NeRF Revisited: Fixing Quadrature Instability in Volume Rendering
This addresses a fundamental numerical issue in NeRF-based 3D reconstruction and view synthesis, offering a drop-in improvement for existing methods, though it is incremental as it builds on the established NeRF paradigm.
The paper tackles the problem of quadrature instability in Neural Radiance Fields (NeRF) volume rendering, which causes unstable results due to numerical approximations, and proposes a reformulation that resolves this by using piecewise linear density, leading to benefits like sharper textures and better geometric reconstruction.
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.