Sparse Reconstruction Using Hyperbolic Tangent as Smooth l1-Norm Approximation

Multiphase flow measurements, based on tomographic imaging techniques, have made substantial progress in the past several years. In particular, ultrasonic tomographic reconstruction has shown its high potential in accurately measuring fluid fractions in a flow cross-section based on direct flow measurements. These types of measurements are very attractive as the system is nonradioactive, has direct contact to fluid, yet without flow obstruction. An ultrasonic tomographic multiphase flow meter (UTM) is being developed to capture the fluid flow rates and phase compositions in producing oil wells. The UTM aims to generate cross-sectional images of the flow inside the pipe. To recover these images, complex computations and high processing time are required, based on a limited amount of available data. For this purpose, compressive sensing (CS) reconstruction methods have shown great potential in overcoming the mentioned data quality and quantity issues. CS builds upon the fact that one can recover certain data of interest from potentially very few measurements. Many CS algorithms have been proposed in the literature. In this paper, we seek to find a suitable algorithm by comparing different CS algorithms in terms of the resulting recovered image quality. Specifically, we survey three algorithms in CS, namely basis pursuit, iterative soft thresholding (IST), and orthogonal matching pursuit (OMP). We also compare the results of the CS algorithms to that of a least-square based conventional algorithm. The numerical results show the superiority of CS recovery algorithms over the conventional approach, both qualitatively and quantitatively. The results also suggest that high image quality could potentially be achieved with less complex CS algorithms, i.e., the IST and OMP algorithms. As presented in the paper, our goal is to adapt CS reconstruction algorithms for images generated by the UTM, for the purpose of accurate quantification of oil, gas, and water phase fractions produced from different wells.

L1 norm estimator has been widely used as a robust parameter estimation method for outlier detection. Different algorithms have been applied for L1 norm minimization among which the linear programming problem based on the simplex method is well known. In the present contribution, in order to solve an L1 norm minimization problem in a linear model, an interior point algorithm is developed which is based on Dikin's method. The method can be considered as an appropriate alternative for the classical simplex method, which is sometimes time-consuming. The proposed method, compared with the simplex method, is thus easier for implementation and faster in performance. Furthermore, a recursive form of the Dikin's method is derived, which resembles the recursive least-squares method. Two simulated numerical examples show that the proposed algorithm gives as accurate results as the simplex method but in considerably less time. When dealing with a large number of observations, this algorithm can thus be used instead of the iteratively reweighted least-squares method and the simplex method.

Towards practical implementation of the compressed sensing framework for multi-element synthetic transmit aperture imaging

The recently proposed compressive sensing (CS) theory provides a new solution for multiple description coding (MDC) with fine granularity, by treating each random CS measurement as a description. The performance of CS-based MDC (CS-MDC) depends on the efficacy of the CS recovery algorithm. Existing CS recovery algorithms recover the signal in a fixed space (e.g., Wavelet, DCT, and gradient spaces) for the entire duration of the signal, even though a typical multimedia signal exhibits sparsity in time/space variant spaces. To rectify this problem and develop a better CS recovery algorithm for CSMDC, we propose a learning-based framework to conduct the CS recovery in locally adaptive spaces, and carry out a case study on image MDC. A set of prior image models are learned offline from a training set to facilitate the CS recovery in local adaptive bases. Experiments show that the learning-based CS recovery algorithm can significantly improve the performance of the previous CS-MDC technique in both PSNR and visual quality.

Compressed sensing (CS) is a novel developed theoretical framework for information acquisition and processing. Taking advantages of the sparsity or compressibility inherent in real world signals, CS can collect compressed data at the sampling rate much lower than that needed in Shannon's theorem. When it has been used for medical imaging techniques, CS theory can fast the scanning speed of MRI, reduce the radiation doses and alleviate the patients' suffering. In order to obtain the effective sparse representation, the nonsubsampled contourlet transform in the frequency domain (NSCT-FD) is adopted, then an improved fast iterative soft thresholding algorithm (FISTA) is applied for CS-MRA reconstruction. This novel method not only inherits the simplicity and effectiveness of the original FISTA, and the sparse curvy representation ability of the contourlet transform, but also has the advantages of the shift-invariant property and much less computational burden. The performance is evaluated qualitatively and quantitatively both in noiseless and noisy situations, compared to the classic wavelet and the sharp frequency localization contourlet transform (SFLCT) method. Three quantitative indices are employed including the peak signal to noise ratio (PSNR), mutual information (MI) and relative l 2 norm error (RLNE) and qualitative performance evaluations use the profiles in horizontal direction and local region magnification comparison. Experimental results demonstrate the superiority of the NSCT-FD algorithm, displaying higher PSNR and MI, and lower RLNE indices in both noiseless and noisy MRA images, which have good reconstruction accuracy with reasonable real time computation speed. It is manifest that the NSCT-FD algorithm can be applied in fast medical imaging fields to achieve quality images in a relatively fast process, which will benefit cardiac and carotid MRI, dynamic MRI and MRA.

