Limiting aspects of non-convex ${TV}^\phi$ models

TL;DR

This paper investigates the limitations of non-convex TV models with concave energies, revealing issues in their extension to BV spaces and proposing remedies through multiscale regularisation and linearisation, supported by experiments.

Contribution

It identifies fundamental difficulties in extending non-convex TV models to BV spaces and proposes a new approach using multiscale regularisation and energy linearisation for better practical performance.

Findings

Non-convex TV models face extension issues to BV spaces.

Linearising energies improves model behavior for high values.

Numerical experiments show enhanced reconstructions with proposed methods.

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 ${TV}^\phi(u) := \int \phi(|\nabla u(x)|) d x$. 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.