Discrete Green's Functions for Harmonic and Biharmonic Inpainting with Sparse Atoms

Recent research has shown that inpainting with the Laplace or biharmonic operator has a high potential for image compression, if the stored data is optimised and sufficiently sparse. The goal of our paper is to connect these linear inpainting methods to sparsity concepts. To understand these relations, we explore the theory of Green’s functions. In contrast to most work in the mathematical literature, we derive our Green’s functions in a discrete setting and on a rectangular image domain with homogeneous Neumann boundary conditions. These discrete Green’s functions can be interpreted as columns of the Moore–Penrose inverse of the discretised differential operator. More importantly, they serve as atoms in a dictionary that allows a sparse representation of the inpainting solution. Apart from offering novel theoretical insights, this representation is also simple to implement and computationally efficient if the inpainting data is sparse.