Homogeneous Particles with Variable Density

Recall from Section 1.1 that we modeled our volume as a collection of minute particles. Max, Hanrahan, and Crawfis [64] proposed the following restriction. Let the density of the particles, $\rho$, vary throughout the volume, but constrain the particles to all have the same properties.

The luminance, $L$, of the volume is a direct property of the volume and is constant. In Section 1.1, I define the attenuation coefficient as $\tau = A \rho$ where $A$ is the cross sectional area of the particles. $A$ is a property of the particles, which is constant when the particles are homogeneous. Thus, $\tau$ is proportional to $\rho$.

It is therefore equivalent to say that $\tau$ varies whereas $L$ does not. We therefore constrain the volume rendering integral by substituting $L$ for $L(s)$ in Equation 1.2.

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

Assuming that $\tau (s)$ is integrable, we can further resolve this equation.

$$I(D) = { I_0 e^{-\int_0^D \tau (t) dt} + { \left. Le^{-\int_s^D \tau (t) dt} \right\vert _{s=0}^D } }$$

$$= { I_0 e^{-\int_0^D \tau (t) dt} + L \left( 1 - e^{-\int_0^D \tau (t) dt} \right) }$$ (15)

Again, we have a more powerful but less than ideal form to the volume rendering integral. Here we are able to vary the density of our cloud in almost any fashion we desire, yet the luminance must remain constant.