Limiting aspects of non-convex ${TV}^\phi$ 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 $\phi$ in the total variation type functional ${TV}^\phi(u) := \int \phi(|\nabla u(x)|) d x$. In this paper, it is demonstrated that for typical choices of $\phi$, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, ${BV}(\Omega)$. In particular, if $\phi(t)=t^q$ for $q \in (0, 1)$ and ${TV}^\phi$ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to ${BV}(\Omega)$, then still ${TV}^\phi(u)=\infty$ for $u$ not piecewise constant. If, on the other hand, ${TV}^\phi$ is defined analogously via continuously differentiable functions, then ${TV}^\phi \equiv 0$, (!). We study a way to remedy the models through additional multiscale regularisation and area strict convergence, provided that the energy $\phi(t)=t^q$ 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. 
Michael Hinterm\"uller, Tuomo Valkonen, and Tao Wu
math.FA math.OC