We consider a bilevel optimisation approach for parameter learning in
higher-order total variation image reconstruction models. Apart from the least
squares cost functional, naturally used in bilevel learning, we propose and
analyse an alternative cost, based on a Huber regularised TV-seminorm.
Differentiability properties of the solution operator are verified and a
first-order optimality system is derived. Based on the adjoint information, a
quasi-Newton algorithm is proposed for the numerical solution of the bilevel
problems. Numerical experiments are carried out to show the suitability of our
approach and the improved performance of the new cost functional. Thanks to the
bilevel optimisation framework, also a detailed comparison between TGV$^2$ and
ICTV is carried out, showing the advantages and shortcomings of both
regularisers, depending on the structure of the processed images and their
noise level.

研究使用双层优化方法在高阶总变分图像重建模型中进行参数学习，提出并分析一种基于 Huber 正则化 TV 半范数的替代代价，验证解算符的可微性，并推导出一种一阶最优系统。基于伴随信息，提出了一个拟牛顿算法来数值解决双层问题。通过数值实验展示了我们的方法的适用性和新代价函数的改进性能。同时，利用双层优化框架，对比了 TGV^2 和 ICTV，在处理图像结构和噪声水平方面的优缺点。