Morphological floodings and optimal cuts in hierarchies

The non-horizontal cuts of a hierarchy and the floodings of an image are well-established tools for image segmenting and filtering respectively. We present definitions of non-horizontal cuts and of floodings in the same framework of hierarchies of partitions. We show that, given a hierarchy, there is a one-to-one correspondence between the non-horizontal cuts and the floodings. This opens the door to optimal image filtering based on non-horizontal cuts and, conversely, to nonhorizontal cuts obtained by morphological floodings, or more generally by connected filterings.