Reconstruction of sparse signals by minimizing a re-weighted approximate ℓ0-norm in the null space of the measurement matrix

2010 ◽  
Author(s):  
Jeevan K. Pant ◽  
Wu-Sheng Lu ◽  
Andreas Antoniou
Keyword(s):  
Null Space ◽  
Sparse Signals ◽  
Measurement Matrix

Compressed Sampling of Spectrally Sparse Signals Using Sparse Circulant Matrices

Frequenz ◽  
2014 ◽  
Vol 68 (11-12) ◽  
Author(s):  
Guangjie Xu ◽  
Huali Wang ◽  
Lei Sun ◽  
Weijun Zeng ◽  
Qingguo Wang
Keyword(s):  
Circulant Matrices ◽  
Noise Robustness ◽  
Sparse Signals ◽  
Measurement Matrix ◽  
Compressed Sampling ◽  
Random Measurement ◽  
Compressive Performance ◽  
Generating Sequence ◽  
Periodic Autocorrelation ◽  
Selection Of

AbstractCirculant measurement matrices constructed by partial cyclically shifts of one generating sequence, are easier to be implemented in hardware than widely used random measurement matrices; however, the diminishment of randomness makes it more sensitive to signal noise. Selecting a deterministic sequence with optimal periodic autocorrelation property (PACP) as generating sequence, would enhance the noise robustness of circulant measurement matrix, but this kind of deterministic circulant matrices only exists in the fixed periodic length. Actually, the selection of generating sequence doesn't affect the compressive performance of circulant measurement matrix but the subspace energy in spectrally sparse signals. Sparse circulant matrices, whose generating sequence is a sparse sequence, could keep the energy balance of subspaces and have similar noise robustness to deterministic circulant matrices. In addition, sparse circulant matrices have no restriction on length and are more suitable for the compressed sampling of spectrally sparse signals at arbitrary dimensionality.

scholarly journals Measurement Matrix Optimization via Mutual Coherence Minimization for Compressively Sensed Signals Reconstruction

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Ziran Wei ◽  
Jianlin Zhang ◽  
Zhiyong Xu ◽  
Yong Liu ◽  
Krzysztof Okarma
Keyword(s):  
Optimization Methods ◽  
Superior Performance ◽  
Gram Matrix ◽  
Sparse Signals ◽  
Two Dimensional ◽  
Measurement Matrix ◽  
Mutual Coherence ◽  
One Dimensional ◽  
Two Dimensional Image ◽  
Matrix Optimization

For signals reconstruction based on compressive sensing, to reconstruct signals of higher accuracy with lower compression rates, it is required that there is a smaller mutual coherence between the measurement matrix and the sparsifying matrix. Mutual coherence between the measurement matrix and sparsifying matrix can be expressed indirectly by the property of the Gram matrix. On the basis of the Gram matrix, a new optimization algorithm of acquiring a measurement matrix has been proposed in this paper. Firstly, a new mathematical model is designed and a new method of initializing measurement matrix is adopted to optimize the measurement matrix. Then, the loss function of the new algorithm model is solved by the gradient projection-based method of Gram matrix approximating an identity matrix. Finally, the optimized measurement matrix is generated by minimizing mutual coherence between measurement matrix and sparsifying matrix. Compared with the conventional measurement matrices and the traditional optimization methods, the proposed new algorithm effectively improves the performance of optimized measurement matrices in reconstructing one-dimensional sparse signals and two-dimensional image signals that are not sparse. The superior performance of the proposed method in this paper has been fully tested and verified by a large number of experiments.

scholarly journals On the Null Space Property oflq-Minimization for0

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Yi Gao ◽  
Jigen Peng ◽  
Shigang Yue ◽  
Yuan Zhao
Keyword(s):  
Compressed Sensing ◽  
Null Space ◽  
Equivalent Form ◽  
Sparse Signals ◽  
Null Space Property ◽  
The Relationship ◽  
Space Property

The paper discusses the relationship between the null space property (NSP) and thelq-minimization in compressed sensing. Several versions of the null space property, that is, thelqstable NSP, thelqrobust NSP, and thelq,probust NSP for0<p≤q<1based on the standardlqNSP, are proposed, and their equivalent forms are derived. Consequently, reconstruction results for thelq-minimization can be derived easily under the NSP condition and its equivalent form. Finally, thelqNSP is extended to thelq-synthesis modeling and the mixedl2/lq-minimization, which deals with the dictionary-based sparse signals and the block sparse signals, respectively.

