NeRF Revisited: Fixing Quadrature Instability in Volume Rendering

Published: 21 Sept 2023, Last Modified: 16 Jan 2024NeurIPS 2023 posterEveryoneRevisionsBibTeX
Keywords: neural radiance fields, volumetric rendering, nerfs, numerical quadrature, importance sampling
TL;DR: We derive the volume rendering equation under a piecewise linear opacity and piecewise constant color assumption for NeRFs.
Abstract: Neural radiance fields (NeRF) rely on volume rendering to synthesize novel views. Volume rendering requires evaluating an integral along each ray, which is numerically approximated with a finite sum that corresponds to the exact integral along the ray under piecewise constant volume density. As a consequence, the rendered result is unstable w.r.t. the choice of samples along the ray, a phenomenon that we dub quadrature instability. We propose a mathematically principled solution by reformulating the sample-based rendering equation so that it corresponds to the exact integral under piecewise linear volume density. This simultaneously resolves multiple issues: conflicts between samples along different rays, imprecise hierarchical sampling, and non-differentiability of quantiles of ray termination distances w.r.t. model parameters. We demonstrate several benefits over the classical sample-based rendering equation, such as sharper textures, better geometric reconstruction, and stronger depth supervision. Our proposed formulation can be also be used as a drop-in replacement to the volume rendering equation of existing NeRF-based methods. Our project page can be found at
Supplementary Material: zip
Submission Number: 4164