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(\barI)\big) such that K(I) (resp. K(\barI)) is a topological and geometrical model representing I (resp. its complementary, \barI). Then, a cell decomposition of a picture I is self-dual Euler well-composed if both K(I) and K(\barI) 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.