Sparse Recovery Methods Research Articles

Highly undersampling dynamic cardiac MRI based on low-rank tensor coding

Dynamic cardiac magnetic resonance imaging (CMRI) is an important tool for the non-invasive assessment of cardiovascular disease. However, dynamic CMRI suffers from long acquisition times due to the need of obtaining images with high temporal and spatial resolution, whole-heart coverage. Conventionally, a multidimensional dataset in dynamic CMRI is treated as a series of two-dimensional matrices, and then various matrix/vector transforms are used to explore the sparsity of MR images. In this paper, we propose a low-rank tensor coding (LRTC) model with tensor sparsity for the application of compressive sensing (CS) in dynamic CMRI. In this framework, each group of 3D similar patches extracted from high-dimensional images is considered to be a low-rank tensor. LRTC can better capture the sparse part of dynamic CMRI and make full use of the redundancy between the feature vectors of adjacent positions. ADMM technique is introduced to tackle the proposed model, where soft threshold operator is used to solving the l1 norm relaxation. Higher-order singular value decomposition (HOSVD) is exploited to process high-dimensional tensors and mine correlations in space-time dimensions. Validations based on cardiac cine and myocardial perfusion datasets indicate that the proposed method achieved comparable reconstruction accuracy with the low-rank matrix recovery methods, and outperformed the conventional sparse recovery methods.

Perturbation analysis of 𝐿1‒2 method for robust sparse recovery

Abstract One of the crucial challenges in applying the theory of compressed sensing in practice is to cope with the perturbation caused by different types of unknown noise, which may arise from the physical implementation or the human mis-modeling. In this paper, we study the robust recovery from a completely perturbed model via a recently popular L 1 - 2 L_{1-2} method. By using the powerful restricted isometry constant of order t ⁢ k tk for t > 1 t>1 , we first obtain a series of perturbation analysis results for this L 1 - 2 L_{1-2} method, which shows that this method is also able to guarantee a robust recovery for any (nearly) sparse signals when both the ideal observations and the exact measurement matrix are perturbed by the unknown noise. Moreover, one of the established recovery conditions under the noise-free settings is also demonstrated to be much better than the state-of-the-art one. Finally, some simulation experiments are further carried out to verify the effectiveness of this L 1 - 2 L_{1-2} method.

A Neighborhood-Based Multiple Orthogonal Least Square Method for Sparse Signal Recovery

A Neighborhood-Based Multiple Orthogonal Least Square Method for Sparse Signal Recovery

Joint Low-Rank and Sparse Tensors Recovery for Video Synthetic Aperture Radar Imaging

Video synthetic aperture radar (SAR) receives more and more attention in recent years because it can provide continuous images of the observed scene. However, the enormous data of video SAR to obtain the multiframe images bring big challenges to its transmission, storage, and processing, especially for small unmanned aerial vehicle (UAV) platform. In this article, we aim at proposing an efficient video formation method for video SAR systems with reduced data. First, the characteristics of video SAR observed scene are analyzed. It is found that the observed scene with multiple frames can be modeled as the sum of a low-rank tensor and a sparse tensor efficiently. After that, the video formation problem for video SAR is modeled as a joint low-rank and sparse tensors recovery problem. Finally, an efficient tensor alternating direction method of multiplier is proposed to obtain the final SAR video. Compared with the traditional frequency- or time-domain imaging methods, the amount of data samples can be greatly reduced. On the other hand, the proposed method outperforms the state-of-the-art SAR imaging methods with reduced samples, including the joint low-rank and sparse matrices recovery method and the low-rank tensor recovery method. Numerical simulations validate the effectiveness of the proposed method.

A Unified Framework for Constructing Nonconvex Regularizations

Over the past decades, many individual nonconvex methods have been proposed to achieve better sparse recovery performance in various scenarios. However, how to construct a valid nonconvex regularization function remains open in practice. In this paper, we fill in this gap by presenting a unified framework for constructing the nonconvex regularization based on the probability density function. Meanwhile, a new nonconvex sparse recovery method constructed via the Weibull distribution is studied.

