Stefan RÃ¶ttger VolumeRendering/Absorption

Absorption

Absorption on a line segment with length $\Delta t$:

For a constant absorption coefficient $\mu$:

$ \mu = const $

Exponential attenuation on the line segment:

$ I = I_0 \cdot e^{âˆ’\mu \cdot \Delta t} $

For piecewise linear absorption coeffizient $\mu(t)$ between two points $\vec{p}_0$ and $\vec{p}_1$ with the scalar value $s_0$ and $s_1$:

$ \mu(t) = (1-t) TF_A(s_0) + t TF_A(s_1) $

$ \mu(t) = (1-t) s_0 \mu_A + t s_1\mu_A $

$ \mu(t) = ( (1-t) s_0 + t s_1 ) \mu_A $

Attenuation approximation by averaging the coefficients $\hat{\mu}=\frac{\mu_0+\mu_1}{2}$:

$ I = I_0 \cdot e^{âˆ’\frac{s_0+s_1}{2} \mu_A \cdot \Delta t} $

For an arbitrary absorption coefficient $\mu(t)$ on the line segment $\vec{p}(t)=(1-t)\vec{p}_0+t\vec{p}_1$:

$s=f(\vec{p}(t))$

$ \mu(t) = TF_A(t) = s(t) \mu_A $

Attenuation is determined by the integral of coefficients $\int_{t=0}^1 \mu(t) dt$:

$ I = I_0 \cdot e^{âˆ’\int_{t=0}^1 \mu(t) dt} $

Problem: no closed form of the line integral!

Numerical integration on the line segment:

$ I = I_0 \cdot e^{âˆ’\sum_{i=0}^n \mu(\frac{i}{n-1}) \Delta t} $

$ I = I_0 \cdot \prod_{i=0}^n e^{-\mu(\frac{i}{n-1}) \Delta t} $

Single step of numerical integration:

$ I' = I \cdot e^{âˆ’\mu(t) \cdot \Delta t} $