@TechReport{CS-TR-DCGI-2014-2,
author = {Vlastimil Havran and Mateu Sbert},
title = {{Optimal Combination of Techniques in Multiple Importance Sampling}},
number = {CS-TR-DCGI-2014-2},
type = {Technical Report Series of DCGI},
institution = {Department of Computer Graphics and Interaction},
address = {Czech Technical University, FEE},
year = {2014},
month = aug,
howpublished = {Available at http://dcgi.fel.cvut.cz/techreps},
abstract = {Since its introduction by Veach, the Multiple Importance Sampling (MIS)
technique has been widely used in Computer Graphics in many rendering
algorithms. MIS is based on weighting several sampling techniques into
a single estimator. When the mixing weights are taken
such that the sample contributions are balanced, i.e. they are the
same for all techniques, it becomes a balance heuristic. It has been
used since its invention almost exclusively on equal sampling for all
techniques, and until now the question whether unequal sampling can give better
variance, has raised little interest, maybe due to its
intrinsic difficulty and also due to the fact that good results were already
obtained with equal sampling. The most interesting cases of the use
of MIS in Computer Graphics, where an environment map is a particular
case, correspond to the integral of a product of functions. Based on
the properties of the balance heuristic MIS as a weighted mixture of
distributions, weights proportional to the number of samples, we
obtain for this kind of integral an implicit closed formula for the
optimal sampling. We also take into account the cost of each sampling
technique. Although this closed formula cannot be written in an
explicit way, we outline an iterative procedure for obtaining the optimal
values. To bypass the combinatorially growing cost of the iterative
procedure, we introduce a sound heuristic approximation based on the
optimal combination of two independent estimators with known
variance. We validate our theory with the results from implementing
1-dimensional function examples and 2-dimensional examples of
environment map illumination.},
}