Wavelets: Theory and Applications

小波：理论与应用

• 批准号: 9706753
• 负责人: Ingrid Daubechies
• 金额: $ 12.9万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-09-01 至 2001-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706753&HistoricalAwards=false
• 关键词: Wavelets; Theory; Applications

Daubechies ABSTRACT Daubechies will concentrate on several aspects of wavelet algorithms and their applications: a) Riesz bases of wavelets on the sphere: Recent constructions of wavelet algorithms on the sphere lead to good numerical performance; so far no proof exists that they correspond to Riesz bases for the square integrable functions on the sphere. Traditional methods for planar constructions exploit translational invariance, absent for triangulations on the sphere. b) Irregular subdivision schemes construct curves or surfaces from irregular grids, placing new grid points in positions that are not equally spaced (even locally) as the grids are refined. Similar to the equally spaced case, preservation of polynomial behavior plays an important role in determining the smoothness of the limit curve or surface. Such irregular subdivision schemes are being explored. c) Approaches to decompose wavelet filters into elementary matrices in an efficient manner, and possibly using only integers. d) Frames of wavelets can be used for multiplexing of signals, similar to CDMA, as well as for sending information efficiently over multiple channels. e) The role of smart coding strategies in realizing the full potential of wavelet-based nonlinear approximation theorems. Wavelets are a mathematical technique that allows us to view a complex structure as consisting of several, increasingly detailed layers. This "multi-resolution analysis" is analogous to viewing a painting from a distance, where only large features can be distinguished, and then, as we approach closer, perceiving more detailed, smaller features, until, when we are very close, we can detect even individual brush strokes. As a mathematical tool that has this potential to "fill in" detail at increasingly small scales, and only where needed, wavelets have turned out to be useful in computer aided graphic design, in image and data compression, and in mathematical analysis and numerical computation, in particular for composite materials. Several of the mathematical problems in this proposal are inspired by such applications of wavelets, and their solutions will widen the scope of settings in which we can use multi-resolution analysis.

Daubechies 摘要 Daubechies将集中讨论小波算法的几个方面 及其应用：a）球面上小波的Riesz基： 最近构造的小波算法的球导致 良好的数值性能;到目前为止，还没有证据表明，他们 对应于球面上平方可积函数的Riesz基。传统的平面 结构利用平移不变性，缺席， 球面上的三角形B）不规则细分格式 从不规则网格构建曲线或曲面，放置新网格 位置上的点不是等距的（即使是局部的）， 网格被细化。与等距情形类似，多项式行为的保持在确定极限曲线或曲面的光滑性方面起着重要作用。等 正在研究不规则细分方案。c）方法 以有效的方式将小波滤波器分解成基本矩阵，并且可能仅使用整数。（d）框架 小波可用于信号的多路复用（类似于CDMA），以及用于在多个信道上有效地发送信息。e）智能编码战略在实现 基于小波的非线性逼近定理的充分潜力。 波函数是一种数学技术，它允许我们观察 复杂的结构，包括几个，越来越详细 层次。这种“多分辨率分析”类似于查看 一幅画从远处看，只有大的特点可以 当我们走近时，我们会发现， 详细的，更小的特征，直到，当我们非常接近时，我们可以 甚至可以检测单个画笔笔划。作为一种数学工具，小波具有在越来越小的尺度上“填充”细节的潜力，并且只在需要的地方，小波具有 在计算机辅助图形设计中， 和数据压缩，以及数学分析和数值 计算，特别是用于复合材料。的几 这个建议中的数学问题受到这样的启发， 小波的应用，他们的解决方案将扩大范围 我们可以使用多分辨率分析。