Accelerated Regularized Estimation of MR Coil Sensitivities Using Augmented Lagrangian Methods

Magnetic resonance imaging (MRI) has an important feature that it provides multiple images with different contrasts for complementary diagnostic information. However, a large amount of data is needed for multi-contrast images depiction, and thus, the scan is time-consuming. Many methods based on parallel magnetic resonance imaging (pMRI) and compressed sensing (CS) are applied to accelerate multi-contrast MR imaging. Nevertheless, the image reconstructed by sophisticated pMRI methods contains residual aliasing artifact that degrades the quality of the image when the acceleration factor is high. Other methods based on CS always suffer the regularization parameter-selecting problem. To address these issues, a new method is presented for joint multi-contrast image reconstruction and coil sensitivity estimation. The coil sensitivities can be shared during the reconstruction due to the identity of coil sensitivity profiles of different contrast images for imaging stationary tissues. The proposed method uses the coil sensitivities as sharable information during the reconstruction to improve the reconstruction quality. As a result, the residual aliasing artifact can be effectively removed in the reconstructed multi-contrast images even if the acceleration factor is high. Besides, as there is no regularization term in the proposed method, the troublesome regularization parameter selection in the CS can also be avoided. Results from multi-contrast in vivo experiments demonstrated that multi-contrast images can be jointly reconstructed by the proposed method with effective removal of the residual aliasing artifact at a high acceleration factor.

Image restoration problems are often solved by finding the minimizer of a suitable objective function. Usually this function consists of a data-fitting term and a regularization term. For the least squares solution, both the data-fitting and the regularization terms are in the ℓ2 norm. In this paper, we consider the least absolute deviation (LAD) solution and the least mixed norm (LMN) solution. For the LAD solution, both the data-fitting and the regularization terms are in the ℓ1 norm. For the LMN solution, the regularization term is in the ℓ1 norm but the data-fitting term is in the ℓ2 norm. The LAD and the LMN solutions are formulated as the solutions of a linear and a quadratic programming problems respectively, and solved by interior point methods. At each iteration of the interior point method, a structured linear system must be solved. The preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to such structured inner systems. Experimental results are used to demonstrate the effectiveness of our approach. We also show the quality of the restored images using the minimization of ℓ1 norm/mixed ℓ1 and ℓ2 norms is better than that using ℓ2 norm approach.

A novel sine transform based preconditioned conjugate gradient (PCG) method for signal restoration is presented. If the bandwidth 2? +1 in the model matrix is a constant, then the asymptotic complexity for the new algorithm is O(N) per iterative step, where Nis the order of the model. This complexity is reduced by an order of magnitude in comparison with the known fast Fourier transform (FFT) technique proposed recently in the literature. Moreover, if ? has a size similar to N, then the new PCG method is accomplished by the fast sine transform (FST) and the fast Hartley transform (FHT). The computational and storage cost per preconditioning step for the new PCG method is reduced by 50% as compared to the FFT approach. To stabilize the computation, a special Tikhonov regularization is introduced. Numerical experimentations show that the new PCG method is of a speedy convergence rate. The new PCG methods maintain less computational complexity per iterative step and have a better convergence rate than the other known PCG methods.

Parallel magnetic resonance imaging (pMRI) which utilizes the redundant information from different coil sensitivities has been widely used. As the number of coils increases, algorithms that reconstruct single combined image are more suitable for pMRI. Most single combined image reconstruction need explicit functions of coil sensitivity. As a result, estimation of the coil sensitivities is important in pMRI. Nowadays, ESPIRiT(Iterative self-consistent parallel imaging reconstruction using eigenvector maps) based on the estimation of coil sensitivities is an outstanding algorithm. However, ESPIRiT has its limitation, which estimates the coil sensitivities not very correctly, while the autocalibration data is insufficient. In this paper, we focus on the pMRI algorithms based on improving the estimation of coil sensitivities and propose a new algorithm called GESPIRiT (ESPIRiT combined with GRAPPA), which is based on combining ESPIRiT with GRAPPA to obtain better estimation of coil sensitivities and final reconstructed image. Experiments are applied to demonstrate its feasibility and efficiency.

