Unitary representations of groups and the implications for wavelet analysis.

群的酉表示及其对小波分析的影响。

• 批准号: 3176-2013
• 负责人: Taylor, Keith
• 金额: $ 1.09万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635238
• 关键词: Unitary; representations; groups; implications; wavelet

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).

这一研究的目的是使用来自数学抽象领域（本地紧凑型组的单一表示理论）的强大工具，以发现有效分析，存储和操纵多维信号的新方法。音频记录是一个维度信号的一个很好的例子，而图像（​​例如照片或CT扫描）是二维信号的一个很好的例子。在过去的30年中，通过引入和开发一系列技术，在信号和图像处理方面进行了重大革命，统称为“小波分析”。影响是深远的。从现在以MP3的普遍压缩或类似格式的数字化音乐到功能性MRI（磁共振成像）到在可搜索的数据库中的数字化指纹，我们的所有生活都在信号处理中感到这次革命。执行信号处理的许多算法都是基于单一表示的属性。例如，为了分析图像，一个人使用仿射运动移动一块易于管理的图像（小波）。这些是翻译，旋转，翻转，拉伸和剪切的组合。所有这些动作形成了所谓的仿射组。当这些动作对所有可能的二维信号（图像）的集体作用时，结果称为仿射组的（单一）表示。统一表示的抽象理论指导我们选择特定较小的仿射运动集合以进行有效分析。最近开发的剪切变换可有效检测图像中的曲线，以这种方式出现。我们刚刚成功地将晶体对称组引入了多维信号分析。此外，我们有一种完全新颖的方法，用于四个维（思考3D运动）信号。现在需要完成工作以使用此基础来开发方法来利用多维信号的固有特征，以进行有效的分析和存储。我们的团队将在这一发展中发挥作用，指导蓝图的系统设计（基础理论）。