Sparse Reconstruction for Radar Imaging Based on Quantum Algorithms

The sparse-driven radar imaging can obtain the high-resolution images about target scene with the down-sampled data. However, the huge computational complexity of the classical sparse recovery method for the particular situation seriously affects the practicality of the sparse imaging technology. In this paper, this is the first time the quantum algorithms are applied to the image recovery for the radar sparse imaging. Firstly, the radar sparse imaging problem is analyzed and the calculation problem to be solved by quantum algorithms is determined. Then, the corresponding quantum circuit and its parameters are designed to ensure extremely low computational complexity, and the quantum-enhanced reconstruction algorithm for sparse imaging is proposed. Finally, the computational complexity of the proposed method is analyzed, and the simulation experiments with the raw radar data are illustrated to verify the validity of the proposed method.

Representation recovery via [formula omitted]-norm minimization with corrupted data

This paper studies the recovery probability of a state-of-the-art sparse recovery method, the optimization problem of YALL1, which has been rigorously used in face recognition, dense error correction, anomaly detection, etc. This work generalizes a theoretical work which is based on a special case of the optimization problem of YALL1. Furthermore, the new results cover more practical cases which do not fulfill the bouquet model proposed in the early work. The results also show that not only the special case but also some other cases of the optimization problem of YALL1; which fulfill certain conditions; can also recover any sufficiently sparse coefficient vector x when the fraction of the support of the error e is bounded away from 1 and the support of x is a very small fraction of its dimension m as m becomes large. The trade-off parameter λ in the optimization problem of YALL1 allows the recovery probability to be optimally tuned than the special case. Experimental results also show that the optimization problem of YALL1 (the Eq. (7)) with primal augmented Lagrangian optimization technique outperforms the state-of-the-art sparse recovery methods using their corresponding optimization techniques in term of the speed.

Scattering Model-Based Frequency-Hopping RCS Reconstruction Using SPICE Methods

RCS reconstruction is an important way to reduce the measurement time in anechoic chambers and expand the radar original data, which can solve the problems of data scarcity and a high measurement cost. The greedy pursuit, convex relaxation, and sparse Bayesian learning-based sparse recovery methods can be used for parameter estimation. However, these sparse recovery methods either have problems in solving accuracy or selecting auxiliary parameters, or need to determine the probability distribution of noise in advance. To solve these problems, a non-parametric Sparse Iterative Covariance Estimation (SPICE) algorithm with global convergence property based on the sparse Geometrical Theory of Diffraction (GTD) model (GTD–SPICE) is employed for the first time for RCS reconstruction. Furthermore, an improved coarse-to-fine two-stage SPICE method (DE–GTD–SPICE) based on the Damped Exponential (DE) model and the GTD model (DE–GTD) is proposed to reduce the computational cost. Experimental results show that both the GTD–SPICE method and the DE–GTD–SPICE method are reliable and effective for RCS reconstruction. Specifically, the DE–GTD–SPICE method has a shorter computational time.

Minimization of the [formula omitted]-ratio sparsity with [formula omitted] for signal recovery

In this paper, we propose a general scale invariant approach for sparse signal recovery via the minimization of the q-ratio sparsity sq(z)=(∥z∥1/∥z∥q)qq−1 with q∈[0,∞]. The properties of the q-ratio sparsity measure are studied and illustrated with examples. For the proposed q-ratio sparsity minimization problem with 1<q≤∞, we establish a verifiable exact reconstruction condition and derive its concise error bounds in terms of q-ratio constrained minimal singular values (CMSV). From an algorithmic point of view, we recognize that the proposed problem belongs to the nonlinear fractional programming and investigate two kinds of methods for solving it including the parametric methods and the change of variable method. Numerical experiments are conducted to demonstrate the advantageous performance of the proposed approaches over the state-of-the-art sparse recovery methods.

Spectral sparse recovery form a single RGB image

An efficient procedure for spectral reflectance recovery using a new compressive sensing-based spectral sparse recovery method is proposed by using the optimized sparse basis functions to recover spectral reflectance from a single RGB image. Unlike the conventional spectral recovery methods, the sparse basis functions of the proposed method are an optimal solution for obtaining the information needed to consider the characteristics of the RGB response values. The novelty of the proposed method is to obtain the optimized sparse basis functions considering the influence of the spectral reflectance dataset and the sample’s colorimetric characteristic, which meets the minimum of the spectral recovery error. Spectral and colorimetric accuracy of the proposed method is studied to verify the effectiveness compared to existing methods. As the spectral recovery results show, the proposed method is extremely effective in recovering spectral reflectance.

