Volume Rendering Digest (for NeRF)

TL;DR

The paper presents a compact, differentiable volume rendering framework for NeRF by formalizing light transport through a density field along rays. It derives the transmittance equation and its exponential solution, provides a probabilistic interpretation of opacity and stopping density, and specializes to piecewise-constant media to recover the NeRF color accumulation as an alpha-compositing weighted sum. An alternative derivation via CDF/PDF relations is provided to corroborate the same results. Together, these derivations clarify the multiplicative structure of transmittance and the role of alpha weights in enabling efficient, differentiable rendering for neural radiance fields.

Abstract

Neural Radiance Fields employ simple volume rendering as a way to overcome the challenges of differentiating through ray-triangle intersections by leveraging a probabilistic notion of visibility. This is achieved by assuming the scene is composed by a cloud of light-emitting particles whose density changes in space. This technical report summarizes the derivations for differentiable volume rendering. It is a condensed version of previous reports, but rewritten in the context of NeRF, and adopting its commonly used notation.