Sparse Recovery Methods Research Articles

A Noise Reduced Hard Thresholding method for Noisy Sparse Reconstruction

The problem of noisy sparse signal recovery has been studied in this paper. The generalized expectation maximization (GEM) algorithm provides solutions over state of the art convex relaxation methods for sparse signal recovery by considering the noisy measurements as missing data. However, in the case of images, artifacts in the reconstructed signal renders it to have low quality, especially based on qualitative performance metrics and so are unsuitable for further application specific processing. The problem may be reduced significantly by minimizing the total variation in the signal considering the constraints created by the statistics of noise in the reconstructed image. A modified iterative GEM for low noise reconstruction is presented and analyzed. Even though time consuming, simulations and numerical analysis show significantly higher performance metrics, especially PSNR and Structural Similarity, which imply reconstructed images may be suitable for further processing.

Three-Dimensional Imaging of Terahertz Circular SAR with Sparse Linear Array.

Due to the non-contact detection ability of radar and the harmlessness of terahertz waves to the human body, three-dimensional (3D) imaging using terahertz synthetic aperture radar (SAR) is an efficient method of security detection in public areas. To achieve high-resolution and all aspect imaging, circular trajectory movement of radar and linear sensor array along the height direction were used in this study. However, the short wavelength of terahertz waves makes it practically impossible for the hardware to satisfy the half-wavelength spacing condition to avoid grating lobes. To solve this problem, a sparse linear array model based on the equivalent phase center principle was established. With the designed imaging geometry and corresponding echo signal model, a 3D imaging algorithm was derived. Firstly, the phase-preserving algorithm was adopted to obtain the 2D image of the ground plane for each sensor. Secondly, the sparse recovery method was applied to accomplish the scattering coefficient reconstruction along the height direction. After reconstruction of all the range-azimuth cells was accomplished, the final 3D image was obtained. Numerical simulations and experiments using terahertz radar were performed. The imaging results verify the effectiveness of the 3D imaging algorithm for the proposed model and validate the feasibility of terahertz radar applied in security detection.

Improved sparse signal recovery method based on matching pursuit revised MAP estimation

The problem of recovering a sparse signal buried in noise from a linear observing system has been studied, via utilising statistic prior information of noise and signal, a fast matching pursuit revised maximum a posterior (MP-RMAP) estimation method is proposed. The proposed method overcomes the none-robust estimation problem that fast Bayesian matching pursuit has in high signal-to-noise ratio cases. Compared with other methods, the MP-RMAP method is able to improve the estimation accuracy with low complexity. Experimental results verify the validity of the proposed algorithm.

Sequential adaptive elastic net approach for single-snapshot source localization

This paper proposes efficient algorithms for accurate recovery of direction-of-arrivals (DoAs) of sources from single-snapshot measurements using compressed beamforming (CBF). In CBF, the conventional sensor array signal model is cast as an underdetermined complex-valued linear regression model and sparse signal recovery methods are used for solving the DoA finding problem. A complex-valued pathwise weighted elastic net (c-PW-WEN) algorithm is developed that finds solutions at the knots of penalty parameter values over a path (or grid) of elastic net (EN) tuning parameter values. c-PW-WEN also computes least absolute shrinkage and selection operator (LASSO) or weighted LASSO in its path. A sequential adaptive EN (SAEN) method is then proposed that is based on c-PW-WEN algorithm with adaptive weights that depend on previous solution. Extensive simulation studies illustrate that SAEN improves the probability of exact recovery of true support compared to conventional sparse signal recovery approaches such as LASSO, EN, or orthogonal matching pursuit in several challenging multiple target scenarios. The effectiveness of SAEN is more pronounced in the presence of high mutual coherence.

