Sparse Signal Reconstruction Research Articles

Binary Compressive Sensing Via Analog Fountain Coding

In this paper, a compressive sensing (CS) approach is proposed for sparse binary signals' compression and reconstruction based on analog fountain codes (AFCs). In the proposed scheme, referred to as the analog fountain compressive sensing (AFCS), each measurement is generated from a linear combination of L randomly selected binary signal elements with real weight coefficients. The weight coefficients are chosen from a finite weight set and L, called measurement degree, is obtained based on a predefined degree distribution function. We propose a simple verification based reconstruction algorithm for the AFCS in the noiseless case. The proposed verification based decoder is analyzed through SUM-OR tree analytical approach and an optimization problem is formulated to find the optimum measurement degree to minimize the number of measurements required for the reconstruction of binary sparse signals. We show that in the AFCS, the number of required measurements is of O(-n log(1-k/n)), where n is the signal length and k is the signal sparsity level. We then consider the signal reconstruction of AFCS in the presence of additive white Gaussian noise (AWGN) and the standard message passing decoder is then used for the signal recovery. Simulation results show that the AFCS can perfectly recover all non-zero elements of the sparse binary signal with a significantly reduced number of measurements, compared to the conventional binary CS and L1-minimization approaches in a wide range of signal to noise ratios (SNRs). Finally, we show a practical application of the AFCS for the sparse event detection in wireless sensor networks (WSNs), where the sensors' readings can be treated as measurements from the CS point of view.

Simultaneous denoising and compression of power system disturbances using sparse representation on overcomplete hybrid dictionaries

This study introduces a novel unified framework for simultaneous denoising and compression of electric power system disturbance signals using sparse signal decomposition and reconstruction on overcomplete hybrid dictionary (OHD) matrix. In the proposed method, the power quality signal is first decomposed into deterministic sinusoidal components and non-deterministic components using the OHD matrix, including discrete impulse dictionary ( I ), cosine dictionary ( C ), sine dictionary ( S ) and the l1-norm optimisation algorithm. Then, the hard-thresholding, uniform threshold dead-zone quantisation, modified index coding and Huffman coding techniques are used for compression of significant detail signal samples and approximation coefficients. To justify the selection of OHD matrix, four compression methods are implemented using the decomposition techniques based on the dictionaries Ψ = [ I C S ] and Ψ = [ I C ], the wavelet transform (WT) and the discrete cosine transform (DCT). The performance of each method is tested and validated using a wide variety of typical power quality disturbance (PQD) signals taken from the IEEE-1159-PQE and GIM–PQE databases and generated using the Microgrid model. The results show that the method with dictionary Ψ = [ I C S ] is capable of effectively compressing the PQD signals as well as suppressing the noise components in the signals.

Heart Rate monitoring during physical exercise using wrist-type photoplethysmographic (PPG) signals.

Heart rate monitoring using wrist-type using photoplethysmographic (PPG) signals during subjects' intensive exercises is a challenging problem, since signals are strongly affected by motion artifacts caused by unexpected movements. This paper presents a method that uses both time and frequency characteristics of signals; using sparse signal reconstruction for high-resolution spectrum estimation. Experimental results on type data sets recorded from 12 subjects during fast running at peak speed of 15 km/hour. The results have a performance with the average absolute error being 1.80 beat per minute.

DOA, power and polarization angle estimation using sparse signal reconstruction with a COLD array

Existing polarized source localization methods mostly rely on subspace technique. In this paper, a very different framework, namely sparse signal reconstruction, is extended to estimate direction-of-arrival (DOA), power and polarization angle parameters using a cocentered orthogonal loop and dipole (COLD) array. The key points of the proposed algorithm can be summarized as three aspects: (i) this paper first constructs a special second-order statistics vector by using sum-average arithmetic, which is only the function of DOA and power parameters; (ii) then this paper constructs another three special second-order statistics vectors, which are only the function of polarization parameters; (iii) this paper exploits Zhang penalty to enforce sparsity, which lead to almost unbiased index and amplitude estimation. This paper also demonstrates how to distinguish two sources using their polarization characteristics. Simulation results validate the effectiveness and superiority of the proposed algorithm.

