Fast Iterative Soft-thresholding Algorithm Research Articles

A fused LASSO operator for fast 3D magnetic particle imaging reconstruction

Objective. Magnetic particle imaging (MPI) is a promising imaging modality that leverages the nonlinear magnetization behavior of superparamagnetic iron oxide nanoparticles to determine their concentration distribution. Previous optimization models with multiple regularization terms have been proposed to achieve high-quality MPI reconstruction, but these models often result in increased computational burden, particularly for dense gridding 3D fields of view. In order to achieve faster reconstruction speeds without compromising reconstruction quality, we have developed a novel fused LASSO operator, total sum-difference (TSD), which effectively captures the sparse and smooth priors of MPI images. Methods. Through an analysis-synthesis equivalence strategy and a constraint smoothing strategy, the TSD regularized model was solved using the fast iterative soft-thresholding algorithm (FISTA). The resulting reconstruction method, TSD-FISTA, boasts low computational complexity and quadratic convergence rate over iterations. Results. Experimental results demonstrated that TSD-FISTA required only 10% and 37% of the time to achieve comparable or superior reconstruction quality compared to commonly used fused LASSO-based alternating direction method of multipliers and Tikhonov-based algebraic reconstruction techniques, respectively. Significance. TSD-FISTA shows promise for enabling real-time 3D MPI reconstruction at high frame rates for large fields of view.

DIDR-Net: a sparse-view CT deep iterative reconstruction network with an independent detail recovery network

Sparse-view CT is an effective method to reduce X-ray radiation dose in clinical CT imaging. However, sparse view image reconstruction is still challenging due to highly undersampled data. To this end, we propose a new deep learning-based reconstruction model of CT images, called DIDR-Net. Unlike existing methods, DIDR-Net employs a dual network structure, including an iterative reconstruction sub-network and a detail recovery sub-network. The iterative reconstruction sub-network expands the FISTA (Fast Iterative Soft Thresholding Algorithm) into a deep network, utilizing learnable nonlinear sparse transform and shrinkage thresholding to improve reconstruction performance. To avoid loss of image details while removing artifacts, we design a detail recovery sub-network. Specifically, this sub-network captures the local details and global information of the initial image through local and global branches, and adaptively fuses the results of these two branches through a fusion module. DIDR-Net generates both the initial reconstruction map and the detail feature map in a parallel manner, and finally fuses the two sides to reconstruct a high-quality CT image. Experimental results on the AAPM public dataset show that DIDR-Net exhibits better performance in both streak artifact removal and detail structure preservation compared to other advanced reconstruction algorithms.

A convergence analysis for projected fast iterative soft-thresholding algorithm under radial sampling MRI

Radial sampling is a fast magnetic resonance imaging technique. Further imaging acceleration can be achieved with undersampling but how to reconstruct a clear image with fast algorithm is still challenging. Previous work has shown the advantage of removing undersampling image artifacts using the tight-frame sparse reconstruction model. This model was further solved with a projected fast iterative soft-thresholding algorithm (pFISTA). However, the convergence of this algorithm under radial sampling has not been clearly set up. In this work, the authors derived a theoretical convergence condition for this algorithm. This condition was approximated by estimating the maximal eigenvalue of reconstruction operators through the power iteration. Based on the condition, an optimal step size was further suggested to allow the fastest convergence. Verifications were made on the prospective in vivo data of static brain imaging and dynamic contrast-enhanced liver imaging, demonstrating that the recommended parameter allowed fast convergence in radial MRI.

A Fast and Scalable Polyatomic Frank-Wolfe Algorithm for the LASSO

We propose a fast and scalable Polyatomic Frank-Wolfe (P-FW) algorithm for the resolution of high-dimensional LASSO regression problems. The latter improves upon traditional Frank-Wolfe methods by considering generalized greedy steps with polyatomic (i.e. linear combinations of multiple atoms) update directions, hence allowing for a more efficient exploration of the search space. To preserve sparsity of the intermediate iterates, we re-optimize the LASSO problem over the set of selected atoms at each iteration. For efficiency reasons, the accuracy of this re-optimization step is relatively low for early iterations and gradually increases with the iteration count. We provide convergence guarantees for our algorithm and validate it in simulated compressed sensing setups. Our experiments reveal that P-FW outperforms state-of-the-art methods in terms of runtime, both for FW methods and optimal first-order proximal gradient methods such as the Fast Iterative Soft-Thresholding Algorithm (FISTA).

