A hybrid hard thresholding algorithm for compressed sensing

Commonly employed reconstruction algorithms in compressed sensing (CS) use the $L_2$ norm as the metric for the residual error. However, it is well-known that least squares (LS) based estimators are highly sensitive to outliers present in the measurement vector leading to a poor performance when the noise no longer follows the Gaussian assumption but, instead, is better characterized by heavier-than-Gaussian tailed distributions. In this paper, we propose a robust iterative hard Thresholding (IHT) algorithm for reconstructing sparse signals in the presence of impulsive noise. To address this problem, we use a Lorentzian cost function instead of the $L_2$ cost function employed by the traditional IHT algorithm. We also modify the algorithm to incorporate prior signal information in the recovery process. Specifically, we study the case of CS with partially known support. The proposed algorithm is a fast method with computational load comparable to the LS based IHT, whilst having the advantage of robustness against heavy-tailed impulsive noise. Sufficient conditions for stability are studied and a reconstruction error bound is derived. We also derive sufficient conditions for stable sparse signal recovery with partially known support. Theoretical analysis shows that including prior support information relaxes the conditions for successful reconstruction. Simulation results demonstrate that the Lorentzian-based IHT algorithm significantly outperform commonly employed sparse reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments. Numerical results also demonstrate that the partially known support inclusion improves the performance of the proposed algorithm, thereby requiring fewer samples to yield an approximate reconstruction.

Compressive radar imaging has attracted considerable attention because it substantially reduces imaging time through directly compressive sampling. However, a problem that must be addressed for compressive radar imaging systems is the high computational complexity of reconstruction of sparse signals. In this paper, a novel algorithm, called two-dimensional (2D) normalized iterative hard thresholding (NIHT) or 2D-NIHT algorithm, is proposed to directly reconstruct radar images in the matrix domain. The reconstruction performance of 2D-NIHT algorithm was validated by an experiment on recovering a synthetic 2D sparse signal, and the superiority of the 2D-NIHT algorithm to the NIHT algorithm was demonstrated by a comprehensive comparison of its reconstruction performance. Moreover, to be used in compressive radar imaging systems, a 2D sampling model was also proposed to compress the range and azimuth data simultaneously. The practical application of the 2D-NIHT algorithm in radar systems was validated by recovering two radar scenes with noise at different signal-to-noise ratios, and the results showed that the 2D-NIHT algorithm could reconstruct radar scenes with a high probability of exact recovery in the matrix domain. In addition, the reconstruction performance of the 2D-NIHT algorithm was compared with four existing efficient reconstruction algorithms using the two radar scenes, and the results illustrated that, compared to the other algorithms, the 2D-NIHT algorithm could dramatically reduce the computational complexity in signal reconstruction and successfully reconstruct 2D sparse images with a high probability of exact recovery.

Compressed sensing (CS) is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper, we apply the iterative hard thresholding (IHT) algorithm for compressed sensing on the speech signal. The interested speech signal is transformed to the frequency domain using Discrete Fourier Transform (DCT) and then compressed sensing is applied to that signal. The compressed signal can be reconstructed using the recently introduced Iterative Hard Thresholding (IHT) algorithm and also by the tradditional \( \ell_{1} \) minimization (basic pursuit) for comparison. It is shown that the compressed sensing can provide better root mean square error (RMSE) than the tradition DCT compression method, given the same compression ratio.

In this letter, the authors propose block normalised iterative hard thresholding (BNIHT) algorithm for the recovery of block sparse signal, in which the non-zero elements are presented in clusters. Based on block restricted isometry property, the sufficient conditions to guarantee the convergence of BNIHT are derived. In addition, the number of required iterations is obtained. The simulation experiment shows that BNIHT algorithm is superior to the block IHT (BIHT) algorithm when the step size satisfies μ < 1 .

