  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.
