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**21 - 25**of**25**### Sparse Models and Multiple Receivers

, 2012

"... Magnetic resonance imaging (MRI) is an increasingly versatile diagnostic tool for a variety of medical purposes. During a conventional MRI scan, samples are acquired along a trajectory in the spatial Fourier transform domain (called k-space) and the image is reconstructed using an inverse discrete F ..."

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Magnetic resonance imaging (MRI) is an increasingly versatile diagnostic tool for a variety of medical purposes. During a conventional MRI scan, samples are acquired along a trajectory in the spatial Fourier transform domain (called k-space) and the image is reconstructed using an inverse discrete Fourier transform. The affordability, availability, and applications of MRI remain limited by the time required to sam-ple enough points of k-space for the desired field of view (FOV), resolution, and signal-to-noise ratio (SNR). GRAPPA, an accelerated parallel imaging method, and compressed sensing (CS) have been successfully employed to accelerate the acquisi-tion process by reducing the number of k-space samples required. GRAPPA leverages the different spatial weightings of each receiver coil to undo the aliasing from the re-duction in FOV induced by undersampling k-space. However, accelerated parallel imaging reconstruction methods like GRAPPA amplify the noise present in the data,

### Running head: An Eigenvalue Approach to Parallel Imaging

"... Purpose: Parallel imaging allows the reconstruction of images from undersampled multi-coil data. The two main approaches are: SENSE, which explicitly uses coil sensitivities, and GRAPPA, which makes use of learned correlations in k-space. The purpose of this work is to clarify their relationship and ..."

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Purpose: Parallel imaging allows the reconstruction of images from undersampled multi-coil data. The two main approaches are: SENSE, which explicitly uses coil sensitivities, and GRAPPA, which makes use of learned correlations in k-space. The purpose of this work is to clarify their relationship and to develop and evaluate an improved algorithm. Theory and Methods: A theoretical analysis shows: 1. The correlations in k-space are encoded in the null space of a calibration matrix. 2. Both approaches restrict the solution to a subspace spanned by the sensitivities. 3. The sensitivities appear as the main eigenvector of a reconstruction operator computed from the null space. The basic assumptions and the quality of the sensitivity maps are evaluated in experimental examples. The

### IEEE TRANSACTIONS ON MEDICAL IMAGING 1 Accelerated Regularized Estimation of MR Coil Sensitivities Using Augmented Lagrangian Methods

"... Abstract—Several magnetic resonance (MR) parallel imaging techniques require explicit estimates of the receive coil sensitivity profiles. These estimates must be accurate over both the object and its surrounding regions to avoid generating artifacts in the reconstructed images. Regularized estimatio ..."

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Abstract—Several magnetic resonance (MR) parallel imaging techniques require explicit estimates of the receive coil sensitivity profiles. These estimates must be accurate over both the object and its surrounding regions to avoid generating artifacts in the reconstructed images. Regularized estimation methods that involve minimizing a cost function containing both a data-fit term and a regularization term provide robust sensitivity estimates. However, these methods can be computationally expensive when dealing with large problems. In this paper, we propose an iterative algorithm based on variable splitting and the augmented Lagrangian method that estimates the coil sensitivity profile by minimizing a quadratic cost function. Our method, ADMM–Circ, reformulates the finite differencing matrix in the regularization term to enable exact alternating minimization steps. We also present a faster variant of this algorithm using intermediate updating of the associated Lagrange multipliers. Numerical experiments with simulated and real data sets indicate that our proposed method converges approximately twice as fast as the preconditioned conjugate gradient method (PCG) over the entire field-of-view. These concepts may accelerate other quadratic optimization problems. Index Terms—Augmented Lagrangian, coil sensitivity, finite differences, parallel imaging, quadratic minimization. I.

### Magnetic Resonance in Medicine 000:000–000 (2011) Parallel Imaging With Nonlinear Reconstruction Using Variational Penalties

"... A new approach based on nonlinear inversion for autocalibrated parallel imaging with arbitrary sampling patterns is presented. By extending the iteratively regularized Gauss–Newton method with variational penalties, the improved reconstruction qual-ity obtained from joint estimation of image and coi ..."

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A new approach based on nonlinear inversion for autocalibrated parallel imaging with arbitrary sampling patterns is presented. By extending the iteratively regularized Gauss–Newton method with variational penalties, the improved reconstruction qual-ity obtained from joint estimation of image and coil sensitivi-ties is combined with the superior noise suppression of total variation and total generalized variation regularization. In addi-tion, the proposed approach can lead to enhanced removal of sampling artifacts arising from pseudorandom and radial sam-pling patterns. This is demonstrated for phantom and in vivo