Mathematical Sciences: Problems in Wavelet and Sampling Theory

Grochenig 9306430 这个项目是关于小波和采样理论中出现的数学问题。将特别强调对多层面问题的分析。 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框架概念，但在信号处理的新发展中没有认识到它的重要性。 工作的一个目标是构造正交和双正交小波基的一般类的伸缩矩阵在更高的维度。 这个建设的一个特别的步骤是所有自相似平铺自然发生在任何试图找到解决方案ofdilation方程的特征。 不可分小波的一般构造将利用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类型的小波基，以在小波的数量、角频率分布和采样几何结构方面获得更多的灵活性。第二个目标是开发从不规则样本重构带限多元函数的算法。 特别强调的是，不规则采样的有限维可实现模型与原始无限维重建问题之间的关系，以及迭代重建的效率和速度的分析。 小波分析的发展已经将谐波分析中相当大的一部分研究转化为寻找有效和快速的程序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。 许多概念已经在数学文献中以各种形式被已知，但是最近的应用和发现已经以计算有效的方式在不同尺度上在时间和频率上对信号进行局部分析的基本目标增加了动力。 ***

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. ***