Fourier Analysis of Correlated Monte Carlo Importance Sampling
Fourier Analysis of Correlated Monte Carlo Importance Sampling
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相关蒙特卡罗重要性采样的傅立叶分析
DOI：
10.1111/cgf.13613
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发表时间：
2019
影响因子：
2.5
通讯作者：
Jarosz, Wojciech
中科院分区：
文献类型：
--
作者：
Singh, Gurprit;Subr, Kartic;Coeurjolly, David;Ostromoukhov, Victor;Jarosz, Wojciech
傅立叶分析作为蒙特卡罗(MC)积分误差分析的一种工具，在图像合成中越来越受欢迎。然而，现有的工具只能在简化的假设(如随机平移)下分析收敛，而这些假设在渲染过程中并没有在实践中应用。我们重新表述了基于抽样的积分器的偏差和方差的表达式，以统一非均匀样本分布[重要性抽样(IS)]以及样本之间的相关性，同时考虑有限的抽样域。我们的统一公式暗示了基于傅立叶的工具在进行MC积分的方差分析时的基本局限性。同时表明，当与相关采样相结合时，IS可以通过在被积函数中引入或抑制不连续来影响收敛速度。证明了多重重要性抽样的收敛是由收敛最慢的策略决定的，并提出了克服这一局限性的几种简单方法。我们表明，平滑灯光边界(在生产中通常用于减少方差)可以改善(M)收敛(以引入少量偏差为代价)，因为它消除了积分域中的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，消除了边界不连续对估计收敛速度的影响。
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 removesC0discontinuities 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.