Adaptive Image Processing: First Order PDE Constraint Regularizers and a Bilevel Training Scheme

Elisa Davoli; Irene Fonseca; Pan Liu

doi:10.1007/s00332-023-09902-4

Adaptive Image Processing: First Order PDE Constraint Regularizers and a Bilevel Training Scheme

J Nonlinear Sci. 2023;33(3):41. doi: 10.1007/s00332-023-09902-4. Epub 2023 Mar 3.

Authors

Elisa Davoli¹, Irene Fonseca², Pan Liu³

Affiliations

¹ Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria.
² Department of Mathematics, Center of Nonlinear Analysis, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213 USA.
³ Department of Radiology, First Medical Center of Chinese PLA General Hospital, Beijing, 100853 China.

Abstract

A bilevel training scheme is used to introduce a novel class of regularizers, providing a unified approach to standard regularizers $T G V^{2}$ and $N s T G V^{2}$ . Optimal parameters and regularizers are identified, and the existence of a solution for any given set of training imaging data is proved by $Γ$ -convergence under a conditional uniform bound on the trace constant of the operators and a finite-null-space condition. Some first examples and numerical results are given.

Keywords: Γ -convergence; First order differential operators; Image processing; Optimal training scheme.