Convergence analysis for anisotropic Monte Carlo sampling spectra

1Dartmouth College

In ACM Transactions on Graphics (Proceedings of SIGGRAPH), 2017

The expected power spectrum of N-rooks with N=256 samples (left) is highly anisotropic, with drastically different radial behavior along different directions (blue vs. red arrows). Fourier analysis using the radially averaged power spectrum (radial mean) cannot detect the good anisotropic properties of the sampler along the canonical axes.


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. jittered, Latin-hypercube (N-rooks), multijittered—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 develop a Fourier-domain mathematical tool to analyze the variance, and subsequently the convergence rate, of Monte Carlo integration using any arbitrary (anisotropic) sampling power spectrum. We also validate and leverage our theoretical analysis, demonstrating that judicious alignment of anisotropic sampling and integrand spectra can improve variance and convergence rates in MC rendering, and that similar improvements can apply to (anisotropic) deterministic samplers.



Video (full-quality)


Gurprit Singh, Wojciech Jarosz. Convergence analysis for anisotropic Monte Carlo sampling spectra. ACM Transactions on Graphics (Proceedings of SIGGRAPH), 36(4), July 2017.
    author   = {Singh, Gurprit and Jarosz, Wojciech},
    title    = {Convergence analysis for anisotropic {{Monte}} {{Carlo}} sampling spectra},
    journal  = {ACM Transactions on Graphics (Proceedings of SIGGRAPH)},
    volume   = {36},
    number   = {4},
    year     = {2017},
    month    = jul,
    doi      = {10/gbxfhj},
    keywords = {stochastic sampling, signal processing, Fourier transform, Power spectrum}
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