Plug-and-play (PnP) method is a recent paradigm for image regularization, where the proximal operator (associated with some given regularizer) in an iterative algorithm is replaced with a powerful denoiser. Algorithmically, this involves repeated inversion (of the forward model) and denoising until convergence. Remarkably, PnP regularization produces promising results for several restoration applications. However, a fundamental question in this regard is the theoretical convergence of the PnP iterations, since the algorithm is not strictly derived from an optimization framework. This question has been investigated in recent works, but there are still many unresolved problems. For example, it is not known if convergence can be guaranteed if we use generic kernel denoisers (e.g. nonlocal means) within the ISTA framework (PnP-ISTA). We prove that, under reasonable assumptions, fixed-point convergence of PnP-ISTA is indeed guaranteed for linear inverse problems such as deblurring, inpainting and superresolution (the assumptions are verifiable for inpainting). We compare our theoretical findings with existing results, validate them numerically, and explain their practical relevance.

中文翻译：

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即插即用的 ISTA 与内核降噪器融合

即插即用 (PnP) 方法是图像正则化的最新范例，其中迭代算法中的近端算子（与某些给定的正则化器相关联）被强大的降噪器取代。从算法上讲，这涉及（前向模型的）重复反演和去噪直到收敛。值得注意的是，PnP 正则化为多个恢复应用程序产生了有希望的结果。然而，这方面的一个基本问题是 PnP 迭代的理论收敛性，因为该算法并非严格源自优化框架。这个问题在最近的工作中有所研究，但仍有许多未解决的问题。例如，如果我们在 ISTA 框架 (PnP-ISTA) 内使用通用内核去噪器（例如非局部均值），是否可以保证收敛是未知的。我们证明，在合理的假设下，PnP-ISTA 的定点收敛确实可以保证线性逆问题，例如去模糊、修复和超分辨率（这些假设对于修复是可验证的）。我们将我们的理论发现与现有结果进行比较，对它们进行数值验证，并解释它们的实际相关性。