Bilevel learning is a powerful optimization technique that has extensively been employed in recent years to bridge the world of model-driven variational approaches with data-driven methods. Upon suitable parametrization of the desired quantities of interest (e.g., regularization terms or discretization filters), such approach computes optimal parameter values by solving a nested optimization problem where the variational model acts as a constraint. In this work, we consider two different use cases of bilevel learning for the problem of image restoration. First, we focus on learning scalar weights and convolutional filters defining a Field of Experts regularizer to restore natural images degraded by blur and noise. For improving the practical performance, the lower-level problem is solved by means of a gradient descent scheme combined with a line-search strategy based on the Barzilai-Borwein rule. As a second application, the bilevel setup is employed for learning a discretization of the popular total variation regularizer for solving image restoration problems (in particular, deblurring and super-resolution). Numerical results show the effectiveness of the approach and their generalization to multiple tasks.

翻译：双层学习是一种强大的优化技术，近年来被广泛用于弥合基于模型驱动的变分方法与数据驱动方法之间的鸿沟。通过对目标量（如正则化项或离散化滤波器）进行合适的参数化，该方法通过求解一个嵌套优化问题——其中变分模型作为约束条件——来计算最优参数值。本文针对图像复原问题，考虑双层学习的两种不同应用场景。首先，聚焦于学习定义专家场正则化器的标量权重与卷积滤波器，以恢复受模糊和噪声退化的自然图像。为提升实际性能，底层问题采用结合Barzilai-Borwein准则的线搜索策略的梯度下降法求解。其次，将双层框架用于学习流行全变差正则化器的离散化形式，以解决图像复原问题（特别是去模糊和超分辨率）。数值结果表明了该方法在多项任务中的有效性及其泛化能力。