Graphs and mathematical morphology

Abstract

In several fields of image processing (e.g. geography, histology, 2-D electrophoretic gels), neighborhood relationships between objects may be modelled by graphs. Now, a graph being a lattice, it can be processed by Mathematical Morphology: this theory provides a great number of powerful tools for studying these graphs. The present paper deals first with some theoretical aspects of Mathematical Morphology on lattices and graphs. It then presents various graphs that can be defined on a given set S of objects, depending on the intensity of the desirable neighborhood relationships (e.g. Delaunay triangulation, Gabriel graph, relative neighborhood graph, etc.). Algorithms for constructing these graphs are also explained. Depending on S, computational geometry techniques or new digital procedures, based on Euclidean distance and zones of influence, will be preferred. In a third section, the main operators of Mathematical Morphology are defined on graphs (erosions and dilations, morphological filters, distance function, skeletons, reconstruction, labelling, geodesic operators, etc.). Their properties and interest are mentioned. Fast algorithms for computing the transforms are also introduced.