Adaptive multiple importance sampling for general functions
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.
KeywordsGlobal 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.
- 1.Bekaert, P., Sbert, M., Halton, J.: Accelerating path tracing by re-using paths. In: Proceedings of EGRW ’02, pp. 125–134. Eurographics Association, Switzerland (2002)Google Scholar
- 4.Csonka, F., Szirmay-Kalos, L., Antal, G.: Cost-driven multiple importance sampling for Monte-Carlo rendering. TR-186-2-01-19, Institute of Computer Graphics and Algorithms, Vienna University of Technology (2001)Google Scholar
- 7.Elvira, V., Martino, L., Luengo, D., Corander, J.: A Gradient adaptive population importance sampler. In: Proceedings of ICASSP 2015, pp. 4075–4079. IEEE (2015)Google Scholar
- 11.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 (2014)Google Scholar
- 13.Korovkin, P.P.: Inequalities. Little Mathematics Library. Mir Publishers, Moscow (1975)Google Scholar
- 14.Lafortune, E.P., Willems, Y.D.: Using the Modified Phong Reflectance Model for Physically Based Rendering. TR CW197, Department of Computers, K.U. Leuven (1994)Google Scholar
- 17.Marin, J.M., Pudlo, P., Sedki, M.: Consistency of the adaptive multiple importance sampling. Preprint arXiv:1211.2548 (2012)
- 18.Owen, A., Zhou, Y.: Safe and effective importance sampling. J. Am. Stat. Assoc. 95, 135–143 (2000)Google Scholar
- 19.Rubinstein, R., Kroese, D.: Simulation and the Monte Carlo Method. Wiley Series in Probability and Statistics. Wiley, New York (2008)Google Scholar
- 21.Tokuyoshi, Y., Ogaki, S., Sebastian, S.: Final Gathering Using Adaptive Multiple Importance Sampling. In: ACM SIGGRAPH ASIA 2010 Posters, SA ’10, pp. 47:1–47:1. ACM, New York, USA (2010)Google Scholar
- 22.Veach, E.: Robust Monte Carlo methods for light transport simulation. Ph.D. thesis, Stanford University (1997)Google Scholar
- 23.Veach, E., Guibas, L.J.: Optimally combining sampling techniques for Monte Carlo rendering. In: Proceedings of SIGGRAPH ’95, pp. 419–428. ACM, New York (1995)Google Scholar