Limiting Aspects of Nonconvex TV theta Models

Type
Article

Authors
Hintermueller, Michael
Valkonen, Tuomo
Wu, Tao

KAUST Grant Number
KUK-I1-007-43

Date
2015

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.

Citation
Hintermüller, M., Valkonen, T., & Wu, T. (2015). Limiting Aspects of Nonconvex Models. SIAM Journal on Imaging Sciences, 8(4), 2581–2621. doi:10.1137/141001457

Acknowledgements
A large part of this work was done while the corresponding author was at the Center for Mathematical Modeling, Escuela Polit´ecnica Nacional, Quito, Ecuador, where his work was supported by a Prometeo scholarship of the Senescyt. In Cambridge, this author was supported by the King Abdullah University of Science and Technology (KAUST) Award KUK-I1-007-43 and the EPSRC first grant EP/J009539/1“Sparse & Higher-order Image Restoration.”

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM JOURNAL ON IMAGING SCIENCES

DOI
10.1137/141001457

arXiv
1412.7572

Additional Links
http://epubs.siam.org/doi/10.1137/141001457

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