A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery

In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including ℓ0, bridge, smoothly clipped absolute deviation, capped ℓ1 and minimax concavity penalty. First, we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinatewise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of active set type which at each step involves first updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in terms of computational efficiency and recovery accuracy compared with the existing sparse recovery methods.

A New Upper Bound for Sampling Numbers

We provide a new upper bound for sampling numbers (g_n)_{nin mathbb {N}} associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that gn2≤Clog(n)n∑k≥⌊cn⌋σk2,n≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g^2_n \\le \\frac{C\\log (n)}{n}\\sum \\limits _{k\\ge \\lfloor cn \\rfloor } \\sigma _k^2,\\quad n\\ge 2, \\end{aligned}$$\\end{document}where (sigma _k)_{kin mathbb {N}} is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding mathrm {Id}:H(K) rightarrow L_2(D,varrho _D). The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H^s_{text {mix}}(mathbb {T}^d) in L_2(mathbb {T}^d) with s>1/2. We obtain the asymptotic bound gn≤Cs,dn-slog(n)(d-1)s+1/2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g_n \\le C_{s,d}n^{-s}\\log (n)^{(d-1)s+1/2}, \\end{aligned}$$\\end{document}which improves on very recent results by shortening the gap between upper and lower bound to sqrt{log (n)}. The result implies that for dimensions d>2 any sparse grid sampling recovery method does not perform asymptotically optimal.

MIMO Radar Super-Resolution Imaging Based on Reconstruction of the Measurement Matrix of Compressed Sensing

This letter proposes a novel sparse recovery method of multiple-input and multiple-output (MIMO) radar compressed sensing (CS) imaging algorithms. This method leverages the prior structure of the measurement matrix to judge targets’ locations and to estimate the sparsity level in the grid roughly and finally inhabits the emergence of false targets in the imaging figure. Explicitly, the algorithm we propose is inspired by the orthogonal matching pursuit (OMP) algorithm. First, the measurement matrix can be divided into some submatrices by column. Then, we estimate which submatrices do not contain signal components by the algorithms we propose in this literature to achieve the reconstruction of the measurement matrix. Finally, we use the sparse Bayesian learning algorithm and the sparsity adaptive matching pursuit algorithm to recover the target location and scattering intensity. Experiments validate that the reconstruction error of the algorithm we propose is much lower than other sparse recovery algorithms, and targets in the imaging are more obvious than other algorithms.

High Angular Resolution for 77GHz FMCW Radar via a Sparse Weighted Quadratic Minimization

Combining the covariance matching criterion with sparse representation, much effort was devoted to improve the angular resolution. However, the requirement of hyperparameters or the high sidelobe makes them unsatisfactory in practical radar applications. In this paper, we propose a Sparse Capon-like Weighted Quadratic Estimator (SCWQE) with hyperparameter-free and apply it to a 77GHz Frequency Modulated Continuous Wave (FMCW) radar for the high-resolution Direction Of Arrival (DOA) estimation. Under the assumption of the uncorrelated sources, SCWQE is formulated by converting the covariance matching criterion to a quadratic function with respect to the unknown power spectrum, in which a symmetrical and non-diagonal matrix called as the Capon-like spectrum is embedded into the quadratic form. The consequent multiple weighting operation would further promote the sparsity because multiple and different weighting values are exerted on each element of the spatial power spectrum. This is fundamentally different from the traditional weighted approach that employs the diagonal weighting matrix and assigns a single weighting value to each element. Therefore, SCWQE could further eliminate the spurious peaks and sharpen the desired peaks. Numerical examples including simulations and actual data collected from a 77GHz FMCW radar sensor demonstrate that, the proposed SCWQE algorithm produces better accuracy of DOA estimation and higher angular resolution compared to some typical sparse recovery methods.

A modulus-based iterative method for sparse signal recovery

Solving large-scale l1 minimization problem is very important and has attracted much attentions in recent years. In this paper, we present an efficient method for solving this problem. To this end, we first reformulate this problem as a nonnegative constrained minimization problem. Then we propose a modulus-based iterative method to solve the nonnegative constrained minimization problem. Convergence analysis and the choice of near optimal parameter are presented. Experimental results are given to illustrate feasibility and effectiveness of our proposed method.

Iterative Weighted Group Thresholding Method for Group Sparse Recovery.

