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Holey Sampling: Topological Analysis of Sampling Patterns for Assessing Error in High-dimensional Quadrature

孔采样：用于评估高维正交误差的采样模式的拓扑分析

基本信息

批准号：
EP/R019606/1
负责人：
Kartic Subr
金额：
$ 12.86万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR019606%2F1
关键词：
Holey Sampling Topological Analysis Patterns

项目摘要

Estimating integrals of functions forms the cornerstone of many general classes of problems such as optimisation, sampling and normalisation; these problems, in turn, are central tools for a plethora of applications across various fields such as computer graphics, computer vision and machine learning. The integrand, or function to be integrated, is complicated and rarely available in closed form. Its domain spans spaces of arbitrarily high dimensionality. Exact integration is hopeless and approximation is unavoidable in practice. An estimate of the integral is typically constructed using evaluations of the integrand at a number of sampled locations in the domain. The set of points where the function is sampled is often referred to collectively as a sampling pattern. For computer graphics applications, a modern animation feature film of length 1.5h typically involves the generation of a total of a few hundreds of trillions of high-dimensional samples that are mapped into light paths.Although a number of strategies have been proposed towards generating samples, measuring the quality of high-dimensional sampling patterns is an open problem. Sampling strategies are currently compared on a case-by-case basis by explicitly computing errors in the context of each application independently. The computation associated with measures such as discrepancy and Fourier analysis scale exponentially with dimensionality and are therefore not practicable for samples in high-dimensional domains. The proposed work seeks to quantify equidistribution of high-dimensional point sets using an alternative measure to discrepancy that is tractable. This project will establish mathematical connections between computational topology, stochastic geometry and error analysis for Monte Carlo integration. The goal is to develop a measure for assessing the quality of sampling-based estimators purely based on the samples used. The derived theory will be evaluated and applied on Monte Carlo rendering for Computer Graphics applications.

函数积分估计构成了许多一般问题的基石，如优化、采样和归一化；这些问题反过来又是计算机图形学、计算机视觉和机器学习等不同领域大量应用的中心工具。被积函数或要积分的函数很复杂，很少以封闭的形式提供。它的域跨越任意高维的空间。在实际应用中，精确积分是无望的，近似是不可避免的。通常使用在域中的多个采样位置处的被积函数的求值来构造积分的估计。对函数进行采样的点集通常统称为采样模式。对于计算机图形学应用，长度为1.5h的现代动画故事片通常涉及总共生成几百万亿个高维样本，这些样本被映射到光路中。尽管已经提出了许多生成样本的策略，但测量高维样本模式的质量是一个悬而未决的问题。采样策略目前是在逐个案例的基础上通过独立地显式计算每个应用程序的上下文中的误差来比较的。与诸如差异和傅立叶分析之类的度量相关的计算与维度成指数关系，因此对于高维域中的样本是不可行的。这项拟议的工作试图使用一种容易处理的差异的替代度量来量化高维点集的均匀分布。该项目将建立计算拓扑学、随机几何学和蒙特卡罗积分误差分析之间的数学联系。其目标是制定一种措施，用于纯粹根据所使用的样本来评估基于抽样的估计者的质量。该理论将被评估并应用于计算机图形学中的蒙特卡罗绘制。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Fourier Analysis of Correlated Monte Carlo Importance Sampling

相关蒙特卡罗重要性采样的傅立叶分析

DOI：
10.1111/cgf.13613
发表时间：
2019
期刊：
Computer Graphics Forum
影响因子：
2.5
作者：
Singh, Gurprit;Subr, Kartic;Coeurjolly, David;Ostromoukhov, Victor;Jarosz, Wojciech
通讯作者：
Jarosz, Wojciech

Learning rewards from exploratory demonstrations using probabilistic temporal ranking

DOI：
10.1007/s10514-023-10120-w
发表时间：
2020-02
期刊：
Autonomous Robots
影响因子：
3.5
作者：
Michael Burke;Katie Lu;Daniel Angelov;Artūras Straižys;Craig Innes;Kartic Subr;S. Ramamoorthy
通讯作者：
Michael Burke;Katie Lu;Daniel Angelov;Artūras Straižys;Craig Innes;Kartic Subr;S. Ramamoorthy

Spectral Coarsening with Hodge Laplacians

使用 Hodge Laplacian 进行光谱粗化