Improved data efficiency for NMR diffusion-relaxation processing

Two dimensional diffusion and transverse (T2) NMR relaxation measurements are effective for a variety of research and industrial processes. Conversion of the measurements into a D-T2 map is performed using an inverse integral transformation. A difficulty with D-T2 estimation from data acquired without pulsed field gradients (using, for example, the inherent static field gradient of a single-sided magnet) is that the diffusion and relaxation kernels are coupled. One commonly used solution is to introduce a time offset to enable the kernels to be decoupled, but this has the undesirable results of causing some of the data, and a large proportion of the signal energy, to become unusable. This paper presents two methods of processing the data that do not require this wastage. Both methods are based on insights that arise from considering the linear operator that describes the forwards integral transformation. One method involves data compression, while the other method is an application (that we call FLINT) of the fast iterative soft thresholding algorithm. Both methods are able to use all of the available data. The paper demonstrates the improved accuracy that results from these methods on simulated data, as well as the improved discovery of important features on measured data.

XCloud-pFISTA: A Medical Intelligence Cloud for Accelerated MRI.

Machine learning and artificial intelligence have shown remarkable performance in accelerated magnetic resonance imaging (MRI). Cloud computing technologies have great advantages in building an easily accessible platform to deploy advanced algorithms. In this work, we develop an open-access, easy-to-use and high-performance medical intelligence cloud computing platform (XCloud-pFISTA) to reconstruct MRI images from undersampled k-space data. Two state-of-the-art approaches of the Projected Fast Iterative Soft-Thresholding Algorithm (pFISTA) family have been successfully implemented on the cloud. This work can be considered as a good example of cloud-based medical image reconstruction and may benefit the future development of integrated reconstruction and online diagnosis system.

A guaranteed convergence analysis for the projected fast iterative soft-thresholding algorithm in parallel MRI.

Sparse sampling and parallel imaging techniques are two effective approaches to alleviate the lengthy magnetic resonance imaging (MRI) data acquisition problem. Promising data recoveries can be obtained from a few MRI samples with the help of sparse reconstruction models. To solve the optimization models, proper algorithms are indispensable. The pFISTA, a simple and efficient algorithm, has been successfully extended to parallel imaging. However, its convergence criterion is still an open question. Besides, the existing convergence criterion of single-coil pFISTA cannot be applied to the parallel imaging pFISTA, which, therefore, imposes confusions and difficulties on users about determining the only parameter - step size. In this work, we provide the guaranteed convergence analysis of the parallel imaging version pFISTA to solve the two well-known parallel imaging reconstruction models, SENSE and SPIRiT. Along with the convergence analysis, we provide recommended step size values for SENSE and SPIRiT reconstructions to obtain fast and promising reconstructions. Experiments on in vivo brain images demonstrate the validity of the convergence criterion.

Imputation and low-rank estimation with Missing Not At Random data

Missing values challenge data analysis because many supervised and unsupervised learning methods cannot be applied directly to incomplete data. Matrix completion based on low-rank assumptions are very powerful solution for dealing with missing values. However, existing methods do not consider the case of informative missing values which are widely encountered in practice. This paper proposes matrix completion methods to recover Missing Not At Random (MNAR) data. Our first contribution is to suggest a model-based estimation strategy by modelling the missing mechanism distribution. An EM algorithm is then implemented, involving a Fast Iterative Soft-Thresholding Algorithm (FISTA). Our second contribution is to suggest a computationally efficient surrogate estimation by implicitly taking into account the joint distribution of the data and the missing mechanism: the data matrix is concatenated with the mask coding for the missing values; a low-rank structure for exponential family is assumed on this new matrix, in order to encode links between variables and missing mechanisms. The methodology that has the great advantage of handling different missing value mechanisms is robust to model specification errors. The performances of our methods are assessed on the real data collected from a trauma registry (TraumaBase $$^{\textregistered }$$ ) containing clinical information about over twenty thousand severely traumatized patients in France. The aim is then to predict if the doctors should administrate tranexomic acid to patients with traumatic brain injury, that would limit excessive bleeding.