The recently proposed compressed sensing theory equips us with methods to recover exactly or approximately, high resolution images from very few encoded measurements of the scene. The traditional ill-posed problem of MRI image recovery from heavily under-sampled &kappa;-space data can be thus solved using CS theory. Differing from the soft thresholding methods that have been used earlier in the case of CS MRI, we suggest a simple iterative hard thresholding algorithm which efficiently recovers diagnostic quality MRI images from highly incomplete &kappa;-space measurements. The new multi-scale redundant systems, curvelets and contourlets having high directionality and anisotropy, and thus best suited for curved-edge representation are used in this iterative hard thresholding framework for CS MRI reconstruction and their performance is compared. The &kappa;-space under-sampling schemes such as the variable density sampling and the more conventional radial sampling are experimented at the same sampling rate and the effect of encoding scheme on iterative hard thresholding compressed sensing reconstruction is studied.

This paper develops a low-complexity channel estimation (CE) scheme based on compressive sensing (CS) for time-domain synchronous (TDS) orthogonal frequency-division multiplexing (OFDM) to overcome the performance loss under doubly selective fading channels. Specifically, an overlap-add method of the time-domain training sequence is first proposed to obtain the coarse estimates of the channel length, path delays and path gains of the wireless channel, by exploiting the channel's temporal correlation to improve the robustness of the coarse CE under the severe fading channel with long delay spread. We then propose the priori-information aided (PA) iterative hard threshold (IHT) algorithm, which utilizes the priori information of the acquired coarse estimate for the wireless channel and therefore is capable of obtaining an accurate channel estimate of the doubly selective fading channel. Compared with the classical IHT algorithm whose convergence requires the $l_2$ norm of the measurement matrix being less than 1, the proposed PA-IHT algorithm exploits the priori information acquired to remove such a limitation as well as to reduce the number of required iterations. Compared with the existing CS based CE method for TDS-OFDM, the proposed PA-IHT algorithm significantly reduces the computational complexity of CE as well as enhances the CE accuracy. Simulation results demonstrate that, without sacrificing spectral efficiency and changing the current TDS-OFDM signal structure, the proposed scheme performs better than the existing CE schemes for TDS-OFDM in various scenarios, especially under severely doubly selective fading channels.

Wearable ECG monitoring devices, often limited by power constraints, require efficient methods to reduce energy consumption and extend battery life for continuous health tracking. This work proposes a digital compressed sensing approach for ECG (DCS-ECG), featuring a suitable sensing matrix, effective sparse signal recovery algorithms, and an optimized number of bits for quantization (BM). The framework is designed to achieve high compression ratios (CR) with minimal signal distortion and parameter extraction without the reconstruction of the original signal. The following five sensing matrices are examined: deterministic block binary diagonal (DBBD) matrix, random binary (RBin) matrix, random Bernoulli (RBen) matrix, random Gaussian (RG) matrix, and random sparse fixed binary (RSB) matrix. These are paired with a sparsifying matrix (ψ), a hybrid of identity matrix (I), discrete sine transform matrix (S), and discrete cosine transform matrix (C) (i.e. ψ = [I S C]). The following five sparse reconstruction algorithms are evaluated: iterative hard thresholding (IHT) algorithm, orthogonal matching pursuit (OMP) algorithm, smoothed ℓ0 (SL0) algorithm, ℓ1 least square (L1LS) algorithm, and subspace matching pursuit (SMP) algorithm. The proposed DCS-ECG system was tested on ECG signals from the MIT-BIH arrhythmia database. The configuration using the DBBD sensing matrix and OMP recovery algorithm with the sparsifying matrix achieved a low average percentage of root mean square difference (PRD) of 3.75 ± 1.43% at CR = 12.5%, which also offers fast reconstruction. For CR = 12.5% and BM = 8, the CS measurement generation on the Arduino Due consumes 2.02 mJ, while reconstruction on the Raspberry Pi requires 1.13 J. The wireless transmission energy is 1.2973 J. Overall, the proposed CS-based system achieves an 86.24% total energy saving compared to the without CS approach, demonstrating its suitability for low-power IoT-enabled wearable edge ECG health monitoring applications.

