Neural Radiance Fields (NeRF) have attracted significant attention due to their ability to synthesize novel scene views with great accuracy. However, inherent to their underlying formulation, the sampling of points along a ray with zero width may result in ambiguous representations that lead to further rendering artifacts such as aliasing in the final scene. To address this issue, the recent variant mip-NeRF proposes an Integrated Positional Encoding (IPE) based on a conical view frustum. Although this is expressed with an integral formulation, mip-NeRF instead approximates this integral as the expected value of a multivariate Gaussian distribution. This approximation is reliable for short frustums but degrades with highly elongated regions, which arises when dealing with distant scene objects under a larger depth of field. In this paper, we explore the use of an exact approach for calculating the IPE by using a pyramid-based integral formulation instead of an approximated conical-based one. We denote this formulation as Exact-NeRF and contribute the first approach to offer a precise analytical solution to the IPE within the NeRF domain. Our exploratory work illustrates that such an exact formulation Exact-NeRF matches the accuracy of mip-NeRF and furthermore provides a natural extension to more challenging scenarios without further modification, such as in the case of unbounded scenes. Our contribution aims to both address the hitherto unexplored issues of frustum approximation in earlier NeRF work and additionally provide insight into the potential future consideration of analytical solutions in future NeRF extensions.

翻译：神经辐射场（NeRF）因其高精度合成新视角场景的能力而受到广泛关注。然而，其基础公式中沿零宽度射线采样点可能导致模糊表示，进而引发渲染伪影，如最终场景中的锯齿现象。为解决这一问题，近期变体mip-NeRF提出了基于锥形视锥体的集成位置编码（IPE）。尽管该编码以积分形式表达，但mip-NeRF将其近似为多元高斯分布的期望值。这一近似在短视锥体下可靠，但在处理大景深下的远处场景物体时，会因高度拉伸区域而退化。本文探索了一种精确计算IPE的方法，采用基于金字塔的积分公式替代近似锥形公式。我们将此公式命名为Exact-NeRF，并首次在NeRF领域为IPE提供了精确解析解。我们的探索性研究表明，这种精确公式Exact-NeRF在精度上与mip-NeRF相当，且能无需额外修改自然扩展到更具挑战性的场景（如无界场景）。本研究旨在解决先前NeRF工作中未探索的视锥体近似问题，并为未来NeRF扩展中解析解的潜在应用提供见解。