压缩感知基本理论：回顾与展望-中国图象图形学报.DOC

压缩感知基本理论：回顾与展望 邵文泽韦志辉(南京理工大学计算机科学与技术学院 南京 210094)摘要 随着信息社会的迅速发展，人们对数字信息的需求越来越大。，对采样速率、传输速度和存储空间的要求越来越高。保持信号信息尽可能地信号采样数Candès在2006年国际数学家大会上了一种压缩感知新颖信号采样理论指出只要远少于传统Nyquist采样定理所要求的采样数即可精确或高概率精确重建原始信号。本文主要围绕压缩感知稀疏字典设计、测量矩阵设计、重建算法设计三个核心问题对基本理论主要法进行了，指出压缩感知有待解决的若干。中图分类号：T：：Compressed Sensing Theory SHAO Wen-Ze WEI Zhi-Hui (School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094) Abstract In the past century, the Shannon sampling theorem has underlain nearly all the modern signal acquisition techniques. It claims that the sampling rate must be at least twice the maximum frequency present in the signal. One inherent disadvantage of the theorem, however, is the large number of data samples particularly in the case of special-purpose applications. The sampling data have to be compressed for efficient storage, transmission and processing. Recently, Candès reported a novel sampling theory called compressed sensing, also known as compressive sampling (CS). The theory asserts that one can recover signals and images from far fewer samples or measurements, not strictly speaking, as long as one adheres to two basic principles: sparsity and incoherence, or sparsity and restricted isometry property. The aim of this article is to survey the advances and perspectives of the CS theory, including the design of sparse dictionaries, the design of measurement matrices, the design of sparse reconstruction algorithms, and our proposal of several important problems to be studied. Keywords Compressed sensing, sparse approximation, incoherence, measurement matrix, sparse optimization 0. 引言 信号采样是模拟的物理世界通向数字的信息世界之必备手段。多年来，指导信号采样的理论基础一直是著名的Nyquist采样定理。定理指出，只有当采样速率达到信号带宽的两倍以上时，才能由采样信号精确重建原始信号。可见，带宽是Nyquist采样定理对采样的本质要求。但是，信号带宽越来越大，人们对信号的采样速率、传输速度和存储空间的要求也变得越来越高。为了缓解对信号传输速度和存储空间的压力，当前常见的解决方案是信号压缩，基于小波变换的JPEG2000标准。代表某种可压缩信号，代表的解压缩信号。但是，信号压缩实际上是一种严重的资源浪费，因为大量采样数据在压缩过程中被丢弃了)，它们对于信号来说是不重要的或者只是冗余信息。从这个意义而言，我们得到以下结论：带宽不能本质地表达信号的信息，基于信号带宽的Nyquist采样机制是冗余的