  Neural radiance field (NeRF) enables the synthesis of cutting-edge realistic
novel view images of a 3D scene. It includes density and color fields to model
the shape and radiance of a scene, respectively. Supervised by the photometric
loss in an end-to-end training manner, NeRF inherently suffers from the
shape-radiance ambiguity problem, i.e., it can perfectly fit training views but
does not guarantee decoupling the two fields correctly. To deal with this
issue, existing works have incorporated prior knowledge to provide an
independent supervision signal for the density field, including total variation
loss, sparsity loss, distortion loss, etc. These losses are based on general
assumptions about the density field, e.g., it should be smooth, sparse, or
compact, which are not adaptive to a specific scene. In this paper, we propose
a more adaptive method to reduce the shape-radiance ambiguity. The key is a
rendering method that is only based on the density field. Specifically, we
first estimate the color field based on the density field and posed images in a
closed form. Then NeRF's rendering process can proceed. We address the problems
in estimating the color field, including occlusion and non-uniformly
distributed views. Afterward, it is applied to regularize NeRF's density field.
As our regularization is guided by photometric loss, it is more adaptive
compared to existing ones. Experimental results show that our method improves
the density field of NeRF both qualitatively and quantitatively. Our code is
available at https://github.com/qihangGH/Closed-form-color-field.
