Mathematical Sciences: Problems in Wavelet and Sampling Theory

本课题涉及小波和采样理论中出现的数学问题。将特别强调对多维问题的分析。将这些理论联系在一起的线索来自于20世纪50年代早期发展起来的更广泛的帧概念，但它在信号处理新发展中的重要性并未得到认可。本文的目的之一是构造一类广义高维膨胀矩阵的正交和双正交小波基。这种构造的一个特殊步骤是描述所有自相似的瓦片，这些瓦片在任何寻找膨胀方程解的尝试中都会自然出现。不可分小波的一般构造将利用其与自相似平铺理论的联系。在编码、压缩和快速传输图像的应用中，需要这种小波基在小波数、角频率分布和采样几何上获得更大的灵活性。第二个目标是从不规则样本中重建带限多元函数的算法的发展。特别强调不规则采样的有限维可实现模型与原无限维重构问题之间的关系，以及迭代重构的效率和速度的分析。小波分析的发展已经将谐波分析的相当一部分研究转变为努力寻找有效和快速的方法来分析和传输不同领域的信号，如断层扫描、地震学、数据压缩和图像重建。许多概念在数学文献中以各种形式已知，但最近的应用和发现增加了动力，以计算有效的方式在不同尺度上对时间和频率的信号进行局部分析的基本目标。＊＊＊

Grochenig 9306430 This project is concerned mathematical problems arising in wavelet and sampling theory. Particular emphasis will be placed on the analysis of multidimensional problems. The thread which ties together the theories comes from the broader concept of frames developed in the early 1950's, but not recognized for its importance in new developments in signal processing. One objective of the work is the construction of orthogonal and biorthogonal wavelet bases for a general class of dilation matrices in higher dimensions. A particular step towards this construction is the characterization of all self-similar tilings which occur naturally in any attempt to find solutions ofdilation equations. The general construction of nonseparable wavelets will exploit the links with the theory of self-similar tilings. This type of wavelet basis is needed in applications to coding, compression and fast transmission images to obtain more flexibility with respect to the number of wavelets, the angular frequency distribution and the sampling geometry. The second objective is the development of algorithms for the reconstruction of band-limited multivariate functions from irregular samples. Particular emphasis will be given to the relation between finite-dimensional implementable models of irregular sampling to the original infinite-dimensional reconstruction problem, and th analysis of efficiency and speed of iterative reconstructions. The growth of wavelet analysis has transformed a sizable portion of research in harmonic analysis into efforts to find efficient and fast procedures for analyzing and transmitting signals in diverse areas such as tomography, seismology, data compression and image reconstruction. Many of the concepts have been known in various forms in the mathematical literature, but recent applications and discoveries have added impetus to the basic goal of local analysis of signals in both time and frequency at different scales in a comput ationally efficient manner. ***