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
elisa.davoli@univie.ac.at
Pan Liu
Cambridge Image Analysis
Department of Applied Mathematics and Theoretical Physics
University of Cambridge, UK
panliu.0923@maths.cam.ac.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.
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