Optimal Parameters and Regularizers for Image Reconstruction Problems - One Dimensional Setting

A new fractional order seminorm, ICTV r , r ∈ R , r ≥ 1 , is proposed in the one-dimensional setting, as a generalization of the standard ICTV k -seminorms, k ∈ N . The fractional ICTV r -seminorms are shown to be intermediate between the standard ICTV k -seminorms of integer order. A bilevel learning scheme is proposed, where under a box constraint a simultaneous optimization with respect to the parameter α and the order r is performed. A numerical implementation of the learning scheme is provided, as well as an example where the optimal reconstruction is achieved for non-integer values of the order of derivation r .