Jackknife Transmittance and MIS Weight Estimation

Abstract

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.