Algebraic Signal Processing Theory: Towards Multiresolution Analysis

代数信号处理理论：走向多分辨率分析

• 批准号: 0634967
• 负责人: Markus Pueschel
• 金额: --
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-12-15 至 2010-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0634967&HistoricalAwards=false
• 关键词: Algebraic; Signal; Processing; Theory; Towards

Signal processing is the enabler and driving force ("brain in the box") behind many everyday technologies including audio/image/video processing (MP3, JPEG, MPEG), communications (cell phones), medical and bioimaging (MRI, fMRI, CAT/PET scan, high-throughput drug screening), sensor networks (cars, surveillance), security (biometrics for financial industry and securing US borders). As a discipline, signal processing makes heavy use of deep mathematics including complex calculus, linear algebra, stochastics and approximation theory. In fact, many of the above advances were made possible by importing and using novel techniques from mathematics. This research connects signal processing to abstract algebra (a major discipline in mathematics) at a fundamental level. In doing so, a whole new set of mathematical tools becomes available to signal processing. In preliminary work, these tools have already produced novel andefficient processing techniques. The investigators will develop many more and aim to tackle problems that have eluded solution with previous methods.The platform for the research is a novel approach to signal processing, called algebraic signal processing theory (ASP), which will be further developed in this work. ASP generalizes the standard linear signal processing to provide novel notions of filtering, Fourier transforms, and others. ASP captures many existing signal processing methods into one common framework and enables the derivation of new ones in one and higher dimensions, separable and nonseparable, shift-invariant and shift-variant. A major goal in this research is to expand ASP to include general notions of filter banks and ultiresolution methods for important applications. At a fundamental level, ASP may provide a more rigorous, axiomatic approach to signal processing and thus impact education. A first course based on ASP will be developed in this project.

信号处理是许多日常技术背后的推动者和驱动力（“盒子中的大脑”），包括音频/图像/视频处理（MP3，JPEG，MPEG），通信（手机），医疗和生物成像（MRI，fMRI，CAT/PET扫描，高通量药物筛选），传感器网络（汽车，监控），安全（金融业的生物识别技术和保护美国边境）。作为一门学科，信号处理大量使用深层数学，包括复微积分，线性代数，随机和近似理论。事实上，上述许多进步都是通过引进和使用数学中的新技术而实现的。这项研究将信号处理与抽象代数（数学中的一门主要学科）在基础层面联系起来。在这样做的过程中，一套全新的数学工具可用于信号处理。在初步工作中，这些工具已经产生了新颖而有效的加工技术。研究人员将开发更多，旨在解决以前的方法无法解决的问题。研究平台是一种新的信号处理方法，称为代数信号处理理论（ASP），将在这项工作中进一步发展。ASP概括了标准的线性信号处理，提供了滤波、傅立叶变换等新概念。ASP将许多现有的信号处理方法捕获到一个通用框架中，并能够在一维和更高维、可分离和不可分离、平移不变和平移变中导出新的信号处理方法。在这项研究中的一个主要目标是扩展ASP，包括一般概念的过滤器银行和多分辨率方法的重要应用。在基本层面上，ASP可以提供更严格的，公理化的信号处理方法，从而影响教育。第一个基于ASP的课程将在这个项目中开发。