Analysis of Sample Correlations for Monte Carlo Rendering

Abstract

Modern physically based rendering techniques critically depend on approximating integrals of high dimensional functions representing radiant light energy. Monte Carlo based integrators are the choice for complex scenes and effects. These integrators work by sampling the integrand at sample point locations. The distribution of these sample points determines convergence rates and noise in the final renderings. The characteristics of such distributions can be uniquely represented in terms of correlations of sampling point locations. Hence, it is essential to study these correlations to understand and adapt sample distributions for low error in integral approximation. In this work, we aim at providing a comprehensive and accessible overview of the techniques developed over the last decades to analyze such correlations, relate them to error in integrators, and understand when and how to use existing sampling algorithms for effective rendering workflows.

Blue Noise Plots

“…The resulting spatial distribution of dots respect some minimum distance giving perceptually pleasing patterns [Yel83]. Consequently, blue noise has been widely adopted in many computer graphics applications including halftoning [Uli88], stippling [Sec02], artistic packing [RRS13], anti‐aliasing [DW85] and variance reduction for Monte Carlo rendering [SOA*19]. Typically, such methods are free to place dots in whatever arrangement, only their correlation and in some cases an importance function is relevant.…”

Antithetic sampling for Monte Carlo differentiable rendering

“…Being a classic variance reduction framework for Monte Carlo estimation, antithetic sampling [Hammersley and Mauldon 1956;Geweke 1988] has been studied in probabilistic inference and machine learning [Ren et al 2019;]. In computer graphics, this technique has been explored by several previous works in forward rendering [Subr et al 2014;Öztireli 2016;Singh et al 2019Singh et al , 2020. In Monte Carlo differentiable rendering, Bangaru et al [2020] have applied antithetic sampling to efficiently handle discontinuity boundaries under the warped-area formulation.…”

Perceptual error optimization for Monte Carlo rendering

“…Since these metrics often do not accurately reflect the visual quality, equal-time visual comparisons are also commonly reported. Various theoretical frameworks have been developed in the spatial [Niederreiter 1992;Kuipers and Niederreiter 1974] and Fourier [Singh et al 2019] domains to understand the error reported through these metrics. Numerous variance reduction algorithms like multiple importance sampling [Veach 1998], and control variates [Loh 1995;Glasserman 2004] improve the error convergence of light transport renderings.…”

“…Sampling in rendering. Sample correlations [Singh et al 2019] directly affect the error in rendering. Quasi-Monte Carlo samplers [Halton 1964;Sobol 1967] preserve correlations (i. e., stratification) well in higher dimensions, which makes them a perfect candidate for rendering problems [Keller 2013].…”