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.

中文翻译：

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即插即用算法的定点和目标收敛

用于图像重建的标准模型包括最小化数据保真度项以及正则化器，其中使用ISTA和ADMM等近端算法来执行优化。在即插即用（PnP）正则化中，ISTA和ADMM中的近端算子（与正则化器关联）被功能强大的图像降噪器取代。尽管PnP正则化在实践中表现出令人惊讶的良好效果，但即使对于简单的线性除噪器（如非局部均值），其理论收敛性（是否保证PnP迭代的收敛性，以及是否使某些目标函数最小化）也无法完全理解。尤其是，尽管有些作品中分别建立了迭代收敛或目标收敛，但对于我们所知的任何降噪器，都无法同时保证迭代和目标收敛。在本文中，我们为一类特殊的线性降噪器建立了两种收敛形式。值得注意的是，与现有工作侧重于对称降噪器不同，我们的分析涵盖了非对称降噪器，例如非局部均值和几乎任何凸数据保真度。在这方面的新颖之处在于，我们利用了平均算子的收敛理论，并且使用了从线性降噪器得到的特殊内积（和范数）。后者要求我们适当定义与数据保真度术语相关的梯度和近端算子。我们使用图像重建实验来验证我们的收敛结果。我们的分析涵盖非对称降噪器，例如非局部均值和几乎任何凸数据保真度。在这方面的新颖之处在于，我们利用了平均算子的收敛理论，并且使用了从线性降噪器得到的特殊内积（和范数）。后者要求我们适当定义与数据保真度术语相关的梯度和近端算子。我们使用图像重建实验来验证我们的收敛结果。我们的分析涵盖非对称降噪器，例如非局部均值和几乎任何凸数据保真度。在这方面的新颖之处在于，我们利用了平均算子的收敛理论，并且使用了从线性降噪器得到的特殊内积（和范数）。后者要求我们适当定义与数据保真度术语相关的梯度和近端算子。我们使用图像重建实验来验证我们的收敛结果。