The total generalized variation extends the total variation by incorporating higher-order smoothness. Thus, it can also suffer from similar discretization issues related to isotropy. Inspired by the success of novel discretization schemes of the total variation, there has been recent work to improve the second-order total generalized variation discretization, based on the same design idea. In this work, we propose to extend this to a general discretization scheme based on interpolation filters, for which we prove variational consistency. We then describe how to learn these interpolation filters to optimize the discretization for various imaging applications. We illustrate the performance of the method on a synthetic data set as well as for natural image denoising.

翻译：总广义变分通过引入高阶光滑性扩展了总变分，因此也可能面临与各向同性相关的类似离散化问题。受新型总变分离散化方案成功经验的启发，基于相同设计思想，近期已有工作改进了二阶总广义变分的离散化。在本研究中，我们提出将其推广为基于插值滤波器的一般离散化方案，并证明了该方案的变分一致性。随后，我们描述了如何通过学习这些插值滤波器来优化不同成像应用中的离散化效果。我们在合成数据集以及自然图像去噪任务中验证了该方法的性能。