Sparse recovery of missing image samples using a convex similarity index

Abstract This paper investigates the problem of recovering missing samples using methods based on sparse representation adapted for visually enhanced quality of reconstruction of image signals. Although, the popular Mean Square Error (MSE) criterion is convex and simple, it may not be entirely consistent with Human Visual System (HVS). Thus, instead of l2-norm or MSE, a new perceptual quality measure is used as the similarity criterion between the original and the reconstructed images. The proposed criterion called Convex SIMilarity (CSIM) index is a modified version of the Structural SIMilarity (SSIM) index, which despite its predecessor, is convex and uni-modal. We derive mathematical properties for the proposed index and show how to optimally choose the parameters of the proposed criterion, investigating the Restricted Isometry (RIP) and error-sensitivity properties. We also propose an iterative sparse recovery method based on a constrained l1-norm minimization problem, incorporating CSIM as the fidelity criterion. The resulting convex optimization problem is solved via an algorithm based on Alternating Direction Method of Multipliers (ADMM). Taking advantage of the convexity of the CSIM index, we also prove the convergence of the algorithm to the globally optimal solution of the proposed optimization problem, starting from any arbitrary point. Simulation results confirm the performance of the new similarity index as well as the proposed algorithm for missing sample recovery of image patch signals.

Hardware Implementation of Iterative Method With Adaptive Thresholding for Random Sampling Recovery of Sparse Signals

Sparse recovery methods find extensive applications in various fields such as image and audio processing, wireless communication, and spectral estimation. In this paper, a hardware architecture of iterative method with adaptive thresholding (IMAT) is presented to recover a sparse signal from its random samples. To demonstrate the effectiveness of IMAT, a comparison is performed between the IMAT algorithm and the compressive sensing recovery method, orthogonal matching pursuit, in terms of complexity and reconstruction quality. In the context of image reconstruction, as a practical case, simulation results show the advantages of IMAT in improving the performance of the signal reconstruction via peak signal-to-noise ratio and structural similarity index metrics. Since IMAT employs discrete transform as a principal operation in each iteration, using fast or fast approximate algorithms allows more efficient implementation. In this paper, two multiplication-free transforms, Walsh–Hadamard transform (WHT) and approximate discrete cosine transform (ADCT), are used to reduce computational complexity of IMAT implementation. The IMAT recovery algorithm is implemented on Virtex6 FPGA using two above transforms, and the results are compared with respect to the hardware resource utilization, power consumption, and recovery time. The results demonstrate that using WHT for IMAT implementation is more efficient than using ADCT in terms of hardware resources. Also, the comparison implies that the performance of the hardware implementation is very close to its floating point simulated counterpart. For a block size of $32 \times 32$ , the IMAT implementation using WHT provides the reconstruction time of $185~\mu \text{s}$ and the dynamic power of 123 mW.

Weighted LASSO for Sparse Recovery With Statistical Prior Support Information

Compressive sensing is used to recover a sparse signal from linear measurements. Without any prior support information (PSI), least absolute shrinkage and selection operator (LASSO) is a useful method for sparse recovery. In some settings, a statistical prior about the support of the sparse signal may be provided. It is critical to optimally incorporate such statistical PSI to enhance the recovery performance. We propose a weighted LASSO algorithm to fully exploit the statistical PSI and optimize the recovery performance. In the proposed algorithm, we exploit the most general statistical PSI model, a multilevel PSI, and incorporate it into the LASSO using a weighted $l_{1}$ norm penalty function. An optimal weight policy is proposed to minimize the asymptotic normalized squared error (aNSE). We also derive the closed-form accurate expression for the minimum aNSE and characterize the minimum number of measurements required to achieve stable recovery. Based on this, we discuss the impact of PSI quality on the aNSE performance of the proposed algorithm. Both theoretical analysis and simulations show the performance advantages of our proposed solution over various baselines.

