Watersheds, mosaics, and the emergence paradigm

In this paper, we investigate the links between the flooding paradigm and the topological watershed. Guided by the analysis of a classical flooding algorithm, we present several notions that lead us to a better understanding of the watershed: minima extension, mosaic, pass value and separation. We first make a detailed examination of the effectiveness of the divide set produced by watershed algorithms. We introduce the mosaic to retrieve the altitude of points along the divide set. A desirable property is that, when two minima are separated by a crest in the original image, they are still separated by a crest of the same altitude in the mosaic. Our main result states that this is the case if and only if the mosaic is obtained through a topological thinning. We investigate the possibility for a flooding to produce a topological watershed, and conclude that this is not feasible. This leads us to reverse the flooding paradigm, and to propose a notion of emergence. An emergence process is a transformation based on a topological criterion, in which points are processed in decreasing altitude order while preserving the number of connected components of lower cross-sections. Our main result states that any emergence watershed is a topological watershed, and more remarkably, that any topological watershed of a given image can be obtained as an emergence watershed of the image.