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Equal-sample comparison (20 per technique per pixel) of direct illumination estimated by an MIS combination of two light sampling techniques (Trained and Uniform, see the paper text for details) with our optimal weights (top row) and the power heuristic (bottom row). The false-color images b) show per-pixel average MIS weight values as determined by the two weighting strategies. Unlike any of the existing MIS weighting heuristics, the optimal weights can have negative values, which provides additional opportunity for variance reduction, leading to an overall 9.6 times lower error per sample taken than the power heuristic in this scene.

Optimal multiple importance sampling

Ivo Kondapaneni Petr Vévoda Pascal Grittmann Tomáš Skřivan Philipp Slusallek Jaroslav Křivánek
Shared first-authorship.

Abstract

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.

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