Fourier analysis is gaining popularity in image synthesis as a tool for the analysis of error in Monte Carlo (MC) integration. Still, existing tools are only able to analyze convergence under simplifying assumptions (such as randomized shifts) which are not applied in practice during rendering. We reformulate the expressions for bias and variance of sampling-based integrators to unify non-uniform sample distributions (importance sampling) as well as correlations between samples while respecting finite sampling domains. Our unified formulation hints at fundamental limitations of Fourier-based tools in performing variance analysis for MC integration. At the same time, it reveals that, when combined with correlated sampling, importance sampling (IS) can impact convergence rate by introducing or inhibiting discontinuities in the integrand. We demonstrate that the convergence of multiple importance sampling (MIS) is determined by the strategy which converges slowest and propose several simple approaches to overcome this limitation. We show that smoothing light boundaries (as commonly done in production to reduce variance) can improve (M)IS convergence (at a cost of introducing a small amount of bias) since it removes C0 discontinuities within the integration domain. We also propose practical integrand- and sample-mirroring approaches which cancel the impact of boundary discontinuities on the convergence rate of estimators.
Supplemental document (includes mathematical derivations and analytic evaluations)
Presentation slides: Keynote PDF
We are grateful to all the anonymous reviewers for their constructive remarks, Tobias Ritschel for suggesting edits in Figure 7. This work was partially supported by the Fraunhofer and Max Planck cooperation program within the German pact for research and innovation (PFI) and NSF grant CNS-1205521. Kartic Subr was supported by a Royal Society University Research Fellowship and Wojciech Jarosz was partially supported by NSF grant ISS-181279.
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