\section{Related Work}
\label{sec:prior}
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\paragraph{Radiative transfer}
Radiative transfer has been used in natural sciences and application areas, including astrophysics, neutron transport, material science, remote sensing and biomedical imaging~\cite{van1957light,chandrasekhar1960radiative,pomraning1973equations,ishimaru1978wave,mishchenko2006multiple}.
This framework was later introduced to computer graphics~\cite{Blinn:1982:LRF,Kajiya:1984:RTV,Rushmeier:1987:ZMC}, where it has been generalized to support light transport effects such as those arising from anisotropic media with organized structures~\cite{Jakob:2010:RTF} and non-exponential media~\cite{Bitterli:2018:RTF}.

\paragraph{Monte Carlo volume rendering}
Monte Carlo methods have been the ``gold standard'' for simulating photon and neutron transport in complex environments~\cite{spanier1969mc}.
In computer graphics, volumetric path tracing and its variations (e.g., \cite{Kajiya:1984:RTV,lafortune1996rendering,cerezo2005survey}) produce unbiased and consistent estimates of radiometric measures.
Later-developed Markov-Chain Monte Carlo (MCMC) methods (e.g., \cite{pauly2000metropolis,kelemen2002simple,Jakob:2012:MEM}) additionally enable efficient simulation of challenging effects such as volumetric caustics.
For a comprehensive survey on Monte Carlo volume rendering techniques, we refer to the survey by No\'vak~et~al.~\shortcite{novak2018monte}.

\paragraph{Derivatives for rendering}
Analytical derivatives have been used in computer graphics to compute pixel footprints~\cite{Igehy:1999:TRD}, handle specular light paths~\cite{Chen:2000:TAS,Jakob:2012:MEM}, use Hamiltonian Monte Carlo in rendering~\cite{Li:2015:AGM}, and enable interactive material editing~\cite{Hasan:2013:IAE}.
These derivatives are specialized for certain types of light transport effects, and usually neglect geometric discontinuities.

Arvo~\shortcite{Arvo:1994:IJP} presented an analytical method for calculating the gradients of irradiance.
This method is specialized for diffuse surfaces and requires clipping triangle meshes, making it difficult to generalize to arbitrary materials or scale to complex scenes.

Ramamoorthi et al.~\shortcite{Ramamoorthi:2007:FAL} introduced a first-order analysis of light transport, focusing on effects such as soft shadows.
Our work can be considered a generalization of this theoretical framework to include other effects, such as volumetric light transport.

\paragraph{Physics-based differentiable rendering}
Differentiable rendering has been used to solve analysis-by-synthesis problems in volumetric scattering~\cite{Gkioulekas:2013:IVR,Gkioulekas2016}, cloth rendering~\cite{Khungurn:2015:MRF}, prefiltering of high-resolution volumes~\cite{Zhao:2016:DSP}, appearance modeling of human teeth~\cite{Velinov:2018:ACM}, fabrication of translucent materials~\cite{Sumin:2019:GSC}, reflectance and lighting estimation~\cite{Azinovic_2019_CVPR}, and 3D reconstruction~\cite{Tsai_2019_CVPR}.
All these methods compute radiance derivatives using algorithms specialized to specific light transport effects.

A few recent works have focused on extending differentiable rendering to more general settings. Che~et~al.~\shortcite{che2018inverse} developed a system capable of computing derivatives with respect to optical material (reflectance, scattering) and local shape (normal) properties.
Unfortunately, this technique cannot differentiate with respect to global changes of the scene geometry: As we discuss in \S\ref{sec:background}, this requires calculating derivatives of integrals with evolving boundaries, which in turn requires evaluating additional boundary integrals.

To overcome these challenges, Li~et~al.~\shortcite{Li:2018:DMC} generalized the work by Ramamoorthi~et~al.~\shortcite{Ramamoorthi:2007:FAL} and presented a technique that uses an edge-sampling process to %capture geometric discontinuities. %for surface-only scenes.
handle the boundary terms.
Despite significantly extending the range of parameters that can be differentiated, this work still neglects all volumetric light transport effects, by essentially differentiating only the rendering equation but not the radiative transfer equation.
Our theory subsumes this work by showing how to differentiate both equations, resulting in the most general differentiable rendering framework to date.

\paragraph{Derivatives for vision}
Having derivatives of rendered images also enables back-propagation through rendering processes, allowing them to be efficiently integrated into probabilistic programming~\cite{Kulkarni_2015_CVPR} and deep learning pipelines (e.g., as the decoder of an auto-encoder architecture~\cite{vincent2010stacked}).
In fact, many recent works~\cite{wu2017neural,meka2018lime,sengupta2018sfsnet,kato2018neural,Li:2018:LRS,che2018inverse} leverage various forms of rendering losses to regularize the training, and improve generalization, of neural networks.
The renderers used in most of these works make restrictive simplifications such as direct-only illumination~\cite{loper2014opendr}.

