How to Make nD Functions Digitally Well-Composed in a Self-dual Way

Abstract

Abstract. Latecki et al. introduced the notion of 2D and 3D wellcomposed images, i.e., a class of images free from the "connectivities paradox" of digital topology. Unfortunately natural and synthetic images are not a priori well-composed. In this paper we extend the notion of "digital well-composedness" to nD sets, integer-valued functions (graylevel images), and interval-valued maps. We also prove that the digital well-composedness implies the equivalence of connectivities of the level set components in nD. Contrasting with a previous result stating that it is not possible to obtain a discrete nD self-dual digitally well-composed function with a local interpolation, we then propose and prove a selfdual discrete (non-local) interpolation method whose result is always a digitally well-composed function. This method is based on a sub-part of a quasi-linear algorithm that computes the morphological tree of shapes.

A Tutorial on Well-Composedness

“…In 2015, Boutry et al [29] have extended the seminal definition of well-composedness to n-D, n ≥ 2, in such a manner that a digital set X ⊂ Z n is said well-composed based on the equivalence of connectivities (EWC) iff the set of (3 n −1)-connected components X (respectively, of X c ) is equal to the set of 2n-connected components of X (respectively, of X c ). This definition will be detailed in Section 5.…”

“…↔ AGWC 2015 [29] 2015 [29] 2013 [127] 2000 [102] 2016 [26] topological boundary of the continuous analog of a set is a manifold is named well-composedness in the continuous sense or CWCness [29]. Well-composedness based on discrete surfaces in Alexandrov spaces is named wellcomposedness in the Alexandrov sense or AWCness [29].…”

“…Well-composedness based on discrete surfaces in Alexandrov spaces is named wellcomposedness in the Alexandrov sense or AWCness [29].…”

“…The natural extension in n-D of well-composedness such that the connected components of a set or of the complementary of this set do not depend on the chosen connectivity is named well-composedness based on the equivalence of connectivities or EWCness [29]. Wellcomposedness based on critical configurations, i.e., such that a set is said well-composed iff it does not contain any critical configuration [29], is named digital wellcomposedness or DWCness [29].…”

“…Wellcomposedness based on critical configurations, i.e., such that a set is said well-composed iff it does not contain any critical configuration [29], is named digital wellcomposedness or DWCness [29]. Finally, well-composedness on arbitrary grids is renamed in this paper under the name of AG-well-composedness.…”