Sparse Signal Reconstruction using BasisPursuit Algorithm

Compressive Sensing acquire sparse signal significantly at very lower rate than Nyquist sampling rate. For this, a low complexity compressed sensing operation is defined and it is the combination of sampling and compression. The signals formed from compressed sensing operation are compressible signals and a set of random linear measurements accurately reconstructs compressible signals with the use of nonlinear or convex reconstruction algorithms. Basis Pursuit algorithm is one of the convex optimization algorithms to reconstruct the sparse signal. The l1 minimization theory for linear programming problems is used to formulate the compressive sensing method. Interior point method is used to solve the basis pursuit algorithm for sparse signal reconstruction. In this paper, the methodology of reconstructing sparse signal using basis pursuit algorithm is discussed.

Sparse‐reconstruction‐based direction of arrival, polarisation and power estimation using a cross‐dipole array

This study demonstrates how the multiple parameters can be exactly obtained in sparse signal reconstruction framework using a cross-dipole array. Instead of using subspace-based methods, first direction of arrival (DOA) estimation of all sources is obtained, by solving a weighted ‘group lasso’ problem in second-order statistics domain. Then a truncated l1-function is utilised to approximate l0-norm, and an unbiased estimator is successively proposed to obtain the polarisation and power estimation. A statistical technique is introduced to select the regularisation parameter properly. Compared with the estimation of signal parameters via rotational invariance techniques-based algorithm, the proposed algorithm can provide improved resolution and estimation accuracy. Furthermore, the proposed algorithm can identify two sources with the same DOA successfully, provided that the polarisation parameters are different.

압축 센싱 기반의 신호 검출 및 추정 방법

Compressive sensing is a new data acquisition method enabling the reconstruction of sparse or compressible signals from a smaller number of measurements than Nyquist rate, as long as the signal is sparse and the measurement is incoherent. In this paper, we consider a simple hypothesis testing in target detection and estimation problems using compressive sensing, where the performance depends on the sparsity level of the signals being detected. We provide theoretical analysis results along with some experiment results.

Grid-free compressive beamforming.

The direction-of-arrival (DOA) estimation problem involves the localization of a few sources from a limited number of observations on an array of sensors, thus it can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve high-resolution imaging. On a discrete angular grid, the CS reconstruction degrades due to basis mismatch when the DOAs do not coincide with the angular directions on the grid. To overcome this limitation, a continuous formulation of the DOA problem is employed and an optimization procedure is introduced, which promotes sparsity on a continuous optimization variable. The DOA estimation problem with infinitely many unknowns, i.e., source locations and amplitudes, is solved over a few optimization variables with semidefinite programming. The grid-free CS reconstruction provides high-resolution imaging even with non-uniform arrays, single-snapshot data and under noisy conditions as demonstrated on experimental towed array data.

Extended‐aperture DOA estimation with unknown number of sources

Most of the existing direction-of-arrival (DOA) estimation methods require a priori knowledge of the source number, and can only estimate M − 1 sources with M sensors. A novel technique for a sparse signal reconstruction framework is proposed, which is unbiased and can extend the array aperture effectively. As a result, the proposed technique can not only estimate 2(M − 1) sources with M sensors, but also provide improved resolution and estimation accuracy. It can even obtain DOA estimation without knowing the number of sources.

Reconfigurable FPGA/GPU-Based Architecture of Block Compressive Sampling Matching Pursuit Algorithm

The signals in reality are sparse signal where a few numbers of samples are non-zero. So, a compression technique must be applied to reduce the overhead of processing, storing, and transmission. Blocking compressive sampling matching pursuit (BCoSaMP) algorithm is a recursive algorithm which provides an accurate reconstruction of sparse signal from a small number of noisy samples. It doesn't assume that the noise is Gaussian or bounded but it uses information about the noise magnitude for stopping criterion. However, BCoSaMP is a computationally intensive algorithm. So, BCoSaMP algorithm has been implemented on both field-programmable gate array (FPGA) and graphic processing units (GPU) by exploiting parallel and pipelining approaches. A new software tool called radar signal processing tool (RSPT) is also presented. It allows the designer to auto-generate fully optimized VHDL representation of BCoSaMP by specifying many user input parameters through graphical user interface (GUI). Moreover, it provides the designer a feedback on various performance parameters. This offer the designer the ability to make any adjustments to the BCoSaMP component until gets the desired performance of the overall system-on-chip (SoC). Our simulation results indicate that the achieved speed-up of FPGA and GPU over the sequential one is improved by up to 14 and 10.7, respectively.

