压缩感知与稀疏信号恢复

项目名称： 压缩感知与稀疏信号恢复

项目编号： No.11471012

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 毕宁

作者单位： 中山大学

项目金额： 65万元

中文摘要： 近年来，压缩感知(Compressed Sensing)成为了信号分析与处理领域最为热门的研究课题之一。由于其理论彻底改变了传统的Nyquist-Shannon 信号采样规则，从而对相关领域的影响和发展产生了新的启示。本项目利用压缩感知思想，对稀疏信号的恢复问题进行深入的研究。主要研究内容是以固定测量矩阵A为前提，鉴于计算测量矩阵A的RIP常数(RIC)是一个NP难问题，所以我们将避免使用RIP，而考虑k稀疏信号x可极小恢复的概率估计，从而达到对任意给定一个测量矩阵A,就能得到稀疏信号可极小恢复概率的分布。最后，对一个有实际意义的测量矩阵A，研究其可极小恢复信号x的支集分布情况。本项目预期研究成果在诸如脑信号(fMRI，EEG，Neural Spike Data等)稀疏表示的分析处理等应用问题中，具有十分重要的意义。

中文关键词： 稀疏表示；逼近论；最优恢复；小波分析；逼近误差

英文摘要： In recent years, one of the hot researching in the field of signal analysis and processing is compressed sensing. According to the theory of compressed sensing, the rule of traditional Nyquist-Shannon sampling theorem is not necessary. Therefore, compressed sensing push forward a series of development with the new idea in many related application fields. In this project, we will focus on sparse signal recovery with an in-depth research for a fixed measurement matrix A. Due to the NP-hardness of computing RIC (Restricted Isometry Constant) for any given measurement matrix A, we will consider the probability of a sparse signal x can be minimization recovery and avoid RIP (Restricted Isometry Property), and achive the aim that there are probability distribution of k sparse signal can be minimization recovery for any fixed measurement matrix A. At last, we will discuss the location of support x，where x can be minimization recovery for a practical measurement matrix A. It is important for brain signal analysis and processing, such as Functional Magnetic Resonance Imaging, Electroencephalogram and Neural Spike Data,etc.

英文关键词： sparse representation;approximation theory;optimal recovery;wavelet analysis;approximate error