Morphological filtering on graphs

We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph G . To this end, we consider two dual adjunctions between the edge set and the vertex set of G . This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G . Afterward, we propose several new openings, closings, granulometries and alternate filters acting (i) on the subsets of the edge and vertex set of G and (ii) on the subgraphs of G . The proposed framework is then extended to functions that weight the vertices and edges of a graph. We illustrate with applications to binary and grayscale image denoising, for which, on the provided images, the proposed approach outperforms the usual one based on structuring elements.