  A fundamental concept in solving inverse problems is the use of regularizers,
which yield more physical and less-oscillatory solutions. Total variation (TV)
has been widely used as an edge-preserving regularizer. However, objects are
often over-regularized by TV, becoming blob-like convex structures of low
curvature. This phenomenon was explained mathematically in the analysis of
Andreau et al. They have shown that a TV regularizer can spatially preserve
perfectly sets which are nonlinear eigenfunctions of the form $\lambda u \in
\partial J_{TV}(u)$, where $\partial J_{TV}(u)$ is the TV subdifferential. For
TV, these shapes are convex sets of low-curvature. A compelling approach to
better preserve structures is to use anisotropic functionals, which adapt the
regularization in an image-driven manner, with strong regularization along
edges and low across them. This follows earlier ideas of Weickert on
anisotropic diffusion, which do not stem directly from functional minimization.
Adaptive anisotropic TV (A$^2$TV) was successfully used in several studies in
the past decade. However, until now there is no theory formulating the type of
structures which can be perfectly preserved. In this study we address this
question. We rely on a recently developed theory of Burger et al on nonlinear
spectral analysis of one-homogeneous functionals. We have that eigenfunction
sets, admitting $\lambda u \in \partial J_{A^2TV}(u)$, are perfectly preserved
under A$^2$TV-flow or minimization with $L^2$ square fidelity. We thus
investigate these eigenfunctions theoretically and numerically. We prove
non-convex sets can be eigenfunctions in certain conditions and provide
numerical results which characterize well the relations between the degree of
local anisotropy of the functional and the admitted maximal curvature....