Debiasing of seismic reflectivity inversion using basis pursuit de-noising algorithm

Seismic reflectivity inversion using basis pursuit de-noising algorithm (BPDN) produces sparse solutions by applying l1-norm regularization. Due to the presence of the regularization term, there is always a deviation between the solution obtained by BPDN and the true one. A debiasing step of “adding back the residual” was introduced into seismic reflectivity inversion using BPDN to reduce the deviation, so that the accuracy of seismic reflectivity inversion using BPDN is improved. The debiasing step is carried out using a moderately large trade-off factor by iteratively performing the fast iterative soft thresholding algorithm (FISTA) to solve the objective function of BPDN with the input updated by adding back the residual to the initial input. The solution obtained by using FISTA with incorporation of the debiasing step of “adding back the residual” is a better approximation of the true solution than the result obtained by using FISTA a single time. In addition, the debiasing step of “adding back the residual” is not limited to the number of nonzero components in the solution and can be used in a low signal-to-noise ratio circumstance. A multichannel inversion method was adopted to recover the acoustic impedance from the inverted reflection coefficient, and it gained good results both in numerical and field examples.

Directional tensor product complex tight framelets for compressed sensing MRI reconstruction

Compressed sensing magnetic resonance imaging (CS-MRI) is an effective way of reducing the sampling data in the k-space and shortening the scanning time. Motivated by the high performance of directional tensor product complex tight framelets (TPCTFs) for the image denoising problem, the authors proposed a novel framework that integrated TPCTF for sparse representation and projected fast iterative soft-thresholding algorithm (pFISTA) for CS-MRI reconstruction. Furthermore, to take advantage of the cross-scale relations in the wavelet tree of frame coefficients, the bivariate shrinkage (BS) function with local variance estimation is proposed to shrink thresholding. Such TPCTFs can provide sparse directional representations very well for MR image. When compared with other the state-of-the-art CS-MRI algorithms in numerical experiments, the proposed TPCTF-BS method achieves a higher reconstruction quality with respect to image edge preservation and the artefact suppression.

Positive‐contrast susceptibility imaging based on first‐order primal‐dual optimization

To achieve faster reconstruction and better imaging quality of positive-contrast MRI based on the susceptibility mapping by incorporating a primal-dual (PD) formulation. The susceptibility-based positive contrast MR technique was applied to estimate arbitrary magnetic susceptibility distributions of the metallic devices using a kernel deconvolution algorithm with a regularized minimization. The regularized positive-contrast inversion problem and its PD formulation were derived. The visualization of the positive contrast and convergence behavior of the PD algorithm were compared with those of the nonlinear conjugate gradient algorithm, fast iterative soft-thresholding algorithm, and alternating direction method of multipliers. These methods were tested and validated on computer simulations and phantom experiments. The PD approach could provide a faster reconstruction time compared with other methods. Experimental results showed that the PD algorithm could achieve comparable or even better visualization and accuracy of the metallic interventional devices in positive-contrast imaging with different SNRs and orientations to the B0 field. A susceptibility-based positive-contrast imaging technique by PD algorithm was proposed. The PD approach has more superior performance than other algorithms in terms of reconstruction time and accuracy for imaging the metallic interventional devices.

Nonstationary acoustic-impedance inversion algorithm via a novel equivalent Q-value estimation scheme and sparse regularizations

Nonstationary seismograms can be mathematically expressed by the nonstationary convolution model (NSCM), which integrates the equivalent [Formula: see text]-value [Formula: see text], source wavelet, and acoustic impedance AI into a single formula. Performing nonstationary AI inversion based on NSCM requires subsurface [Formula: see text] field distribution and a source wavelet, which are all challenging to perform. The constant [Formula: see text] field and the extracted wavelet obtained from the early part of the seismograms are commonly used as alternatives; consequently, inaccurate AI inversion results will be easily yielded. To deal with this problem, we decompose nonstationary seismic inversion into three subproblems. First, [Formula: see text] curves are estimated at well locations using a novel scheme. Because the Gabor transform can induce approximate factorization of NSCM, the time-varying wavelet amplitude spectrum (TVWAS) can be obtained by dividing the time-frequency amplitude spectrum (TFAS) of the seismic signal by the TFAS of the reflection coefficient. Stable [Formula: see text] curves at well locations then can be estimated by the compensation-based [Formula: see text]-analysis method, whose input data are TVWAS. Second, having an estimation of [Formula: see text] and AI logging information at well locations, the source-wavelet inversion objective function with wavelet transform-domain sparse constraint is proposed based on the time-domain NSCM. This inverse problem is solved efficiently using projected fast iterative soft-thresholding algorithm. We have developed the AI inversion objective function with the low-frequency constraint and isotropic total variation transform-domain sparse constraint based on frequency-domain NSCM and the split Bregman method for minimizing this objective function. Benchmark model synthetic tests and field-data application indicate the effectiveness of our nonstationary AI inversion algorithm by demonstrating the improved accuracy and resolution of the AI inversion results.

