  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.
