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总览 评价 廖安平 ， 田玉帅 ， 杨晓波 （ 湖南大学数学与计量经济学院,长沙 410082 ； ） 摘要： 压缩感知是近年来所研究的一种关于信号传输的新的理论,~信号的稀疏表示、编码测量和重构算法等构成了压缩感知理论的三个主要方面.~信号的稀疏表示为压缩感知
廖安平， 田玉帅， 杨晓波
（
湖南大学数学与计量经济学院,长沙 410082 ； ）
摘要：
压缩感知是近年来所研究的一种关于信号传输的新的理论,~信号的稀疏表示、编码测量和重构算法等构成了压缩感知理论的三个主要方面.~信号的稀疏表示为压缩感知的先决条件.~为了不改变信号的原始结构,~编码测量中测量矩阵应当满足相应的等距约束性条件,~然后根据原始信号与测量矩阵的乘积进而得到原始信号的线性投影测量.~最后,~根据所得到的测量值及测量矩阵通过重构算法重构原始信号.~本文主要建立了稀疏信号恢复等距约束性常数一个新的估计,~对于高维稀疏信号的恢复,~本文主要考虑约束的$ell_1$极小化方法下无噪音的情形.~当测量矩阵$A$满足相应的$RIP$条件时,~证明了所有的$k$稀疏信号$eta$能通过约束的$ell_1$极小化方法基于$y=Aeta$精确恢复.~所得结果改进了T. Cai 和 A. Zhang2014 年所给出的相应结果.~
关键词：
计算数学,压缩感知,稀疏信号恢复,等距约束,$ell_1$极小化
LIAO AnPing， TIAN YuShuai， YANG XiaoBo
（
College of Mathematics and Econometrics, Hunan University, Chang Sha 410082 ； ）
Abstract：
Compressed sensing is put forward in recent years as a new type of signal transmission theory framework.~Compressed sensing theorymainly includes three aspects:~the sparse representation of signal,~encoding measuring and reconstruction algorithm.~Sparse representation of signal is a priori condition of compressed sensing.~In the measurement of coding,~In order to keep the original structure of the signal,~projection matrix must satisfy restricted isometry conditions,~and then obtain linear projection measurement of the original signal through the product of original signal and measure matrix.~Finally,~reconstruct the original signal by the measured value and the projection matrix using the reconstruction algorithm.~In this paper, a new bound on the restricted isometry conditions for sparse signals recovery is established. For the recovery of high-dimensional sparse signals, this paper considers constraint $ell_1$ minimization methods in the noiseless. It is shown that if the sensing matrix $A$ satisfies the corresponding $RIP$ condition, then all $k$-$sparse$ signals $eta$ can be recovered exactly via the constrained $ell_{1}$ minimization based on $y=Aeta$, which has improved the bound that was established by T. Cai and A. Zhang (IEEE Trans. Inf. Theory, 2014).
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