Generalized Coprime Array Configurations for Direction-of-Arrival Estimation

A coprime array uses two uniform linear subarrays to construct an effective difference coarray with certain desirable characteristics, such as a high number of degrees-of-freedom for direction-of-arrival (DOA) estimation. In this paper, we generalize the coprime array concept with two operations. The first operation is through the compression of the inter-element spacing of one subarray and the resulting structure treats the existing variations of coprime array configurations as well as the nested array structure as its special cases. The second operation exploits two displaced subarrays, and the resulting coprime array structure allows the minimum inter-element spacing to be much larger than the typical half-wavelength requirement, making them useful in applications where a small interelement spacing is infeasible. The performance of the generalized coarray structures is evaluated using their difference coarray equivalence. In particular, we derive the analytical expressions for the coarray aperture, the achievable number of unique lags, and the maximum number of consecutive lags for quantitative evaluation, comparison, and design of coprime arrays. The usefulness of these results is demonstrated using examples applied for DOA estimations utilizing both subspace-based and sparse signal reconstruction techniques.

Convolutional Sparse Coding for Trajectory Reconstruction.

Trajectory basis Non-Rigid Structure from Motion (NRSfM) refers to the process of reconstructing the 3D trajectory of each point of a non-rigid object from just their 2D projected trajectories. Reconstruction relies on two factors: (i) the condition of the composed camera & trajectory basis matrix, and (ii) whether the trajectory basis has enough degrees of freedom to model the 3D point trajectory. These two factors are inherently conflicting. Employing a trajectory basis with small capacity has the positive characteristic of reducing the likelihood of an ill-conditioned system (when composed with the camera) during reconstruction. However, this has the negative characteristic of increasing the likelihood that the basis will not be able to fully model the object's "true" 3D point trajectories. In this paper we draw upon a well known result centering around the Reduced Isometry Property (RIP) condition for sparse signal reconstruction. RIP allow us to relax the requirement that the full trajectory basis composed with the camera matrix must be well conditioned. Further, we propose a strategy for learning an over-complete basis using convolutional sparse coding from naturally occurring point trajectory corpora to increase the likelihood that the RIP condition holds for a broad class of point trajectories and camera motions. Finally, we propose an l1 inspired objective for trajectory reconstruction that is able to "adaptively" select the smallest sub-matrix from an over-complete trajectory basis that balances (i) and (ii). We present more practical 3D reconstruction results compared to current state of the art in trajectory basis NRSfM.

A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing

SummaryIn this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley & Sons, Ltd.

Decoupled 2D direction-of-arrival estimation based on sparse signal reconstruction

A new two-dimensional direction-of-arrival estimation algorithm called 2D- l 1-singular value decomposition (SVD) and its improved version called enhanced-2D- l 1-SVD are proposed in this paper. They are designed for rectangular arrays and can also be extended to rectangular arrays with faulty or missing elements. The key idea is to represent direction-of-arrival with two decoupled angles and then successively estimate them. Therefore, two-dimensional direction finding can be achieved by applying several times of one-dimensional sparse reconstruction-based direction finding methods instead of directly extending them to two-dimensional situation. Performance analysis and simulation results reveal that the proposed method has a much lower computational complexity and a similar statistical performance compared with the well-known l 1-SVD algorithm, which has several advantages over conventional direction finding techniques due to the application of sparse signal reconstruction. Moreover, 2D- l 1-SVD has better robustness to the assumed number of sources over l 1-SVD.

Robust reconstruction of complex networks from sparse data.

Reconstructing complex networks from measurable data is a fundamental problem for understanding and controlling collective dynamics of complex networked systems. However, a significant challenge arises when we attempt to decode structural information hidden in limited amounts of data accompanied by noise and in the presence of inaccessible nodes. Here, we develop a general framework for robust reconstruction of complex networks from sparse and noisy data. Specifically, we decompose the task of reconstructing the whole network into recovering local structures centered at each node. Thus, the natural sparsity of complex networks ensures a conversion from the local structure reconstruction into a sparse signal reconstruction problem that can be addressed by using the lasso, a convex optimization method. We apply our method to evolutionary games, transportation, and communication processes taking place in a variety of model and real complex networks, finding that universal high reconstruction accuracy can be achieved from sparse data in spite of noise in time series and missing data of partial nodes. Our approach opens new routes to the network reconstruction problem and has potential applications in a wide range of fields.

