Morphological filtering on graphs

Jean Cousty (1) , Laurent Najman (1) , Fabio Dias (1) , Jean Serra (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Mots clés: Graphs; mathematical morphology; adjunctions; spatially variant morphology

Domaines: Traitement des images [eess.IV]

Jean Cousty, Laurent Najman, Fabio Dias, Jean Serra. Morphological filtering on graphs. Computer Vision and Image Understanding, 2012, pp.1-20. ⟨hal-00700784v1⟩