A data-dependent regularization method based on the graph Laplacian

Abstract

We investigate a variational method for ill-posed problems, named $\text{graphLa+}\hat{I}¨$, 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 $\hat{I}¨$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\text{graphLa+}\hat{I}¨$ 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 $\text{graphLa+}\hat{I}¨$ method with various reconstruction techniques $\hat{I}¨$, including Filter Back Projection ($\text{graphLa+FBP}$), standard Tikhonov ($\text{graphLa+Tik}$), Total Variation ($\text{graphLa+TV}$), and a trained deep neural network ($\text{graphLa+Net}$). The $\text{graphLa+}\hat{I}¨$ approach significantly enhances the quality of the approximated solutions for each method $\hat{I}¨$. Notably, $\text{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.