Papers › A data-dependent regularization method based on the graph Laplacian
A data-dependent regularization method based on the graph Laplacian
Davide Bianchi, Davide Evangelista, Stefano Aleotti, Marco Donatelli, Elena Loli Piccolomini, Wenbin Li
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We investigate a variational method for ill-posed problems, named graphLa+Ψ, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method Ψ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that graphLa+Ψ is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in 2D computerized tomography, wherein we integrate the graphLa+Ψ method with various reconstruction techniques Ψ, including Filter Back Projection (graphLa+FBP), standard Tikhonov (graphLa+Tik), Total Variation (graphLa+TV), and a trained deep neural network (graphLa+Net). The graphLa+Ψ approach significantly enhances the quality of the approximated solutions for each method Ψ. Notably, graphLa+Net is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.
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