Superresolution radar imaging based on fast inverse-free sparse Bayesian learning for multiple measurement vectors

Compressive sensing has been successfully applied to inverse synthetic aperture radar (ISAR) imaging of moving targets. By exploiting the block sparse structure of the target image, sparse solution for multiple measurement vectors (MMV) can be applied in ISAR imaging and a substantial performance improvement can be achieved. As an effective sparse recovery method, sparse Bayesian learning (SBL) for MMV involves a matrix inverse at each iteration. Its associated computational complexity grows significantly with the problem size. To address this problem, we develop a fast inverse-free (IF) SBL method for MMV. A relaxed evidence lower bound (ELBO), which is computationally more amiable than the traditional ELBO used by SBL, is obtained by invoking fundamental property for smooth functions. A variational expectation–maximization scheme is then employed to maximize the relaxed ELBO, and a computationally efficient IF-MSBL algorithm is proposed. Numerical results based on simulated and real data show that the proposed method can reconstruct row sparse signal accurately and obtain clear superresolution ISAR images. Moreover, the running time and computational complexity are reduced to a great extent compared with traditional SBL methods.

Off-Grid Localization in MIMO Radars Using Sparsity

In this letter, we propose a new accurate approach for target localization in multiple-input multiple-output (MIMO) radars, which exploits the sparse spatial distribution of targets to reduce the sampling rate. We express the received signal of a MIMO radar in terms of the deviations of target parameters from the grid points in the form of a block sparse signal using the expansion around all the neighbor points. Applying a block sparse recovery method, we can estimate both the grid-point locations of targets and these deviations. The proposed approach can yield more accurate localization with higher detection probability compared with its counterparts. Moreover, the proposed approach can reduce the computational complexity.

MIMO Radar Imaging With Nonorthogonal Waveforms Based on Joint-Block Sparse Recovery

Multiple-input multiple-output (MIMO) radar imaging is a new technique which may solve the motion compensation problem in inverse synthetic aperture radar (ISAR). However, the imaging result in MIMO radar using matched filtering is usually poor, since the waveforms with the same frequency cannot be fully orthogonal. Sparse signal recovery has the potential to restrain the mutual interference of nonorthogonal waveforms by exploiting the sparsity of targets. However, because the range profile is not as sparse as the 2-D or 3-D image, the sparse recovery result of target range profiles is usually unsatisfactory. In this paper, a joint-block sparsity of range profiles is explored and exploited to improve the range profile quality. And then, the 2-D target image is recovered from the refined range profiles. Furthermore, a robust joint-block sparse recovery algorithm is proposed. The ascent searching direction, the parameter selection method, and the computational complexity of the proposed algorithm are also discussed. Simulation results show that the proposed algorithm is superior to algorithms which just consider sparsity, block sparsity, or joint sparsity. And the quality of the simulated MIMO radar images and real data ISAR images obtained using the new imaging method is better than that of the conventional correlation method and sparse signal recovery method.

Fast Frequency Estimation With Prior Information

A fast gridless method for sparse frequency estimation in the presence of prior information is presented. It allows arbitrarily sampled data and has a significantly smaller complexity than the existing gridless sparse recovery methods. The proposed method achieves good frequency estimation accuracy and resolution performance in numerical simulations. It is also useful where no prior knowledge is available.

A new accelerated alternating minimization method for analysis sparse recovery

This paper proposes a new method based on accelerated alternating minimization (AAM) for analysis sparse recovery. This method is extremely attractive as (1) it is very simple and computationally efficient, (2) it exhibits a fast convergence rate, (3) it is flexible and amenable to many kinds of reconstruction problems. We establish the connection between the classical alternating minimization (AM) method and the well-known proximal gradient (PG) method. Thus combining the accelerated proximal gradient (APG) method with the Moreau proximal smoothing technique, a new smoothing-based AAM (SAAM) method, which can obtain an ϵ-optimal solution within O(1/ϵ) iterations, is designed. Numerical experiments on randomly generated data and real image reconstruction show that this method compares favorably with several state-of-the-art methods.

Jittered chirp sequence waveform in combination with CS‐based unambiguous Doppler processing for automotive frequency‐modulated continuous wave radar

The chirp sequence waveform is one of the most commonly used waveforms for automotive frequency-modulated continuous wave radars. When a chirp sequence waveform is applied, the range-Doppler processing method based on two-dimensional fast Fourier transform can be used to extract the range and velocity parameters of multiple targets from the beat signals. However, conventional chirp sequence waveforms suffer from the Doppler ambiguity problem, which results in ambiguities in velocity measurement. In this study, the authors propose a novel jittered chirp sequence waveform. In the proposed waveform, the initial times of the chirps are randomly jittered. According to the compressed sensing theory, they establish the parameter design principle for the new waveform, and apply sparse recovery method to achieve unambiguous Doppler processing. Subspace pursuit algorithm is recommended as the suitable sparse recovery method for Doppler processing due to its low complexity and stable recovery performance. The effectiveness of the proposed waveform and its corresponding signal processing method are verified by simulation and experimental results.

