  We introduce a class of higher-order anisotropic total variation
regularisers, which are defined for possibly inhomogeneous, smooth elliptic
anisotropies, that extends the Total Generalized Variation (TGV) regulariser
and its variants. We propose a primal-dual hybrid gradient approach to
approximate numerically the associated gradient flow. This choice of
regularisers allows to preserve and enhance intrinsic anisotropic features in
images. This is illustrated on various examples from different imaging
applications: image denoising, wavelet-based image zooming, and reconstruction
of surfaces from scattered height measurements.
