Unitary representations of groups and the implications for wavelet analysis.

这项研究的目的是使用来自抽象数学领域的强大工具(局部紧群的么正表示理论)来发现有效地分析、存储和处理多维信号的新方法。音频记录是一维信号的一个很好的例子，而图像，如照片或CT扫描是二维信号的一个很好的例子。在过去的30年里，随着一系列统称为小波分析的技术的引入和发展，信号和图像处理领域发生了一场重大革命。其影响是深远的。从现在普遍压缩成MP3或类似于功能磁共振成像(MRI)格式的数字化音乐到可搜索数据库中的数字化指纹，我们所有人都感受到了信号处理方面的这场革命。许多执行信号处理的算法都基于么正表示的性质。例如，要分析一幅图像，可以使用仿射运动来移动一小块易于管理的图像(小波)。这些是平移、旋转、翻转、拉伸和剪切的组合。所有这些运动形成了所谓的仿射群。当这些运动作用于所有可能的二维信号(图像)的集合时，结果被称为仿射群的(么正)表示。么正表示的抽象理论指导我们选择特定的更小的仿射运动集合来进行有效的分析。最近发展起来的Shearlet变换就是以这种方式产生的，它可以有效地检测图像中的曲线边缘。我们刚刚成功地将晶体对称群引入到多维信号分析中。此外，我们有一种全新的方法来处理四维(想象3D运动)信号。现在需要做的工作是利用这一基础开发方法，以利用多维信号的固有特征进行有效的分析和存储。我们的团队将在这一开发中发挥作用，通过蓝图(基本理论)的系统设计来指导方向。

The purpose of this line of research is to use powerful tools from an abstract area of mathematics (the theory of unitary representations of locally compact groups) to discover novel methods for efficiently analyzing, storing and manipulating multidimensional signals. An audio recording is a good example of a one dimensional signal while an image, such as a photograph or CT scan is a good example of a two dimensional signal. In the last 30 years, there has been a major revolution in signal and image processing with the introduction and development of a family of techniques collectively called "wavelet analysis". The impact has been profound. From digitized music now universally compressed in MP3 or a similar format to functional MRI (magnetic resonance imaging) to digitized finger prints in a searchable database, all of our lives have felt this revolution in signal processing. Many of the algorithms which execute the signal processing are based on properties of unitary representations. For example, to analyze an image one moves a small, easily managed piece of image (the wavelet) around using affine motions. These are combinations of translations, rotations, flips, stretches and shears. All these motions form what is called the affine group. When these motions act on the collective of all possible 2-dimensional signals (images), the result is called a (unitary) representation of the affine group. The abstract theory of unitary representations guides us to select particular smaller collections of affine motions to use for efficient analysis. The recently developed shearlet transform which is effective in detecting curved edges in images arose in exactly this fashion. We have just succeeded in introducing crystal symmetry groups into multi-dimensional signal analysis. Moreover, we have a completely novel method for four dimensional (think 3D in motion) signals. Work now needs to be done to use this basis to develop methods to take advantage of inherent features of multi-dimensional signals for efficient analysis and storage. Our team will play a part in that development, guiding the way through the systematic design of the blueprints (underlying theory).