Watershed by image foresting transform, tie- zone, and theoretical relationships with other watershed definitions

To better understand the numerous solutions related to watershed transform (WT), this paper shows the relationships between some discrete definitions of the WT: the watersheds based on image foresting transform (IFT), topographic distance (TD), local condition (LC), and minimum spanning forest (MSF). We demonstrate that the tie-zone (TZ) concept, that unifies the multiple solutions of a given WT, when applied to the IFT-WT, includes all the solutions predicted by the other paradigms: the watershed line of TD-WT is contained in the TZ of the IFT-WT, while the catchment basins of the former contain the basins of the latter; any solution of LCWT or MSF-WT is also solution of the IFT-WT. Furthermore, the TD-WT can be seen as the TZ transform of the LC-WT.