  We present the Bayesian Neural Radiance Field (NeRF), which explicitly
quantifies uncertainty in geometric volume structures without the need for
additional networks, making it adept for challenging observations and
uncontrolled images. NeRF diverges from traditional geometric methods by
offering an enriched scene representation, rendering color and density in 3D
space from various viewpoints. However, NeRF encounters limitations in relaxing
uncertainties by using geometric structure information, leading to inaccuracies
in interpretation under insufficient real-world observations. Recent research
efforts aimed at addressing this issue have primarily relied on empirical
methods or auxiliary networks. To fundamentally address this issue, we propose
a series of formulational extensions to NeRF. By introducing generalized
approximations and defining density-related uncertainty, our method seamlessly
extends to manage uncertainty not only for RGB but also for depth, without the
need for additional networks or empirical assumptions. In experiments we show
that our method significantly enhances performance on RGB and depth images in
the comprehensive dataset, demonstrating the reliability of the Bayesian NeRF
approach to quantifying uncertainty based on the geometric structure.
