Monte Carlo Methods for Volumetric Light Transport Simulation

Section 2. Optical Properties

Collision Coefficient

1. density per unit volume: $\rho (m^{-3})$
2. cross sectional area: $\sigma_{s}, \sigma_{a} (m^2)$
3. scattering(absorption) coefficient: $\mu_{s}(\mu_{a})=\sigma_{s}(\sigma_{a}) \cdot \rho (m^{-1})$
4. extinction coefficient: $\mu_{t}=\mu_{s}+\mu_{a}$
5. single-scattering albedo: $\alpha=\frac{\mu_{s}}{\mu_{t}}$

Phase Function

Examples: Henyey-Greenstein, Rayleigh, Lorenz-Mie.

For more details in Handbook of Digital Image Synthesis: Scientific Foundations of Rendering

Section 3. Volumetric Rendering Equation

Radiative Transfer Equation (RTE)

$$(\omega\cdot\nabla)\cdot L(x,\omega)=-\mu_{t}(x)L(x,\omega)+\mu_{a}(x)L_{e}(x,\omega)+\mu_{s}(x)L_{s}(x,\omega)$$

Then, this differential equation can yield

$$L(x,\omega)=\int_{0}^{\infty}T(x,y)[\mu_{a}(y)L_{e}(y,\omega)+\mu_{s}(y)L_{s}(y,\omega)]dt$$

Here, $y=x-t\cdot\omega$, $T(x,y)=e^{-\int_{x}^{y}\mu_{t}(z)dz}$(Beer Lambert Law)

Volume Rendering Equation

$$L(x,\omega)=\int_{0}^{dist}T(x,y)[\mu_{a}(y)L_{e}(y,\omega)+\mu_{s}(y)L_{s}(y,\omega)]dt+T(x,z)L(z,\omega)$$

Here,$z=x-dist\cdot \omega$

Path Integral Formulation

$\overline{x_{k}}={x_0,x_1,\dots,x_k}$, the $x_0$ is the point on the sensor

$$f(\overline{x_{k}})=W(x_0,x_1)G(x_0,x_1)T(x_0,x_1)\cdot L_{e}(x_k\rightarrow x_{k-1})\cdot \prod_{i=1}^{i=k-1}f_{s}(x_{i+1}\rightarrow x_{i}\rightarrow x_{i-1})G(x_{i},x_{i+1})T(x_{i},x_{i+1})$$

$W$ is response of the point on the sensor

$G(x,y)=\frac{D(x,y)\cdot D(y,x)}{\Vert x-y\Vert ^2}$

$D(x,y)=\begin{cases} \vert n(x)\cdot \omega_{x\rightarrow y}\vert & \text{x is on a surface} \newline 1 & \text{x is in a medium} \end{cases}$

$f_{x\rightarrow y\rightarrow z}=\begin{cases} f_{r}(x\rightarrow y\rightarrow z) & \text{y is on a surface} \newline \mu_{s}(y)\cdot f_{p}(x\rightarrow y\rightarrow z) & \text{y is in a medium} \end{cases}$

$L_{e}(x\rightarrow y)=\begin{cases} L_{e}(x,\omega_{x\rightarrow y}) & \text{x is on a surface} \newline \mu_{a}(x)L_{e}(x,\omega_{x\rightarrow y}) & \text{x is in a medium}\end{cases}$

Section 4. Distance Sampling