Variance Analysis for Monte Carlo Integration

Adrien Pilleboue 1 Gurprit Singh 1 David Coeurjolly 2 Michael Kazhdan 3 Victor Ostromoukhov 1
1 R3AM - Rendu Réaliste pour la Réalité Augmentée Mobile
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
2 M2DisCo - Geometry Processing and Constrained Optimization
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : We propose a new spectral analysis of the variance in Monte Carlo integration, expressed in terms of the power spectra of the sampling pattern and the integrand involved. We build our framework in the Euclidean space using Fourier tools and on the sphere using spherical harmonics. We further provide a theoretical background that explains how our spherical framework can be extended to the hemi-spherical domain. We use our framework to estimate the variance convergence rate of different state-of-the-art sampling patterns in both the Euclidean and spherical domains, as the number of samples increases. Furthermore, we formulate design principles for constructing sampling methods that can be tailored according to available resources. We validate our theoretical framework by performing numerical integration over several integrands sampled using different sampling patterns.
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Adrien Pilleboue, Gurprit Singh, David Coeurjolly, Michael Kazhdan, Victor Ostromoukhov. Variance Analysis for Monte Carlo Integration. ACM Transactions on Graphics, Association for Computing Machinery, 2015, SIGGRAPH 2015, 34 (4), pp.14. ⟨hal-01150268v2⟩

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