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Publication 16-CNA-027

The Optimal Parameters And Regularizers For Image Reconstruction Problems - One Dimensional Setting

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

Pan Liu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Abstract: A new fractional order seminorm, $ICTV^r$, $r\in \mathbb R$, $r\geq 1$, is proposed in the one-dimensional setting, as a generalization of the standard $ICTV^k$-seminorms, $k\in\mathbb{N}$. 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 $\alpha$ 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 $r$.

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