A core operation in Monte Carlo volume rendering is transmittance estimation: Given a segment along a ray, the goal is to estimate the fraction of light that will pass through this segment without encountering absorption or out-scattering. A naive approach is to estimate optical depth τ using unbiased ray marching and to then use exp(-τ) as transmittance estimate. However, this strategy systematically overestimates transmittance due to Jensen's inequality. On the other hand, existing unbiased transmittance estimators either suffer from high variance or have a cost governed by random decisions, which makes them less suitable for SIMD architectures. We propose a biased transmittance estimator with significantly reduced bias compared to the naive approach and a deterministic and low cost. We observe that ray marching with stratified jittered sampling results in estimates of optical depth that are nearly normal-distributed. We then apply the unique minimum variance unbiased (UMVU) estimator of exp(-τ) based on two such estimates (using two different sets of random numbers). Bias only arises from violations of the assumption of normal-distributed inputs. We further reduce bias and variance using a variance-aware importance sampling scheme. The underlying theory can be used to estimate any analytic function of optical depth. We use this generalization to estimate multiple importance sampling (MIS) weights and introduce two integrators: Unbiased MIS with biased MIS weights and a more efficient but biased combination of MIS and transmittance estimation.

Physically based differentiable rendering has established itself as key to inverse rendering, in which scenes are recovered from images through gradient-based optimization. Taking the derivative of the rendering equation is made difficult by the presence of discontinuities in the integrand at object silhouettes. To obtain correct derivatives w.r.t. changing geometry, accounting e.g. for changing penumbras or silhouettes in glossy reflections, differentiable renderers must compute an integral over these silhouettes. Prior work proposed importance sampling of silhouette edges for a given shading point. The main challenge is to efficiently reject parts of the mesh without silhouettes during sampling, which has been done using top-down traversal of a tree. Inaccuracies of this existing rejection procedure result in many samples with zero contribution. Thus, variance remains high and subsequent work has focused on alternatives such as area sampling or path space differentiable rendering. We propose an improved rejection test. It reduces variance substantially, which makes edge sampling in a unidirectional path tracer competitive again. Our rejection test relies on two approximations to the triangle planes of a mesh patch: A bounding box in dual space and dual quadrics. Additionally, we improve the heuristics used for stochastic traversal of the tree. We evaluate our method in a unidirectional path tracer and achieve drastic improvements over the original edge sampling and outperform methods based on area sampling.