# An unbiased ray-marching transmittance estimator

arXiv: https://arxiv.org/abs/2102.10294 · HTML: https://mlsanity.com/paper/2102.10294/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2021.

## Built on by (most cited first, 5 of 5)

- [Don't Splat your Gaussians: Volumetric Ray-Traced Primitives for Modeling and Rendering Scattering and Emissive Media](https://mlsanity.com/paper/2405.15425/) (2024)
- [Neural Implicit Surface Reconstruction using Imaging Sonar](https://mlsanity.com/paper/2209.08221/) (2022)
- [Objects as volumes: A stochastic geometry view of opaque solids](https://mlsanity.com/paper/2312.15406/) (2023)
- [Deep Appearance Prefiltering](https://mlsanity.com/paper/2211.05932/) (2022)
- [Gabor Fields: Orientation-Selective Level-of-Detail for Volume Rendering](https://mlsanity.com/paper/2602.05081/) (2026)

## Abstract (arXiv)

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.