An analog hardware solution for compressive sensing reconstruction using gradient-based method

Compressive sensing is a sub-Nyquist sampling technique for efficient signal acquisition and reconstruction of sparse or compressible signals. In order to account for the sparsity of the underlying signal of interest, it is common to use sparsifying priors such as Bernoulli–Gaussian-inverse Gamma (BGiG) and Gaussian-inverse Gamma (GiG) priors on the components of the signal. With the introduction of variational Bayesian inference, the sparse Bayesian learning (SBL) methods for solving the inverse problem of compressive sensing have received significant interest as the SBL methods become more efficient in terms of execution time. In this paper, we consider the sparse signal recovery problem using compressive sensing and the variational Bayesian (VB) inference framework. More specifically, we consider two widely used Bayesian models of BGiG and GiG for modeling the underlying sparse signal for this problem. Although these two models have been widely used for sparse recovery problems under various signal structures, the question of which model can outperform the other for sparse signal recovery under no specific structure has yet to be fully addressed under the VB inference setting. Here, we study these two models specifically under VB inference in detail, provide some motivating examples regarding the issues in signal reconstruction that may occur under each model, perform comparisons and provide suggestions on how to improve the performance of each model.

Periodic Non-uniform sampling in shift-invariant spaces is an effective method for lowrate sampling of continuous time sparse signals, but, recovery of sampled time sparse signals must be unstable in general. In the paper, the possibility of stable recovery under a combination of sufficient sparsity is proposed. The sampling and reconstruction of sparse signals were transformed into matrix and vector operations by using minimum L1 normal. Necessary condition of reconstruction is analyzed. Finally, we are taking multiband sinusoidal signals as an example to prove that the method can achieve the sampling and reconstruction of sparse signals from recovery successful rate and integer of system.

In this paper, we propose a new test response acquisition technique for high-speed devices-based on dual-frequency incoherent sub-sampling and sparse signal reconstruction. The proposed technique enables reconstruction of spectrally sparse wideband signals such as multi-tone signals and short pseudo-random bit sequences (PRBS) with enhanced time/frequency resolution as opposed to current methods. The sampling hardware utilizes dual analog-to-digital converters (ADCs) and dedicated sampling frequency synthesizers with a common frequency reference. As compared to other compressive sampling architectures [1], the proposed hardware architecture is easy to implement at low cost since it does not require accurate sampling clock phase adjustment or random timing generation. For digital signal reconstruction, the proposed technique requires less number of waveform samples than conventional equivalent-time sampling techniques. In addition, the use of an resolution-enhanced discrete Fourier transform (DFT) frame and basis pursuit algorithms minimizes spectral leakage of incoherently sub-sampled signals. This co-design of sampling hardware and signal reconstruction algorithms enables testing of spectrally sparse wideband signals with enhanced time/frequency resolution.

In this paper, we consider the problem of reconstructing sparse multiband wavelet signals of finite levels from their samples in Fourier domain. We show that those sparse signals are determined by their Fourier measurements on a set of size proportional to their sparsity.

Sparse signal reconstruction methods are used in a wide range of applications such as compressive sensing, denoising, signal separation and general linear inverse problems. The numerical algorithms used for the sparse signal recovery frequently involve finding solutions to the least squares optimization problem with l1-norm regularization. As the l1-norm penalty is not differentiable, so it rules out the possibility of using the efficient optimization techniques that call for the derivative of the objective function. This paper presents a hyperbolic tangent based surrogate function to closely approximate the l1-norm regularization. Simultaneously, we also develop an iterative algorithm for sparse signal reconstruction using the gradient of the proposed smooth function. The algorithm can be used to recover the compressively sampled images from a reduced set of measurements. Various numerical and imaging experiments are used to illustrate the performance of the promising recovery method. It has been shown that the algorithm can be used to reconstruct the compressively sampled images from less number of acquired data, which makes fast imaging possible without compromising high spatial resolutions. The results are validated using phantom as well as original MR and microwave SAR images.

Sampling and reconstruction of sparse signals on circulant graphs – an introduction to graph-FRI

The conditions under which Compressed sensing (CS) succeeds depend on the structure of the measurement matrix. Researches indicated that matrices whose entries are drawn independently from certain probability distributions satisfy the restricted isometry property (RIP) and guarantee exact recovery of a sparse signal with high probability. Motivated by signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. In order to achieve reduction of sampling rate in impulse radio Ultra-Wideband (IR-UWB) communication systems, the output streams of UWB channel were periodically down-sampled. Sub-sampling at the receiver leads to quasi-Toeplitz-structured measurement matrices, whose entries are UWB channel coefficients. In this paper, the feasibility of this kind of quasi-Toeplitz matrices as measurement matrices is discussed. It is shown that the quasi-Toeplitz matrices with entries drawn from logarithm normal distribution satisfy RIP with high probability and also ensures the exact reconstruction of the sparse signals. Simulation results validate their performance.

