Image restoration remains a challenging task in image processing. Numerous methods tackle this problem, which is often solved by minimizing a nonsmooth penalized co-log-likelihood function. Although the solution is easily interpretable with theoretic guarantees, its estimation relies on an optimization process that can take time. Considering the research effort in deep learning for image classification and segmentation, this class of methods offers a serious alternative to perform image restoration but stays challenging to solve inverse problems. In this work, we design a deep network, named DeepPDNet, built from primal-dual proximal iterations associated with the minimization of a standard penalized co-log-likelihood with an analysis prior, allowing us to take advantage of both worlds. We reformulate a specific instance of the Condat-Vũ primal-dual hybrid gradient (PDHG) algorithm as a deep network with fixed layers. Each layer corresponds to one iteration of the primal-dual algorithm. The learned parameters are both the PDHG algorithm step-sizes and the analysis linear operator involved in the penalization (including the regularization parameter). These parameters are allowed to vary from a layer to another one. Two different learning strategies: “Full learning” and “Partial learning” are proposed, the first one is the most efficient numerically while the second one relies on standard constraints ensuring convergence of the standard PDHG iterations. Moreover, global and local sparse analysis prior are studied to seek a better feature representation. We apply the proposed methods to image restoration on the MNIST and BSD68 datasets and to single image super-resolution on the BSD100 and SET14 datasets. Extensive results show that the proposed DeepPDNet demonstrates excellent performance on the MNIST dataset compared to other state-of-the-art methods and better or at least comparable performance on the more complex BSD68, BSD100, and SET14 datasets for image restoration and single image super-resolution task.

中文翻译：

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图像还原的深原始对偶近端网络

图像恢复在图像处理中仍然是一项艰巨的任务。许多方法解决了这个问题，通常可以通过最小化不平滑的惩罚共对数似然函数来解决该问题。尽管可以通过理论上的保证轻松地解释该解决方案，但是其估计依赖于可能需要花费时间的优化过程。考虑到深度学习中用于图像分类和分割的研究成果，此类方法为执行图像还原提供了一种严肃的选择，但解决逆问题仍然充满挑战。在这项工作中，我们设计了一个名为DeepPDNet的深度网络，该深度网络是通过原始对偶近端迭代建立的，该迭代与标准罚分共对数似然率的最小化和先验分析相关联，从而使我们能够同时利用两个世界。我们将Condat-Vũ原始-双重混合梯度（PDHG）算法的特定实例重新构造为具有固定层的深层网络。每一层对应于原始对偶算法的一次迭代。学习的参数既是PDHG算法的步长，也是惩罚中涉及的分析线性算子（包括正则化参数）。这些参数允许从一层到另一层变化。提出了两种不同的学习策略：“完全学习”和“部分学习”，第一种是数值上最有效的，而第二种则依赖于确保标准PDHG迭代收敛的标准约束。此外，先验全局和局部稀疏分析以寻求更好的特征表示。我们将所提出的方法应用于MNIST和BSD68数据集上的图像恢复，以及应用于BSD100和SET14数据集上的单图像超分辨率。大量结果表明，与其他最新方法相比，拟议的DeepPDNet在MNIST数据集上表现出出色的性能，在更复杂的BSD68，BSD100和SET14数据集上，在图像恢复和单图像超级上表现出更好或至少可比的性能解决任务。