MRI currently is the most powerful medical diagnostic tomography system. It provides high resolution with high detail medical image. Its magnetization sensing is also free from radiation impact hence safe for the patient. However, the data acquisition process takes quite a long time with high-rate sampling. The sensing patient should stay still in the MRI sensing device for 20 to 40 minutes. It can affect claustrophobia for some patients. The high-rate sampling will also linearly affect the cost, which is considerably high. Hence, the research on enabling under-sampling MRI imaging has been raised. The main objective of under-sampling MRI Imaging is reducing the scanning time by under-sampling the data acquisition while at the same time the resulting qualified reconstructed image can be maintained. Compressive Sensing (CS) framework is one of the approaches that can be utilized on solving projection with less measurement. It can recover signals which have sparsity property. The signal is sparse if the signal being sensed has low information rate. Thus, the framework of Compressive Sensing matches under-sampling MRI Imaging. The sparsity requirement is closely related to the design of sensing matrix and dictionary on the Compressive Sensing framework. This paper presents analysis of two proposed sensing matrix designs for Under-sampling MRI Imaging: square sensing matrix and sorting-based sensing matrix designs. The proposed sensing matrix design is intended to appropriately sample the data measurement on certain sampling rate in frequency domain to maintain the quality of the reconstructed MRI image closely to the ground truth (full sampling MRI Imaging). The simulations show that both proposed sensing matrix designs are able to conduct under-sampling MRI Imaging for at least at 40% sampling rate with average of PSNR 27.72 and 33.07 respectively subject to the ground truth.

The application of compressed sensing (CS) to magnetic resonance (MR) images utilizes the transformed domain sparsity to enable the reconstruction from an under-sampled k-space (Fourier) data using a non-linear recovery algorithm. In order to estimate the missing k-space data from the partial Fourier samples, the reconstruction algorithms minimize an objective function based on mixed l1 − l2 norms. Iterative-shrinkage algorithms, such as parallel coordinate descent (PCD) and separable surrogate functional, provide an efficient numerical technique to minimize the l1-regularized least square optimization problem. These algorithms require a thresholding step to induce sparsity in the solution, which is an essential requirement of the CS recovery. This paper introduces a novel soft-thresholding method based on the hyperbolic tangent function. It has been shown that by using the proposed thresholding function in the sparsifying domain and a data consistency step in the k-space, the iterative-shrinkage algorithms can be used effectively to recover the under-sampled MR images. For the purpose of demonstration, we use the proposed soft-thresholding and data consistency with the PCD algorithm and compare its performance with the conventional PCD, projection onto convex sets and low-resolution reconstruction methods. The metrics used to compare the various algorithms are the artifact power, the peak signal-to-noise ratio, the correlation and the structural similarity index. The experimental results are validated using Shepp–Logan phantom image as well as real human head MR images taken from the MRI scanner at St. Mary’s Hospital, London.

We study classic streaming and sparse recovery problems using deterministic linear sketches, including ℓ1/ℓ1 and ℓ ∞ /ℓ1 sparse recovery problems, norm estimation, and approximate inner product. We focus on devising a fixed matrix A ∈ ℝ m×n and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. We contribute several improved bounds for these problems. A proof that ℓ ∞ /ℓ1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices can be used to solve both problems. Our upper bound for the number of measurements is m = O(ε − 2 min {logn, (logn / log(1/ε))2}). We can also obtain fast sketching and recovery algorithms by making use of the Fast Johnson-Lindenstrauss transform. Both our running times and number of measurements improve upon previous work. We can also obtain better error guarantees than previous work in terms of a smaller tail of the input vector. A new lower bound for the number of linear measurements required to solve ℓ1/ℓ1 sparse recovery. We show Ω(k/ε 2 + klog(n/k)/ε) measurements are required to recover an x′ with ∥ x − x′ ∥ 1 ≤ (1 + ε) ∥ x tail(k) ∥ 1, where x tail(k) is x projected onto all but its largest k coordinates in magnitude. A tight bound of m = Θ(ε − 2log(ε 2 n)) on the number of measurements required to solve deterministic norm estimation, i.e., to recover ∥ x ∥ 2 ±ε ∥ x ∥ 1. For all the problems we study, tight bounds are already known for the randomized complexity from previous work, except in the case of ℓ1/ℓ1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems.

Sparse recovery is ubiquitous in machine learning and signal processing. Due to the NP-hard nature of sparse recovery, existing methods are known to suffer either from restrictive (or even unknown) applicability conditions, or high computational cost. Recently, iterative regularization methods have emerged as a promising fast approach because they can achieve sparse recovery in one pass through early stopping, rather than the tedious grid-search used in the traditional methods. However, most of those iterative methods are based on the l1 norm which requires restrictive applicability conditions and could fail in many cases. Therefore, achieving sparse recovery with iterative regularization methods under a wider range of conditions has yet to be further explored. To address this issue, we propose a novel iterative regularization algorithm, IRKSN, based on the k-support norm regularizer rather than the l1 norm. We provide conditions for sparse recovery with IRKSN, and compare them with traditional conditions for recovery with l1 norm regularizers. Additionally, we give an early stopping bound on the model error of IRKSN with explicit constants, achieving the standard linear rate for sparse recovery. Finally, we illustrate the applicability of our algorithm on several experiments, including a support recovery experiment with a correlated design matrix.

