This math is a bit over my head, but I'll hazard a guess as to how conservation of energy might lead to a proof that this series converges: the space of possible distributions of radiance throughout a scene is a normed vector space, with the norm being related to the integral of all radiances across the entire scene. Conservation of energy says that neither reflection nor light propagation can increase this integral; mathematically, this means the light transport operator has norm $|K| \leq 1$.

According to Wikipedia, the Neumann series of a operator with norm *less than* 1 over a *complete* vector space always converges. In a real scene, some radiance is necessarily absorbed and turned into heat (at the very least, the camera itself is absorbing light!), which gives us the strict inequality we need. I'm not sure about the second requirement, though -- I've never understood what it means in practice for a vector space to be complete.

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And just to make this all more complicated, radiance actually increases when passing through the boundary from a medium with a lower index of refraction and to one with a higher index of refraction! (Essentially, differential cones of rays are squeezed down into smaller differential cones; this cancels out when going the other way.)

This doesn't violate energy conservation, but it does complicate the analysis of the operator norms. See Chapter 4 of Eric Veach's thesis for more info...

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So with each bounce the radiance of each photon should decrease which means there should be an optimal number of bounces where adding more bounces doesn't noticeably increase the quality of the image. Is finding this limit just a matter of trial and error and eyeballing it or can we predict the optimal number of bounces ahead of time?

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My understanding is that Russian roulette tries to solve this problem by estimating the contribution of each path and discarding it with high probability if the contribution is low. This figures out the decrease in radiance you mention by taking into account the number of bounces and the BSDF of the surface at each bounce in the estimation of a path's potential contribution. Doing this on a per-sample basis rather than globally across the whole scene makes sense because some parts of the scene will require more bounces than others (look e.g. at the lamp at the top of the picture--it requires four bounces to even appear transparent, whereas the rest of the scene looks reasonably good even with two).

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