Strong separating (k, k)−surfaces on Z3

For each adjacency pair (k, k) != (6, 6), k, k ∈ {6, 18, 26}, we introduce a new family Skk of surfaces in the discrete space Z3 that strictly contains several families of surfaces previously defined, and other objects considered as surfaces, in the literature. Actually, Skk characterizes the strongly k−separating objects of the family of digital surfaces, defined by means of continuous analogues, of the universal (k, k)−spaces introduced in [6]

A unified topological framework for digital imaging

International audienceIn this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on Z^n and equipped with a standard pair of adjacencies) as an image defined in the space of cubical complexes (F^n). In particular, it is shown that all the standard pairs of adjacencies in Z^n can then be correctly modelled in F^n. Moreover, it is established that the digital fundamental group of a digital image in Z^n is isomorphic to the fundamental group of its corresponding image in F^n, thus proving the topological correctness of the proposed approach. From these results, it becomes possible to establish links between topology-oriented methods developed either in classical digital spaces (Z^n) or cubical complexes (F^n)

Digital homotopy with obstacles

AbstractIn (Ayala et al. (Discrete Appl. Math. 125 (1) (2003) 3) it was introduced the notion of a digital fundamental group π1d(O/S;σ) for a set of pixels O in relation to another set S which plays the role of an “obstacle”. This notion intends to be a generalization of the digital fundamental groups of both digital objects and their complements in a digital space. However, the suitability of this group was only checked for digital objects in that paper. As a sequel, we extend here the results in Ayala et al. (2003) for complements of objects. More precisely, we prove that for arbitrary digital spaces the group π1d(O/S;σ) maps onto the usual fundamental group of the difference of continuous analogues |AO∪S|−|AS|. Moreover, this epimorphism turns to be an isomorphism for a large class of digital spaces including most of the examples in digital topology

Weak lighting functions and strong 26-surfaces

AbstractThe goal of this paper is to introduce the notion of weak lighting function in order to replicate the “continuous perception” associated with strong 26-surfaces. As a consequence, the continuous analogue defined ad hoc by Malgouyres and Bertrand only for these surfaces is extended for arbitrary objects, and the local characterization of finite strong 26-surfaces given in (Malgouyres and Bertrand, Int. J. Pattern Recognition Art. Intell. 13(4) (1999) 465–484) is generalized to possibly infinite surfaces. Moreover, weak lighting functions also replicate the “continuous perception” associated with (α,β)-surfaces, (α,β)≠(6,6), since they are generalizing the lighting functions previously defined by the authors