Stable recovery of sparse signals with coherent tight frames via lp-analysis approach

2016 ◽  
Vol 15 (04) ◽  
pp. 505-520
Author(s):  
Dekai Liu ◽  
Song Li
Keyword(s):  
Compressed Sensing ◽  
Minimization Problem ◽  
Tight Frame ◽  
Complex Case ◽  
Closed Form Expression ◽  
Sparse Signals ◽  
Tight Frames ◽  
Orthonormal Matrix ◽  
Redundant Dictionaries ◽  
True Signal

In this paper, we consider to recover a signal which is sparse in terms of a tight frame from undersampled measurements via [Formula: see text]-minimization problem for [Formula: see text]. In [Compressed sensing with coherent tight frames via [Formula: see text]-minimization for [Formula: see text], Inverse Probl. Imaging 8 (2014) 761–777], Li and Lin proved that when [Formula: see text] there exists a [Formula: see text], depending on [Formula: see text] such that for any [Formula: see text], each solution of the [Formula: see text]-minimization problem can approximate the true signal well. The constant [Formula: see text] is referred to as the [Formula: see text]-RIP constant of order [Formula: see text] which was first introduced by Candès et al. in [Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal. 31 (2011) 59–73]. The main aim of this paper is to give the closed-form expression of [Formula: see text]. We show that for every [Formula: see text]-RIP constant [Formula: see text], if [Formula: see text] where [Formula: see text] then the [Formula: see text]-minimization problem can reconstruct the true signal approximately well. Our main results also hold for the complex case, i.e. the measurement matrix, the tight frame and the signal are all in the complex domain. It should be noted that the[Formula: see text]-RIP condition is independent of the coherence of the tight frame (see [Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal. 31 (2011) 59–73]). In particular, when the tight frame reduces to an identity matrix or an orthonormal matrix, the conclusions in our paper coincide with the results appeared in [Stable recovery of sparse signals via [Formula: see text]-minimization, Appl. Comput. Harmon. Anal. 38 (2015) 161–176].

scholarly journals Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yao Wang ◽  
Jianjun Wang
Keyword(s):  
Compressed Sensing ◽  
Sufficient Conditions ◽  
Tight Frame ◽  
Restricted Isometry Property ◽  
Tight Frames ◽  
Analysis Method

This paper establishes new sufficient conditions on the restricted isometry property (RIP) for compressed sensing with coherent tight frames. One of our main results shows that the RIP (adapted to D) condition δk+θk,k<1 guarantees the stable recovery of all signals that are nearly k-sparse in terms of a coherent tight frame D via the l1-analysis method, which improves the existing ones in the literature.

THE RESTRICTED ISOMETRY PROPERTY FOR SIGNAL RECOVERY WITH COHERENT TIGHT FRAMES

2015 ◽  
Vol 92 (3) ◽  
pp. 496-507 ◽  
Author(s):  
FEN-GONG WU ◽  
DONG-HUI LI
Keyword(s):  
Compressed Sensing ◽  
Orthonormal Basis ◽  
Natural Extension ◽  
Tight Frame ◽  
Restricted Isometry Property ◽  
Tight Frames ◽  
Signal Recovery ◽  
Measurement Matrix

In this paper, we consider signal recovery via $l_{1}$-analysis optimisation. The signals we consider are not sparse in an orthonormal basis or incoherent dictionary, but sparse or nearly sparse in terms of some tight frame $D$. The analysis in this paper is based on the restricted isometry property adapted to a tight frame $D$ (abbreviated as $D$-RIP), which is a natural extension of the standard restricted isometry property. Assuming that the measurement matrix $A\in \mathbb{R}^{m\times n}$ satisfies $D$-RIP with constant ${\it\delta}_{tk}$ for integer $k$ and $t>1$, we show that the condition ${\it\delta}_{tk}<\sqrt{(t-1)/t}$ guarantees stable recovery of signals through $l_{1}$-analysis. This condition is sharp in the sense explained in the paper. The results improve those of Li and Lin [‘Compressed sensing with coherent tight frames via $l_{q}$-minimization for $0<q\leq 1$’, Preprint, 2011, arXiv:1105.3299] and Baker [‘A note on sparsification by frames’, Preprint, 2013, arXiv:1308.5249].

scholarly journals A Note on Block-Sparse Signal Recovery with Coherent Tight Frames

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yao Wang ◽  
Jianjun Wang ◽  
Zongben Xu
Keyword(s):  
Sufficient Conditions ◽  
Tight Frame ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Sparse Signals ◽  
Tight Frames ◽  
Signal Recovery ◽  
Analysis Method ◽  
Measurement Matrix ◽  
Undersampled Data

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 frameD. By introducing the notion of blockD-restricted isometry property (D-RIP), we establish several sufficient conditions for the proposed mixedl2/l1-analysis method to guarantee stable recovery of nearly block-sparse signals in terms ofD. One of the main results of this note shows that if the measurement matrix satisfies the blockD-RIP with constantsδk<0.307, then the signals which are nearly blockk-sparse in terms ofDcan be stably recovered via mixedl2/l1-analysis in the presence of noise.

Compressed-Sensing MRI Based on Adaptive Tight Frame in Gradient Domain

2018 ◽  
Vol 49 (5) ◽  
pp. 465-477 ◽  
Author(s):  
Xiaoyu Fan ◽  
Qiusheng Lian ◽  
Baoshun Shi
Keyword(s):  
Compressed Sensing ◽  
Tight Frame ◽  
Gradient Domain

Nonuniform recovery of fusion frame structured sparse signals

2017 ◽  
Vol 15 (03) ◽  
pp. 333-352
Author(s):  
Yu Xia ◽  
Song Li
Keyword(s):  
Probability Distribution ◽  
Compressed Sensing ◽  
Fourier Coefficients ◽  
A Priori ◽  
Sparse Recovery ◽  
A Priori Knowledge ◽  
Sparse Signals ◽  
Fusion Frame ◽  
Redundant Representation ◽  
Vector Valued

This paper considers the nonuniform sparse recovery of block signals in a fusion frame, which is a collection of subspaces that provides redundant representation of signal spaces. Combined with specific fusion frame, the sensing mechanism selects block-vector-valued measurements independently at random from a probability distribution [Formula: see text]. If the probability distribution [Formula: see text] obeys a simple incoherence property and an isotropy property, we can faithfully recover approximately block sparse signals via mixed [Formula: see text]-minimization in ways similar to Compressed Sensing. The number of measurements is significantly reduced by a priori knowledge of a certain incoherence parameter [Formula: see text] associated with the angles between the fusion frame subspaces. As an example, the paper shows that an [Formula: see text]-sparse block signal can be exactly recovered from about [Formula: see text] Fourier coefficients combined with fusion frame [Formula: see text], where [Formula: see text].

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