Variance analysis for Monte Carlo integration

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 hemispherical 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.

An adaptive point sampler on a regular lattice

“…Theoretical analysis on Monte Carlo integration error/variance, by different groups [Durand 2011;Öztireli 2016;Pilleboue et al 2015;Ramamoorthi et al 2012;Subr and Kautz 2013], leads to the intuition that a sampling pattern with a blue noise spectrum is a good choice for area-light sampling. Yet, low-discrepancy sequences continue to prevail as the favorite sampler for this purpose [Pharr and Humphreys 2010].…”

Low-discrepancy blue noise sampling

“…On the one hand, the variance in integration is directly dependent on the product of the power spectra of the integrand and the sampling pattern [Pilleboue et al 2015]. Consequently, for integrands whose spectral content can be estimated, one can considerably reduce variance by choosing the parameters of the blue-noise sampling distribution in such a way that its vanished low-frequency spectral part contains the major part of the spectral content of the integrand.…”

“…Thus, for bounded-variation integrands whose spectral content cannot be easily estimated, the LD property will still guarantee low variance, at least asymptotically. Finally, the absence of spectral peaks will avoid aliasing [Durand 2011;Pilleboue et al 2015].…”

“…We also compare timings with optimized fast approximation techniques of Poisson disk sampling [Dunbar and Humphreys 2006]. However, spectral profiles of Poisson disk point patterns lead to slower variance convergence rates ( [Pilleboue et al 2015] and Figure 8 and 9). The zoneplates also show how state-of-the-art LD samplers, such as Sobol or rank-1, perform poorly for this reconstruction, due to the spectral peaks in their Fourier spectra.…”

“…To predict the integration error with respect to the number of samples, the discrepancy is a first critical value, as discussed earlier. The shape of the point set power spectra can also be explicitly related to variance or integration error [Pilleboue et al 2015].…”