Multiple Importance Sampling (MIS) is a key technique for achieving robustness of Monte Carlo estimators in computer graphics and other fields. We derive optimal weighting functions for MIS that provably minimize the variance of an MIS estimator, given a set of sampling techniques. We show that the resulting variance reduction over the balance heuristic can be higher than predicted by the variance bounds derived by Veach and Guibas, who assumed only non-negative weights in their proof. We theoretically analyze the variance of the optimal MIS weights and show the relation to the variance of the balance heuristic. Furthermore, we establish a connection between the new weighting functions and control variates as previously applied to mixture sampling. We apply the new optimal weights to integration problems in light transport and show that they allow for new design considerations when choosing the appropriate sampling techniques for a given integration problem.

Direct illumination calculation is an important component of any physically-based Tenderer with a substantial impact on the overall performance. We present a novel adaptive solution for unbiased Monte Carlo direct illumination sampling, based on online learning of the light selection probability distributions. Our main contribution is a formulation of the learning process as Bayesian regression, based on a new, specifically designed statistical model of direct illumination. The net result is a set of regularization strategies to prevent over-fitting and ensure robustness even in early stages of calculation, when the observed information is sparse. The regression model captures spatial variation of illumination, which enables aggregating statistics over relatively large scene regions and, in turn, ensures a fast learning rate. We make the method scalable by adopting a light clustering strategy from the Lightcuts method, and further reduce variance through the use of control variates. As a main design feature, the resulting algorithm is virtually free of any preprocessing, which enables its use for interactive progressive rendering, while the online learning still enables super-linear convergence.