Krueger, among other analytic BRDFs. Our method first decomposes the real-valued differential BRDF into a sum of single-signed functions, eliminating variance from a change in sign. Next, we importance sample each of the resulting single-signed functions separately. The first decomposition, positivization, partitions the real-valued function based on its sign, and is effective at variance reduction when applicable. However, it requires analytic knowledge of the roots of the differential BRDF, and for it to be analytically integrable too. Our key insight is that the single-signed functions can have overlapping support, which significantly broadens the ways we can decompose a real-valued function. Our product and mixture decompositions exploit this property, and they allow us to support several BRDF derivatives that positivization could not handle. For a wide variety of BRDF derivatives, our method significantly reduces the variance (up to 58x in some cases) at equal computation cost and enables better recovery of spatially varying textures through gradient-descent-based inverse rendering.

翻译：我们提出了一组高效重要性采样多种BRDF模型导数的技术。在可微渲染中，BRDF被替换为其微分BRDF对应物，这些对应物是实值函数且可能取负值，这导致了由符号变化引入的新方差源。实值函数无法被正值的概率密度函数完美重要性采样，直接应用BRDF采样会导致高方差。先前尝试的对偶采样仅解决了各向同性微面元BRDF粗糙度参数的导数问题。我们的工作将BRDF导数采样推广至各向异性微面元模型、混合BRDF、Oren-Nayar、Hanrahan-Krueger及其他解析BRDF。该方法首先将实值微分BRDF分解为单符号函数之和，消除由符号变化引起的方差；随后分别对每个单符号函数进行重要性采样。第一种分解——正化法——基于符号划分实值函数，在适用时能有效降低方差，但要求解析掌握微分BRDF的根并具备解析可积性。我们的关键见解在于：单符号函数可具有重叠支撑域，这显著拓宽了实值函数的分解路径。乘积分解与混合分解利用了这一性质，使我们能够支持正化法无法处理的多种BRDF导数。对于各类BRDF导数，我们的方法在相同计算代价下显著降低了方差（某些情况下可达58倍），并通过基于梯度下降的逆渲染实现了空间变化纹理的更优恢复。