A Data-Dependent Regularization Method Based on the Graph Laplacian

We investigate a variational method for ill-posed problems, named graphLa+Psi, 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 Psi 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+Psi is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in two-dimensional computed tomography, wherein we integrate the GraphLa+Psi method with various reconstruction techniques Psi, including filtered back projection (graphLa+FBP), standard Tikhonov (graphLa+Tik), total variation (graphLa+TV), and a trained deep neural network (graphLa+Net). The GraphLa+Psi approach significantly enhances the quality of the approximated solutions for each method Psi. Notably, graphLa+Net outperforms, offering a robust and stable application of deep neural networks in solving inverse problems.