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

New fractional $r$-order seminorms, $TG{V}^r$, $r\in \mathbb{R}$, $r\ge 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TG{V}^k$-seminorms, $k\in \mathbb{N}$. The fractional $r$-order $TG{V}^r$-seminorms are shown to be intermediate between the integer order $TG{V}^k$-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 $\mathrm{\Gamma }$-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.