Keywords: Di erent optimal structures: minimum cuts, minimum spanning forests and shortest-path forests, have been used as the basis for powerful image segmentation procedures. The well-known notion of watershed also falls into this category. In this paper, we present some new results about the links which exist between these di erent approaches. Especially, we show that min-cuts coincide with watersheds for some particular weight functions. min-cuts, spanning forests, watersheds, shortest-path forests.

This paper analyzes the robustness issue in three segmentation approaches: the iterative relative fuzzy object extraction, the watershed transforms (WT) by image foresting transform and by minimum spanning forest. These methods need input seeds, which can be source of variability in the segmentation result. So, the robustness of these segmentation methods in relation to the input seeds is focused. The core of each seed is defined as the region where the seed can be moved without altering the segmentation result. We demonstrate that the core is identical for the three methods providing that the tie-zone transform has previously been applied on these methods. Indeed, as the two WT approaches do not return unique solution, the set of possible solutions has to be considered in a unified solution. So does the tie-zone transform. As opposed to what we could think, we show that the core is included in but different from the catchment basin. We also demonstrate that the tie-zone transforms of these WTs are always identical. Furthermore, the framework of minimal sets of seeds, an inverse problem of segmentation, is extended to the pixel level and related to the cores. A new algorithm for the computation of minimal seed sets is finally proposed. 1.