Jaroslav Křivánek

Variance-Aware Multiple Importance Sampling

Pascal Grittmann
Saarland University
Iliyan Georgiev
Philipp Slusallek
Saarland University
Jaroslav Křivánek
Charles University, Prague
Chaos Czech a. s.

Equal-time comparison of bidirectional path tracing (BPT) with different MIS heuristics. The balance (b) and power (c) heuristics perform visibly worse than using only the unidirectional path tracing samples that BPT includes (b). The error reduction in parentheses is w.r.t. the balance heuristic combination; lower is better. Our variance-aware balance heuristic significantly improves the result (e), especially the direct illumination component (bottom row).


Many existing Monte Carlo methods rely on multiple importance sampling (MIS) to achieve robustness and versatility. Typically, the balance or power heuristics are used, mostly thanks to the seemingly strong guarantees on their variance. We show that these MIS heuristics are oblivious to the effect of certain variance reduction techniques like stratification. This shortcoming is particularly pronounced when unstratified and stratified techniques are combined (e.g., in a bidirectional path tracer). We propose to enhance the balance heuristic by injecting variance estimates of individual techniques, to reduce the variance of the combined estimator in such cases. Our method is simple to implement and introduces little overhead.


Pascal Grittmann, Iliyan Georgiev, Philipp Slusallek, and Jaroslav Křivánek. Variance-Aware Multiple Importance Sampling. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia 2019), 38(6), 2019.
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We thank the anonymous reviewers for their valuable feedback. The test scenes are slightly modified versions of those in the scene repositories of PBRT and Benedikt Bitterli. This work was supported by the Czech Science Foundation Grant 19-07626S and Charles University Grant SVV-2017-260452.