Parallel MR image reconstruction using augmented Lagrangian methods

Abstract

Magnetic resonance image (MRI) reconstruction using SENSitivity Encoding (SENSE) requires regularization to suppress noise and aliasing effects. Edge-preserving and sparsity-based regularization criteria can improve image quality, but they demand computation-intensive nonlinear optimization. In this paper, we present novel methods for regularized MRI reconstruction from undersampled sensitivity encoded data--SENSE-reconstruction--using the augmented Lagrangian (AL) framework for solving large-scale constrained optimization problems. We first formulate regularized SENSE-reconstruction as an unconstrained optimization task and then convert it to a set of (equivalent) constrained problems using variable splitting. We then attack these constrained versions in an AL framework using an alternating minimization method, leading to algorithms that can be implemented easily. The proposed methods are applicable to a general class of regularizers that includes popular edge-preserving (e.g., total-variation) and sparsity-promoting (e.g., l(1)-norm of wavelet coefficients) criteria and combinations thereof. Numerical experiments with synthetic and in vivo human data illustrate that the proposed AL algorithms converge faster than both general-purpose optimization algorithms such as nonlinear conjugate gradient (NCG) and state-of-the-art MFISTA.

Fig. 1

Experiment with synthetic data: (a) Noise-free T2-weighted MR image used for the experiment; (b) Poisson-disk-based sampling pattern (on a Cartesian grid) in the phase-encode plane with 80% undersampling (black spots represent sample locations); (c) SoS of sensitivity maps (S^HS) of coils; (d) Square-root of SoS of coil images (SNR = 9.52 dB) obtained by taking inverse Fourier transform of the undersampled data after filling the missing k-space samples with zeros (also the initial guess x⁽⁰⁾); (e) the solution x^(∞) to PO in (2) obtained by running MFISTA-20; (f) Absolute difference between (a) and (e). The goal of this work is to converge to the image x^(∞) in (e) quickly.

Fig. 2

Experiment with synthetic data: Plot of ξ(j) as a function time t_j for NCG, MFISTA, and AL-P1 and AL-P2. Both AL algorithms converge much faster than NCG and MFISTA.

Fig. 3

Experiment with in-vivo human brain data (Slice 38): (a) Body-coil image corresponding to fully-sampled phase-encode; (b) Poisson-disk-based k-space sampling pattern (on a Cartesian grid) with 84% undersampling (black spots represent sample locations); (c) Square-root of SoS of coil images obtained by taking inverse Fourier transform of the undersampled data after filling the missing k-space samples with zeros (also the initial guess x⁽⁰⁾); (d) the solution x^(∞) to P0 in (2) obtained by running MFISTA-20; (e) Absolute difference between (a) and (d) indicates that aliasing artifacts and noise have been suppressed considerably in the reconstruction (d).

Fig. 4

Experiment with in-vivo human brain data (Slice 90): (a) Body-coil image corresponding to fully-sampled phase-encodes; (b) Square-root of SoS of coil images obtained by taking inverse Fourier transform of the undersampled data after filling the missing k-space samples with zeros (also the initial guess x⁽⁰⁾); (c) the solution x^(∞) to P0 in (2) obtained by running MFISTA-20; (d) Absolute difference between (a) and (c) indicates that aliasing artifacts and noise have been suppressed considerably in the reconstruction (c).

Fig. 5

Experiment with in-vivo human brain data: Plot of ξ(j) as a function time t_j for NCG, MFISTA, AL-P1, and AL-P2 for the reconstruction of (a) Slice 38, and (b) Slice 90. The AL penalty parameter μ was manually tuned for fast convergence of AL-P1 for reconstructing Slice 38, while the same μ-value was used in AL-P1 for reconstructing Slice 90. For AL-P2, the “universal” setting (40)–(41) was used for reconstructing both slices. It is seen that the AL algorithms converge much faster than NCG and MFISTA in both cases. These results also indicate that the proposed condition-number-setting (40)–(41) provides agreeably fast convergence of AL-P2 for reconstructing multiple slices of a 3-D volume.