On optimality of parallel MRI reconstruction in k-space.

Parallel MRI reconstruction in k-space has several advantages, such as tolerance to calibration data errors and efficient non-Cartesian data processing. These benefits largely accrue from the approximation that a given unsampled k-space datum can be synthesized from only a few local samples. In this study, several aspects of parallel MRI reconstruction in k-space are studied: the design of optimized reconstruction kernels, the effect of regularization on image error, and the accuracy of different k-space-based parallel MRI methods. Reconstruction of parallel MRI data in k-space is posed as the problem of approximating the pseudoinverse with a sparse matrix. The error of the approximation is used as an optimization criterion to find reconstruction kernels optimized for the given coil setup. An efficient algorithm for automatic selection of reconstruction kernels is described. Additionally, a total error metric is introduced for validation of the reconstruction kernel and choice of regularization parameters. The new methods yield reduced reconstruction and noise errors in both simulated and real data studies when compared with existing methods. The new methods may be useful for reduction of image errors, faster data processing, and validation of parallel MRI reconstruction design for a given coil system and k-space trajectory.