One dimensional fractional order $TGV$: gamma-convergence and bilevel training scheme

New fractional $r$-order seminorms, $TGV^r , r \in \mathbb{R} , r \geq 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TGV^k$-seminorms, $k \in \mathbb{N}$. The fractional $r$-order $TGV^r$-seminorms are shown to be intermediate between the integer order $TGV^k$-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by $\Gamma$-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.

Keywords

total generalized variation, fractional derivatives, optimization and control, computer vision and pattern recognition

2010 Mathematics Subject Classification

06B30, 47J20, 94A08

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P. Liu is partially funded by the National Science Foundation under Grant No. DMS-1411646. E. Davoli is supported by the Austrian Science Fund (FWF) projects P27052 and F65. The authors wish to thank Irene Fonseca for suggesting this project and for many helpful discussions and comments. The authors also thank Giovanni Leoni for comments on the subject of Section 2.

Received 15 December 2016

Received revised 8 November 2017

Published 29 March 2018