scholarly journals A Unified Study on L1 over L2 Minimization

2021 ◽  
Author(s):  
Min Tao ◽  
Xiao-Ping Zhang
Keyword(s):  
State Of The Art ◽  
Null Space ◽  
Linear Convergence ◽  
Computational Time ◽  
Measurement Matrix ◽  
Unconstrained Model ◽  
Local Linear Convergence ◽  
Alternating Direction ◽  
Unified Study ◽  
The Solution Existence

<div>In this paper, we carry out a unified study for L_1 over L_2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signal. First, we provide the exact recovery condition on both the constrained and the unconstrained models for a broad set of signals. Next, we prove the solution existence of these L_{1}/L_{2} models under the assumption that the null space of the measurement matrix satisfies the $s$-spherical section property. Then by deriving an analytical solution for the proximal operator of the L_{1} / L_{2} with nonnegative constraint, we develop a new alternating direction method of multipliers based method (ADMM$_p^+$) to solve the unconstrained model. We establish its global convergence to a d-stationary solution (sharpest stationary) and its local linear convergence under certain conditions. Numerical simulations on two specific applications confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.</div>

scholarly journals A Unified Study on L1 over L2 Minimization

2021 ◽  
Author(s):  
Min Tao ◽  
Xiao-Ping Zhang
Keyword(s):  
State Of The Art ◽  
Null Space ◽  
Linear Convergence ◽  
Computational Time ◽  
Measurement Matrix ◽  
Unconstrained Model ◽  
Local Linear Convergence ◽  
Alternating Direction ◽  
Unified Study ◽  
The Solution Existence

<div>In this paper, we carry out a unified study for L_1 over L_2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signal. First, we provide the exact recovery condition on both the constrained and the unconstrained models for a broad set of signals. Next, we prove the solution existence of these L_{1}/L_{2} models under the assumption that the null space of the measurement matrix satisfies the $s$-spherical section property. Then by deriving an analytical solution for the proximal operator of the L_{1} / L_{2} with nonnegative constraint, we develop a new alternating direction method of multipliers based method (ADMM$_p^+$) to solve the unconstrained model. We establish its global convergence to a d-stationary solution (sharpest stationary) and its local linear convergence under certain conditions. Numerical simulations on two specific applications confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.</div>

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.

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 Reconstruction Based on Multiparameter Approximation Function with Smoothedl0Norm

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Xiao-Feng Fang ◽  
Jiang-She Zhang ◽  
Ying-Qi Li
Keyword(s):  
Optimization Problem ◽  
Null Space ◽  
Signal Reconstruction ◽  
Iteration Process ◽  
Reconstruction Algorithm ◽  
Measurement Matrix ◽  
Hyperbolic Tangent ◽  
Reconstruction Quality ◽  
And Performance ◽  
Quasi Newton

The smoothedl0norm algorithm is a reconstruction algorithm in compressive sensing based on approximate smoothedl0norm. It introduces a sequence of smoothed functions to approximate thel0norm and approaches the solution using the specific iteration process with the steepest method. In order to choose an appropriate sequence of smoothed function and solve the optimization problem effectively, we employ approximate hyperbolic tangent multiparameter function as the approximation to the big “steep nature” inl0norm. Simultaneously, we propose an algorithm based on minimizing a reweighted approximatel0norm in the null space of the measurement matrix. The unconstrained optimization involved is performed by using a modified quasi-Newton algorithm. The numerical simulation results show that the proposed algorithms yield improved signal reconstruction quality and performance.

scholarly journals Efficient Tuning-Free l1-Regression of Nonnegative Compressible Signals

2021 ◽  
Vol 7 ◽  
Author(s):  
Hendrik Bernd Petersen ◽  
Bubacarr Bah ◽  
Peter Jung
Keyword(s):  
Null Space ◽  
General Assumption ◽  
Basis Pursuit ◽  
Measurement Matrix ◽  
Linear Measurements ◽  
Absolute Deviation ◽  
Tuning Parameters ◽  
Null Space Property ◽  
Regular Bipartite Graph ◽  
Space Property

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the M+-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

Sign in / Sign up

Export Citation Format

Share Document