Differentiating General Integrals #
by Shuang Zhao
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 $\Omega$
associated with measure $\mu$
:
\begin{equation}
\label{eqn:I}
I(\theta) = \int_{\Omega} f(\bx, \,\theta) \,\D\mu(\bx).
\end{equation}
For practical rendering problems, the domain $\Omega$
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).
Assuming that $f(\bx, \theta)$
is piecewise continuous with a zero-measure extended boundary $\overline{\partial\Omega}$
comprising the domain boundary $\partial\Omega$
and all jump discontinuity points denoted as $\Delta\Omega$
:
Although this assumption may seem restrictive, it is the case for most, if not all, problems in rendering.
Reynolds Transport Theorem #
TBD.
Computing Boundary Change Rates #
TBD.