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模型驱动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）进行适当参数化，该方法通过求解一个以变分模型为约束的嵌套优化问题来计算最优参数值。本文针对图像复原问题，探讨了双层学习的两种不同应用场景。首先，我们聚焦于通过学习标量权重和卷积滤波器来定义专家场（Field of Experts）正则化器，以恢复受模糊和噪声退化的自然图像。为提升实际性能，下层问题通过结合基于Barzilai-Borwein规则的线搜索策略的梯度下降方案求解。其次，将双层框架用于学习流行全变分正则化器的离散化形式，以解决图像复原问题（尤其是去模糊与超分辨率）。数值结果表明了该方法的有效性及其对多任务的泛化能力。