Fusion Graphs: Merging Properties and Watersheds

Abstract Region merging methods consist of improving an initial segmentation by merging some pairs of neighboring regions. In this paper, we consider a segmentation as a set of connected regions, separated by a frontier. If the frontier set cannot be reduced without merging some regions then we call it a cleft, or binary watershed. In a general graph framework, merging two regions is not straightforward. We define four classes of graphs for which we prove, thanks to the notion of cleft, that some of the difficulties for defining merging procedures are avoided. Our main result is that one of these classes is the class of graphs in which any cleft is thin. None of the usual adjacency relations on ℤ2 and ℤ3 allows a satisfying definition of merging. We introduce the perfect fusion grid on ℤn, a regular graph in which merging two neighboring regions can always be performed by removing from the frontier set all the points adjacent to both regions.