This article proposes a novel iterative weighted group thresholding method for group sparse recovery of signals from underdetermined linear systems. Based on an equivalent weighted group minimization problem with lpp -norm ( ), we derive closed-form solutions for a subproblem with respect to some specific values of p when using the proximal gradient method. Then, we design the corresponding algorithmic framework, including stopping criterion and the method of nonmonotone line search, and prove that the solution sequence generated by the proposed algorithm converges under some mild conditions. Moreover, based on the proposed algorithm, we develop a homotopy algorithm with an adaptively updated group threshold. Extensive computational experiments on the simulated and real data show that our approach is competitive with state-of-the-art methods in terms of exact group selection, estimation accuracy, and computation time.

Phase Imbalance Estimation for Azimuth Multi-Channel ScanSAR System

The new concept of combining the azimuth multichannel synthetic aperture radar (SAR) system with the Scan mode was put forward to meet the requirements of future developments in high-resolution and ultrawide-swath SAR design. However, the phase imbalance between channels is inevitable, which will greatly reduce the quality of the image. This article proposed an approach of phase imbalance estimation for azimuth multichannel ScanSAR system. For this, the particularity of the target Doppler spectrum in Scan mode is analyzed firstly. Then, the theoretical derivation reveals the equivalence of scene ambiguity and Doppler spectrum ambiguity. Subsequently, the main idea of the proposed approach is to determine the index of the ambiguous component of the target in the Doppler spectrum by the sparse recovery method. The equivalent multichannel ScanSAR real measured data experiment demonstrates the effectiveness of the proposed approach. Compared with the estimation results of the signal subspace comparison method, the proposed method shows better performance. Finally, the details in the implementation process of the proposed algorithm are described by combining theoretical analysis with simulation experiments.

Robust Sparse Representation in Quaternion Space.

Sparse representation has achieved great success across various fields including signal processing, machine learning and computer vision. However, most existing sparse representation methods are confined to the real valued data. This largely limit their applicability to the quaternion valued data, which has been widely used in numerous applications such as color image processing. Another critical issue is that their performance may be severely hampered due to the data noise or outliers in practice. To tackle the problems above, in this work we propose a robust quaternion valued sparse representation (RQVSR) method in a fully quaternion valued setting. To handle the quaternion noises, we first define a new robust estimator referred as quaternion Welsch estimator to measure the quaternion residual error. Compared to the conventional quaternion mean square error, it can largely suppress the impact of large data corruption and outliers. To implement RQVSR, we have overcome the difficulties raised by the noncommutativity of quaternion multiplication and developed an effective algorithm by leveraging the half-quadratic theory and the alternating direction method of multipliers framework. The experimental results show the effectiveness and robustness of the proposed method for quaternion sparse signal recovery and color image reconstruction.

DOA Refinement Through Complex Parabolic Interpolation of a Sparse Recovered Signal

This letter considers the design of a two-stage direction of arrival (DOA) scheme for radar systems. Precisely, at the first stage a sparse recovery approach is used to obtain both DOA and complex amplitude estimates of the incoming signal. Since the DOA is evaluated on a predefined grid of bins sampling the antenna azimuth mainbeam, at the second stage, a closed-form complex-valued parabolic interpolation is performed to refine it. By doing so, the angle accuracy is improved, but at the same time maintaining fixed the overall computational complexity. Numerical results show the enhancement provided by the proposed procedure to the initial sparse recovery method.

High Dimensional Channel Estimation Using Deep Generative Networks

This paper presents a novel compressed sensing (CS) approach to high dimensional wireless channel estimation by optimizing the input to a deep generative network. Channel estimation using generative networks relies on the assumption that the reconstructed channel lies in the range of a generative model. Channel reconstruction using generative priors outperforms conventional CS techniques and requires fewer pilots. It also eliminates the need of a priori knowledge of the sparsifying basis, instead using the structure captured by the deep generative model as a prior. Using this prior, we also perform channel estimation from one-bit quantized pilot measurements, and propose a novel optimization objective function that attempts to maximize the correlation between the received signal and the generator's channel estimate while minimizing the rank of the channel estimate. Our approach significantly outperforms sparse signal recovery methods such as Orthogonal Matching Pursuit (OMP) and Approximate Message Passing (AMP) algorithms such as EM-GM-AMP for narrowband mmWave channel reconstruction, and its execution time is not noticeably affected by the increase in the number of received pilot symbols.