Publication 23-CNA-007

Dyadic Partition-Based Training Schemes for TV/TGV Denoising

Elisa Davoli
Institute of Analysis and Scientific Computing
TU Wien
Wiedner Hauptstrasse 8-10
1040 Vienna, Austria
elisa.davoli@tuwien.ac.at

Rita Ferreira
King Abdullah University of Science and Technology (KAUST)
CEMSE Division
Thuwal 23955-6900
Saudi Arabia
rita.ferreira@kaust.edu.sa

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

José A. Iglesias
Department of Applied Mathematics
University of Twente
The Netherlands
jose.iglesias@utwente.nl

Abstract: Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the bestknown methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters.

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