The Visual Computer

, Volume 33, Issue 6–8, pp 845–855 | Cite as

Adaptive multiple importance sampling for general functions

  • Mateu SbertEmail author
  • Vlastimil Havran
Original Article


We propose a mathematical expression for the optimal distribution of the number of samples in multiple importance sampling (MIS) and also give heuristics that work well in practice. The MIS balance heuristic is based on weighting several sampling techniques into a single estimator, and it is equal to Monte Carlo integration using a mixture of distributions. The MIS balance heuristic has been used since its invention almost exclusively with an equal number of samples from each technique. We introduce the sampling costs and adapt the formulae to work well with them. We also show the relationship between the MIS balance heuristic and the linear combination of these techniques, and that MIS balance heuristic minimum variance is always less or equal than the minimum variance of the independent techniques. Finally, we give one-dimensional and two-dimensional function examples, including an environment map illumination computation with occlusion.


Global illumination Rendering equation analysis Multiple importance sampling Monte Carlo 



This work has been partially funded by Czech Science Foundation research program GA14-19213S and by Grant TIN2016-75866-C3-3-R from the Spanish Government.

Supplementary material

371_2017_1398_MOESM1_ESM.pdf (239 kb)
Supplementary material 1 (pdf 238 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Tianjin University, School of Computer Science and TechnologyTianjinChina
  2. 2.Girona University, Institute of Informatics and ApplicationsGironaSpain
  3. 3.Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic

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