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

提案ID：DMS-0140698，DMS-0140540，DMS-0140587，DMS-0140623 PI：Donoho，Candes，Huo和Jones标题：多尺度几何分析的重点研究组-理论，工具，应用摘要一个跨学科的团队，桥接谐波分析，统计学，图像分析和天文学，提出了一个联合的努力，利用和扩展计算谐波分析的最新突破。该团队将把几个分散的进展整合到一个统一的理论和方法中，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，能够隔离和操纵中维现象。简单的例子包括二维图像中的边缘，三维点目录中的细丝和薄片。描述这种中维现象对于在广泛的问题领域（包括2-d和3-d成像）取得根本性进展是必不可少的，传统的多尺度方法现在已经在这些领域发挥作用。我们的联合努力有三个主要成果。1.理论连贯，全面的知识，显示什么可以和不可以完成与计算谐波分析。2.工具.广泛的实用MGA算法和统一的公共软件环境- BeamLab -部署它们。3.应用.我们最初的主要重点将放在天文学中的三维点目录的分析上，例如斯隆数字巡天即将上线的那些。我们先验地知道，数据中包含着细丝和薄片，它们嵌入在分散的背景中，而MGA提供了一套决定性的工具来解析这类星表的结构属性，它比目前可用于此类分析的任何工具都要敏感得多，也更容易理解。在我们这个时代向数据丰富社会的革命性过渡中，许多新型的海量数据库正在创建。 许多这些数据库包含几何结构，如表面和细丝，虽然有时隐藏的方式。这类数据库可以包括天文学中的星系目录-其中细丝和薄片是由各种宇宙形成理论预测的-面部和其他物体的三维扫描-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引起的。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，以帮助从这些数据库中提取几何结构和意义。 参与者擅长多尺度分析，这将是一个核心的组织原则。我们期望多尺度方法在这里产生的影响可以与小波在过去二十年中在科学和技术中的许多其他问题中产生的影响相媲美。

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.