An unbiased ray-marching transmittance estimator

TL;DR

This paper introduces a new unbiased volumetric transmittance estimator that significantly reduces variance and computational cost, achieving near-zero variance for rays with constant extinction, by combining novel statistical techniques and sampling strategies.

Contribution

The authors develop a universally optimal estimator that combines U-statistics, reduced expansion order, and control variates for efficient and low-variance transmittance estimation.

Findings

Up to several orders of magnitude lower variance at the same computational cost

Zero variance for rays with non-varying extinction

Effective combination of statistical and sampling techniques

Abstract

We present an in-depth analysis of the sources of variance in state-of-the-art unbiased volumetric transmittance estimators, and propose several new methods for improving their efficiency. These combine to produce a single estimator that is universally optimal relative to prior work, with up to several orders of magnitude lower variance at the same cost, and has zero variance for any ray with non-varying extinction. We first reduce the variance of truncated power-series estimators using a novel efficient application of U-statistics. We then greatly reduce the average expansion order of the power series and redistribute density evaluations to filter the optical depth estimates with an equidistant sampling comb. Combined with the use of an online control variate built from a sampled mean density estimate, the resulting estimator effectively performs ray marching most of the time while using rarely-sampled higher order terms to correct the bias.