Time-Frequency Signal Representations Using Interpolations in Joint-Variable Domains

Time-frequency (TF) representations are a powerful tool for analyzing Doppler and micro-Doppler signals. These signals are frequently encountered in various radar applications. Data interpolators play a unique role in TF signal representations under missing samples. When applied in the instantaneous autocorrelation domain over the time variable, the low-pass filter characteristic underlying linear interpolators lends itself to cross-terms reduction in the ambiguity domain. This is in contrast to interpolation performed over the lag variable or a direct interpolation of the raw data. We demonstrate the interpolator performance in both the time domain and the time-lag domain and compare it with sparse signal reconstruction, which exploits the local sparsity property assumed by most Doppler radar signals.

Compressive Sampling for Energy Spectrum Estimation of Turbulent Flows

Recent results from compressive sampling (CS) have demonstrated that accurate reconstruction of sparse signals often requires far fewer samples than suggested by the classical Nyquist--Shannon sampling theorem. Typically, signal reconstruction errors are measured in the $\ell^2$ norm and the signal is assumed to be sparse or compressible. Our spectrum estimation by sparse optimization (SpESO) method uses a priori information about isotropic homogeneous turbulent flows with power law energy spectra and applies the methods of CS to one- and two-dimensional turbulence signals to estimate their energy spectra with small logarithmic errors. SpESO is distinct from existing energy spectrum estimation methods which are based on sparse support of the signal in Fourier space. SpESO approximates energy spectra with an order of magnitude fewer samples than needed with Shannon sampling. Our results demonstrate that SpESO performs much better than lumped orthogonal matching pursuit, and as well as or better than wavelet-based best $M$-term or $M/2$-term methods, even though these methods require complete sampling of the signal before compression.

On the Uniqueness of the Sparse Signals Reconstruction Based on the Missing Samples Variation Analysis

An approach to sparse signals reconstruction considering its missing measurements/samples as variables is recently proposed. Number and positions of missing samples determine the uniqueness of the solution. It has been assumed that analyzed signals are sparse in the discrete Fourier transform (DFT) domain. A theorem for simple uniqueness check is proposed. Two forms of the theorem are presented, for an arbitrary sparse signal and for an already reconstructed signal. The results are demonstrated on illustrative and statistical examples.

基于等角紧框架的稀疏信号重构算法

由于信号稀疏表示的优良特性，已被用于信号处理很多领域，但计算复杂成为其实际应用中一大障碍。框架理论是一个新的研究方向，框架可以更为灵活的表示信号。本文结合稀疏信号和框架的特点，提出一种基于等角紧框架(Equiangular Tight Frames, ETF)的稀疏信号重构算法，并通过相应仿真验证。 As sparse representation of signals has excellent characteristics, it has been applied in several fields of signal processing. However, the computational complexity has become a major obstacle in practical application. Frame theory is a new research direction and can be more flexible repre-sentation signal. In this paper, with the characteristics of sparse signal and frameworks, we propose a sparse signal reconstruction algorithm based on ETF, and then simulate and verify it.

MIMO Radar Imaging Based on Smoothedl0Norm

For radar imaging, a target usually has only a few strong scatterers which are sparsely distributed. In this paper, we propose a compressive sensing MIMO radar imaging algorithm based on smoothedl0norm. An approximate hyperbolic tangent function is proposed as the smoothed function to measure the sparsity. A revised Newton method is used to solve the optimization problem by deriving the new revised Newton directions for the sequence of approximate hyperbolic tangent functions. In order to improve robustness of the imaging algorithm, main value weighted method is proposed. Simulation results show that the proposed algorithm is superior to Orthogonal Matching Pursuit (OMP), smoothedl0method (SL0), and Bayesian method with Laplace prior in performance of sparse signal reconstruction. Two-dimensional image quality of MIMO radar using the new method has great improvement comparing with aforementioned reconstruction algorithm.