Communications in Mathematical Sciences
Volume 13 (2015)
Exact solutions of one-dimensional total generalized variation
Pages: 171 – 202
Total generalized variation regularization has been introduced by Bredies, Kunisch, and Pock [K. Bredies, K. Kunisch, and T. Pock, SIAM J. Imaging Sci., 3(3), 492–526, 2010]. This regularization method requires careful tuning of two regularization parameters. The focus of this paper is to derive analytical results, which allow for characterizing parameter settings, which make this method in fact different from total variation regularization (that is the Rudin-Osher-Fatmi model [L.I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60(1–4), 259–268, 1992]) and the second order variation model [O. Scherzer, Computing, 60(1), 1–27, 1998] regularization, respectively. In this paper we also provide explicit solutions of total generalized variation denoising for particular one-dimensional function data.
Fenchel duality, total variation, total generalized variation, bounded Hessian, $G$-norm, convex optimization
2010 Mathematics Subject Classification
Primary 46N10. Secondary 49M29.