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.
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The authors acknowledge the comments by anonymous reviewers that helped to improve a preliminary version of the paper.
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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.
Appendix
Appendix
Further justification for the heuristics \(\alpha _k \propto \frac{1}{c_k v_k}\) used in Sect. 5.3 comes from the following. Observe that for \(\alpha _k \propto \frac{1}{c_k v_k}\), the second term of Eq. 13 becomes
while for \(\alpha _k = \frac{1}{M}\) the second term of Eq. 13 becomes
where \(A(\{v_k\})\) stands for arithmetic mean of \(\{v_k\}\). We want to see that the expression in Eq. 15 is less than the expression in Eq. 16. A necessarily unfavorable case is when \(c_i \propto 1/v_i\). Let us consider without loss of generality that the proportionality constant is 1, i.e., \(c_i = 1/v_i\). In that case \(H(\{v_k\}) = A(\{c_k\})^{-1}\), \(H(\{c_k v_k\}) = 1\), and both expressions in Eqs. 15 and 16 are equal. In the necessarily favorable case \(c_i = v_i\), and the expression in Eq. 15 becomes \(\frac{H(\{v_k^2\})^2}{H(\{v_k\})^2}\), but \(\frac{H(\{v_k^2\})}{H(\{v_k\}) } \le H(\{v_k\})\), as \(H(\{v_k\})\) corresponds to a power mean with exponent \(-1\) while \(\sqrt{H(\{v_k^2\})}\) to exponent − 2. Thus \(\frac{H(\{v_k^2\})^2}{H(\{v_k\})^2} \le H(\{v_k\})^2 \le A(\{v_k\})^2\).
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Sbert, M., Havran, V., Szirmay-Kalos, L. et al. Multiple importance sampling characterization by weighted mean invariance. Vis Comput 34, 843–852 (2018). https://doi.org/10.1007/s00371-018-1522-x
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DOI: https://doi.org/10.1007/s00371-018-1522-x