Graph Laplacian and Neural Networks for Inverse Problems in Imaging: GraphLaNet

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.