Multi-echo reconstruction from partial K-space scans via adaptively learnt basis

In multi-echo imaging, multiple T1/T2 weighted images of the same cross section is acquired. Acquiring multiple scans is time consuming. In order to accelerate, compressed sensing based techniques have been proposed. In recent times, it has been observed in several areas of traditional compressed sensing, that instead of using fixed basis (wavelet, DCT etc.), considerably better results can be achieved by learning the basis adaptively from the data. Motivated by these studies, we propose to employ such adaptive learning techniques to improve reconstruction of multi-echo scans. This work will be based on two basis learning models – synthesis (better known as dictionary learning) and analysis (known as transform learning). We modify these basic methods by incorporating structure of the multi-echo scans. Our work shows that we can indeed significantly improve multi-echo imaging over compressed sensing based techniques and other unstructured adaptive sparse recovery methods.

Applications of sparse recovery and dictionary learning to enhance analysis of ambulatory electrodermal activity data

Electrodermal Activity (EDA) − an index of sympathetic nervous system arousal − is one of the primary methods used in psychophysiology to assess the autonomic nervous system [1]. While many studies collect EDA data in short, laboratory-based experiments, recent developments in wireless biosensing have enabled longer, ‘out-of-lab’ ambulatory studies to become more common [2]. Such ambulatory methods are beneficial in that they facilitate more longitudinal and environmentally diverse EDA data collection. However, they also introduce challenges for efficiently and accurately identifying discrete skin conductance responses (SCRs) and measurement artifacts, which complicate analyses of ambulatory EDA data. Therefore, interest in developing automated systems that facilitate analysis of EDA signals has increased in recent years. Ledalab is one such system that automatically identifies SCRs and is currently considered a gold standard in the field of ambulatory EDA recording. However, Ledalab, like other current systems, cannot distinguish between SCRs and artifacts. The present manuscript describes a novel technique to accurately and efficiently identify SCRs and artifacts using curve fitting and sparse recovery methods We show that our novel approach, when applied to expertly labeled EDA data, detected 69% of the total labeled SCRs in an EDA signal compared to 45% detection ability of Ledalab. Additionally, we demonstrate that our system can distinguish between artifact and SCR shapes with an accuracy of 74%. This work, along with our previous work [3], suggests that matching pursuit is a viable methodology to quickly and accurately identify SCRs in ambulatory collected EDA data, and that artifact shapes can be separated from SCR shapes.

A Joint Indoor WLAN Localization and Outlier Detection Scheme Using LASSO and Elastic-Net Optimization Techniques

In this paper, we introduce two indoor Wireless Local Area Network (WLAN) positioning methods using augmented sparse recovery algorithms. These schemes render a sparse user's position vector, and in parallel, minimize the distance between the online measurement and radio map. The overall localization scheme for both methods consists of three steps: 1) coarse localization, obtained from comparing the online measurements with clustered radio map. A novel graph-based method is proposed to cluster the offline fingerprints. In the online phase, a Region Of Interest (ROI) is selected within which we search for the user's location; 2) Access Point (AP) selection; and 3) fine localization through the novel sparse recovery algorithms. Since the online measurements are subject to inordinate measurement readings, called outliers, the sparse recovery methods are modified in order to jointly estimate the outliers and user's position vector. The outlier detection procedure identifies the APs whose readings are either not available or erroneous. The proposed localization methods have been tested with Received Signal Strength (RSS) measurements in a typical office environment and the results show that they can localize the user with significantly high accuracy and resolution which is superior to the results from competing WLAN fingerprinting localization methods.

