How to properly use scattering coefficient during volumetric path tracing?

The increased radiance during sampling is known by:

$$L_s = L_e + \sigma_s \int p(w, w')f_p(w, w')dw'$$

The latter part, which accounts for in-scattering radiance bears a $$\sigma_s$$ in it.

• The first question is, how to explain the physical functionality of $$\sigma_s$$ in a simple way? I think taking a look at the original differential form for volumetric RTE will be clearer? Since differential RTE clearly indicates "increase" and "decrease" of radiance.

• The second problem is, since I can use (almost) any sampling strategy I want for importance sampling, and for the original exponential sampling, I can derive (during mfp-sampling): $$\frac{f(x)}{p(x)} = \frac{\sigma_s e^{-\sigma_t d}}{\sigma_t e^{-\sigma_t d}} = \text{scattering albedo } \alpha$$ Therefore, for high albedo medium, multiple scattering actually won't introduce much energy loss since it would only be $$\alpha^{n-1}$$ (n is the number of scattering events). However, when I choose to use other sampling distribution instead of exponential, the estimator tends to yield small values, since $$p(x)$$ no-longer cancels the exponent in $$f(x)$$, therefore I can't get $$\alpha$$ (which should be close to 1 for high-scattering medium). In the extreme case, where I use deterministic method to add a mfp sample, for example:

It seems that without $$p(x)$$ as the denominator, placing a deterministic sample here is increasing the radiance by a large amount (if distance is small enough and $$\sigma_s$$ is large). So why would this happend?. And, is this correct that whenever a scattering event occurs (no matter stochatic / deterministic sampling), we should multiply a $$\sigma_s$$ somewhere during computation?

• The third confusion of mine: deterministically placing a sample (like, during distance sampling) pdf is 1 (right?), but sampling in a extremely small interval results in a huge pdf value (for example, uniform distribution will have $$1 / (\max - \min)$$ which is pretty big), so what happened here? Any help will be appreciated!

• For the third question, could it be that there is a change of measure? from the continuous volume measure to a discrete measure... whatever it is. It occurred to me that I've seen discrete measure (a tag for different measures) in mitsuba but I never got the chance to understand it... Commented Jul 9, 2023 at 12:42