Stefan Röttger 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:

$I = I_0 \cdot e^{−\mu \cdot \Delta t}$

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 $\hat{\mu}=\frac{\mu_0+\mu_1}{2}$ :

$I = I_0 \cdot e^{−\frac{s_0+s_1}{2} \mu_A \cdot \Delta t}$

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 $\int_{t=0}^1 \mu(t) dt$ :

$I = I_0 \cdot e^{−\int_{t=0}^1 \mu(t) dt}$

Problem: no closed form of the line integral!

Numerical integration on the line segment:

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

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

Single step of numerical integration:

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