This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnP methods are efficient iterative algorithms for solving image inverse problems formulated as the minimization of the sum of a data-fidelity term and a regularization term. PnP methods perform regularization by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD). To ensure convergence of PnP schemes, many works study specific parametrizations of deep denoisers. However, existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem. Observing that these limitations can be due to the proximal algorithm in use, we study a relaxed version of the PGD algorithm for minimizing the sum of a convex function and a weakly convex one. When plugged with a relaxed proximal denoiser, we show that the proposed PnP-$\alpha$PGD algorithm converges for a wider range of regularization parameters, thus allowing more accurate image restoration.

翻译：本文提出了一种新的收敛型即插即用（PnP）算法。PnP方法是高效迭代算法，用于求解将图像反问题表述为数据保真项与正则化项之和的最小化问题。PnP方法通过将预训练去噪器嵌入到近端算法（如近端梯度下降法（PGD））中来实现正则化。为确保PnP方案的收敛性，许多研究致力于深度去噪器的特定参数化设计。然而，现有结果要么要求去噪器满足不可验证或次优的假设，要么对反问题参数施加限制性条件。基于这些局限性可能源于所采用的近端算法的观察，我们研究了用于最小化凸函数与弱凸函数之和的松弛版PGD算法。当嵌入松弛近端去噪器时，我们证明所提出的PnP-αPGD算法在更宽的正则化参数范围内具有收敛性，从而能够实现更精确的图像复原。