# Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Preprint 2015-01

## Preprint 2015-01

Michael Hintermüller, Tuomo Valkonen and Tao Wu

Abstract: 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) := R \phi (|ru(x)|) dx. 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) = tq for q 2 (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) = 1 for u not piecewise constant. If, on the other hand, TV^\phi is defined analogously via continuously differentiable functions, then TV^\phi 0, (!). We study a way to remedy the models through additional multiscale regularisation and area strict convergence, provided that the energy \phi (t) = tq 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.

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2015-01

MSC 2000:26B30, 49Q20, 65J20

21 pp.