A Fixed-Point Continuation Method for L_1-Regularization with Application to Compressed Sensing

Joint estimation of spin density $R_{2}^{\ast }$ decay and OFF-resonance frequency maps is very useful in many magnetic resonance imaging applications. The standard multi-echo approach can achieve high accuracy but requires a long acquisition time for sampling multiple $k$ -space frames. There are many approaches to accelerate the acquisition. Among them, single-shot or multi-shot trajectory-based sampling has recently drawn attention due to its fast data acquisition. However, this sampling strategy destroys the Fourier relationship between $k$ -space and images, leading to a great challenge for the reconstruction. In this paper, we present two trust region methods based on two different linearization strategies for the nonlinear signal model. A trust region is defined as a local area in the variable space where a local linear approximation is trustable. In each iteration, the method minimizes a local approximation within a trust region so that the step size can be kept in a suitable scale. A continuation scheme is applied to reduce the regularization gradually over the parameter maps and facilitates convergence from poor initializations. The two trust region methods are compared with the two other previously proposed methods—the nonlinear conjugate gradients and the gradual refinement algorithm. Experiments based on various synthetic data and real phantom data show that the two trust region methods have a clear advantage in both speed and stability.

Trust Region Methods for the Estimation of a Complex Exponential Decay Model in MRI With a Single-Shot or Multi-Shot Trajectory

${R}_{ {2}}^{*}$ mapping is a useful tool in blood-oxygen-level dependent fMRI due to its quantitative nature. However, like ${T}_{ {2}}^{*}$ -weighted imaging, standard ${R}_{{2}}^{*}$ mapping based on multi-echo EPI suffers from geometric distortion, due to strong off-resonance near the air–tissue interface. Joint mapping of ${R}_{{2}}^{*}$ and off-resonance can correct the geometric distortion and is less susceptible to motion artifacts. Single-shot joint mapping of ${R}_{{2}}^{*}$ and off-resonance is possible with a rosette trajectory due to its frequent sampling of the $k$ -space center. However, the corresponding reconstruction is nonlinear, ill-conditioned, large-scale, and computationally inefficient with current algorithms. In this paper, we propose a novel algorithm for joint mapping of ${R}_{{2}}^{*}$ and off-resonance, using rosette $k$ -space trajectories. The new algorithm, based on the alternating direction method of multipliers, improves the reconstruction efficiency by simplifying the original complicated cost function into a composition of simpler optimization steps. Compared with a recently developed trust region algorithm, the new algorithm achieves the same accuracy and an acceleration of threefold to sixfold in reconstruction time. Based on the new algorithm, we present simulation and in vivo data from single-shot, double-shot, and quadruple-shot rosettes and demonstrate the improved image quality and reduction of distortions in the reconstructed ${R}_{{2}}^{*}$ map.

An Efficient Reconstruction Algorithm Based on the Alternating Direction Method of Multipliers for Joint Estimation of ${R}_{{2}}^{*}$ and Off-Resonance in fMRI

Quantification of spin density, R∗ 2 decay and off-resonance frequency maps is very important in some applications of magnetic resonance imaging (MRI). To reconstruct these parameter maps, a time-varying model such as mono-exponentials must be used to represent the signal from each voxel. When only a single-shot trajectory is adopted, the underlying reconstruction problem is significantly nonlinear and therefore requires an iterative algorithm. The regularized trust region method previously proposed to address this problem is stable but lacks speed. In this paper, we propose a novel auxiliary variable method that is very efficient in solving the underlying optimization problem. This method introduces an auxiliary variable in the spatial-temporal domain that separates the data fidelity term and the structure fidelity term. The algorithm then alternately optimizes the data fidelity and the structure fidelity to reach the solution. The data fidelity optimization has a closed-form solution and can be solved very efficiently. The structure fidelity optimization fits the exponential model with the auxiliary variable and can also be rapidly computed. Some preliminary comparisons between the auxiliary variable method and the trust region method show that the new method is 10 times faster than the trust region method at a reasonable reconstruction precision.

An efficient auxiliary variable method for quantification of spin density, R ∗ 2 decay and field inhomogeneity maps in magnetic resonance imaging

Joint estimation of spin density, R∗ 2 decay and offresonance frequency maps is very useful in many magnetic resonance imaging (MRI) applications. The standard multi-echo approach can achieve high accuracy but requires a long acquisition time for sampling multiple k-space frames. There are many approaches to accelerate the acquisition. Among them, singleor multi-shot trajectory based sampling has recently drawn attention due to its fast data acquisition. However, this sampling strategy destroys the Fourier relationship between k-space and images, leading to a great challenge for the reconstruction. In this work, we present two trust region methods based on two different linearization strategies for the nonlinear signal model. A trust region is defined as a local area in the variable space where a local linear approximation is trustable. In each iteration, the method minimizes a local approximation within a trust region so that the step size can be kept in a suitable scale. A continuation scheme is applied to gradually reduce the regularization over the parameter maps and facilitate convergence from poor initializations. The two trust region methods are compared to two other previously proposed methods—the nonlinear conjugate gradients and the gradual refinement algorithm. Experiments based on various synthetic data and real phantom data show that the two trust region methods have a clear advantage in both speed and stability.

Trust region methods for estimation of a complex exponential decay model in MRI with a single-shot or multi-shot trajectory (in review)

Joint reconstruction of three parameter images — spin magnitude, T 2 ∗ decay and off-resonance field map — is an important technique in some applications of magnetic resonance imaging (MRI). Despite its importance, the reconstruction problem is very challenging due to its nonlinearity and ill-conditioning. The standard conjugate gradient (CG) method with regularization is usually very slow for this problem. In this work, we propose a novel regularized trust region (TR) method that reconstructs the three images by solving a constrained linear sub-problem in each iteration. In particular, this method employs a change-of-variable technique, making the sub-problem linear yet keeping the convergence fast. To solve the sub-problem, we use a penalty method with preconditioned conjugate gradient (PCG) algorithm. Based on a synthesized data set, we demonstrate that our algorithm stably converges even with a poor initial guess, and the convergence rate is much higher than that of the CG algorithm.

A regularized trust region method for joint reconstruction of spin magnitude, T 2 ∗ decay, and off-resonance field map

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Papers

An efficient auxiliary variable method for quantification of spin density, R ∗ 2 decay and field inhomogeneity maps in magnetic resonance imaging

A regularized trust region method for joint reconstruction of spin magnitude, T 2 ∗ decay, and off-resonance field map

Trust region methods for estimation of a complex exponential decay model in MRI with a single-shot or multi-shot trajectory (in review)