Anchors of morphological operators and algebraic openings

Opening and closing operators play an important role in the field of mathematical morphology,
mainly because of their useful property of idempotence, which is similar to the notion of ideal
filter in linear filtering. From a theoretical point of view, the study of openings has focused
on the algebraic characterization of the operators themselves. Morphological filters have been
studied for more than 30 years; the effects of the first filters (erosions, openings, and so on) are
known in depth. In discussing the effects of a filter, it is not only the operator that is studied but
also its relationship with the processed function or image and, in the particular case of mathematical
morphology, the structuring element. In addition, there are several approaches to this
analysis. For example, the analysis can consider the whole function or some subparts of the
function, as in Van Droogenbroeck and Buckley (2005), who introduced the notion of morphological
anchors. Anchors were defined in the context of morphological openings, as defined
by the cascade of an erosion followed by a dilation. The extension to other kinds of openings is
not straightforward. Despite the fact that all morphological openings induce the appearance of
anchors, some opening operators (like the quantization opening defined in this chapter) might
have no anchors. This chapter presents the theory of anchors related to morphological erosions
and openings, and establishes some properties for the extended scope of algebraic openings.
It is shown under which circumstances anchors exist for algebraic openings and how to locate
some anchors; for example, it suffices for an opening to be spatial or shift-invariant to guarantee
the existence of anchors. As for morphological openings, the existence of anchors may
help clarify some algorithms or lead to new algorithms to compute algebraic openings.