To improve the image reconstruction for prospective motion correction (PMC) of simultaneous multislice (SMS) EPI of the brain, an update of receiver phase and resampling of coil sensitivities are proposed and evaluated. A camera-based system was used to track head motion (3 translations and 3 rotations) and dynamically update the scanposition and orientation. We derived the change in receiver phase associated with a shifted field of view (FOV) and applied it in real-time to each k-space line of the EPI readout trains. Second, for the SMS reconstruction, we adapted resampled coil sensitivity profiles reflecting the movement of slices. Single-shot gradient-echo SMS-EPI scans were performed in phantoms and human subjects for validation. Brain SMS-EPI scans in the presence of motion with PMC and no phase correction for scan plane shift showed noticeable artifacts. These artifacts were visually and quantitatively attenuated when corrections were enabled. Correcting misaligned coil sensitivity maps improved the temporal SNR (tSNR) of time series by 24% (p = 0.0007) for scans with large movements (up to ˜35 mm and 30°). Correcting the receiver phase improved the tSNR of a scan with minimal head movement by 50% from 50 to 75 for a United Kingdom biobank protocol. Reconstruction-induced motion artifacts in single-shot SMS-EPI scans acquired with PMC can be removed by dynamically adjusting the receiver phase of each line across EPI readout trains and updating coil sensitivity profiles during reconstruction. The method may be a valuable tool for SMS-EPI scans in the presence of subject motion.

We propose an adaptive total generalized variation (TGV) based model, aiming at achieving a balance between edge preservation and region smoothness for image denoising. The variable splitting (VS) and the classical augmented Lagrangian method (ALM) are used to solve the proposed model. With the proposed adaptive model and ALM, the regularization parameter, which balances the data fidelity and the regularizer, is refreshed with a closed form in each iterate, and the image denoising can be accomplished without manual interference. Numerical results indicate that our method is effective in staircasing effect suppression and holds superiority over some other state‐of‐the‐art methods both in quantitative and in qualitative assessment.

Integrated variable projection approach (IVAPA) for parallel magnetic resonance imaging

We present an Augmented Lagrangian Method (ALM) for solving image reconstruction problems with a cost function consisting of multiple regularization functions with a data fidelity constraint. The presented technique is used to solve inverse problems related to image reconstruction, including compressed sensing formulations. Our contributions include an improvement for reducing the number of computations required by an existing ALM method, an approach for obtaining the proximal mapping associated with p-norm based regularizers, and lastly a particular ALM for the constrained image reconstruction problem with a hybrid cost function including a weighted sum of the p-norm and the total variation of the image. We present examples from Synthetic Aperture Radar imaging and Computed Tomography.

—GeoFEM (Iizuka et al., 1999) is a parallel finite element analysis system intended for multi-physics, multi-scale problems of solid earth field phenomena. Very large linear elastic problems have already been resolved by parallel computation with GeoFEM. The next stage is to examine large-scale nonlinear problems using GeoFEM. The analysis of large-scale contact problems for fault zones is particularly important in the development of models that simulate the occurrence and cycle of earthquakes.¶This paper proposes a parallel FEM using an iterative solver with the augmented Lagrange method for solving large-scale contact problems. Direct solvers are presently applied to contact problem analysis because the matrix for such problems is ill-conditioned. However, direct solvers are not suitable for large-scale matrices. The augmented Lagrange method can improve the matrix conditions. The present study evaluates a parallel FEM using an iterative solver with the augmented Lagrange method. Analysis of a contact problem with the augmented Lagrange method revealed that an optimal penalty parameter exists and that large-scale parallel contact analysis using the iterative solver with localized preconditioning is promising.

