Iterative Soft-thresholding Method Research Articles

Fast Acquisition of High-Quality Nuclear Magnetic Resonance Pure Shift Spectroscopy via a Deep Neural Network.

Pure shift methods improve the resolution of proton nuclear magnetic resonance spectra at the cost of time. The pure shift yielded by chirp excitation (PSYCHE) method is a promising pure shift method. We propose a method of reconstructing the undersampled PSYCHE spectra based on deep learning to accelerate the spectra acquisition. It only takes 17 s to obtain a high-quality pure shift spectrum. The network can completely remove undersampling artifacts and chunking sidebands and improve the signal-to-noise ratio, obtaining completely clean pure shift spectra. The reconstruction quality is better than the iterative soft thresholding method. In addition, the network can differentiate low-level signals and chunking sidebands with similar intensities in the mixture, remove sidebands, and retain signals, promoting correct mixture analysis.

Least-squares reverse-time migration with sparsity constraints

Abstract Least-squares reverse-time migration (RTM) works with an inverse operation, rather than an adjoint operation in a conventional RTM, and thus produces an image with a higher resolution and more balanced amplitude than the conventional RTM image. However, least-squares RTM introduces two side effects: sidelobes around reflectors and high-wavenumber migration artifacts. These side effects are caused mainly by the limited bandwidth of seismic data, the limited coverage of receiver arrays and the inaccuracy of the modeling kernel. To mitigate these side effects and to further boost resolution, we employed two sparsity constraints in the least-squares inversion operation, namely the Cauchy and L1-norm constraints. For solving the Cauchy-constrained least-squares RTM, we used a preconditioned nonlinear conjugate-gradient method. For solving the L1-norm constrained least-squares RTM, we modified the iterative soft thresholding method. While adopting these two solution methods, the Cauchy-constrained least-squares RTM converged faster than the L1-norm constrained least-squares RTM. Application examples with synthetic data and laboratory modeling data demonstrated that the constrained least-squares RTM methods can mitigate the side effects and promote image resolution.

Simultaneous seismic interpolation and denoising based on sparse inversion with a 3D low redundancy curvelet transform

The simultaneous seismic interpolation and denoising problem can be solved as a sparse inversion problem by using the sparseness of seismic data in a transformed domain as the a priori information, where the properties of the sparse transform will significantly influence the numerical results and computational efficiency. Curvelet transform has nearly optimal sparse expression for seismic data, thus seismic signal processing based on this transform tends to result in preferable results. However, the redundancy of this transform can be 24–32 for three dimensional data which is not computationally cost efficient. This paper introduces a low redundancy curvelet transform to simultaneously interpolate and denoise. The redundancy of the proposed transform can be reduced to 10 for three dimensional data, and this property will improve the computational efficiency of the curvelet transform-based signal processing. The iterative soft thresholding method was chosen to solve the sparse inversion problem. Some practical principles on how to choose the regularisation parameters are discussed, due to the crucial nature of the regularisation parameter for simultaneous interpolation and denoising. Numerical experiments on synthetic and field data demonstrate that the low redundancy transform can provide reliable results while at the same time improving the computational efficiency.

Fast implementation for compressive recovery of highly accelerated cardiac cine MRI using the balanced sparse model.

Sparsity-promoting regularizers can enable stable recovery of highly undersampled magnetic resonance imaging (MRI), promising to improve the clinical utility of challenging applications. However, lengthy computation time limits the clinical use of these methods, especially for dynamic MRI with its large corpus of spatiotemporal data. Here, we present a holistic framework that utilizes the balanced sparse model for compressive sensing and parallel computing to reduce the computation time of cardiac MRI recovery methods. We propose a fast, iterative soft-thresholding method to solve the resulting ℓ1-regularized least squares problem. In addition, our approach utilizes a parallel computing environment that is fully integrated with the MRI acquisition software. The methodology is applied to two formulations of the multichannel MRI problem: image-based recovery and k-space-based recovery. Using measured MRI data, we show that, for a 224 × 144 image series with 48 frames, the proposed k-space-based approach achieves a mean reconstruction time of 2.35 min, a 24-fold improvement compared a reconstruction time of 55.5 min for the nonlinear conjugate gradient method, and the proposed image-based approach achieves a mean reconstruction time of 13.8 s. Our approach can be utilized to achieve fast reconstruction of large MRI datasets, thereby increasing the clinical utility of reconstruction techniques based on compressed sensing. Magn Reson Med 77:1505-1515, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

A family of second-order methods for convex $$\ell _1$$-regularized optimization

This paper is concerned with the minimization of an objective that is the sum of a convex function f and an $$\ell _1$$l1 regularization term. Our interest is in active-set methods that incorporate second-order information about the function f to accelerate convergence. We describe a semismooth Newton framework that can be used to generate a variety of second-order methods, including block active set methods, orthant-based methods and a second-order iterative soft-thresholding method. The paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration, and employs a mechanism for correcting these estimates, when needed. This corrective mechanism is also evaluated in an orthant-based method. Numerical tests comparing the performance of three active set methods are presented.

