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即插即用（PnP）算法。PnP方法是一种高效的迭代算法，用于求解图像逆问题，该问题被表述为数据保真项与正则化项之和的最小化。PnP方法通过在邻近算法（如邻近梯度下降（PGD））中插入预训练的去噪器来实现正则化。为确保PnP方案的收敛性，许多研究关注深度去噪器的特定参数化。然而，现有结果要么对去噪器提出不可验证或次优的假设，要么对逆问题的参数施加限制性条件。鉴于这些局限性可能源于所使用的邻近算法，我们研究了一种用于最小化凸函数与弱凸函数之和的松弛版PGD算法。当插入松弛邻近去噪器时，我们证明所提出的PnP-αPGD算法在更广泛的正则化参数范围内收敛，从而实现更精确的图像恢复。