Communications in Mathematical Sciences

Volume 13 (2015)

Number 1

Exact solutions of one-dimensional total generalized variation

Pages: 171 – 202



Christiane Pöschl (Department of Mathematics, Alpen Adria Universiät, Klagenfurt, Austria)

Otmar Scherzer (Computational Science Center, University of Vienna, Austria; Johann Radon Institute of Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria)


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.

Published 16 July 2014

Article revised on 6 August 2013 as follows: the typographical error on the first page, bottom -- "June 2103" -- has been corrected.