\section{Discussion and Conclusion}
\label{sec:conclusion}
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\paragraph{Limitations and future work}
Our derivations rely on a few simplifying assumptions stated at the start of \S\ref{sec:diff_rte}.
Although some of them (e.g., no zero-measure light source) are relaxed in the supplemental material, others require more sophisticated algorithms to handle.
For instance, with ideal specular surfaces, discontinuities of the radiance $\Lv(\bx, \bom)$ in $\bom$ need to also include mirror images of the geometric edges, which can be nontrivial to identify.
Also, to support media where the transmittance is non-exponential, some of our derivations will need to be adjusted.

Another exciting direction for future research is to improve the computational efficiency of our Monte Carlo estimator (Algorithm~\ref{alg:diffRadiances}).
Developing more sophisticated estimators (e.g., those analogous to bidirectional path tracing) is an interesting future topic and necessitates the establishment of a path-integral form of Eq.~\eqref{eqn:rte3d_dLv_full}.

\paragraph{Conclusion}
We introduced a differential theory of radiative transfer by showing how the radiative transfer equation~(RTE) can be differentiated with respect to arbitrary scene parameterizations.
To this end, we derived the derivatives for individual terms of the RTE given by the transport, collision, and interfacial scattering operators.
Our theory enjoys the generality to handle a large variety of radiative transfer phenomena including volumetric absorption, single and multiple scattering, anisotropic phase functions, and heterogeneity.

To numerically estimate these derivatives, we presented an unbiased Monte Carlo method analogous to volumetric path tracing (VPT) to compute scene derivatives of interior and interfacial radiances.
Our technique symbolically differentiates contributions of light transport paths and traces additional side paths toward geometric edges to account for the change of discontinuity boundaries.

We validated the correctness of our theory and implementation by comparing our derivative estimations to finite-difference results.
Further, we demonstrated the practical usefulness of our technique via a few synthetic examples inspired by real-world applications in imaging and design.
