ONE DIMENSIONAL FRACTIONAL ORDER TGV: GAMMA-CONVERGENCE AND BILEVEL TRAINING SCHEME*

New fractional r-order seminorms, TGV(r), r is an element of R, r >= 1, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order TGV(k)-seminorms, k is an element of 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.