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

Stanford University, Stanford CA

Investigators

Abstract

Proposal IDs: DMS-0140698, DMS-0140540, DMS-0140587, DMS-0140623 PIs: Donoho, Candes, Huo and Jones TITLE: A Focused Research Group on Multiscale Geometric Analysis--Theory, Tools, Applications Abstract An 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.