The Visual Computer

, Volume 34, Issue 6–8, pp 843–852 | Cite as

Multiple importance sampling characterization by weighted mean invariance

  • Mateu Sbert
  • Vlastimil HavranEmail author
  • László Szirmay-Kalos
  • Víctor Elvira
Original Article


In this paper, we examine the linear combination of techniques and multiple importance sampling for Monte Carlo integration from a new perspective of quasi-arithmetic weighted means. The invariance property of these means allows us to define a new family of heuristics. We illustrate our results with several rendering examples, including environment mapping and path tracing.


Global illumination Rendering equation analysis Multiple importance sampling Monte Carlo 



The authors acknowledge the comments by anonymous reviewers that helped to improve a preliminary version of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Research grants from funding agencies

The authors are funded in part by Czech Science Foundation research program GA14-19213S, by Grant TIN2016-75866-C3-3-R from the Spanish Government, by National Natural Science Foundation of China Grants Nos. 61471261 and 61771335, and by Grant OTKA K-124124 and VKSZ-14 PET/MRI 7T. V.E. acknowledges support from the Agence Nationale de la Recherche of France under PISCES project (ANR-17-CE40-0031-01), the Fulbright program, and the Marie Curie Fellowship (FP7/2007-2013) under REA grant agreement n. PCOFUND-GA-2013-609102, through the PRESTIGE program.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyTianjin UniversityTianjinChina
  2. 2.Institute of Informatics and ApplicationsGirona UniversityGironaSpain
  3. 3.Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic
  4. 4.Budapest University of Technology and EconomicsBudapestHungary
  5. 5.IMT Lille Douai & CRIStAL laboratoryLilleFrance

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