Abstract
In this paper, we define a new flavour of well-composedness, called Euler well-composedness, in the general setting of regular cell complexes: A regular cell complex is Euler well-composed if the Euler characteristic of the link of each boundary vertex is $1$. A cell decomposition of a picture $I$ is a pair of regular cell complexes $\big(K(I),K(\bar{I})\big)$ such that $K(I)$ (resp. $K(\bar{I})$) is a topological and geometrical model representing $I$ (resp. its complementary, $\bar{I}$). Then, a cell decomposition of a picture $I$ is self-dual Euler well-composed if both $K(I)$ and $K(\bar{I})$ are Euler well-composed. We prove in this paper that, first, self-dual Euler well-composedness is equivalent to digital well-composedness in dimension 2 and 3, and second, in dimension 4, self-dual Euler well-composedness implies digital well-composedness, though the converse is not true.