Optimal weights for bidirectional ray tracing with photon maps while mixing 3 strategies

Abstract

Bidirectional stochastic ray tracing with photon maps is a powerful method but suffers from noise. It can be reduced by the Multiple Importance Sampling which combines results of different “strategies”. The “optimal weights” minimize the noise functional thus providing the best quality of the results. In the paper we derive and solve the system of integral equations that determine the optimal weights. It has several qualitative differences from the previously investigated case of mixing two strategies, but further increase of their number beyond 3 retains the qualitative features of the system. It can be solved in a closed form i.e. as an algebraic formula that include several integrals of the known functions that can be calculated in ray tracing.

Optimal MIS weights in case of mixing 3 strategies for bidirectional MCRT with photon maps

“…This product can also reach , but only in the area where the two kernels overlap. Their arguments are random and the probability of intersection is , so the average of the product of kernels with different arguments is [27]. Due to the above, for calculations of averages of for small we can retain only the squares of kernels: while the omitted terms are inessential since have much smaller average .…”

“…Due to the above, for calculations of averages of for small we can retain only the squares of kernels: while the omitted terms are inessential since have much smaller average . Averaging this approximation to over the FMCRT ensemble and then averaging over the BMCRT ensemble (details are in [27]) we arrive at (3) where and denote the vertices of the joint path counted from camera; n(y) is the local normal at the point y.…”

“…Averaging (1) over the BMCRT ensemble and then averaging over the FMCRT ensemble (details are in [27]) we have Taking into account that we arrive at (4) where , and denote the vertices of the joint path counted from camera.…”

“…Since is independent of weight (provided that their sum is 1 [27]), (2) implies that the variation of the pixel RMS is the sum of variations of the three averages:…”

“…The optimal weights are those for which for an arbitrary . Due to the weight normalization conditions [27], there are only three independent weights: one from the family and two from the family . We decided to choose When calculating the variation against it should be remembered that when is varied while is kept, then since the sum of weights is fixed also varies:…”