Adaptive Anisotropic Total Variation - A Nonlinear Spectral Analysis

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....