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

提案编号：dms-0140698，dms-0140540，dms-0140587，dms-0140623pis：Donoho，Candes，Huo和Jones TITLE：一个关于多尺度几何分析的重点研究小组--理论，工具，应用摘要一个跨学科的团队，桥接调和分析，统计学，图像分析和天文学，提出一个联合的努力来开发和扩展计算调和分析的最新突破。该团队将把几个分散的进展整合成一个统一的理论和方法体系，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点目录中的细丝和薄片。表征这种中维现象对于在传统多尺度方法已经走到尽头的广泛问题领域(包括2-D和3-D成像)取得根本性进展至关重要。我们的共同努力有三个主要结果。1.理论。连贯、全面的知识，展示了计算调和分析可以实现什么，不可以实现什么。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.申请。我们最初的主要关注点将是天文学中的3-d点星表的分析，比如那些即将从斯隆数字天空勘测上线的星表。我们先验地知道，数据包含细丝和薄片，嵌入在分散的背景中，MGA提供了一套决定性的工具来解析此类星表的结构属性，比目前可用于此类分析的任何工具都更敏感、更易于理解。在我们这个时代向数据丰富的社会的革命性过渡中，许多新型的海量数据库正在被创建。其中许多数据库包含表面和细丝等几何结构，尽管有时是隐藏的。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造成的--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。多尺度几何分析重点研究小组将数学家、统计学家、图像分析员和天文学家聚集在一起，创造新的工具和理论，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，这将是组织的核心原则。我们预计，多尺度方法在这里的影响可以与小波在过去20年中在许多其他科学和技术问题上的影响相媲美。

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.