\subsection{Completing $\dLv$}
\label{ssec:together}
%
The final term in Figure~\ref{fig:outline} is the derivative of $\Ldir = \trans\Le$.
Since Eq.~\eqref{eqn:rte3d_kernelS_diff} has already provided $\sdev\trans\Lr$, $\dLdir$ can be obtained by respectively replacing $\Lr$ and $\dLr$ in Eq.~\eqref{eqn:rte3d_kernelS_diff} with $\Le$ and $\dLe$, yielding
%
\begin{myMathBox}
	\begin{align}
		& \dLdir(\bx, \bom)	= \sdev[ \trans(\bx_0, \bx) \, \Le(\bx_0, \bom)] \label{eqn:rte3d_Ldir_diff}\\
		%=\;& \trans(\bx_0, \bx) \, \dLe(\bx_0, \bom) + \dot{\trans}(\bx_0, \bx) \, \Le(\bx_0, \bom)\\
		=\;& \trans(\bx_0, \bx) \left[ -\left( \intSigT(\bx, \bom, \dist) + \ddist \, \sigT(\bx_0) \right) \Le(\bx_0, \bom) + \dLe(\bx_0, \bom) \right]. \nonumber
	\end{align}
\end{myMathBox}
%
\noindent
Similar to $\dphase$ and $\dbsdf$, $\dLe(\bx_0, \bom)$ can be obtained using symbolic differentiation.

\paragraph{Putting together}
We now combine Eqs.~(\ref{eqn:rte3d_ind_diff}, \ref{eqn:rte3d_kernelS_diff}, \ref{eqn:rte3d_Ldir_diff}) to get $\dLv$, the partial scene derivative of the interior radiance $\Lv$.

Since $\dLs = \dLr + \dLe$, adding Eq.~\eqref{eqn:rte3d_kernelS_diff} and \eqref{eqn:rte3d_Ldir_diff} effectively yields the derivative of the source term $\src$:
%
\begin{align}
	& \dsrc(\bx, \bom) = \left(\sdev\,\kernelS\,\Lv + \dLdir\right)(\bx, \bom) \label{eqn:rte3d_src_diff}\\
	=\;& \trans(\bx_0, \bx) \left[ -\left( \intSigT(\bx, \bom, \dist) + \ddist \, \sigT(\bx_0) \right) \Ls(\bx_0, \bom) + \dLs(\bx_0, \bom) \right]. \nonumber
\end{align}
%
Adding Eq.~\eqref{eqn:rte3d_ind_diff} and \eqref{eqn:rte3d_src_diff} completes the derivation of $\dLv$, as shown in Eq.~\eqref{eqn:rte3d_dLv_full} below.
The relations between the quantities on which $\dLv$ depends are visualized in Figure~\ref{fig:dependency}.

\begingroup
\addtolength{\stripsep}{-5pt}
\begin{strip}
	\begin{myTitledBox}{Scene derivative of the interior radiance $\Lv$}
		\begin{equation}
		\label{eqn:rte3d_dLv_full}
			\begin{aligned}
				\dLv(\bx, \bom)	=\;& \tint_0^{\dist} \trans(\bx', \bx) \big[ \sigS(\bx') \,\dLins(\bx', \bom) + \left(\dot{\sigS}(\bx') - \intSigT(\bx, \bom, \tau)\,\sigS(\bx')\right) \Lins(\bx', \bom) \big] \,\D\tau\\
				& + \trans(\bx_0, \bx) \big[ - \left( \intSigT(\bx, \bom, \dist) + \ddist \,\sigT(\bx_0) \right) \Ls(\bx_0, \bom) + \dLs(\bx_0, \bom) + \ddist \,\sigS(\bx_0) \,\Lins(\bx_0, \bom) \big],
			\end{aligned}
		\end{equation}
		%
		where $\intSigT$ is defined in Eq.~\eqref{eqn:rte3d_int_sigT}, $\dLins$ follows Eq.~\eqref{eqn:rte3d_Lins_diff}, and $\dLs = \dLr + \dLe$ with $\dLr$ given by Eq.~\eqref{eqn:rte3d_Lr_diff}.
	\end{myTitledBox}
\end{strip}
\endgroup

\subsection{Summary and Discussion}
\label{ssec:diff_rte_discuss}
%
We derived the (partial) scene derivative of the interior radiance~$\Lv$ by differentiating individual components of  the RTE~\eqref{eqn:rte3d_Lonly} using \thmref{reynolds}.
The key to the application of this theorem is to properly handle all the boundary integral terms which arise due to Eq.~\eqref{eqn:thm1} and contribute significantly to the final derivatives, as shown in Figure~\ref{fig:per_component} (middle vs. bottom).

When differentiating the transport operator $\kernelT$ that involves a line integral, as expressed in Eq.~\eqref{eqn:rte3d_ind_diff}, this boundary term reduces to evaluating the integrand $\sigS\,\Lins$ at the boundary~(Eq.~\eqref{eqn:rte3d_Lins_boundary}) multiplied by the change rate~$\ddist$ of the boundary.

Besides the boundary term, according to \thmref{reynolds}, we also need to differentiate the integrand $\sigS\,\Lins$ itself.
To this end, we in turn need the derivative of the in-scattered radiance $\Lins$ that is defined in Eq. \eqref{eqn:rte3d_Lins} and involves a spherical integral of interior radiance~$\Lv$ modulated by the phase function~$\phase$.
The boundary term $\Bins$ in this case takes the form of an integral over all 3D edges capable of creating geometric discontinuities, as expressed in Eq.~\eqref{eqn:rte3d_Lins_edge}.

The differentiation of the interfacial scattering operator~$\kernelS$ of Eq.~\eqref{eqn:rte3d_kernelS} requires the derivative of the reflected/refracted radiance $\Lr$ (Eq.~\eqref{eqn:rte3d_re}), as expressed in Eq.~\eqref{eqn:rte3d_kernelS_diff}.
This can be done in a similar manner as the calculation of $\dLins$ in Eq.~\eqref{eqn:rte3d_Lr_diff}, given that both of them comprise spherical integrals of the interior radiance~$\Lv$.

\paragraph{Relation to prior work}
When differentiating radiance, Li~et~al. \shortcite{Li:2018:DMC} only considers the RE~\eqref{eqn:rte3d_re}, which is equivalent to our handling of the reflected/refracted radiance~$\Lr$ in Eq.~\eqref{eqn:rte3d_Lr_diff}, and neglects the RTE~\eqref{eqn:rte3d_Lonly}.
Thus, their technique cannot handle the volumetric phenomena governed by the RTE and is a special case of ours.