This paper deals with sparse signal reconstruction by designing a discrete-time projection neural network. Sparse signal reconstruction can be converted into an L1 -minimization problem, which can also be changed into the unconstrained basis pursuit denoising problem. To solve the L1 -minimization problem, an iterative algorithm is proposed based on the discrete-time projection neural network, and the global convergence of the algorithm is analyzed by using Lyapunov method. Experiments on sparse signal reconstruction and several popular face data sets are organized to illustrate the effectiveness and performance of the proposed algorithm. The experimental results show that the proposed algorithm is not only robust to different levels of sparsity and amplitude of signals and the noise pixels but also insensitive to the diverse values of scalar weight. Moreover, the value of the step size of the proposed algorithm is close to 1/2, thus a fast convergence rate is potentially possible. Furthermore, the proposed algorithm achieves better classification performance compared with some other algorithms for face recognition.

In this paper, we present an approach to the reconstruction of signals exhibiting sparsity in a transformation domain, having some heavily disturbed samples. This sparsity-driven signal recovery exploits a carefully suited random sampling consensus (RANSAC) methodology for the selection of a subset of inlier samples. To this aim, two fundamental properties are used: A signal sample represents a linear combination of the sparse coefficients, whereas the disturbance degrades the original signal sparsity. The properly selected samples are further used as measurements in the sparse signal reconstruction, performed using algorithms from the compressive sensing framework. Besides the fact that the disturbance degrades signal sparsity in the transformation domain, no other disturbance-related assumptions are made—there are no special requirements regarding its statistical behavior or the range of its values. As a case study, the discrete Fourier transform is considered as a domain of signal sparsity, owing to its significance in signal processing theory and applications. Numerical results strongly support the presented theory. In addition, the exact relation for the signal-to-noise ratio of the reconstructed signal is also presented. This simple result, which conveniently characterizes the RANSAC-based reconstruction performance, is numerically confirmed by a set of statistical examples.

Robust sensing matrix design for the Orthogonal Matching Pursuit algorithm in compressive sensing

It is significant to solve \(l_1\) minimization problems efficiently and reliably in compressed sensing (CS) since the \(l_1\) minimization is essential for the recovery of sparse signals. In view of this, a neurodynamic optimization approach is proposed for solving the \(l_1\)-minimization problems for reconstruction of sparse signals based on a projection neural network (PNN). The proposed neurodynamic optimization approach differs from most \(l_1\)-solvers in that it operates in continuous time rather than being specified by discrete iterations; i.e., it evolves according to deterministic neurodynamics. The proposed PNN is designed based on subgradient projection methods. The neural network has a simple structure, giving it a potential to be implemented as a large-scale analog circuit. It is proved that under appropriate conditions on the measurement matrix, every neuronal state of the proposed neural network is convergent to the optimal solution of the \(l_1\)-minimization problem under study. Simulation results are provided to substantiate the effectiveness of the proposed approach.

Sparse signals can be reconstructed from a reduced set of signal samples using compressive sensing (CS) methods. The discrete cosine transform (DCT) can provide highly concentrated representations of audio signals. This property implies the DCT as a good sparsity domain for the audio signals. In this paper, the DCT is studied within the context of sparse audio signal processing using the CS theory and methods. The DCT coefficients of a sparse signal, calculated with a reduced set of available samples, can be modeled as random variables. It has been shown that the statistical properties of these variables are closely related to the unique reconstruction conditions. The main result of this paper is in an exact formula for the mean-square reconstruction error in the case of approximately sparse and nonsparse noisy signals reconstructed under the sparsity assumption. Based on the presented analysis, a simple and computationally efficient reconstruction algorithm is proposed. The presented theoretical concepts and the efficiency of the reconstruction algorithm are verified numerically, including examples with synthetic and recorded audio signals with unavailable or corrupted samples. Random disturbances and disturbances simulating clicks or inpainting in audio signals are considered. Statistical verification is done on a dataset with experimental signals. Results are compared with some classical and recent methods used in similar signal and disturbance scenarios.

A fast compressed sensing reconstruction using least squares method with the signal correlation is presented in this paper. It is well known that the complexity of l1-minimisation is very high and is undesirable for many practical applications. The least squares method, on the other hand, has a much lower complexity. However, least squares does not promote the sparsity of signal and therefore cannot provide acceptable reconstructed results. The main contribution of this paper is to show that by exploiting signal correlation, the reconstruction error of least squares is greatly improved. Moreover, the correlated reference used in this method is very flexible, and can contain many kinds of correlation, such as spatial or temporal correlation. Experimental results show that the performance of this method is comparable to the state-of-the-art algorithms, whilst having a much lower complexity. It also shows that this method can be applied to both sparse and redundant signal reconstruction.

Reconstructing signals that have a sparse representation has been studied in the recent years. However, most of the work dealing with this problem requires a low-coherent dictionary matrix. This article presents a novel procedure for sparse signal reconstruction with high coherent dictionary by partitioning the dictionary in the preprocessing step and addressing the reconstruction of hierarchical sparse signals via a new matching pursuit algorithm. We analyse the performance of the proposed algorithm in the noiseless case and show that given the same conditions as required for OMP, it achieves at least the same reconstruction performance as OMP. Numerical simulation and experimental results show that by exploiting the hierarchical sparse structure of the signal, the proposed method outperforms those traditional methods.

Compressive sensing (CS) idea enables the reconstruction of a sparse signal from a small set of measurements. CS approach has applications in many practical areas. One of the areas is radar systems. In this article, the radar ambiguity function is denoised within the CS framework. A new denoising method on the projection onto the epigraph set of the convex function is also developed for this purpose. This approach is compared to the other CS reconstruction algorithms. Experimental results are presented1.