  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.
