An Unrolled Implicit Regularization Network for Joint Image and Sensitivity Estimation in Parallel MR Imaging with Convergence Guarantee

Parallel imaging (PI), relying on multicoils to sense k-space data, is an effective technique to accelerate magnetic resonance imaging by exploiting spatial sensitivity coding of multiple coils, with an integrated compressive sensing (CS) technology to achieve higher acceleration. In this paper, we propose a novel nonconvex reconstruction model and its proximal alternating linearized minimization (PALM) algorithm for PI in a blind setting that MR image and multichannel sensitivity maps are jointly estimated, regularized by image and sensitivity regularizers. Instead of hand-crafting the image and sensitivity regularizers, we propose unrolling the PALM algorithm to be a deep network for Blind Parallel MRI, dubbed as BPMRI-Net, with two learnable subnetworks to substitute the proximal operators of the image and sensitivity regularizers. We theoretically prove the linear convergence of BPMRI-Net as an iterative algorithm, which alternately updates two variables based on the learnable proximal operators. The learned BPMRI-Net can simultaneously output the MR image and sensitivity maps from undersampled multichannel k-space data even when the number of low-frequency sampling lines in the center of k-space is small. Numerical results demonstrate the effectiveness of our method with state-of-the-art reconstruction accuracy.