Domain of $\Psi$

In the previous section, I show that it is most prudent to build a two dimensional table with entries corresponding to $({\tau }_{\,\mathrm{b}}D,{\tau }_{\,\mathrm{f}}D)$ pairs (Equation 1.8). However, valid values for $\tau {}D$ lay in the range $[0,\infty)$. We must choose a subset of this range that provides good resolution for the most frequently encountered values of $\tau {}D$.

Figure 1.1: Plot of $\Psi$ (Equation 1.8) against ${\tau }_{\,\mathrm{b}}D$ and ${\tau }_{\,\mathrm{f}}D$ for values in the range $[0,4]$.

Figure 1.1 shows $\Psi_{{\tau }_{\,\mathrm{b}}D,{\tau }_{\,\mathrm{f}}D}$ plotted for parameters in the range $[0,4]$. Despite the number of terms generated when solving Equation 1.8, we see that our plot is quite smooth. The absence of high frequencies makes storing the function over a finite domain in a table practical.

However, the partial pre-integration tables I have described thus far are not complete. For example, the domain for $\Psi_{{\tau }_{\,\mathrm{b}}D,{\tau }_{\,\mathrm{f}}D}$ shown in Figure 1.1 is not sufficient. Notice that $\Psi_{4,0}=0.6$, whereas $\lim_{{\tau }_{\,\mathrm{b}}D \rightarrow \infty} \Psi_{{\tau }_{\,\mathrm{b}}D,0} = 0$. Therefore, the assumption that $\Psi_{4,{\tau }_{\,\mathrm{f}}D} \approx \Psi_{\infty,{\tau }_{\,\mathrm{f}}D}$ is incorrect. Using a table based on the plot in Figure 1.1 can cause extreme and incorrect fluctuations in color space, an effect I dub $\Psi$ clipping.

So what can we do about $\Psi$ clipping? One approach is to structure the application such that no integrated ray segment ever has a value of $\tau {}D$ that exceeds $4$. For many applications, such an approach is perfectly reasonable. After all, providing a larger attenuation does not make the model look more opaque, so why support higher attenuations? On the other hand, if rendering a model with cells varying wildly in size, it may not be possible to pick a range for $\tau$ that allows for smaller cells to be rendered opaque while maintaining the constraint that $\tau {}D < 4$.

Another straightforward approach to eliminate $\Psi$ clipping is to increase the domain of $\Psi_{{\tau }_{\,\mathrm{b}}D,{\tau }_{\,\mathrm{f}}D}$ until the $\Psi$ clipping effect is below a tolerable threshold.

Figure 1.2: Plot of $\Psi$ (Equation 1.8) against ${\tau }_{\,\mathrm{b}}D$ and ${\tau }_{\,\mathrm{f}}D$ for values in the range $[0,16]$.

Figure 1.2 demonstrates why this approach will not work. In it, we have significantly increased the range over which $\Psi_{{\tau }_{\,\mathrm{b}}D,{\tau }_{\,\mathrm{f}}D}$ is plotted. The area plotted increased 16 fold, and the surface is far less smooth, thereby necessitating a significant increase in lookup table resolution. Yet $\Psi_{16,0} = 0.3$, which is still significantly greater than $\Psi_{\infty,0}$.

I propose a means of eliminating $\Psi$ clipping by changing the variables we use to index $\Psi$, much like we did to reduce the dimensionality of the $\Psi$ table as described in Section 1.2.2. First, we define the variable $\gamma$ as

$$\gamma \equiv \frac{\tau {}D}{\tau {}D+1} \index{$\gamma$}\index{gamma~($\gamma$)}$$ (60)

Solving Equation 1.9 for $\tau {}D$,

$$\tau {}D = \gamma/(1-\gamma)$$

and substituting into Equation 1.8, we get the following.

$$\Psi_{{\gamma}_{\,\mathrm{b}},{\gamma}_{\,\mathrm{f}}} = { \int_0... ... \frac{{\gamma}_{\,\mathrm{f}}}{1-{\gamma}_{\,\mathrm{f}}} t \right) } dt} ds }$$ (61)

Figure 1.3: Plot of $\Psi$ against ${\gamma}_{\,\mathrm{b}}$ and ${\gamma}_{\,\mathrm{f}}$ (Equation 1.10).

The principle advantage of using $\gamma$ over $\tau {}D$ is that valid values of $\gamma$ range only over $[0,1)$. It is therefore possible to store the entire domain of $\Psi$ into a single table. Figure 1.3 shows a plot of $\Psi_{{\gamma}_{\,\mathrm{b}},{\gamma}_{\,\mathrm{f}}}$ over its entire domain.

Appendix 1 contains instructions on computing tables of $\Psi$ values with Mathematica. Appendix 1.4 lists Cg code that uses such a table to compute the volume rendering integral.