Pseudo-inverses of difference matrices and their application to sparse signal approximation

We derive new explicit expressions for the components of Moore–Penrose inverses of symmetric difference matrices. These generalized inverses are applied in a new regularization approach for scattered data interpolation based on partial differential equations. The columns of the Moore–Penrose inverse then serve as elements of a dictionary that allow a sparse signal approximation. In order to find a set of suitable data points for signal representation we apply the orthogonal matching pursuit (OMP) method.