How is the beam transmittance calculated in PBRT V3?

Question

In pbrt v3, the book gives this description of beam transmittance, but I don't know how to solve the differential equation like it says to get Tr , can someone please tell me how to solve the differential equation? Thanks a lot.

Answer 1

It's a version of the standard derivation of the exponential function from its differential equation, by separation of variables.

To review that standard derivation: suppose we want a function $y(x)$ obeying the differential equation $\mathrm{d}y/\mathrm{d}x = -ky$, for some constant $k$. Then we can solve the equation as follows: $$ \begin{aligned} \mathrm{d}y &= -ky \, \mathrm{d}x \\ \frac{\mathrm{d}y}{y} &= -k \, \mathrm{d}x \\ \int \frac{\mathrm{d}y}{y} &= -\int k \, \mathrm{d}x \\ \ln y &= -kx \\ y &= e^{-kx} \end{aligned} $$ (there should actually be some constants of integration in there, but I left them out since they're not important for this answer.)

Now, suppose we generalize and make the constant $k$ into a function $k(x)$. Then we can repeat this derivation, but we will not be able to do the integral on the right side, since $k(x)$ is unspecified. The result will then be: $$ y = e^{-\int k(x) \, \mathrm{d}x} $$

The derivation for the transmittance along a ray is just the same, but with some variables renamed: $y$ becomes $L$, $x$ is now the parameter $t$ along the ray, and $k(x)$ is called $\sigma_\mathrm{t}(\mathrm{p})$.