Linearized dynamic warping with ℓ1-norm constraint for multi-component registration

Registration of the travel time of PP waves and PS waves on the same temporal coordinate is critical for joint interpretation of multi-component seismic data. We propose an ℓ1-norm minimization approach to register the corresponding amplitude of reflection events in the PP waves and PS waves data. The dynamic warping function is linearized in the ℓ2-norm fitting term, and a sparsity constrained is used to estimate smooth warping functions. The latter is possible by defining the warping function in terms of a cosine basis. Sparsity is used to minimize the number of basis functions that are used to model the warping function. We have also applied a fast iterative soft thresholding algorithm for solving the non-linear registration optimization problem. We apply the linearized dynamic warping approach with ℓ1-norm regularization to a synthetic example and to a field data set from the Shengli Oilfield, China.

Conjugate gradient acceleration of iteratively re-weighted least squares methods

Iteratively re-weighted least squares (IRLS) is a method for solving minimization problems involving non-quadratic cost functions, perhaps non-convex and non-smooth, which however can be described as the infimum over a family of quadratic functions. This transformation suggests an algorithmic scheme that solves a sequence of quadratic problems to be tackled efficiently by tools of numerical linear algebra. Its general scope and its usually simple implementation, transforming the initial non-convex and non-smooth minimization problem into a more familiar and easily solvable quadratic optimization problem, make it a versatile algorithm. However, despite its simplicity, versatility, and elegant analysis, the complexity of IRLS strongly depends on the way the solution of the successive quadratic optimizations is addressed. For the important special case of compressed sensing and sparse recovery problems in signal processing, we investigate theoretically and numerically how accurately one needs to solve the quadratic problems by means of the conjugate gradient (CG) method in each iteration in order to guarantee convergence. The use of the CG method may significantly speed-up the numerical solution of the quadratic subproblems, in particular, when fast matrix-vector multiplication (exploiting for instance the FFT) is available for the matrix involved. In addition, we study convergence rates. Our modified IRLS method outperforms state of the art first order methods such as Iterative Hard Thresholding (IHT) or Fast Iterative Soft-Thresholding Algorithm (FISTA) in many situations, especially in large dimensions. Moreover, IRLS is often able to recover sparse vectors from fewer measurements than required for IHT and FISTA.

Compressed Sensing MRI Using Discrete Nonseparable Shearlet Transform and FISTA

We propose a new compressed sensing MRI approach that uses the discrete nonseparable shearlet transform (DNST) as a sparsifying transform and the fast iterative soft thresholding algorithm (FISTA) for reconstruction. FISTA has a simple design and has shown good convergence behavior. The DNST transform has excellent localization properties within the space domain and excellent directional selectivity. We utilize the frequency representation of the DNST canonical dual filters to obtain a memory efficient modified FISTA based algorithm with a simple and efficient way of calculating the update, tuned to the non tight frame DNST transform. The proposed approach shows improved performance and similar execution time when compared with other state of the art reconstruction approaches.

Stability of Over-Relaxations for the Forward-Backward Algorithm, Application to FISTA

This paper is concerned with the convergence of over-relaxations of the forward-backward algorithm (FB) (in particular the fast iterative soft thresholding algorithm (FISTA)) in the case when proximal maps and/or gradients are computed with a possible error. We show that, provided these errors are small enough, the algorithm still converges to a minimizer of the functional, and with a speed of convergence (in terms of values of the functional) that remains the same as in the noise-free case. We also show that larger errors can be allowed, using a lower over-relaxation than FISTA. This still leads to the convergence of iterates and with ergodic convergence speed faster than the classical FB and FISTA.

