Limiting aspects of non-convex ${TV}^ϕ$ models
Recently, non-convex regularisation models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies $ϕ$ in the total variation type functional ${TV}^ϕ(u) := \int ϕ(|\nabla u(x)|) d x$. In this paper, it is demonstrated that for typical choices of $ϕ$, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, ${BV}(Ω)$. In particular, if $ϕ(t)=t^q$ for $q \in (0, 1)$ and ${TV}^ϕ$ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to ${BV}(Ω)$, then still ${TV}^ϕ(u)=\infty$ for $u$ not piecewise constant. If, on the other hand, ${TV}^ϕ$ is defined analogously via continuously differentiable functions, then ${TV}^ϕ\equiv 0$, (!). We study a way to remedy the models through additional multiscale regularisation and area strict convergence, provided that the energy $ϕ(t)=t^q$ is linearised for high values. The fact, that this kind of energies actually better matches reality and improves reconstructions, is demonstrated by statistics and numerical experiments.