Differentiating General Integrals

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 #


Computing Boundary Change Rates #