Plug-and-play (PnP) denoising is a popular iterative framework for solving imaging inverse problems using off-the-shelf image denoisers. Their empirical success has motivated a line of research that seeks to understand the convergence of PnP iterates under various assumptions on the denoiser. While a significant amount of research has gone into establishing the convergence of the PnP iteration for different regularity conditions on the denoisers, not much is known about the asymptotic properties of the converged solution as the noise level in the measurement tends to zero, i.e., whether PnP methods are provably convergent regularization schemes under reasonable assumptions on the denoiser. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with linear denoisers and propose a novel spectral filtering technique to control the strength of regularization arising from the denoiser. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we rigorously show that PnP with linear denoisers leads to a convergent regularization scheme. More specifically, we prove that in the limit as the noise vanishes, the PnP reconstruction converges to the minimizer of a regularization potential subject to the solution satisfying the noiseless operator equation. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.

翻译：即插即用（PnP）去噪是一种利用现成图像去噪器求解成像反问题的流行迭代框架。其经验成功推动了一系列研究，这些研究试图理解在不同去噪器假设条件下PnP迭代的收敛性。尽管已有大量研究致力于在去噪器的不同正则性条件下建立PnP迭代的收敛性，但对于当测量噪声趋于零时收敛解的渐近性质（即PnP方法在去噪器合理假设下是否能被证明是收敛正则化方案）仍知之甚少。本文有两个目的：首先，我们概述了反问题中的经典正则化理论，并回顾了近期若干可证明为收敛正则化方案的数据驱动方法。随后，我们继续讨论PnP算法及其已有的收敛性保证。接着，我们考虑采用线性去噪器的PnP算法，并提出一种新的谱滤波技术来控制由去噪器引入的正则化强度。此外，通过将去噪器的隐式正则化与显式正则化泛函联系起来，我们严格证明了线性去噪器PnP能够导出收敛正则化方案。更具体地，我们证明当噪声趋于零时，PnP重建结果收敛于满足无噪声算子方程的正则化势函数的极小化点。理论分析通过经典反问题——层析图像重建的数值实验得到了验证。