  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.