High-fidelity adaptive curvelet domain primary-multiple separation

In this paper, we propose an adaptive scheme for primary-multiple separation whereby the multiples are first estimated from the seismic data and then removed using the curvelet transform. Because of the sparseness of seismic data in the curvelet domain, the primary-multiple separation problem is formulated by incorporating L1- and L2-norms, based on the framework of the Bayesian Probability Maximization theory. An iterative soft-thresholding method is used for solving the optimization problem. Prior to removal, the predicted multiples are preconditioned to match the actual multiples in the seismic data by least-squares matched filtering. We show that such an adaptive implementation is more robust and has a superior performance to the conventional least-squares method for attenuating multiples. To improve the effectiveness of primary-multiple separation for complex data, we develop a frequency-regularized adaptive curvelet domain separation approach. The method is optimized for different frequencies to improve attenuation in the presence of noise and in areas where multiple models are less accurate (e.g. narrowing of frequency bandwidth due to the convolution process in SRME). Accordingly, this extension provides more flexibility and leads to higher separation fidelity than its original form. We demonstrate the application of our approach on synthetic and field data. The results obtained from our approaches show significant improvement over those obtained from conventional least-squares methods.

Compressed-sensing multispectral imaging of the postoperative spine.

To apply compressed sensing (CS) to in vivo multispectral imaging (MSI), which uses additional encoding to avoid magnetic resonance imaging (MRI) artifacts near metal, and demonstrate the feasibility of CS-MSI in postoperative spinal imaging. Thirteen subjects referred for spinal MRI were examined using T2-weighted MSI. A CS undersampling factor was first determined using a structural similarity index as a metric for image quality. Next, these fully sampled datasets were retrospectively undersampled using a variable-density random sampling scheme and reconstructed using an iterative soft-thresholding method. The fully and undersampled images were compared using a 5-point scale. Prospectively undersampled CS-MSI data were also acquired from two subjects to ensure that the prospective random sampling did not affect the image quality. A two-fold outer reduction factor was deemed feasible for the spinal datasets. CS-MSI images were shown to be equivalent or better than the original MSI images in all categories: nerve visualization: P = 0.00018; image artifact: P = 0.00031; image quality: P = 0.0030. No alteration of image quality and T2 contrast was observed from prospectively undersampled CS-MSI. This study shows that the inherently sparse nature of MSI data allows modest undersampling followed by CS reconstruction with no loss of diagnostic quality.

Two-point step-size iterative soft-thresholding method for sparse reconstruction

Many problems in signal processing involve minimizing a quadratic error term combined with an ℓ1 norm term. Iterative soft-thresholding (IST) algorithm is a basic method for these problems. Despite previous explanations of IST, this study presents it as a method of constructing a local model to approximate the objective function. It only uses the approximation of the quadratic term while keeping the ℓ1 norm term unchanged. Based on this, we propose a modified IST (MIST), using a general strictly convex quadratic function to approximate the quadratic part. IST uses the identity matrix to approximate the Hessian matrix of the quadratic term, while we adopt an adaptive matrix by using the information of current and former iterates. This strategy results in the two-point step-size IST, including MISTBB1 and MISTBB2. Various experiments on compressed sensing show that MISTBB1 is much faster than competing codes and insensitive to the regularization parameter.

A review on restoration of seismic wavefields based on regularization and compressive sensing

Restoration of seismic data as an ill-posed inverse problem means to recover the complete wavefields from sub-sampled data. Since seismic data are typically sparse in the curvelet domain, this problem can be solved based on the compressive sensing theory. Meanwhile three major problems are modelling, sampling and solving methods. We first construct l0 and l1 minimization models and then develop fast projected gradient methods to solve the restoration problem. For seismic data interpolation/restoration, the regular sub-sampled data will generate coherence aliasing in the frequency domain, while the random sub-sampling cannot control the largest sampling gap. Therefore, we consider a new sampling technique in this article which is based on the controlled piecewise random sub-sampling scheme. Numerical simulations are made and compared with the iterative soft thresholding method and the spectral gradient-projection method. It reveals that the proposed algorithms have the advantages of high precision, robustness and fast calculation.

Two-step version of fixed point continuation method for sparse reconstruction

l1-regularized problems have a wide application in various areas such as signal processing. It minimizes a quadratic function combined with an l1 norm term. Iterative soft-thresholding method (IST) is originally proposed to deal with these problems, and fixed point continuation algorithm (FPC) was proposed recently as an improved version of IST. This paper obtains a two-step version of FPC (TwFPC) by combining the new iterate of FPC with its previous two iterates. We also provide an analysis for the convergence of FPC and TwFPC. Various numerical experiments on image deconvolution and compressed sensing show that TwFPC improves IST significantly and is much faster than other competing codes. What is more important, it is very robust to the involved parameters and the regularization parameter.

A forward–backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space

We consider the task of computing an approximate minimizer of the sum of a smooth and a non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward–backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman–Taylor-distance estimates, rates of convergence for the forward–backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation-based image restoration in higher dimensions are presented.