http://renderman.pixar.com/view/implementing-a-skin-bssrdf
In this paper on subsurface scattering, I'm trying to understand how importance sampling is used to compute single scattering.
It says that the outscattered radiance for the single term, $L_o$, can be rewritten as the product of the exponential falloff in $s'_o$ with another function:
$L_o(x_o,\vec{\omega}_o) = \bigg[ L_i(x_i,\vec{\omega}_i)\cfrac{\sigma_s(x_o)F_p(\vec{\omega}'_i \cdotp \vec{\omega}'_o )}{\sigma_{tc}}e^{-s'_i \sigma_t(x_i)} \bigg] e^{-s_o\sigma(x_o)}$
By picking samples according to $x \sim \sigma_t e^{-\sigma_t x}$, the integration in $L_o$ can be approximated by a summation by applying importance sampling. Also by choosing $s'_o = \cfrac{-\mathbb{log(random())}}{\sigma_{tr}}$, you can sum up all contributions without computing the falloff term in $s_o$.
I understand you can approximate an integral using importance sampling, but I didn't follow how it applies to the rendering equation. Please could you describe how this works in greater detail?
$E[f(x)]=\int f(x)p(x) = \int f(x)\cfrac{p(x)}{q(x)}q(x) \approx \cfrac{1}{n}\sum \limits_{i=1}^n f(x_i)\cfrac{p(x_i)}{q(x_i)}$
Also, where does single scattering fit into the general rendering equation?
$L_{\text{o}}(\mathbf x,\, \omega_{\text{o}}) \,=\, L_e(\mathbf x,\, \omega_{\text{o}}) \ +\, \int_\Omega f_r(\mathbf x,\, \omega_{\text{i}},\, \omega_{\text{o}})\, L_{\text{i}}(\mathbf x,\, \omega_{\text{i}})\, (\omega_{\text{i}}\,\cdot\,\mathbf n)\, \operatorname d \omega_{\text{i}}$