Graph Laplacian in ℓ2 – ℓq regularization for image reconstruction

The use of the Laplacian of a properly constructed graph for denoising images has attracted a lot of attention in the last years. Recently, a way to use this instrument for image deblurring has been proposed. Even though the previously proposed method was able to provide extremely accurate reconstructions, it had several limitations, namely it was only applicable when periodic boundary conditions were employed, the regularization parameter had to be hand-tuned, and only convex regularization terms were allowed. In this paper, we propose two automatic methods that do not need the tuning of any parameter and that can be used for different imaging problems. Moreover, thanks to the projection into properly constructed subspaces of fairly small dimension, the proposed algorithms can be used for solving large scale problems.