A Frequency-Dependent Regularization for Autocalibrating Parallel MRI Using the Generalized Discrepancy Principle

This paper quantitatively evaluates regularized GRAPPA SPIRiT reconstruction in autocalibrating parallel MRI and develops a frequency-dependent regularization (FDR) aimed at achieving higher SNR with reduced penalty on image resolution. The latter is achieved by imposing an upper limit on the extent of regularization for each k -space location. The procedure starts with selection of a suitable truncation parameter, followed by perturbation of the AutoCalibrating Signal (ACS) lines using addition of white Gaussian noise samples of a predetermined variance. The noise variance chosen is such that first-order perturbation rules apply. In each perturbation step, the regularized solutions are estimated using a first-order update of singular values, and conditions stipulated by the generalized discrepancy principle (GDP) with regards to the norms of calibration and perturbation errors. The procedure is stopped at the crossover limit, once the GDP conditions are violated. The crossover information yields an upper limit for perturbation, enabling computation of an error bound as the difference between k-spaces estimated with filter weights from calibrations performed on original and the perturbed ACS lines at cross-over. Imposition of an upper limit on the extent of regularization is achieved by matching this error bound to the difference in k-spaces reconstructed with varying offsets of regularization from that of a reference. Furthermore, the crossover distance defined in terms of difference between minimum singular values of the initial and crossover calibration matrices serves as a proxy indicator of the achievable reduction in reconstruction error using regularization.