Strong Euler well-composedness

Abstract

In this paper, we define a new flavour of well-composedness, called strong Euler well-composedness. In the general setting of regular cell complexes, a regular cell complex of dimension n is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is 1, which is the Euler characteristic of an $$(n-1)$$ ( n - 1 ) -dimensional ball. Working in the particular setting of cubical complexes canonically associated with $$n$$ n D pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension $$n\ge 2$$ n ≥ 2 and that the converse is not true when $$n\ge 4$$ n ≥ 4 .

On the Topological Disparity Characterization of Square-Pixel Binary Image Data by a Labeled Bipartite Graph

“…The calculation of Euler numbers is well-known in literature, and several papers provide algorithms to make fast calculations on large images, as can be seen in [3,6,20], not only for 2D-images, but also for 3D-images [19]. Moreover, there exists a strong relation between these properties and the "well-composedness" problem in image analysis [1,2].…”

“…The connected components (CC, for short) of these sets (and of their complements) do not depend on the chosen connectivity. In others words, in the case of rectangular pixels, a set D = I [1] ⊂ I is wellcomposed if and only if 8-adjacency (vertex-connectedness) implies 4-adjacency (edge-connectedness). It is clear that the two patterns consisting of four mutually 8-adjacent square pixels having four (Foreground or Background) 4-CCs are the only ones which are forbidden in well-composedness.…”