NeRF Revisited: Fixing Quadrature Instability in Volume Rendering

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 for existing methods like NeRFs. Our project page can be found at pl-nerf.github.io. brings up a model specific issue on z-aliasing, where their model struggles under this setting. Similar to z-aliasing observed by ZipNeRF, we consider the setting of having conflicting supervision when presented with training views at different distances from the scene. While they may appear similar on the surface, the phenomena we study is different in that it is general and independent of the model, on having conflicting ray supervision from camera views, e.g. different camera-to-scene distances and the grazing angle setup.