Closed Forms of the Volume Rendering Integral

In Section 1.1, I provide a model for light transmission through volumes and derive the volume rendering integral (Equation 1.2), provided here for reference.

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

However, this equation has no closed form, and we cannot solve it without further information about $L(s)$ and $\tau (s)$. In this section, I will impose various restrictions on $L(s)$ and $\tau (s)$ that will allow us to solve the volume rendering integral.