Provable Preconditioned Plug-and-Play Approach for Compressed Sensing MRI Reconstruction

Model-based methods play a key role in the reconstruction of compressed sensing (CS) MRI. Finding an effective prior to describe the statistical distribution of the image family of interest is crucial for model-based methods. Plug-and-play (PnP) is a general framework that uses denoising algorithms as the prior or regularizer. Recent work showed that PnP methods with denoisers based on pretrained convolutional neural networks outperform other classical regularizers in CS MRI reconstruction. However, the numerical solvers for PnP can be slow for CS MRI reconstruction. This paper proposes a preconditioned PnP $\left({\text{P}}^2\text{nP}\right)$ method to accelerate the convergence speed. Moreover, we provide proofs of the fixed-point convergence of the ${\text{P}}^2\text{nP}$ iterates. Numerical experiments on CS MRI reconstruction with non-Cartesian sampling trajectories illustrate the effectiveness and efficiency of the ${\text{P}}^2\text{nP}$ approach.