Fractional graph Laplacian for image reconstruction
Stefano Aleotti, Alessandro Buccini, Marco Donatelli
Abstract
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 ℓ2 term and an ℓq term with 0<q≤1. The first penalizes the distance between the measured data and the reconstructed one, the latter imposes sparsity on some features of the computed solution. In this work, we propose to use the fractional Laplacian of a properly constructed graph in the ℓq term to compute extremely accurate reconstructions of the desired images. A simple model with a fully automatic method, i.e., that does not require the tuning of any parameter, 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 the fractional Laplacian is a global operator, i.e., its matrix representation is completely full, it cannot be formed and stored. We propose to replace it with an approximation in an appropriate Krylov subspace. We show that the algorithm is a regularization method under some reasonable assumptions. Some selected numerical examples in image deblurring and computer tomography show the performance of our proposal.