Inverse synthetic aperture radar phase adjustment and cross-range scaling based on sparsity

Due to inherent sparsity of ISAR images, compressive sensing theory has been used to obtain a high resolution image. However, before applying sparse recovery methods, the phase error due to the translational motion of target is compensated by autofocusing algorithms and the target rotation rate is estimated by cross-range scaling methods. In this paper, a comprehensive matrix model for a uniformly rotating target that includes the phase error and chirp-rate of the target is derived. Then by using sparsity and minimum entropy criterion, the estimation of residual phase error and the rotation rate is refined. In order to reduce the computational load, we simplify the model and by an iterative method based on adaptive dictionary, the phase error and chirp-rate are estimated separately. Actually, by exploiting a two-dimensional (2D) optimization method and the Nelder–Mead algorithm the phase adjustment is performed and the chirp-rate is estimated by solving a 1D optimization method for dominant range cells of the target. Finally, both simulation and practical data have been used to verify the validity of the proposed approach.

Fast Inverse-Free Sparse Bayesian Learning via Relaxed Evidence Lower Bound Maximization

Sparse Beyesian learning is a popular approach for sparse signal recovery, and has demonstrated superior performance in a series of experiments. Nevertheless, the sparse Bayesian learning algorithm involves a matrix inverse at each iteration. Its associated computational complexity grows significantly with the problem size, which hinders its application to many practical problems even with moderately large datasets. To address this issue, in this letter, we develop a fast inverse-free sparse Bayesian learning method. Specifically, by invoking a fundamental property for smooth functions, we obtain a relaxed evidence lower bound (relaxed-ELBO) that is computationally more amiable than the conventional ELBO used by sparse Bayesian learning. A variational expectation-maximization (EM) scheme is then employed to maximize the relaxed-ELBO, which leads to a computationally efficient inverse-free sparse Bayesian learning algorithm. Simulation results show that the proposed algorithm has a fast convergence rate and achieves lower reconstruction errors than other state-of-the-art fast sparse recovery methods in the presence of noise.

Sparse Bayesian learning using correlated hyperparameters for recovery of block sparse signals

Traditional sparse Bayesian learning applies independent hyperparameters to control the sparsity of each coefficient in the sparse signal. For recovery of block-sparse signals with unknown block partitioning structure, pattern coupled sparse Bayesian learning (PCSBL) method considers the sparsity patterns of the neighboring coefficients to be related to each other and uses a predefined parameter to control the relevance between the hyperparameters. However, for most real signals with block sparsity, the degree of relevance between neighboring coefficients is inhomogeneous, and should be data-dependent. This motivates us to modify PCSBL by replacing the single predefined parameter with a set of data-dependent coupling parameters (DDCP) to capture the relevance. In this paper, by performing a linear transformation to the independent hyperparameters, a set of correlated hyperparameters is generated to build the hierarchical Gaussian prior model for recovery of block sparse signals. The coupling parameters which compose the transformation matrix are estimated by the expectation maximization (EM) principle. The relevance between the hyperparameters is data dependent so the sparsity patterns of neighboring coefficients are related in an adaptive way. Compared with existing block sparse recovery methods, our approach is able to encourage the block sparse structure in a more flexible and more reliable way.

Joint sparsity‐driven three‐dimensional imaging method for multiple‐input multiple‐output radar with sparse antenna array

Three-dimensional (3D) imaging of a moving target can be achieved via a wideband multiple-input multiple-output (MIMO) radar with planar array in a single snapshot. The sparse recovery (SR) method has been introduced to deal with the MIMO imaging problem with sparse array recently. For sparse linear array, SR method can successfully achieve the high cross-range resolution. For sparse planar array, however, it is not effective to use a SR algorithm directly in 2D cross-range direction, because the final 2D imagery will be blurred by the SR-induced range cell migration (RCM) in the former cross-range direction. To remove the RCM, a joint sparsity-driven method is proposed in this study by exploiting the joint sparsity of echo signal. An improved imagery can be obtained with fewer antennas and higher calculation efficiency. Comparative experiments demonstrate the superiority of the proposed method.