Plug and play augmented HQS: Convergence analysis and its application in MRI reconstruction

Sparse recovery in the context of the inverse problem has become an enormously popular technique in reconstructing various degraded images in various applications. One of the well-known techniques in modularizing these particular inverse problems is the Plug and Play Half-Quadratic-Splitting (PnP-HQS). This method has been demonstrated to achieve good results in the literature; however, it is still too plain to be fully exploited for reconstruction. In this regard, we introduce an augmented version of this technique dubbed "PnP-AugHQS" to efficiently utilize its capabilities in image reconstruction. We provide a comprehensive convergence analysis of the proposed algorithm to ensure its effectiveness in image reconstruction. We then exploit the new parameters to further modify the procedure of the conventional PnP in order to account for the noise in the measurement. The PnP-AugHQS is equipped with a compact deep Convolutional Neural Network denoising regularization to maximize its power in image recovery. As a special case, we further modified the algorithm to be used in the application of MRI reconstruction. Various experiments evaluated on the proposed algorithm showed the superiority of the PnP-AugHQS compared to the PnP-HQS and other state-of-the-art techniques in MRI reconstruction.