Optical Model for Volume Rendering

Optical models for direct volume rendering view the volume as a cloud of particles [30]. Light from a source can either be scattered or absorbed by particles. In practice, models that take into account all the phenomena tend to be very complicated. Therefore, practical models use several simplifications. A common approximation for the volume rendering integral is given by [32]:

$${\rm I}_\lambda (x,r) = \int_0^L {C_\lambda (s)\mu (s)e^{ - \int_0^s {\mu (t)dt} } ds}$$ (2.1)

Hereby, $I_\lambda$ is the amount of light of wavelength $\lambda$ coming from a ray direction $r$ that is received at location $x$ on the image plane. $L$ is the length of the ray $r$ and $\mu$ is the density of volume particles which receive light from the light sources and reflect it towards the observer according to their material properties. $C_\lambda$ is the light of wavelength $\lambda$ reflected and/or emitted at location $s$ in the direction of $r$. The equation takes into account emission and absorbtion effects, but discards more advanced effects such as scattering and shadows.

In general, Equation 2.1 cannot be computed analytically. Hence, most volume rendering algorithms use a numeric solution of the equation. This results in the common compositing equation:

$${\rm I}_\lambda (x,r) = \sum\limits_{i = 0}^{L/\Delta s} {C_... ..._i )} \cdot \prod\limits_{j = 0}^{i - 1} {(1 - \alpha (s_j ))}$$ (2.2)

Here $\alpha (s_i )$ are the opacity samples along a ray and $C_\lambda (s_i)$ are the local color values derived from the illumination model. $C$ and $\alpha$ are referred to as transfer functions. These functions assign color and opacity to each intensity value in the volume.