压缩感知基本理论:回顾与展望

Advances and perspectives on compressed sensing theory

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。同时,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。如何在保持信号信息的同时尽可能地减少信号的采样数量?Candès在2006年的国际数学家大会上介绍了一种称为压缩感知的新颖信号采样理论,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。围绕压缩感知的稀疏字典设计、测量矩阵设计、重建算法设计这3个核心问题,对其基本理论和主要方法进行了系统阐述,同时指出了压缩感知有待解决的若干理论问题与关键技术。

In the past century, the Shannor sampling theorem has underlain nearly all the modem 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, howew;r, is the large number of data samples particularly in the case of specialpurpose applications. The sampling data have to be compressed for efficient storage, transmission and processing. Recently, Cand^s 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.