Regularization by Denoising via Fixed-Point Projection (RED-PRO)

Inverse problems in image processing are typically cast as optimization tasks, consisting of data fi-delity and stabilizing regularization terms. A recent regularization strategy of great interest utilizes the power of denoising engines. Two such methods are the plug-and-play prior (PnP) and regu-larization by denoising (RED). While both have shown state-of-the-art results in various recovery tasks, their theoretical justification is incomplete. In this paper, we aim to bridge RED and PnP, enriching the understanding of both frameworks. Toward that end, we reformulate RED as a convex optimization problem utilizing a projection (RED-PRO) onto the fixed-point set of demicontractive denoisers. We offer a simple iterative solution to this problem, by which we show that under certain conditions the PnP proximal gradient method is a special case of RED-PRO, while providing guarantees for the convergence of both frameworks to globally optimal solutions. In addition, we present relaxations of RED-PRO that allow for handling denoisers with limited fixed-point sets. Finally, we demonstrate RED-PRO for the tasks of image deblurring and superresolution, showing improved results with respect to the original RED framework.