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.

scholarly journals Sparse Signal Recovery via ECME Thresholding Pursuits

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Heping Song ◽  
Guoli Wang
Keyword(s):  
Optimization Problems ◽  
Greedy Algorithms ◽  
Experimental Studies ◽  
Superior Performance ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Sparse Signals ◽  
Signal Recovery ◽  
Support Set ◽  
Reconstructed Signal

The emerging theory of compressive sensing (CS) provides a new sparse signal processing paradigm for reconstructing sparse signals from the undersampled linear measurements. Recently, numerous algorithms have been developed to solve convex optimization problems for CS sparse signal recovery. However, in some certain circumstances, greedy algorithms exhibit superior performance than convex methods. This paper is a followup to the recent paper of Wang and Yin (2010), who refine BP reconstructions via iterative support detection (ISD). The heuristic idea of ISD was applied to greedy algorithms. We developed two approaches for accelerating the ECME iteration. The described algorithms, named ECME thresholding pursuits (EMTP), introduced two greedy strategies that each iteration detects a support setIby thresholding the result of the ECME iteration and estimates the reconstructed signal by solving a truncated least-squares problem on the support setI. Two effective support detection strategies are devised for the sparse signals with components having a fast decaying distribution of nonzero components. The experimental studies are presented to demonstrate that EMTP offers an appealing alternative to state-of-the-art algorithms for sparse signal recovery.

Sparse signal recovery with OMP algorithm using sensing measurement matrix

2011 ◽  
Vol 8 (5) ◽  
pp. 285-290 ◽  
Author(s):  
Guan Gui ◽  
Abolfazl Mehbodniya ◽  
Qun Wan ◽  
Fumiyuki Adachi
Keyword(s):  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Signal Recovery ◽  
Measurement Matrix

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.

scholarly journals A Simple Gaussian Measurement Bound for Exact Recovery of Block-Sparse Signals

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhi Han ◽  
Jianjun Wang ◽  
Jia Jing ◽  
Hai Zhang
Keyword(s):  
Lower Bound ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Sparse Signals ◽  
Signal Recovery ◽  
Minimization Method ◽  
Numerical Examples ◽  
Norm Minimization ◽  
Minimization Methods ◽  
Theoretical Results

We present a probabilistic analysis on conditions of the exact recovery of block-sparse signals whose nonzero elements appear in fixed blocks. We mainly derive a simple lower bound on the necessary number of Gaussian measurements for exact recovery of such block-sparse signals via the mixedl2/lq  (0<q≤1)norm minimization method. In addition, we present numerical examples to partially support the correctness of the theoretical results. The obtained results extend those known for the standardlqminimization and the mixedl2/l1minimization methods to the mixedl2/lq  (0<q≤1)minimization method in the context of block-sparse signal recovery.

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 New Smoothed L0 Regularization Approach for Sparse Signal Recovery

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Jianhong Xiang ◽  
Huihui Yue ◽  
Xiangjun Yin ◽  
Linyu Wang
Keyword(s):  
Signal Reconstruction ◽  
Real Data ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Sparse Signals ◽  
Signal Recovery ◽  
Sparse Signal Reconstruction ◽  
System Equation ◽  
Norm Minimization ◽  
Better Than

Sparse signal reconstruction, as the main link of compressive sensing (CS) theory, has attracted extensive attention in recent years. The essence of sparse signal reconstruction is how to recover the original signal accurately and effectively from an underdetermined linear system equation (ULSE). For this problem, we propose a new algorithm called regularization reweighted smoothed L0 norm minimization algorithm, which is simply called RRSL0 algorithm. Three innovations are made under the framework of this method: (1) a new smoothed function called compound inverse proportional function (CIPF) is proposed; (2) a new reweighted function is proposed; and (3) a mixed conjugate gradient (MCG) method is proposed. In this algorithm, the reweighted function and the new smoothed function are combined as the sparsity promoting objective, and the constraint condition y-Φx22 is taken as a deviation term. Both of them constitute an unconstrained optimization problem under the Tikhonov regularization criterion and the MCG method constructed is used to optimize the problem and realize high-precision reconstruction of sparse signals under noise conditions. Sparse signal recovery experiments on both the simulated and real data show the proposed RRSL0 algorithm performs better than other popular approaches and achieves state-of-the-art performances in signal and image processing.

Fast Thresholding Algorithms with Feedbacks and Partially Known Support for Compressed Sensing

2020 ◽  
Vol 37 (03) ◽  
pp. 2050013
Author(s):  
Kaiyan Cui ◽  
Zhanjie Song ◽  
Ningning Han
Keyword(s):  
Numerical Experiments ◽  
Prior Information ◽  
Null Space ◽  
Sufficient Conditions ◽  
Sparse Signal Recovery ◽  
Sparse Signals ◽  
Signal Recovery ◽  
Hard Thresholding ◽  
Property Condition ◽  
Approximate Reconstruction

Some works in modified compressive sensing (CS) show that reconstruction of sparse signals can obtain better results than traditional CS using the partially known support. In this paper, we extend the idea of these works to the null space tuning algorithm with hard thresholding, feedbacks ([Formula: see text]) and derive sufficient conditions for robust sparse signal recovery. The theoretical analysis shows that including prior information of partially known support relaxes the preconditioned restricted isometry property condition comparing with the [Formula: see text]. Numerical experiments demonstrate that the modification improves the performance of the NST+HT+FB, thereby requiring fewer samples to obtain an approximate reconstruction. Meanwhile, a systemic comparison with different methods based on partially known support is shown.

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