Thinned Array Beampattern Synthesis by Iterative Soft-Thresholding-Based Optimization Algorithms

In this paper, we cast thinned array beampattern synthesis as a linear inverse problem (LIP) and apply three iterative soft-thresholding based optimization algorithms to solve it effectively. We combine synthesis algorithms with beampattern error reweighting and beampattern phase adjustment to achieve accurate approximation of the desired reference pattern. The proposed methods can be used to synthesize arbitrarily shaped beam-patterns, including multibeam forming and radiation suppression within several angular regions. These methods offer a number of advantages in regard to array scale, beampattern specifics and computational time compared to existing methods. Since this paper focuses on synthesizing receiver arrays, there are no explicit constraints applied to the excitation weights. Numerical results demonstrate the efficiency of three proposed approaches for designing non-uniformly spaced arrays. In particular, the fast iterative soft-thresholding algorithm can synthesize of up to a few hundred antennas.

Nonuniform update for sparse target recovery in fluorescence molecular tomography accelerated by ordered subsets.

Fluorescence molecular tomography (FMT) is a promising imaging modality and has been actively studied in the past two decades since it can locate the specific tumor position three-dimensionally in small animals. However, it remains a challenging task to obtain fast, robust and accurate reconstruction of fluorescent probe distribution in small animals due to the large computational burden, the noisy measurement and the ill-posed nature of the inverse problem. In this paper we propose a nonuniform preconditioning method in combination with L (1) regularization and ordered subsets technique (NUMOS) to take care of the different updating needs at different pixels, to enhance sparsity and suppress noise, and to further boost convergence of approximate solutions for fluorescence molecular tomography. Using both simulated data and phantom experiment, we found that the proposed nonuniform updating method outperforms its popular uniform counterpart by obtaining a more localized, less noisy, more accurate image. The computational cost was greatly reduced as well. The ordered subset (OS) technique provided additional 5 times and 3 times speed enhancements for simulation and phantom experiments, respectively, without degrading image qualities. When compared with the popular L (1) algorithms such as iterative soft-thresholding algorithm (ISTA) and Fast iterative soft-thresholding algorithm (FISTA) algorithms, NUMOS also outperforms them by obtaining a better image in much shorter period of time.

A discrepancy-based parameter adaptation and stopping rule for minimization algorithms aiming at Tikhonov-type regularization

We present a discrepancy-based parameter choice and stopping rule for iterative algorithms performing approximate Tikhonov-functional minimization which adapts the regularization parameter value during the optimization procedure. The suggested parameter choice and stopping rule can be applied to a wide class of penalty terms and iterative algorithms which aim at Tikhonov regularization with a fixed parameter value. It leads, in particular, to computable guaranteed estimates for the regularized exact discrepancy in terms of numerical approximations. Based on these estimates, convergence to a solution is shown. As an example, the developed theory and the algorithm is applied to the case of sparse regularization. We prove order optimal convergence rates in the case of sparse regularization, i.e. weighted ℓp norms, which turn out to be the same as for the a priori parameter choice rule already obtained in the literature as well as for Morozov’s principle applied to exact regularized solutions. Finally, numerical results for two different minimization techniques, iterative soft thresholding algorithm and monotone fast iterative soft thresholding algorithm, are presented, confirming, in particular, the results from the theory.

Wavelet-based edge correlation incorporated iterative reconstruction for undersampled MRI

Undersampling k-space is an effective way to decrease acquisition time for MRI. However, aliasing artifacts introduced by undersampling may blur the edges of magnetic resonance images, which often contain important information for clinical diagnosis. Moreover, k-space data is often contaminated by the noise signals of unknown intensity. To better preserve the edge features while suppressing the aliasing artifacts and noises, we present a new wavelet-based algorithm for undersampled MRI reconstruction. The algorithm solves the image reconstruction as a standard optimization problem including a ℓ 2 data fidelity term and ℓ 1 sparsity regularization term. Rather than manually setting the regularization parameter for the ℓ 1 term, which is directly related to the threshold, an automatic estimated threshold adaptive to noise intensity is introduced in our proposed algorithm. In addition, a prior matrix based on edge correlation in wavelet domain is incorporated into the regularization term. Compared with nonlinear conjugate gradient descent algorithm, iterative shrinkage/thresholding algorithm, fast iterative soft-thresholding algorithm and the iterative thresholding algorithm using exponentially decreasing threshold, the proposed algorithm yields reconstructions with better edge recovery and noise suppression.