On the probabilities of hierarchical watersheds

International audienceHierarchical watersheds are obtained by iteratively merging the regions of a watershed segmentation. In the watershed segmentation of an image, each region contains exactly one (local) minimum of the original image. Therefore, the construction of a hierarchical watershed of any image I can be guided by a total order ≺ on the set of minima of I. The regions that contain the least minima according to the order ≺ are the first regions to be merged in the hierarchy. In fact, given any image I, for any hierarchical watershed H of I, there exists more than one total order on the set of minima of I which could be used to obtain H. In this article, we define the probability of a hierarchical watershed H as the probability of H to be the hierarchical watershed of I for an arbitrary total order on the set of minima of I. We introduce an efficient method to obtain the probability of hierarchical watersheds and we provide a characterization of the most probable hierarchical watersheds