Numerical Linear Algebra Days (Due giorni di Algebra Lineare Numerica)

Fractional graph Laplacian for image reconstruction

Not scheduled

20m

Gran Sasso Science Institute, L'Aquila

Poster

Speaker

Stefano Aleotti (Università degli studi dell'Insubria)

Description

Image reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization.
A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an $\ell^2$ term and an $\ell^q$ term with $0

In this work, we propose to use the fractional Laplacian of a properly constructed graph in the $\ell^q$ term to compute extremely accurate reconstructions of the desired images.
A simple model with a fully plug-and-play method is used to construct the graph and enhanced diffusion on the graph is achieved with the use of a fractional exponent in the Laplacian operator. Since this is a global operator, we propose to replace it with an approximation in an appropriate Krylov subspace.

[1] D. Bianchi, A. Buccini, M. Donatelli, E. Randazzo,
"Graph Laplacian for image deblurring", Electronic Transactions on Numerical Analysis, 2021, 55, pp. 169-186.
[2] A. Buccini, M. Donatelli, "Graph Laplacian in $\ell^2-\ell^q$ regularization for image reconstruction", Proceedings - 2021 21st International Conference on Computational Science and Its Applications, ICCSA 2021, 2021, pp. 29-38.
[3] S. Aleotti, A. Buccini, M. Donatelli, "Fraction Graph Laplacian for image reconstruction", in progress.