Some Equivalence Relation between Persistent Homology and Morphological Dynamics

In mathematical morphology, connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from persistent homology and Morse theory that represents stability function. Since these two concepts seem be closely related, in this paper we examine their relationship, prove they equal n-D functions, $$n\ge 1$$ . More exactly, pairing minimum with 1-saddle by or same leads exactly pairing, assuming critical values studied function unique. This result step further show how much topological data analysis morphology paving way for more in-depth study relations between research fields.