SIAM Journal on Imaging Sciences, Volume 14, Issue 3, Page 1374-1406, January 2021.
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 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.

中文翻译：

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通过定点投影（RED-PRO）去噪正则化

SIAM 成像科学杂志，第 14 卷，第 3 期，第 1374-1406 页，2021 年 1 月。
图像处理中的逆问题通常被视为优化任务，包括数据保真度和稳定正则化项。最近一个非常有趣的正则化策略利用了去噪引擎的力量。两种这样的方法是即插即用先验（PnP）和去噪正则化（RED）。虽然两者都在各种恢复任务中显示了最先进的结果，但他们的理论依据是不完整的。在本文中，我们旨在桥接 RED 和 PnP，丰富对这两个框架的理解。为此，我们将 RED 重新表述为凸优化问题，利用投影 (RED-PRO) 到非收缩降噪器的定点集上。我们为这个问题提供了一个简单的迭代解决方案，通过它我们表明在某些条件下 PnP 近端梯度方法是 RED-PRO 的一个特例，同时为两个框架收敛到全局最优解提供了保证。此外，我们展示了 RED-PRO 的松弛，允许处理具有有限定点集的降噪器。最后，我们展示了用于图像去模糊和超分辨率任务的 RED-PRO，显示了相对于原始 RED 框架的改进结果。