(L) [2014/03/25] [tby beason] [Question about "Five Common Misconceptions about Bias in..."] Wayback!In [LINK http://cs.au.dk/~toshiya/misc.pdf]
the author mentions the "geometry term" (in particular from a shadow ray) as example of variance being NOT finite and existing.
1) Is the geometry term "G" from Veach's thesis, namely: visibility*cos(theta1)*cos(theta2)/r^2?
2) Why would this be either not finite or existing? When r=0, or something else?
(L) [2014/03/26] [tby bouliiii] [Question about "Five Common Misconceptions about Bias in..."] Wayback!Yes this is the geometric term from Veach PhD.
As Toshiya said, you need to importance sample the geometric term i.e. 1/r^2 has to "go away" in the estimator. This is trivially the case in path tracing or less trivially, with multiple important sampling for bidirectional path tracing.
With Instant Radiosity non-clamped estimator, the geometric term indeed "causes" the unbounded variance (when r tends to 0). This is just the way the estimator is built.
Interestingly, you still have some convergence property with unbounded variance with the weak law of large numbers (convergence in probability) but as far as I know, this is useless for numerical purposes (exactly as Instant Radiosity is with no clamp) since you cannot rely anymore on the central limit theorem that provides the convergence rate.