Fourier Analysis of Correlated Monte Carlo Importance Sampling,Computer Graphics Forum

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Fourier Analysis of Correlated Monte Carlo Importance Sampling
Computer Graphics Forum ( IF 2.7 ) Pub Date : 2019-04-05 , DOI: 10.1111/cgf.13613
Gurprit Singh _{1,

2} , Kartic Subr ₃ , David Coeurjolly ₄ , Victor Ostromoukhov ₄ , Wojciech Jarosz ₂

Affiliation

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 analyse 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 (IS)] 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, 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.

中文翻译：

相关蒙特卡罗重要性采样的傅里叶分析

傅立叶分析在图像合成中作为一种分析蒙特卡罗 (MC) 积分误差的工具越来越受欢迎。尽管如此，现有工具只能在简化的假设（例如随机移位）下分析收敛性，而这些假设在渲染过程中并未在实践中应用。我们重新制定了基于采样的积分器的偏差和方差的表达式，以在尊重有限采样域的同时统一非均匀样本分布 [重要性采样 (IS)] 以及样本之间的相关性。我们的统一公式暗示了基于傅立叶的工具在执行 MC 积分方差分析时的基本局限性。同时，它揭示了当与相关采样相结合时，IS 可以通过引入或抑制被积函数中的不连续性来影响收敛速度。我们证明了多重重要性采样（MIS）的收敛是由收敛最慢的策略决定的，并提出了几种简单的方法来克服这个限制。我们表明平滑光边界（通常在生产中进行以减少方差）可以改善 (M)IS 收敛（以引入少量偏差为代价），因为它消除了积分域内的 C0 不连续性。我们还提出了实用的被积函数和样本镜像方法，它们消除了边界不连续性对估计器收敛速度的影响。我们表明平滑光边界（通常在生产中进行以减少方差）可以改善 (M)IS 收敛（以引入少量偏差为代价），因为它消除了积分域内的 C0 不连续性。我们还提出了实用的被积函数和样本镜像方法，它们消除了边界不连续性对估计器收敛速度的影响。我们表明平滑光边界（通常在生产中进行以减少方差）可以改善 (M)IS 收敛（以引入少量偏差为代价），因为它消除了积分域内的 C0 不连续性。我们还提出了实用的被积函数和样本镜像方法，它们消除了边界不连续性对估计器收敛速度的影响。

更新日期：2019-04-05

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