Iterative hard thresholding for compressed sensing

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.

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.

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 κ-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 κ-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 κ-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.

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.

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 .

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.

Various deconvolution algorithms for acoustic source are developed to improve spatial resolution and suppress sidelobe of the conventional beamforming. To improve the computational efficiency and solution convergence of deconvolution, this paper proposes a Fourier-based improved fast iterative shrinkage thresholding algorithm. Simulations and experiments show that Fourier-based improved fast iterative shrinkage thresholding algorithm can achieve excellent acoustic identification performance, with high computational efficiency and good convergence. For Fourier-based improved fast iterative shrinkage thresholding algorithm, the larger the weight coefficient, the narrower the mainlobe width, and the better the convergence, but the spurious source also increases. The recommended weight coefficient for the array described herein is 3. In addition, like other Fourier-based deconvolution algorithms, Fourier-based improved fast iterative shrinkage thresholding algorithm using irregular focus grid can obtain better acoustic source identification performance than using the conventional regular focus grid.

Iterative hard thresholding algorithm (IHT) is a novel and efficient method to solve signal and image reconstruction in compressed sensing, but it is sensitive to the initial point and converges to a local optimal solution. Therefore, to overcome its shortcoming, in this paper a hybrid hard thresholding algorithm (HHT) is derived by introducing the simulated annealing algorithm (SA) into the IHT. And a series of experiments are provided on signal and image reconstruction to assess performance of the algorithm. The experiments and applications show that the proposed algorithm uses less sampling to construct the signal and image and is more stable, as compared with IHT.

When sampling signals below the Nyquist rate, efficient and accurate reconstruction is nevertheless possible, whenever the sampling system is well behaved and the signal is well approximated by a sparse vector. This statement has been formalised in the recently developed theory of compressed sensing, which developed conditions on the sampling system and proved the performance of several efficient algorithms for signal reconstruction under these conditions. In this paper, we prove that a very simple and efficient algorithm, known as Iterative Hard Thresholding, has near optimal performance guarantees rivalling those derived for other state of the art approaches.

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

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.

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.

Most signals can be represented sparsely in a basis. Recently, Compressive Sensing Theorem which offers convex optimization algorithms based on l 1 -minimization for sparse signal recovery is often being used. In this paper, some of the iterative signal recovery algorithms alternative to l 1 -minimization solution which are Orthogonal Matching Pursuit (OMP), Compressive Sampling Matching Pursuit (CoSaMP), Iterative Hard Thresholding (IHT) and Lipschitz Iterative Hard Theresholding (LIHT) are compared in noisy and noiseless conditions with various tests. Iterative algorithms alternative to the l 1 optimization method with similar performance are verified. OMP algorithm that works at higher true reconstruction rates in noisy and noiseless conditions can be preferred instead of convex optimization methods.

Magnetic resonance imaging systems usually feature linear and shift-invariant (stationary) transform characteristics. The point spread function or equivalently the modulation transfer function may thus be used for an objective quality assessment of imaging modalities. The recently introduced theory of compressed sensing, however, incorporates nonlinear and nonstationary reconstruction algorithms into the magnetic resonance imaging process which prohibits the usage of the classical point spread function and therefore the according evaluation. In this work, a local point spread function concept was applied to assess the quality of magnetic resonance images which were reconstructed by an iterative soft thresholding algorithm for compressed sensing. The width of the main lobe of the local point spread function was used to perform studies on the spatial and temporal resolution properties of both numerical phantom and in vivo images. The impact of k-space sampling patterns as well as additional sparsifying transforms on the local spatial image resolution was investigated. In addition, the local temporal resolution of image series, which were reconstructed by exploiting spatiotemporal sparsity, was determined. Finally, the dependency of the local resolution on the thresholding parameter of the algorithm was examined. The sampling patterns as well as the additional sparsifying transform showed a distinct impact on the local image resolution of the phantom image. The reconstructions, which were using x-f-space as a sparse transform domain showed slight temporal blurring for dynamic parts of the imaged object. The local image resolution had a dependence on the thresholding parameter, which allowed for optimizing the reconstruction. Local point spread functions enable the evaluation of the local spatial and temporal resolution of images reconstructed with the nonlinear and nonstationary iterative soft thresholding algorithm. By determining the influence of thresholding parameter and sampling pattern chosen on this model-based reconstruction, the method allows selecting appropriate acquisition parameters and thus improving the results.