Algebraic Signal Processing Theory: Towards Multiresolution Analysis

信号处理是许多日常技术背后的推动力和推动力，这些技术包括音频/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(金融行业和美国边境安全的生物识别)。作为一门学科，信号处理大量使用深奥的数学，包括复微积分、线性代数、随机学和逼近理论。事实上，上述许多进步都是通过引进和使用来自数学的新技术而成为可能的。这项研究将信号处理与抽象代数(数学的一门主要学科)在基础层面上联系起来。这样一来，一套全新的数学工具就可以用于信号处理。在初步工作中，这些工具已经产生了新颖而高效的加工技术。研究人员将开发更多，旨在解决以前方法无法解决的问题。研究的平台是一种新的信号处理方法，称为代数信号处理理论(ASP)，它将在本工作中得到进一步发展。ASP泛化了标准的线性信号处理，提供了滤波、傅里叶变换等新概念。ASP将许多现有的信号处理方法归结到一个公共框架中，并允许在一维和更高维上推导出新的信号处理方法，包括可分离和不可分离、移位不变和移位可变。这项研究的一个主要目标是扩展ASP，使其包括滤波器组的一般概念和重要应用的多分辨方法。从根本上说，ASP可能会为信号处理提供一种更严格、更公理的方法，从而影响教育。该项目将开发一门基于ASP的第一门课程。

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.