Calculation Methods

In the course of implementing a volume tracing algorithm, it is necessary to evaluate the integral equations for absorption, emission and scattering. The problem is, that for all but the most trivial scenes, this can not be done analytically. The only possible solution is to evaluate the equations by means of numerical methods. The simplest numerical approximation to an integral $\int_0^D{h(x)dx}$ is the Riemann sum $\sum_{i=0}^n{h(x_i)\Delta x}$. The interval $[0,D]$ is divided up into $n$ equal segments and for each segment a sample $x_i$ is choosen. The length of a segment is $\Delta x=D/n$. If shaded rendering is used, it has to be considered, that the sourceterm $g$ will also include a Riemann sum to approximate the absorption and emission properties of the ``shadow feeler'' between the sample point and the lightsource. If the number of segments is chosen too low, aliasing effects may occur due to undersampling of the underlying density functions. Undersampling will result in striped images (similar to Mach-Bands) and loss of detail. Participating media may also be incorporated into rendering systems using global illumination, yielding some of the most impressive computer generated images produced so far. Important work has been presented by Jensen [JC98], [Jen96], as well as Lafortune [LW96].