I am studying computer graphics and have read several references in this field on the topic of volume rendering/rendering participating medium. They generally start from the Radiative Transfer Equation which I know is provided in S. Chandrasekhar's book "Radiative Transfer". I have access to the book and have been through the first 10 pages in which he introduces the equation using a notation that's different than the notation used in the CG literature. Namely, I have been looking at a book called PBRT and this online reference: Production Volume Rendering p.9.
The RTE equation in these references is presented as:
$$(\omega \nabla)L(x,\omega)=-\sigma_t(x) L(x, \omega) + \sigma_s(x)\int_{S^2} f_p(x, \omega, \omega')L(x,\omega')d\omega' $$
Where $\sigma_t$ is the extinction coefficient ($\sigma_t = \sigma_a + \sigma_s$) and $\sigma_a$ and $\sigma_s$ are the absorption and scattering coefficient respectively. The second term on the right hand side(also called the source term) accounts for in-scattering (incoming light coming for the entire sphere of directions indicated by $S^2$) and $f_p$ is the phase function.
This equation makes more or less sense to me (I omitted the emission term here for brevity) and if I am correct, describes the change of radiance/intensity along a beam of light in the direction vector $\omega$.
Now in these references, they then introduce another equation called the "volume rendering equation" or VRE (I can't find the origin of the term btw if someone knows) as follows:
$$L(x,\omega)=\int_{t=0}^d T(d) [\sigma_s L_s(x, \omega)]dt$$
where:
$$ T(t) = exp(-\int_{s=0}^{t} \sigma_t(x_s) ds)$$
and:
$$L_s(x, \omega) = \int_{S^2} f_p(x, \omega, \omega')L(x,\omega')d\omega'$$
The VRE instead gives you the radiance at x, where d, is the distance over which you are integrating (along the light ray with direction vector $\omega$).
In both references they write something like this (Pixar Research p. 9):
We can formally integrate both sides of the RTE, turning the gradient (ω · ∇) on the left side of the RTE equation into an integral on the right side, giving an explicit equation for L(x,ω) and arriving at the volume rendering equation (VRE).
and (PBRT)
With suitable boundary conditions, this equation can be transformed into a purely integral equation.
They go on to write that the VRE can intuitively be understood as tracking the light backward from a potential surface from which light is emitted towards the point p (with a participating medium between the back surface and the point $x$). As you "backtrack", you collect light along the line at various locations and each time "extinguish" this result by the transmittance term (T(t)) up to that point. I got that too.
However, none of the references I found, actually describe the steps to go from the first equation (RTE) to the second (VRE). And I don't see how integrating the first equation would effectively lead to the second equation? All references, including Chandrasekhar's, indicate that this can be done if you set the boundaries (second quote) so it seems like boundaries play a role here, but this doesn't help me much. It seems VRE is the same equation as RTE but where the path of integration is reversed but I'd like to understand this formally. I'd be very grateful if someone with greater knowledge could provide the steps to go from the RTE to the VRE and/or point me in the right direction.
(let me know if this question would be better suited in the Physics section).
Edit
I found a derivation available on this website / scratchapixel.
So from what I see there, they "simply" apply the "general solution to a non-homogeneous ordinary differential equation" to the RTE equation (using the integrating factor method). I won't reproduce the steps here.