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）依赖体绘制来合成新视角。体绘制需要沿每条射线计算积分，该积分通过有限求和进行数值逼近，该求和对应于分段恒定体密度下沿射线的精确积分。由此，渲染结果关于射线采样点的选择不稳定，我们将此现象称为求积不稳定性。我们提出一种数学上严谨的解法：重新表述基于样本的渲染方程，使其对应于分段线性体密度下的精确积分。这同时解决了多个问题：不同射线间样本的冲突、不精确的分层采样、以及射线终止距离分位数关于模型参数的非可微性。我们展示了相对于经典基于样本的渲染方程的多个优势，例如更清晰的纹理、更好的几何重建以及更强的深度监督。我们提出的公式也可作为现有基于NeRF方法的体绘制方程的即插即用替代方案。我们的项目页面位于pl-nerf.github.io。