Iterative algorithms for compressed sensing with partially known support

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.

Cognitive Radio (CR) is one of the most promising techniques for optimizing the spectrum usage. However, the large amount of data of spectral information that must be processed to identify and assign spectral resources increases the channel assignment times, therefore worsening the quality of service for the devices using the spectrum. Compressive Sensing (CS) is a digital processing technique that allows the reconstruction of sparse or compressible signals using fewer samples than those required traditionally. This paper presents a model that addresses the Spectral Sensing problem in Cognitive Radio using Compressive Sensing as an effective way of decreasing the number of samples required in the sensing process. This model is based on Compressive Spectral Imaging (CSI) architectures where a centralized spectrum manager selects what power data must be delivered by the different wireless devices using binary patterns, and builds a multispectral data cube image with the geographical and spectral data power information. The results show that this multispectral data cube can be built with only a 50% of the samples generated by the devices and, therefore reducing the data traffic dramatically.

Fast acquisition in magnetic resonance imaging (MRI) is considered in this paper. Often, fast acquisition is achieved using parallel imaging (pMRI) techniques. It has been shown recently that the combination of pMRI and compressed sensing (CS), which enables exact reconstruction of sparse or compressible signals from a small number of random measurements, can accelerate the speed of MRI acquisition because the number of measurements are much smaller than that by pMRI per se. Also recently in CS, chaos filters were designed to obtain chaotic measurements. This chaotic CS approach potentially offers simpler hardware implementation. In this paper, we combine chaotic CS and pMRI. However, instead of using chaos filters, the measurements are obtained by chaotically undersampling the k-space. MRI image reconstruction is then performed by using nonlinear conjugate gradient optimization. For pMRI, we use the well-known approach SENSE - sensitivity encoding -, which requires an estimation of the sensitivity maps. The performance of the proposed method is analyzed using the point spread function, the transform point spread function, and the reconstruction error measure.

Noncoherent ultra-wideband (UWB) receivers are suboptimal but have advantage in low-complexity and low-power consumption. The theory of compressed sensing (CS) enables the reconstruction or approximation of sparse or compressible signals from a small set of incoherent projections. This paper presents a noncoherent detection approach based on CS for pulse ultra-wideband systems. The Matching Pursuit (MP) algorithm with implicit denoising operation is used as the signal reconstruction method of CS for pulse UWB. The dictionary of the MP algorithm is weighted with the prior knowledge of the average power delay profile (APDP) of the channel to achieve higher performance. Simulations show that the proposed noncoherent detection approach is resilient to additive noise and outperforms the conventional energy detection method.

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.

In this paper, we study the recovery of a signal from a collection of unlabeled and possibly noisy measurements via a measurement matrix with random i.i.d. Gaussian components. We call the measurements unlabeled since their order is missing, namely, it is not known a priori which elements of the resulting measurements correspond to which row of the measurement matrix. We focus on the special case of ordered measurements, where only a subset of the measurements is kept and the order of the taken measurements is preserved. We identify a duality between this problem and the traditional Compressed Sensing, where we show that the unknown support (location of the nonzero elements) of a sparse signal in Compressed Sensing corresponds in a natural way to the unknown location of the measurements kept in unlabeled sensing. While in Compressed Sensing it is possible to recover a sparse signal from an under-determined set of linear equations (less equations than the dimension of the signal), successful recovery in unlabeled sensing requires taking more samples than the dimension of the signal. We develop a low-complexity alternating minimization algorithm to recover the target signal from the set of its unlabeled samples. We also study the behavior of the proposed algorithm for different signal dimensions and number of measurements empirically via numerical simulations. The results are a reminiscent of the phasetransition similar to that occurring in Compressed Sensing.

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.

The compressive sensing(CS) and sparse representation(SR) technique provide new way to improve the ISAR image resolution. At present, in most traditional CS and SR based ISAR imaging methods, the target is regarded as a set of isolated scattering centers. As a matter of fact, adjacent scattering centers compose a lot of small sets which can reflect the structure of the target. It is no doubt that these small sets can be exploited to improve the ISAR image quality, but the traditional CS and SR ISAR imaging methods ignore this. So in this paper, a block sparse signal recovery algorithm-Block Sparse Bayesian Learning(BSBL) method is introduced to modify the traditional method, and the simulate experiment show that this method can get better ISAR image than the other methods.

Electrocorticography (ECoG) is a type of electrophysiological monitoring useful for recording the activity from the cerebral cortex. It has emerged as a promising recording technique in brain-computer interfaces (BCI). Compression of these signals is essential for saving power and bandwidth in the novel application scenarios of Health-based IoT and Body Area Networks. However, this task is particularly challenging since, ECoG signals are not compressible either in time domain or in frequency domain. To that end, Block Sparse Bayesian Learning (BSBL) techniques were suggested for the reconstruction of compressed EEG and ECG signals, which is however, computationally demanding. Furthermore, given the heterogeneity in modern computing systems, careful design partitioning is required to most effectively evaluate the particular resources available on the deployed architecture. In this paper, we propose to utilise a combination of compressive sensing and neural network for the compression and reconstruction of ECoG signals, respectively. For the choice of the neural network, a multi-layer perceptron regressor with a stochastic gradient descent solver is developed. For a sample system, we show that the network has a compression ratio of 50%, and reconstruction accuracy of 89.85% after training with a practical, medium-sized dataset. In general, the results show that the most efficient system implementation is a heterogeneous architecture combining a CPU and a fieldprogrammable gate array (FPGA).

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.

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.

Making use of the sparsity of targets, three novel direction of arrival (DOA) models based on compressed sensing (CS) theory are proposed. Covariance matrix CS, interpolated array CS and beam space CS carry out compressive sampling on covariance matrix, interpolated array and beam space, respectively. High-resolution DOA estimations are obtained through reconstruction of sparse signal by convex optimization problem resolution. The proposed methods are conceptually different from subspace-based methods and provide high resolution using a uniform linear array without restricting requirements on the spatial and temporal stationary and correlation properties of the sources and the noise. Results of both simulated data and measured data show that these methods are superior to conventional DOA methods in angular estimation 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 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 work, we study the problem of reconstructing a sparse signal from a limited number of its linear projections when the following knowledge is available. (1) We are given partial, and partly erroneous, knowledge of the signal’s support, denoted by T . (2) We are also given an erroneous estimate of the signal values on T . Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant. We presented algorithms by modifying Compressive Sensing (CS) using the partly erroneous support and also the erroneous signal estimate for both noiseless and noisy measurements. The idea of our proposed solution is to solve a convex relaxation of the following problem: find the signal that is sparsest outside the set T , while being “close enough” to signal estimate on T and satisfying the data constraint. We obtain sufficient conditions for exact reconstruction using modified-CS and regularized modified-BP. These are much weaker than those needed for CS when the size of the unknown part of the support is small compared to the support size. We also propose solutions modified-BPDN and regularized modified-BPDN for noisy measurements using the similar idea. We obtain the computable and tighter bounds without any sufficient conditions for the reconstruction error. Simulation comparisons for both sparse and compressible signals are shown. In this work, we also study the application of CS based approaches for blood oxygenation level dependent (BOLD) contrast functional MR imaging (fMRI). In particular, we show, via exhaustive experiments on actual MR scanner data for brain fMRI, that our recently proposed approach for recursive reconstruction of sparse signal sequences, modifiedCS-residual, outperforms other existing CS based approaches.