%0 Journal Article
%T A Note on Block-Sparse Signal Recovery with Coherent Tight Frames
%A Yao Wang
%A Jianjun Wang
%A Zongben Xu
%J Discrete Dynamics in Nature and Society
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/905027
%X This note discusses the recovery of signals from undersampled data in the situation that such signals are nearly block sparse in terms of an overcomplete and coherent tight frame . By introducing the notion of block -restricted isometry property ( -RIP), we establish several sufficient conditions for the proposed mixed -analysis method to guarantee stable recovery of nearly block-sparse signals in terms of . One of the main results of this note shows that if the measurement matrix satisfies the block -RIP with constants , then the signals which are nearly block -sparse in terms of can be stably recovered via mixed -analysis in the presence of noise. 1. Introduction Compressed sensing (CS) [1, 2] has attracted great interests in a number of fields including information processing, electrical engineering, and statistics. In principle, CS theory states that it is possible to recover an unknown signal from considerably few information if the unknown signal has a sparse or nearly sparse representation in an orthonormal basis. However, a large number of applications in signal and image processing point to problems where signals are not sparse in an orthogonal basis but in an overcomplete and tight frame; see, for example, [3, 4] and the reference therein. Examples include the reflected radar and sonar signals (Gabor frames) and the images with curves (curvelet frames). In such contexts, the signal can be expressed as , where ( ) is a matrix whose columns form a tight frame and is sparse or nearly sparse. Then one acquires via the observed few linear measurements , where is a known matrix ( ), are available measurements, and is a vector of measurements error. There are two common ways to recover based on and . One natural way is first solving an -minimization problem: to find the sparse transform coefficients ; here is a bounded set determined by the noise structure and then reconstructing the signal by a synthesis operation; that is, . This is the so-called -synthesis or synthesis based method [5, 6]. Since the entries of are correlated when is highly coherent, may no longer satisfy the standard restricted isometry property (RIP) and the mutual incoherence property (MIP) which are commonly used in the standard CS framework. Therefore, it is not easy to study the theoretical performance of the -synthesis method. An alternative to -synthesis is the -analysis method, which finds the estimator directly by solving the following -minimization problem: It has been shown that there is a remarkable difference between the two methods despite their apparent similarity
%U http://www.hindawi.com/journals/ddns/2013/905027/