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Publication 16-CNA-027

One dimensional fractional order TGV: Gamma-convergence and bi-level training scheme

Elisa Davoli
Department of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria

Pan Liu
Cambridge Image Analysis
Department of Applied Mathematics and Theoretical Physics
University of Cambridge, UK

Abstract: 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.

Get the paper in its entirety as  16-CNA-027.pdf

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