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.

翻译：神经辐射场（NeRF）依赖体渲染来合成新视角图像。体渲染需要计算沿每条光线的积分，该积分通过有限求和进行数值近似，该求和对应于分段常数体密度假设下沿光线的精确积分。因此，渲染结果对光线采样点的选择不稳定，我们称此现象为求积不稳定性。我们提出一种数学上严谨的解决方案，将基于样本的渲染方程重新表述为分段线性体密度假设下的精确积分形式。这同时解决了多个问题：不同光线间样本的冲突、不精确的分层采样，以及光线终止距离分位数相对于模型参数的非可微性。我们证明了相对于经典基于样本渲染方程的多个优势，如更清晰的纹理、更好的几何重建和更强的深度监督。我们提出的公式可替代现有基于NeRF方法中的体渲染方程。我们的项目页面位于pl-nerf.github.io。