Compressive sensing (CS) is a new technique that give an approach to reconstruct the signal with few numbers of observations or measurements. The CS is based on L1 -norm minimizations to find the sparse solutions and it is known as basis pursuit. In this investigation, CS scheme with different transforms is proposed. The three utilized transform techniques are: Discrete Fourier Transform(DFT), Discrete Cosine Transform(DCT) and Discrete Wavelet Transform(DWT) with daubechies1(DB1) and coiflets1(coif1) basis. The proposed system is tested by employing the following signals: Blocks, Heavy Sine, ‘Bumps’ and ‘Doppler’ which cover wide range of applications. The four testing signals are represented in sparse domain using different transforms. In order to threshold the coefficients of the signals in sparse domain, the universal threshold is utilized in the case of CS with FFT and DWT whereas, the universal threshold is modified to prune the DCT coefficients. The main aim of this study is to investigate the differences among CS with DFT, CS with DCT and CS with DWT, and consequently a suitable transform domain used with CS to be selected. The comparative study is established by assessing the performance of the proposed system using Root Mean Square Error (RMSE), output SNR, and the time required to reconstruct approximated signals. Simulation results have shown that the CS with DWT outperforms the CS with FFT and DCT. CS with DWT has achieved good RMSE values about (0.0014 to 3.359e-8) even when half of the signal elements are removed. CS with FFT and DCT enhanced the noisy Blocks and Bumps signals by 3dB and 1dB respectively, while it is failed to enhance noisy Heavy Sine and Doppler signals. CS with DWT of two basis and for single decomposition level have improved the noisy Blocks, Bumps, Heavy Sine and Doppler signals by 5dB, 4dB, 3dB, and 3dB respectively.

In compressive sensing (CS), a challenge is to find a space in which the signal is sparse and, hence, faithfully recoverable. Since many natural signals such as images have locally varying statistics, the sparse space varies in time/spatial domain. As such, CS recovery should be conducted in locally adaptive signal-dependent spaces to counter the fact that the CS measurements are global and irrespective of signal structures. On the contrary, existing CS reconstruction methods use a fixed set of bases (e.g., wavelets, DCT, and gradient spaces) for the entirety of a signal. To rectify this problem, we propose a new framework for model-guided adaptive recovery of compressive sensing (MARX) and show how a 2-D piecewise autoregressive model can be integrated into the MARX framework to make CS recovery adaptive to spatially varying second order statistics of an image. In addition, MARX offers a mechanism of characterizing and exploiting structured sparsities of natural images, greatly restricting the CS solution space. Simulation results over a wide range of natural images show that the proposed MARX technique can improve the reconstruction quality of existing CS methods by 2-7 dB.

The main contribution of this paper is using compressive sensing (CS) theory for crypto steganography system to increase both the security and the capacity and preserve the cover image imperceptibility. For CS implementation, the discrete Cosine transform (DCT) as sparse domain and random sensing matrix as measurement domain are used. We consider 7 MRI images as the secret and 7 gray scale test images as cover. In addition, three sampling rates for CS are used. The performance of seven CS recovery algorithms in terms of image imperceptibility, achieved peak signal to noise ratio (PSNR), and the computation time are compared with other references. We showed that the proposed crypto steganography system based on CS works properly even though the secret image size is greater than the cover image.

Distributed continuous-time accelerated neurodynamic approaches for sparse recovery via smooth approximation to [formula omitted]-minimization

A review of the most recent contributions in the framework of Compressive Sensing (CS) strategies as applied to Electromagnetics is presented in this work. Towards this end, the CS fundamentals are firstly discussed in the light of their impact in Electromagnetics technologies. A set of CS applicative scenarios is then discussed, taking into account problems coming from antenna array design and diagnosis, direction-of-arrival estimation, and microwave imaging. An assessment of the effectiveness and potentialities of CS techniques in Electromagnetics is provided by selected numerical examples.

EEG is sensitive to certain irrelevant sources as well as artifacts. Various studies have been carried out to detect and denoise the EEG artifacts while retaining the useful information from the original signal but still more improved analysis is required for suppression of artifacts in corrupted EEG data. This paper utilizes a blind source separation algorithm based on the second-order blind identification (SOBI) and then wavelet denoising with soft thresholding is carried out. Experimental observations show that the present method yields better suppression of artifacts in terms of the evaluation parameters of RMSE (Root Mean Square Error) and PSNR (Peak signal to noise ratio).