Differentiating General Integrals #

In the previous section, we have discussed the differentiation of 1D Riemann integrals.

In what follows, we discuss the differentiation of a general Lebesgue integral $I(\theta)$ over some bounded and open domain $\domain$ associated with measure $\mu$ :

\begin{equation} \label{eqn:I} I(\theta) = \int_{\domain} f(\bx, \,\theta) \,\D\mu(\bx). \end{equation}

For practical rendering problems, the domain $\domain$ can be:

The surface of the unit sphere $\sph := \{ \bx \in \real^3 : \| \bx \| = 1 \}$ ;
The union $\calM$ of all object surfaces;
The path space (under Veach’s path-integral formulation).

We assume that $f(\bx, \theta)$ is piecewise continuous with a zero-measure extended boundary $\boundary = \partial\domain \cup \Delta\domain$ comprising the domain boundary $\partial\domain$ and all jump discontinuity points denoted as $\Delta\domain$ —which are illustrated below as black and orange curves, respectively:

domain

Further, we assume the extended boundary $\boundary$ to be associated with a unit-normal field $\bn$ . That is, for any point $\bx$ on the extended boundary, we have the boundary normal $\bn(\bx)$ specified. We note that, although this assumption may seem restrictive, it is the case for most, if not all, problems of our interest.

Reynolds Transport Theorem #

In general, when the domain $\domain$ of integration depends on the parameter $\theta$ , so does the extended boundary $\boundary$ . In this case, the derivative of Eq. \eqref{eqn:I} with respect to $\theta$ is given by Reynolds transport theorem. This theorem—which is a generalization of the result from the previous section—states that:

\begin{equation} \label{eqn:dI_reynolds} \boxed{ \frac{\D}{\D\theta} I(\theta) = \underbrace{\int_{\domain(\theta)} \frac{\D}{\D\theta} f(\bx, \,\theta) \,\D\mu(\bx)}_{\text{interior}} \;+\; \underbrace{\int_{\boundary(\theta)} \Delta f(\bx, \theta) \,\vel(\bx) \,\D\dmu(\bx)}_{\text{boundary}} \,. } \end{equation}

In this equation, the interior integral is obtained by exchanging differentiation and integration operations.

The Boundary Integral #

We now examine the boundary component of Eq. \eqref{eqn:dI_reynolds} more closely.

Domain of Integration #

The boundary integral is over the extended boundary $\boundary(\theta)$ . In practice:

When the domain $\domain(\theta) = (a(\theta), b(\theta)) \subset \real$ of the ordinary integral \eqref{eqn:I} is a 1D interval (with $\mu$ being the Borel measure), $\boundary(\theta)$ is a discrete set of jump discontinuity points including the interval’s endpoints $a(\theta)$ and $b(\theta)$ . Further, the boundary integral reduces to the sum over all these discontinuity points—as presented in the full derivative from the previous section.
When $\domain(\theta) \subset \real^2$ is a 2D region with $\mu$ being the area measure, the extended boundary $\boundary(\theta)$ comprises a set of curves (as illustrated in the figure above) with $\dmu$ being the curve-length measure.
When $\domain(\theta) \subseteq \sph$ is a curved 2D region within the surface $\sph$ of the unit sphere and $\mu$ the solid-angle measure, the extended boundary $\boundary(\theta)$ comprises a set of spherical curves with $\dmu$ being the curve-length measure.
When $\domain(\theta) \subset \real^3$ is a 3D volume and $\mu$ the volume measure, the extended boundary $\boundary(\theta)$ becomes 2D surfaces with $\dmu$ being the surface-area measure.

Definition of $\Delta f$ #

In Eq. \eqref{eqn:dI_reynolds}, another important component of the boundary integral is $\Delta f(\bx, \theta)$ that captures the difference in $f(\bx, \theta)$ across a discontinuity boundary at $\bx$ .

Precisely, let $\bx_{\bot}(\epsilon)$ be a regular curve parameterized by $\epsilon$ (illustrated as the gray dashed line in the figure above) that resides inside the domain $\domain(\theta)$ and satisfies $\bx_{\bot}(0) = \bx$. When the domain $\domain(\theta)$ is “flat”, for example, we have

$$ \bx_{\bot}(\epsilon) := \bx + \epsilon\,\bn(\bx), $$

where $\bn(\bx)$ is the unit normal of the extended boundary at $\bx$ . Then,

\begin{equation} \label{eqn:delta_f} \Delta f(\bx, \theta) = \lim_{\epsilon \uparrow 0} f(\bx_{\bot}(\epsilon), \,\theta) - \lim_{\epsilon \downarrow 0} f(\bx_{\bot}(\epsilon), \,\theta). \end{equation}

Normal Change Rate $\vel$ #

Another key component of the boundary integral in Eq. \eqref{eqn:dI_reynolds} is the normal change rate term $\vel$ which, at a high level, captures how fast the extended boundary $\boundary(\theta)$ evolves along the normal $\bn$ (with respect to $\theta$ ). Precisely, for any $\bx \in \boundary(\theta)$ , we have

\begin{equation} \label{eqn:vel} \vel(\bx) = \frac{\D\bx}{\D\theta} \cdot \bn(\bx), \end{equation}

where “ $\cdot$ ” denotes the dot (or scalar) product of two vectors.

Parameterizing Discontinuity Boundaries #

Evaluating the derivative $\D\bx/\D\theta$ on the right-hand side of Eq. \eqref{eqn:vel} requires the extended boundary $\boundary(\theta)$ to be locally parameterized. Specifically, for each $\theta$ and $\bx \in \boundary(\theta)$ , we assume without loss of generality that $\alphaBnd(\boldsymbol{\xi}, \theta)$ is a differentiable and bijective function satisfying that:

$\alphaBnd(\cdot, \theta)$ maps an open ball centered at the origin $\boldsymbol{0}$ to an open neighborhood around $\bx$ within the extended boundary $\boundary(\theta)$ ;
$\alphaBnd(\boldsymbol{0}, \theta) = \bx$ .

Then,

\begin{equation} \label{eqn:vel_raw} \frac{\D\bx}{\D\theta} = \frac{\D}{\D\theta} \alphaBnd(\boldsymbol{0}, \theta). \end{equation}

Parameterization Independency #

The derivative $\D\bx/\D\theta$ defined in Eq. \eqref{eqn:vel_raw} is parameterization-dependent. That is, it depends on the choice of the local parameterization specified by the function $\alphaBnd$.

On the contrary, the normal change rate $\vel(\bx)$ from Eq. \eqref{eqn:vel} is known to be parameterization-independent. In other words, regardless to the choice of $\alphaBnd$, $\vel(\bx)$ remains identical.