A standard model for image reconstruction involves the minimization of a data-fidelity term along with a regularizer, where the optimization is performed using proximal algorithms such as ISTA and ADMM. In plug-and-play (PnP) regularization, the proximal operator (associated with the regularizer) in ISTA and ADMM is replaced by a powerful image denoiser. Although PnP regularization works surprisingly well in practice, its theoretical convergence -- whether convergence of the PnP iterates is guaranteed and if they minimize some objective function -- is not completely understood even for simple linear denoisers such as nonlocal means. In particular, while there are works where either iterate or objective convergence is established separately, a simultaneous guarantee on iterate and objective convergence is not available for any denoiser to our knowledge. In this paper, we establish both forms of convergence for a special class of linear denoisers. Notably, unlike existing works where the focus is on symmetric denoisers, our analysis covers non-symmetric denoisers such as nonlocal means and almost any convex data-fidelity. The novelty in this regard is that we make use of the convergence theory of averaged operators and we work with a special inner product (and norm) derived from the linear denoiser; the latter requires us to appropriately define the gradient and proximal operators associated with the data-fidelity term. We validate our convergence results using image reconstruction experiments.

翻译：图像重建的标准模型涉及将数据- 忠实化术语和常规化词一起与常规化词一起最小化,在常规化中,优化使用如ISTA和ADMM等近似算法进行。在插接和游戏(PnP)正规化(PnPTA和ADMM)中,将ISTA和ADMM的近似操作者(与正规化者相关)替换为强大的图像代谢器。虽然PnPP的正规化在实践中效果令人惊讶,但其理论趋同 -- -- 是否保证PnP Iterates的趋同,如果它们最大限度地减少某种客观功能 -- -- 即使在使用非本地手段等简单线性直线式隐隐含器的情况下,这种优化也是不完全理解的。 特别是,在进行循环化或客观趋同的趋同(我们非本地手段和几乎所有相联的)数据趋同性趋同性趋同性操作者的工作, 与我们正统化的正统化和正统性变相的变现性操作者使用后期 。