Traditional Monte Carlo (MC) integration methods use point samples to numerically approximate the underlying integral. This approximation introduces variance in the integrated result, and this error can depend critically on the sampling patterns used during integration. Most of the well known samplers used for MC integration in graphics, e.g. jitter, Latin hypercube (n-rooks), multi-jitter, are anisotropic in nature. However, there are currently no tools available to analyze the impact of such anisotropic samplers on the variance convergence behavior of Monte Carlo integration. In this work, we propose a mathematical tool in the Fourier domain that allows analyzing the variance, and subsequently the convergence rate, of Monte Carlo integration using any arbitrary (anisotropic) sampling power spectrum. We apply our analysis to common anisotropic point sampling strategies in Monte Carlo integration, and extend our analysis to recent Monte Carlo approaches relying on line samples which have inherently anisotropic power spectra. We validate our theoretical results with several experiments using both point and line samples.

#### Publication

@techreport{singh16monte, author = "Singh, Gurprit and Jarosz, Wojciech", title = "{{Monte}} {{Carlo}} convergence analysis for anisotropic sampling power spectra", institution = "Dartmouth College, Computer Science", address = "Hanover, NH", number = "TR2016-816", year = "2016", month = aug, abstract = "Traditional Monte Carlo (MC) integration methods use point samples to numerically approximate the underlying integral. This approximation introduces variance in the integrated result, and this error can depend critically on the sampling patterns used during integration. Most of the well known samplers used for MC integration in graphics, e.g. jitter, Latin hypercube (n-rooks), multi-jitter, are anisotropic in nature. However, there are currently no tools available to analyze the impact of such anisotropic samplers on the variance convergence behavior of Monte Carlo integration. In this work, we propose a mathematical tool in the Fourier domain that allows analyzing the variance, and subsequently the convergence rate, of Monte Carlo integration using any arbitrary (anisotropic) sampling power spectrum. We apply our analysis to common anisotropic point sampling strategies in Monte Carlo integration, and extend our analysis to recent Monte Carlo approaches relying on line samples which have inherently anisotropic power spectra. We validate our theoretical results with several experiments using both point and line samples." }