Initial Approximation

Linearly interpolated opacity results in an unwieldy form for $\zeta$. Instead of trying to calculate $\zeta$ directly, we can use an approximation similar to that given by Wilhelms and van Gelder [102]. We assume that $\tau (s)$ is constant in Equation 1.3. In this case, $\zeta_{D,\tau } = e^{-D \tau }$. To get a value for $\tau$, we average the opacity ( $\alpha (s) \approx \frac{1}{2}\left({\alpha }_{\,\mathrm{b}}+{\alpha }_{\,\mathrm{f}}\right)$) and then convert that to an attenuation coefficient (via Equation 1.12). Figure 1.6 demonstrates that this approximation is quite close.

Figure 1.6: Approximation of $\zeta$ for linearly interpolated opacity ($\alpha$). Both computations are with unit length segments.

(a) $\zeta$ with linearly interpolated opacity.	(b) $\zeta$ approximated with average opacity.

When $\alpha$ varies linearly ($\tau$ varies logarithmically), $\Psi$ (Equation 1.4) does not have a closed form. We can approximate $\Psi$ in the same manner as we approximate $\zeta$: by averaging $\alpha$. If we constrain the opacity and attenuation to be constant, Equation 1.4 reduces to

$$\Psi_{D,\tau } = \frac{1-e^{-D\tau }}{D\tau } = \frac{1-\zeta}{D\tau }$$

Figure 1.7 shows us that this approximation is also reasonable.

Figure 1.7: Approximation of $\Psi$ for linearly interpolated opacity ($\alpha$). Both computations are with unit length segments.

(a) $\Psi$ with linearly interpolated opacity.	(b) $\Psi$ approximated with average opacity.

Appendix 1.5 lists Cg code that we can use to perform this volume rendering integral approximation. However, under rare circumstances when the opacity changes drastically over a large cell, errors can become noticeable.