The Strong Property of Morphological Connected Alternated Filters

This paper studies connectivity aspects that arise in image operators that process connected components. The focus is on morphological image analysis (i.e., on increasing image operators) and, in particular, on a robustness property satisfied by certain morphological filters that is denominated the strong property. The behavior of alternated compositions of openings and closings is investigated under certain assumptions, particularly connectedness and a connected component preserving condition. It is shown that these conditions cannot in general guarantee the strong property of certain connected alternated filters because of issues related to the locality of the filters. As treated in the paper, there have been a series of misunderstandings in the literature concerning this topic, and it is important to clarify them. The root cause of those problems is discussed, and a solution is indicated. The class of connected openings and closings used to build connected alternated filters should therefore be defined to avoid such situations, since the strong property of alternated filters should be a distinctive characteristic of this class.