Advertisement

The Visual Computer

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

Multiple importance sampling characterization by weighted mean invariance

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

Abstract

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.

Keywords

Global illumination Rendering equation analysis Multiple importance sampling Monte Carlo 

Notes

Acknowledgements

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.

References

  1. 1.
    Belzunce, F., Martinez-Riquelme, C., Mulero, J.: An Introduction to Stochastic Orders. Academic Press (2016).  https://doi.org/10.1016/B978-0-12-803768-3.09977-4
  2. 2.
    Bugallo, M.F., Elvira, V., Martino, L., Luengo, D., Míguez, J., Djuric, P.M.: Adaptive importance sampling: the past, the present, and the future. IEEE Signal Process. Mag. 34(4), 60–79 (2017)CrossRefGoogle Scholar
  3. 3.
    Bullen, P.: Handbook of Means and Their Inequalities. Springer, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cappé, O., Guillin, A., Marin, J.M., Robert, C.P.: Population Monte Carlo. J. Comput. Graph. Stat. 13(4), 907–929 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cornuet, J.M., Marin, J.M., Mira, A., Robert, C.P.: Adaptive multiple importance sampling. Scand. J. Stat. 39(4), 798–812 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Douc, R., Guillin, A., Marin, J.M., Robert, C.P.: Minimum variance importance sampling via population Monte Carlo. ESAIM Probab. Stat. 11, 424–447 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Elvira, V., Martino, L., Luengo, D., Bugallo, M.F.: Efficient multiple importance sampling estimators. IEEE Signal Process. Lett. 22(10), 1757–1761 (2015).  https://doi.org/10.1109/LSP.2015.2432078 CrossRefGoogle Scholar
  8. 8.
    Elvira, V., Martino, L., Luengo, D., Bugallo, M.F.: Generalized multiple importance sampling. ArXiv e-prints (2015). https://arxiv.org/abs/1511.03095
  9. 9.
    Elvira, V., Martino, L., Luengo, D., Bugallo, M.F.: Heretical multiple importance sampling. IEEE Signal Process. Lett. 23(10), 1474–1478 (2016)CrossRefGoogle Scholar
  10. 10.
    Elvira, V., Martino, L., Luengo, D., Bugallo, M.F.: Multiple importance sampling with overlapping sets of proposals. In: Statistical Signal Processing Workshop (SSP), 2016 IEEE, pp. 1–5. IEEE (2016)Google Scholar
  11. 11.
    Elvira, V., Martino, L., Luengo, D., Bugallo, M.F.: Improving population Monte Carlo: alternative weighting and resampling schemes. Sig. Process. 131(12), 77–91 (2017)CrossRefGoogle Scholar
  12. 12.
    Havran, V., Sbert, M.: Optimal combination of techniques in multiple importance sampling. In: Proceedings of the 13th ACM SIGGRAPH International Conference on Virtual-Reality Continuum and Its Applications in Industry, VRCAI ’14, pp. 141–150. ACM, New York, NY, USA (2014).  https://doi.org/10.1145/2670473.2670496
  13. 13.
    He, H.Y., Owen, A.B.: Optimal Mixture Weights in Multiple Importance Sampling. ArXiv preprint arXiv:1411.3954 (2014)
  14. 14.
    Kajiya, J.T.: The rendering equation. In: Evans, D.C., Athay, R.J. (eds.) Computer Graphics (SIGGRAPH ’86 Proceedings), vol. 20, pp. 143–150 (1986)Google Scholar
  15. 15.
    Lu, H., Pacanowski, R., Granier, X.: Second-order approximation for variance reduction in multiple importance sampling. Comput. Graph. Forum 32(7), 131–136 (2013).  https://doi.org/10.1111/cgf.12220 CrossRefGoogle Scholar
  16. 16.
    Martino, L., Elvira, V., Luengo, D., Corander, J.: Layered adaptive importance sampling. Stat. Comput. 27(3), 599–623 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Neumann, L., Neumann, A., Szirmay-Kalos, L.: Compact metallic reflectance models. Comput. Graph. Forum (Eurographics’99) 18(3), 161–172 (1999)CrossRefGoogle Scholar
  18. 18.
    Owen, A.B., Maximov, Y., Chertkov, M.: Importance Sampling the Union of Rare Events with an Application to Power Systems Analysis. ArXiv preprint arXiv:1710.06965 (2017)
  19. 19.
    Sbert, M., Havran, V.: Adaptive multiple importance sampling for general functions. Vis. Comput. (2017).  https://doi.org/10.1007/s00371-017-1398-1 Google Scholar
  20. 20.
    Sbert, M., Havran, V., Szirmay-Kalos, L.: Variance analysis of multi-sample and one-sample multiple importance sampling. Comput. Graph. Forum 35(7), 451–460 (2016).  https://doi.org/10.1111/cgf.13042 CrossRefGoogle Scholar
  21. 21.
    Sbert, M., Havran, V., Szirmay-Kalos, L.: Multiple importance sampling revisited: breaking the bounds. EURASIP J. Adv. Signal Process. 2018(1), 15 (2018).  https://doi.org/10.1186/s13634-018-0531-2 CrossRefGoogle Scholar
  22. 22.
    Sbert, M., Poch, J.: A necessary and sufficient condition for the inequality of generalized weighted means. J. Inequal. Appl. 2016(1), 292 (2016).  https://doi.org/10.1186/s13660-016-1233-7 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shaked, M., Shanthikumar, G.: Stochastic Orders. Springer, New York (2007).  https://doi.org/10.1007/978-0-387-34675-5 CrossRefzbMATHGoogle Scholar
  24. 24.
    Veach, E.: Robust Monte Carlo Methods for Light Transport Simulation. Ph.D. thesis, Stanford University (1997)Google Scholar
  25. 25.
    Veach, E., Guibas, L.J.: Optimally combining sampling techniques for Monte Carlo rendering. In: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’95, pp. 419–428. ACM, New York, NY, USA (1995).  https://doi.org/10.1145/218380.218498

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

Personalised recommendations