Multivariate Wavelet Analysis: Constructions and Specific Applications

本课题主要研究小波的构造和应用。我们对一般尺度矩阵的不可分离多元小波特别感兴趣。使用一般缩放提供了一些主要的优点。这个项目的目的之一是调查这些优势在实际应用中是否真的有用，或者它们是否被技术困难的开销所浪费。因此，我们希望以经济的方式发展产生合适的小波/多小波基的构造原理。我们将关注具有一些附加特征的小波，例如某些插值特性。此外，我们打算研究一般缩放矩阵是否可以在信号/图像分析的某些应用中有所帮助，特别是关于一般类型噪声的去噪算法。分析将伴随着有效的数据结构的发展，以适应小波算法的具体要求。

This project is concerned with the construction and the applications of wavelets. We are especially interested in nonseparable multivariate wavelets with respect to general scaling matrices. The use of general scalings provides some principal advantages. One aim of this project is to investigate if these superiorities really count when it comes to practical applications, or if they are wasted by an overhead of technical difficulties. We therefore want to develop construction principles which yield suitable wavelet/multiwavelet bases in an economical way. We shall focus on wavelets with some additional features such as certain interpolation properties. Furthermore, we intend to investigate if general scaling matrices can be helpful in certain applications from signal/image analysis, especially concerning denoising algorithms for general kinds of noise. The analysis will be accompanied by the development of efficient data structures which are adapted to the specific requirements of wavelet algorithms.