Only Attenuation

Our first restriction on the volumetric cloud is that it only attenuates light. The cloud emits no light of its own. The only light, if any, reaching the front of the volume comes from light that enters from the back. Mathematically, $L(s) = 0$. Under this restriction, the volume rendering integral reduces to

$$I(D) = I_0 e^{-\int_0^D \tau (t) dt}$$ (11)

The major advantage of this form is its simplicity. The integral $\int_0^1 \tau (t) dt$ can be solved for almost any desired interpolation of attenuation. In addition, when rendering a rectilinear grid with this optical model, we can use the Fourier projection-slice theorem and the fast Fourier transform to render a grid of size $O(N^3)$ in $O(N^2 \log N)$ time [60].

Furthermore, consider what happens when we segment the volume rendering integral to perform piecewise integration.

$$I(D) = I_0 e^{-\int_0^{x_1} \tau (t) dt - \int_{x_1}^{x_2} \tau (t) dt - \cdots - \int_{x_{n-1}}^{x_n} \tau (t) dt}$$

$$= {I_0 e^{-\int_0^{x_1} \tau (t) dt} e^{-\int_{x_1}^{x_2} \tau (t) dt} \cdots e^{-\int_{x_{n-1}}^{x_n} \tau (t) dt}}$$ (12)

Because multiplication is commutative, Equation 1.15 shows that we can compute and then combine in any order the integrals for each segment to obtain a correct image. This allows us to render the cells in an order independent fashion. Because cell visibility sorting can be a challenging and computationally intensive process [11,52,54,64,90,91,92,103], this can be a great advantage when using projective methods.

Unfortunately, the only-attenuation model has major weaknesses. Because the model emits no light, the amount of lighting effects is severely limited. The order independence property that makes this model so easy to render means also that there are no depth cues. There is no way to tell, given a fixed viewpoint, if one feature is in front of another. Totsuka and Levoy [93] offer techniques to introduce visual cues, but the cues are only moderately effective and are not realistic.