Augmented Lagrangian with Variable Splitting for Faster Non-Cartesian ${\rm L}_{1}$-SPIRiT MR Image Reconstruction

SPIRiT (iterative self-consistent parallel imaging reconstruction), and its sparsity-regularized variant
${\rm L}_{1}$
-SPIRiT, are compatible with both Cartesian and non-Cartesian magnetic resonance imaging sampling trajectories. However, the non-Cartesian framework is more expensive computationally, involving a nonuniform Fourier transform with a nontrivial Gram matrix. We propose a novel implementation of the regularized reconstruction problem using variable splitting, alternating minimization of the augmented Lagrangian, and careful preconditioning. Our new method based on the alternating direction method of multipliers converges much faster than existing methods because of the preconditioners' heightened effectiveness. We demonstrate such rapid convergence substantially improves image quality for a fixed computation time. Our framework is a step forward towards rapid non-Cartesian
${\rm L}_{1}$
-SPIRiT reconstructions.