NeRF Revisited: Fixing Quadrature Instability in Volume Rendering

TL;DR

This paper addresses the instability in volume rendering for NeRFs caused by quadrature approximation errors, proposing a mathematically grounded reformulation that improves rendering quality and geometric accuracy.

Contribution

It introduces a new formulation of the volume rendering equation that ensures exact integral approximation under piecewise linear density, enhancing NeRF performance.

Findings

Sharper textures achieved

Improved geometric reconstruction

Enhanced depth supervision

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 pl-nerf.github.io.