Mathematical Sciences: Wavelets: Theory and Application

数学科学：小波：理论与应用

• 批准号: 9401785
• 负责人: Ingrid Daubechies
• 金额: $ 7.5万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-09-01 至 1997-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401785&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Wavelets; Theory; Application

Daubechies 9401785 Work supported by this award focuses on the mathematical development of wavelet analysis and its applications. This is a branch of harmonic analysis closely allied with applications to many areas of science. The underlying goals of wavelet analysis are (i) to determine functions whose integer translates and dyadic dilates form orthogonal (or biorthogonal) bases for square-integrable functions (ii) to ensure that the basis elements localize both time and frequency and (iii) that they have the requisite smoothness dictated by the problem. Normally, wavelets are also expected to have compact support. In this project five goals are to be addressed. They include analysis of the smoothness of infinitely supported wavelets, the construction of redundant families of analytic wavelets associated with a multiresolution analysis and a study of the consequences of truncating wavelet filter coefficients and finding ways of doing the trunction in a stable manner. Work will also be done in applying wavelets, with new filters, and a new nonlinear squeezing technique, to speech analysis. Finally, effort will be made to construct a special family of multifractal functions for which the so-called thermodynamic formalism can be verified explicitly, and which can be used as a mathematical laboratory. The theory of wavelets, as it has become known, is actually a body of ideas which has developed dramatically over the past decade, following several important discoveries by physicists, computer scientists and engineers concerned with signal processing and data compression. It evolved into a synthesis of many existing techniques into a framework which offers possibilities for improved applications and challenging mathematical ideas which will require years to reach what might be considered a mature stage. Many areas of science are now adapting wavelet constructs to important problems currently under investigation. These included statisticians lo oking for patterns in large data sets, compression of fingerprint data, edge reconstruction of images and the generation of fractal sets with prescribed characteristics.

多贝西9401785 该奖项支持的工作集中在小波分析及其应用的数学发展。 这是谐波分析的一个分支，与许多科学领域的应用密切相关。 小波分析的基本目标是（i）确定其整数平移和二进膨胀形成平方可积函数的正交（或双正交）基的函数（ii）以确保基元素在时间和频率上都本地化，以及（iii）它们具有由问题决定的必要平滑度。 通常，小波也被期望具有紧支撑。 在这个项目中，要实现五个目标。 它们包括分析的光滑性无限支持小波，建设冗余家庭的分析小波与多分辨率分析和研究的后果截断小波滤波器系数和寻找方法做的trunction在一个稳定的方式。 工作也将在应用小波，新的过滤器，和一个新的非线性压缩技术，语音分析。 最后，将努力构建一个特殊的家庭的多重分形功能，所谓的热力学形式主义可以明确地验证，并可用作数学实验室。 小波理论，因为它已经成为众所周知的，实际上是一个机构的想法，已显着发展，在过去十年中，以下几个重要的发现，物理学家，计算机科学家和工程师关心的信号处理和数据压缩。 它演变成一个综合了许多现有的技术到一个框架，提供了可能性，改进的应用程序和挑战数学的想法，这将需要数年时间才能达到什么可能被认为是一个成熟的阶段。 许多科学领域现在正在调整小波结构，以解决目前正在研究的重要问题。 其中包括统计学家在大数据集中寻找模式，指纹数据的压缩，图像的边缘重建和具有规定特征的分形集的生成。