Abstract
In imaging problems, the graph Laplacian is proven to be a very effective regularization operator when a good approximation of the image to restore is available. In this paper, we study a Tikhonov method that embeds the graph Laplacian operator in a \(\ell _1\)–norm penalty term. The novelty is that the graph Laplacian is built upon a first approximation of the solution obtained as the output of a trained neural network. Numerical examples in 2D computerized tomography demonstrate the efficacy of the proposed method.
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Acknowledgments
We want to thank the authors of [6] to share the dataset and the authors of [4] to share the code. Davide Bianchi is supported by NSFC (grant no. 12250410253). Wenbin Li is supported by Natural Science Foundation of Shenzhen (grant no. JCYJ20190806144005645) and NSFC (grant no. 41804096). Marco Donatelli is partially supported by GNCS (project 2022 “Tecniche numeriche per lo studio dei problemi inversi e l’analisi delle reti complesse”).
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Bianchi, D., Donatelli, M., Evangelista, D., Li, W., Piccolomini, E.L. (2023). Graph Laplacian and Neural Networks for Inverse Problems in Imaging: GraphLaNet. In: Calatroni, L., Donatelli, M., Morigi, S., Prato, M., Santacesaria, M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009. Springer, Cham. https://doi.org/10.1007/978-3-031-31975-4_14
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