Wavelets: Theory and Applications

本文将集中讨论小波算法及其应用的几个方面：a)小波在球体上的Riesz基：最近在球体上构建的小波算法具有良好的数值性能；到目前为止还没有证据证明它们对应于球面上平方可积函数的Riesz基。平面构造的传统方法利用平移不变性，而球面上的三角剖分则不存在平移不变性。b)不规则细分方案从不规则网格构建曲线或曲面，在网格细化时将新的网格点放置在间距不等的位置（甚至局部位置）。与等间距情况类似，多项式行为的保留在确定极限曲线或曲面的平滑度方面起着重要作用。目前正在探讨这种不规则的细分方案。c)有效地将小波滤波器分解为初等矩阵的方法，并且可能只使用整数。d)小波帧可以用于信号的多路复用，类似于CDMA，以及在多个通道上有效地发送信息。e)智能编码策略在实现基于小波的非线性逼近定理的全部潜力中的作用。小波是一种数学技术，它使我们能够把一个复杂的结构看作是由几个越来越详细的层组成的。这种“多分辨率分析”类似于从远处观看一幅画，只有大的特征可以区分，然后，当我们走近时，感知更多细节，更小的特征，直到，当我们非常接近时，我们甚至可以检测到单个的笔触。作为一种数学工具，它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，小波已经被证明在计算机辅助图形设计，图像和数据压缩，数学分析和数值计算，特别是复合材料方面非常有用。本提案中的几个数学问题受到小波应用的启发，它们的解决方案将扩大我们可以使用多分辨率分析的设置范围。

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.