\section{Introduction}
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A fundamental task of physics-based light transport simulation is to compute the radiant power (generally measured using radiance) at certain 3D locations and directions in a virtual scene, e.g., those corresponding to radiometric sensors.
Such \emph{forward} evaluations of light transport have been a focus of research efforts in computer graphics since the field's inception.
These efforts have resulted in mature forward rendering algorithms, including Monte Carlo techniques, that can efficiently and accurately simulate complex light transport effects such as interreflections and subsurface scattering.

Mathematically, %it is typically convenient to be capable of performing evaluations of various transformations of a given function, besides just forward evaluations. An important such transformation is the function's derivatives.
it is convenient to be capable of evaluating not only a given function but also its various transformations.
One such transformation of great importance is the function's derivatives.
In the context of light transport simulation, this corresponds to \emph{differentiable rendering}, the process of computing the derivatives of radiometric measurements with respect to differential changes of virtual scenes.
Differentiable rendering techniques can enable, for example, \emph{gradient-based optimization} when solving high-dimensional \emph{inverse rendering} problems, where exhaustive search is typically impractical.
Additionally, these techniques facilitate the efficient integration of physics-based light transport simulation in machine learning and probabilistic inference pipelines, where differentiability is a key requirement.

Unfortunately, and unlike their forward counterparts, derivatives of radiometric measurements remain challenging to estimate.
On one hand, general-purpose numerical tools for computing derivatives such as finite differences are either too noisy or too biased when combined with forward Monte Carlo simulations, and scale poorly to high-dimensional gradients.
On the other hand, existing differentiable rendering algorithms are not sufficiently general to support differentiation with respect to arbitrary parameters in the presence of nontrivial scene geometry and complex light transport effects.

At the core of the challenge is the lack of a comprehensive mathematical framework for differential radiometric quantities that encompasses all types of high-order light transport.
This is in contrast with the forward rendering case, where this role is fulfilled by the radiative transfer and rendering equations, which have provided the foundation for the development of Monte Carlo forward rendering algorithms.
We seek to fix this discrepancy by developing a \emph{differential} theory of radiative transfer of the same generality as its forward form.
Additionally, we use this theory to derive Monte Carlo algorithms capable of differentiating arbitrary radiative transfer effects (e.g., volumetric absorption and scattering, anisotropic phase functions, and heterogeneity) with respect to arbitrary scene parameterizations.
This significantly extends prior differentiable rendering techniques that are restricted to differentiating only surface-based light transport~\cite{Li:2018:DMC}.

Concretely, our contributions include:
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\begin{itemize}[noitemsep,topsep=4pt,parsep=4pt,partopsep=0pt,leftmargin=*]
	\item The derivation of differential forms of the radiative transfer and rendering equations, supporting differentiation with respect to arbitrary scene parameterizations (\S\ref{sec:diff_rte}).
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	\item An unbiased Monte Carlo algorithm for estimating the derivatives of radiometric measures in the presence of arbitrarily complex light transport (\S\ref{sec:mc}).
\end{itemize}
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We validate our theory and algorithms by comparing our estimated derivatives with those produced using finite differences (Figure~\ref{fig:validate_v2}).
Additionally, we show applications of our framework in synthetic examples of several inverse rendering problems (Figure~\ref{fig:mainres_generic}), including examples relevant to non-line-of-sight (NLOS) and biomedical imaging (Figure~\ref{fig:mainres_imaging}), as well as design tasks (Figure~\ref{fig:mainres_design}).
