  Inverse problems in image processing are typically cast as optimization
tasks, consisting of data-fidelity 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
Regularization 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 between RED and PnP, enriching the
understanding of both frameworks. Towards 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 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 super-resolution, showing improved results with respect to the
original RED framework.
