Jaroslav Křivánek

Optimal multiple importance sampling

Ivo Kondapaneni
Charles University, Prague
Petr Vévoda
Charles University, Prague
Chaos Czech a.s.
Pascal Grittmann
Saarland University

Tomáš Skřivan
IST Austria
Philipp Slusallek
Saarland University
Jaroslav Křivánek
Charles University, Prague
Chaos Czech a.s.

* Ivo Kondapaneni and Petr Vévoda share the first authorship of this work.

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.


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.


Ivo Kondapaneni, Petr Vévoda, Pascal Grittmann, Tomáš Skřivan, Philipp Slusallek, and Jaroslav Křivánek. Optimal Multiple Importance Sampling. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2019), 38(4), 2019.
DOI | BibTeX

Links and Downloads

pdf (10 MB)
pdf (1.6 MB)
zip (32 MB)
pptx (11 MB)
pdf (3 MB)


Here we provide the additional rendering results to complete the main article. The thumbnails below lead to individual pages for the 4 scenes used in the article: Staircase I, Staircase II, Dining room, Veach's scene. They present full-size images and rendering statistics for both equal-time and equal-sample comparisons of different combination strategies and sampling techniques as discussed in the main text.

Staircase IStaircase IIDining roomVeach
Reference.png Reference.png Reference.png Reference.png


Many thanks to Benedikt Bitterli for the PBRT test scenes. The work was supported by Charles University Grant Agency project GAUK 996218, by the grant SVV-2017-260452, and by the Czech Science Foundation grants 16-18964S and 19-07626S. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642841 (DISTRO).