Completely Homogeneous

Another simple approximation is to assume that the volume is homogeneous. That is, the attenuation and luminance parameters do not vary. We can model this by substituting the constants $L$ and $\tau$ for $L(s)$ and $\tau (s)$, respectively, in Equation 1.2.

$$I(D) = { I_0 e^{-\int_0^D \tau dt} + { \int_0^D L\tau e^{-\int_s^D \tau dt} ds } }$$

$$= { I_0 e^{-\tau D} + { \int_0^D L\tau e^{-\tau (D-s)} ds } }$$

$$= { I_0 e^{-\tau D} + {L \left. e^{-\tau (D-s)} \right\vert _{s=0}^D } }$$

$$= { I_0 e^{-\tau D} + L\left(1 - e^{-\tau D}\right) }$$ (14)

It is, of course, not practical or interesting to limit our volume to be completely homogeneous. Instead (as alluded to in Section 1.2.4), you can use Equation 1.17 in a Riemann sum to accurately estimate the volume rendering integral by breaking it up into small enough pieces [63].

If we sample the volume uniformly, $D$ in Equation 1.17 is constant. In this case, we can convert the attenuation parameter ($\tau$) to an opacity ($\alpha$) offline and use Porter and Duff blending as described in Section 1.2.3 to perform this Riemann sum. Stein, Becker, and Max [92] demonstrate how to use 2D texture hardware to convert the attenuation and distance to opacity before blending for volumes not sampled uniformly.

Although, technically, we could subdivide our volume fine enough for any amount of accuracy (although quantization errors become a problem), more subdivisions result in more computational overhead. Less constrained forms of the volume rendering integral can lead to greater accuracy with fewer subdivisions.