Rendering Equation for photons carrying flux I am trying to understand the mathematical/physical foundations of photon mapping better.

In the forward photon tracing step it is established, that the rays are carrying portions of total power (flux) rather than radiance as it is typically done in other types of ray-tracing algorithms.

However, in all articles and tutorials I could never find a precise explanation how such flux-carrying path interacts with materials? In other words - what is the rendering equation for interaction between flux-path and a surface?

What I typically found is an algorithm, based on Russian Roulette, for building a light path. But not a foundation equation that lead to that algorithm.

In particular, I am interested in:


Should the equation include the cosine of the incident angle or not? I suspect it shouldn't be there, as the power remains constant, but it is spread over a bigger area (thus reducing the radiance at each point, but not total flux).
Should the BRDF function be somehow (re)normalized? For example, for a perfect diffusive material, the BRDF is a constant not greater than 1/pi, so that it is energy conserving. But for a flux-based equation, assuming it is different, energy conservation may set a different maximum value.




Note, the very similar question:

Rendering equation for Photon Mapping

refers to a different step of the photon mapping algorithm. It asks for the rendering equation where the radiance is being estimated based on the nearest stored photons. Instead, I am asking for the flux-surface interaction in an earlier stage of the algorithm, when photons are being generated and propagated.
It'd be hell a lot easier if this were on graphicsexchange, since I can't use latex here but anyways.

In the first pass of Photon Mapping you don't need to use the Flux form of rendering equation. You just divide the original flux coming from the light source among the N photons, then for each photon you use Russian Roulette to determine whether it reflects, transmits or gets absorbed and store the incoming flux.

In the second pass you need to find radiance but you have information in Flux form. The rendering equation can be simply transformed. 





So all you need to do is divide the flux stored for each photon by the area. Note that the Area is often denoted as A= pi r^2 Since we are concerned with a hemisphere and the underlying area is just a circle.

About your 2nd point, I don't think the BRDF needs to be re-normalized or whatsoever since the equation is essentially the same. Note that the second term in the last equation is still a radiance written in flux form. (BRDF has units of inverse solid angle thus making the whole term radiance).