Quantitative SENSE-MRSI of the human brain

The use of multiple independent simultaneous radio-frequency (RF) transmitters and coils, known as parallel transmission, has the potential to make multidimensional excitation applicable to a wide range of magnetic resonance imaging applications. The sensitivity profile of the RF coils in a parallel transmission system determines the performance of the system. We present a theoretical framework, allowing the evaluation of the performance of a coil array for parallel transmission. We show through theoretical analysis and Monte Carlo simulation that the proposed framework predicts the fidelity of excitation that can be achieved by a given coil configuration in the presence of noise in the measured coil sensitivity profiles. We evaluate the fidelity of excitation achieved by four candidate coil configurations for a four-channel parallel transmission system with noisy coil sensitivity estimates. Theoretical results are confirmed with Monte Carlo simulation. The results give insight into the design of coil configurations for parallel transmission. In particular, optimal fidelity of excitation for subsampled Cartesian excitation k -space is achieved with a coil sensitivity profile having uniform amplitude and increasing linear phase for each channel. Such sensitivity profiles may be achieved with twisted birdcage coil designs.

Blind deconvolution, which comprises simultaneous blur and image estimations, is a strongly ill-posed problem. It is by now well known that if multiple images of the same scene are acquired, this multichannel (MC) blind deconvolution problem is better posed and allows blur estimation directly from the degraded images. We improve the MC idea by adding robustness to noise and stability in the case of large blurs or if the blur size is vastly overestimated. We formulate blind deconvolution as an l(1) -regularized optimization problem and seek a solution by alternately optimizing with respect to the image and with respect to blurs. Each optimization step is converted to a constrained problem by variable splitting and then is addressed with an augmented Lagrangian method, which permits simple and fast implementation in the Fourier domain. The rapid convergence of the proposed method is illustrated on synthetically blurred data. Applicability is also demonstrated on the deconvolution of real photos taken by a digital camera.

We propose a new fast algorithm for solving one of the standard formulations of frame-based image deconvolution: an unconstrained optimization problem, involving an $\ell_2$ data-fidelity term and a non-smooth regularizer. Our approach is based on using variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The resulting algorithm efficiently uses a regularized version of the Hessian of the data fidelity term, thus exploits second order information. Experiments on a set of image deblurring benchmark problems show that our algorithm is clearly faster than previous state-of-the-art methods.

In this paper, we denoise a given noisy image by minimizing a smoothness-promoting function over a set of local similarity measures which compare the mean of the given image and some candidate image on a large collection of subboxes. The associated convex optimization problem possesses a huge number of constraints which are induced by extended real-valued functions stemming from the Kullback–Leibler divergence. Alternatively, these nonlinear constraints can be reformulated as affine ones, which makes the model seemingly more tractable. For the numerical treatment of both formulations of the model (i.e., the original one as well as the one with affine constraints), we propose a rather general augmented Lagrangian method which is capable of handling the huge amount of constraints. A self-contained, derivative-free, global convergence theory is provided, allowing an extension to other problem classes. For the solution of the resulting subproblems in the setting of our suggested image denoising models, we make use of a suitable stochastic gradient method. Results of several numerical experiments are presented in order to compare both formulations and the associated augmented Lagrangian methods.

In this paper, we present a new regularization paradigm for inverse based regularized image reconstruction techniques. These methods usually attempt to minimize a cost function expressed as the sum of a data-fitting term and a regularization term. The trade-off between both terms is determined by a weighting parameter that has to be set by the user since this trade-off is data dependent. In the approach we present here, we first concentrate on finding a set of eligible candidates for the data fitting term minimization and then select the most appropriate candidate according to the regularization criterion. The main advantage of this method is that it does not require any weighting parameter, and guarantees that no over-regularization can occur. We illustrate this method with a super-resolution reconstruction technique to show its efficiency compared to other competitive methods. Comparisons are carried out with simulated and real data.