SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 474-496, January 2020.
We analyze a new notion of total anisotropic higher-order variation which, differently from total generalized variation in [K. Bredies, K. Kunisch, and T. Pock, SIAM J. Imaging Sci., 3 (2010), pp. 492--526], quantifies for possibly nonsymmetric tensor fields their variations at arbitrary order weighted by possibly inhomogeneous, smooth elliptic anisotropies. We prove some properties of this total variation and of the associated spaces of tensors with finite variations. We show the existence of solutions to a related regularity-fidelity optimization problem. We also prove a decomposition formula which appears to be helpful for the design of numerical schemes, as shown in a companion paper, where several applications to image processing are studied.

中文翻译：

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高阶总方向变化：分析

SIAM影像科学杂志，第13卷，第1期，第474-496页，2020年1月。
我们分析了总各向异性高阶变化的新概念，它不同于[K]中的总广义变化。Bredies，K. Kunisch和T. Pock，SIAM J. Imaging Sci。，3（2010），pp。492--526]，量化了可能不对称的张量场在任意阶上的变化，并通过可能不均匀，光滑的椭圆各向异性加权。我们证明了这种总变化以及张量的相关空间具有有限变化的一些性质。我们显示了相关的正则保真优化问题的解决方案的存在。我们还证明了分解公式似乎对数字方案的设计很有帮助，如随附文件中所示，其中研究了图像处理的几种应用。