NeRF Revisited: Fixing Quadrature Instability in Volume Rendering

TL;DR

The paper identifies quadrature instability in NeRFs arising from the common piecewise constant opacity assumption and proposes PL-NeRF, which uses piecewise linear opacity to align the rendering integral with the exact solution. This reformulation yields closed-form expressions for transmittance and interval probabilities, enables invertible continuous CDFs for precise inverse transform sampling, and improves texture sharpness, geometry, and depth supervision while acting as a drop-in replacement for existing NeRF-based methods. Empirically, PL-NeRF improves performance on Blender and LLFF datasets, enhances geometric reconstruction, and strengthens depth supervision, with compatible gains when applied to Mip-NeRF and DIVeR. The approach offers a principled, numerically stable alternative that addresses sampling conflicts and gradient issues, advancing accurate neural volumetric rendering in practice.

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.

Figures (12)

• Figure 1: Ray Conflicts: Grazing Angle. (Left) Illustration of conflicting ray supervision at the grazing under the piecewise constant opacity. For the constant setting, to render perpendicular rays (yellow) correctly, the model has to store the associated optical properties at a region in front of the surface as a sample takes the values of the left bin boundary. In the presence of a ray near the grazing angle, it will be crossing this region of high opacity (the gradient in front of the surface), associating it with conflicting opacity/color signals. (Middle) This results in fuzzier surfaces as shown along the side of the microphone as there is a conflict in ray supervision between the perpendicular and grazing angle rays. Our piecewise linear opacity assumption alleviates this issue and results in a clearer rendered view. (Right) As shown, the resulting PDF is peakier and the CDF is sharper for our linear setting, where the plotted distributions correspond to the ray from the marked pixel in red.
• Figure 2: Ray Conflicts: Different Camera-to-Scene Distances. (Left) Rendered views from cameras at different distances from the object. At all distances, the rendered output for linear have sharper texture than constant because of the latter's sensitivity to the choice of samples. We also highlight the instability of the constant model as shown by the noisier texture of the middle view compared to the closer and further views. (Right) An illustration that moving the camera to different distances from the object result in different samples that lead to conflicts.
• Figure 3: Illustration of opacities $\tau$ along a ray under the piecewise constant (green) and piecewise linear (orange) assumptions.
• Figure 4: Qualitative Results for Blender and Real Forward Facing.
• Figure 5: Geometry Extraction Qualitative Examples.