One More Step Towards Well-Composedness of Cell Complexes over nD Pictures

Abstract

An nD pure regular cell complex K is weakly well-composed (wWC) if, for each vertex v of K, the set of n-cells incident to v is face-connected. In previous work we proved that if an nD picture I is digitally well composed (DWC) then the cubical complex Q(I) associated to I is wWC. If I is not DWC, we proposed a combinatorial algorithm to "locally repair" Q(I) obtaining an nD pure simplicial complex PS(I) homotopy equivalent to Q(I) which is always wWC. In this paper we give a combinatorial procedure to compute a simplicial complex PS( Ī) which decomposes the complement space of |PS(I)| and prove that PS( Ī) is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the nD repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is an (n − 1)manifold.Authors' names listed in alphabetical order.

Continuous Well-Composedness Implies Digital Well-Composedness in n-D

“…As the sequel of [7] where we prove thanks to a counter-example that DWCness does not imply CWCness in 4D, we prove in this paper that CWCness implies DWCness in n-D. Some other flavors of well-composednesses exist like well-composedness in the Alexandrov sense [20,2,9,8], well-composedness on arbitrary grids [23,2], weak wellcomposedness [5], or Euler well-composedness [6], but we will not go further into details here.…”