How to make nD functions digitally well-composed in a self-dual way

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.