Collaborative Research: a Focused Research Group on Multiscale Geometric Analysis -- Theory, Tools, and Applications

协作研究：多尺度几何分析的重点研究小组——理论、工具和应用

• 批准号: 0140540
• 负责人: Emmanuel Candes
• 金额: $ 14.35万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-15 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140540&HistoricalAwards=false
• 关键词: Collaborative; Research; Focused; Group; Multiscale

Proposal IDs: DMS-0140698, DMS-0140540, DMS-0140587, DMS-0140623PIs: Donoho, Candes, Huo and JonesTITLE: A Focused Research Group on Multiscale Geometric Analysis--Theory,Tools, ApplicationsAbstractAn interdisciplinary team, bridging Harmonic Analysis, Statistics, Image Analysis and Astronomy, proposes a united effort to exploit and extend recent breakthroughs in computational harmonic analysis. The team will consolidate several scattered advances into a unified body of theory and methods to be called Multiscale Geometric Analysis, which can probe the fine structure of 2-, 3- and higher- dimensional functions and point clouds, able to isolate and manipulate intermediate-dimensional phenomena. Simple examples include edges in 2-d images, filaments and sheets in 3-d point catalogs. Characterizing such intermediate-dimensional phenomena is essential for fundamental progress in a wide range of problem areas (including 2-d and 3-d imaging) where traditional multiscale methods have now run their course. Our united effort has three main outcomes. 1. Theory. Coherent, comprehensive knowledge, showing what can and cannot be accomplished with computational harmonic analysis. 2. Tools. A wide range of practical MGA algorithms and a unified, publicly available software environment - BeamLab - deploying them. 3. Applications. Our main initial focus will be on the analysis of 3-d point catalogs in astronomy, such as those coming on-line soon from the Sloan Digital Sky Survey. We know a priori that the data contain filaments and sheets, embedded in a scattered background, and MGA provides a decisive set of tools to resolve the structural properties of such catalogs, far more sensitive and more comprehensible than any tools currently available for such analysis.Many new kinds of massive databases are being created in our era's revolutionary transition to a data-rich society. Many of these databases contain geometric structures such as surfaces, and filaments, albeit in a sometimes hidden manner. Databases of this sort can include galaxy catalogs in astronomy -- in which the filaments and sheets are predicted by various theories of universe formation -- 3-D scanning of faces and other objects -- in which filaments and surfaces are caused by structures such as skin and hair -- and statistical databases in which curves and surfaces arise from existence of predictable patterns. This Focused Research Group on Multiscale Geometric Analysis is a research effort bringing together mathematicians, statisticians, image analysts and astronomers to create new tools and theories to help extract geometric structure and meaning from such databases. The participants are skilled at multiscale analysis, which will be a central organizational principle. We expect multiscale methods to have an impact here comparable to what wavelets have had over the last twenty years in many other problems in science and technology.

提案 ID：DMS-0140698、DMS-0140540、DMS-0140587、DMS-0140623PI：Donoho、Candes、Huo 和 Jones 标题：多尺度几何分析重点研究小组 - 理论、工具、应用摘要一个跨学科团队，桥接调和分析、统计学、图像分析和天文学， 提议共同努力利用和扩展计算调和分析的最新突破。该团队将把一些分散的进展整合成一个统一的理论和方法，称为多尺度几何分析，它可以探测2维、3维和更高维函数和点云的精细结构，能够分离和操纵中间维现象。简单的例子包括 2D 图像中的边缘、3D 点目录中的细丝和片材。表征这种中间维度现象对于在传统的多尺度方法现已发展的广泛问题领域（包括 2 维和 3 维成像）取得根本性进展至关重要。我们的共同努力取得了三个主要成果。 1.理论。连贯、全面的知识，展示了计算调和分析可以完成什么，不能完成什么。 2. 工具。广泛的实用 MGA 算法和统一的、公开可用的软件环境 - BeamLab - 部署它们。 3. 应用程序。我们最初的主要重点将是分析天文学中的 3D 点目录，例如斯隆数字巡天即将上线的那些。我们先验地知道，数据包含细丝和薄片，嵌入分散的背景中，MGA 提供了一套决定性的工具来解决此类目录的结构属性，比目前可用于此类分析的任何工具都更加敏感和易于理解。在我们这个时代向数据丰富的社会的革命性转变中，正在创建许多新型的大规模数据库。 这些数据库中的许多都包含几何结构，例如表面和细丝，尽管有时是以隐藏的方式。这类数据库可以包括天文学中的星系目录——其中细丝和薄片是通过宇宙形成的各种理论预测的——面部和其他物体的3D扫描——其中细丝和表面是由皮肤和头发等结构引起的——以及统计数据库，其中曲线和表面是由可预测模式的存在产生的。这个多尺度几何分析重点研究小组是一项研究工作，汇集了数学家、统计学家、图像分析师和天文学家，以创建新的工具和理论，以帮助从此类数据库中提取几何结构和含义。 参与者擅长多尺度分析，这将是一个中心组织原则。我们预计多尺度方法在这里所产生的影响可与过去二十年来小波在许多其他科学技术问题上所产生的影响相媲美。