Cross-validation and predicted risk estimation for nonlinear iterative reweighted least-squares MRI reconstruction

Regularization is an effective means of reducing noise and artifacts in MR image reconstruction from undersampled k-space data. Proper application of regularization demands appropriate selection of the associated regularization parameter. Generalized cross-validation (GCV) is a popular parameter tuning technique especially for linear reconstruction methods, but its application to nonlinear iterative MRI reconstruction is more involved as it demands the evaluation of the Jacobian matrix of the reconstruction algorithm with respect to complex-valued data. We derive analytical expressions for recursively updating this Jacobian matrix for an iterative reweighted least-squares reconstruction algorithm. Our method can also be used to calculate a predicted risk estimate (PSURE) for MRI based on Stein´s principle. We demonstrate with simulations and experiments with real data that regularization parameter selection based on GCV and PSURE provides near-MSE-optimal results for nonlinear MRI reconstruction from undersampled k-space data using ℓ₁-regularization.