As an essential task for enabling dynamic spectrum sharing, spectrum sensing in wideband cognitive radio (CR) networks becomes quite challenging due to high sampling pressure, huge data transmission overheads and serious channel fading. To overcome these challenges, this paper proposes a collaborative spectrum sensing scheme based on 1-bit compressed sensing (CS). Each SU performs local 1-bit CS and obtains its own estimated support set via signal reconstruction. To make use of joint sparsity, all the support sets are fused in the fusion center (FC) and the fused result is used as priori information to guide the next local signal reconstruction. The weighted binary iterative hard thresholding (BIHT) algorithm with support estimate vector is proposed to implement the second reconstruction. Simulation results show that our proposed scheme can achieve effective spectrum detection at low communication and computation load.

The paper proposes a novel signal reconstruction algorithm through substituting the gradient descent method in the iterative hard thresholding algorithm with a faster sparse randomized Kaczmarz method. By designing a series of gradually attenuated weights for the matrix rows whose indexes lie outside of the support set of the original sparse signal, we can focus the iterations on the effective support rows of the measurement matrix. The experiment results show that the proposed algorithm presents a faster convergence rate and more accurate reconstruction accuracy than the state-of-the-art algorithms. Meanwhile, the successful reconstruction probability of the proposed algorithm is higher than that of other algorithms. Moreover, the characteristics of the proposed signal reconstruction algorithm are also analyzed in detail through numerical experiments.

In this paper, we generalize Huber's criterion to multichannel sparse recovery problem of complex-valued measurements where the objective is to find good recovery of jointly sparse unknown signal vectors from the given multiple measurement vectors which are different linear combinations of the same known elementary vectors. This requires careful characterization of robust complex-valued loss functions as well as Huber's criterion function for the multivariate sparse regression problem. We devise a greedy algorithm based on simultaneous normalized iterative hard thresholding (SNIHT) algorithm. Unlike the conventional SNIHT method, our algorithm, referred to as HUB-SNIHT, is robust under heavy-tailed non-Gaussian noise conditions, yet has a negligible performance loss compared to SNIHT under Gaussian noise. Usefulness of the method is illustrated in source localization application with sensor arrays.

Generalized [formula omitted]-sparse algorithms for constructing fault tolerant RBF networks

In this paper, we propose an accelerated iterative hard thresholding algorithm for solving the $$l_0$$ regularized box constrained regression problem. We substantiate that there exists a threshold, if the extrapolation coefficients are chosen below this threshold, the proposed algorithm is equivalent to the accelerated proximal gradient algorithm for solving a corresponding constrained convex problem after finite iterations. Under some proper conditions, we get that the sequence generated by the proposed algorithm is convergent to a local minimizer of the $$l_0$$ regularized problem, which satisfies a desired lower bound. Moreover, when the data fitting function satisfies the error bound condition, we prove that both the iterate sequence and the corresponding sequence of objective function values are R-linearly convergent. Finally, we use several numerical experiments to verify our theoretical results.

Iterative hard thresholding for compressed sensing

There has been a growing interest in compressed wideband spectrum sensing. One significant challenge in this field is how to effectively overcome the sparsity dependence. To solve this, two adaptive sparsity methods based on one-bit compressed sensing are explored for wideband spectrum sensing. One method (ASBIHT) is proposed by introducing adaptive sparsity into the binary iterative hard threshold method (BIHT). Using the magnitude of energy, it can accurately reconstruct the signals with unknown sparsity by learning the signal and noise. The other method is constructed by combining the binary iterative hard threshold algorithm with the pinball loss function (PIHT), and adaptive sparsity (ASPIHT). Utilizing the fixed step, the hard threshold parameter is adjusted to approach the signal sparsity gradually. Simulation results verify the efficacies of two proposed methods in the context of wideband spectrum sensing.

Computational complexity reduction in nonuniform compressed sensing by multi-coset emulation