# Higher-Order Total Directional Variation: Analysis

## TL;DR

This paper introduces a new anisotropic higher-order variation measure for tensor fields, analyzing its properties, associated function spaces, and solution existence for related optimization problems, with applications in image processing.

## Contribution

It proposes a novel total variation concept for tensor fields with anisotropic weights, expanding the theoretical framework beyond existing models.

## Key findings

- Proves properties of the new total variation measure.
- Establishes existence of solutions for related optimization problems.
- Provides a decomposition formula useful for numerical schemes.

## Abstract

We analyse a new notion of total anisotropic higher-order variation which, differently from the Total Generalized Variation by Bredies et al., quantifies for possibly non-symmetric 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 optimisation 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.