Limiting Aspects of Nonconvex TV theta Models

Abstract

Recently, nonconvex regularization 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 TVφ (u):= ∫φ(|∇u(x)|) dx. 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) = tq for q ∈ (0, 1), and TVφis defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to BV(Ω), then it still holds that TVφ (u) = ∞ for u not piecewise constant. If, on the other hand, TVφ is defined analogously via continuously differentiable functions, then TVφ ≡ 0 (!). We study a way to remedy the models through additional multiscale regularization and area strict convergence, provided that the energy φ(t) = tq is linearized for high values. The fact that such energies actually better match reality and improve reconstructions is demonstrated by statistics and numerical experiments.