Angel R. Francés

Digital Lighting Functions

In this paper a notion of lighting function is introduced as an axiomatized formalization of the “face membership rules” suggested by Kovalevsky. These functions are defined in the context of the framework for digital topology previously developed by the authors. This enlarged framework provides the (α, β)-connectedness (α, β e {6,18, 26}) defined on ℤ3 within the graph-based approach to digital topology. Furthermore, the Kong-Roscoe (α, β)-surfaces, with (α, β) ≠ (6, 6), (18, 6), are also found as particular cases of a more general notion of digital surface.

Determining the components of the complement of a digital (n-1)-manifold in Zn

The goal of this paper is to determine the components of the complement of digital manifolds in the standard cubical decomposition of Euclidean spaces for arbitrary dimensions. Our main result generalizes the Morgenthaler-Rosenfelds one for (26, 6)-surfaces in ℤ3 [9]. The proof of this generalization is based on a new approach to digital topology sketched in [5] and developed in [2].

Determining whether a simplicial 3-complex collapses to a 1-complex is NP-complete

We show that determining whether or not a simplicial 2- complex collapses to a point is deterministic polynomial time decidable. We do this by solving the problem of constructively deciding whether a simplicial 2-complex collapses to a 1-complex. We show that this proof cannot be extended to the 3D case, by proving that deciding whether a simplicial 3-complex collapses to a 1-complex is an NP-complete problem.

A digital index theorem

This paper is devoted to state and prove a Digital Index Theorem for digital (n - 1)-manifolds in a digital space (Rn, f), where f belongs to a large family of lighting functions on the standard cubical decomposition Rn of the n-dimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)-surfaces of Kong–Roscoe, with α, β ∈ {6, 18, 26} and (α, β) ≠ (6, 6), (18, 26), (26, 26), as well as for the strong 26-surfaces of Bertrand–Malgouyres.

Separation Theorems for Simplicity 26-Surfaces

The main goal of this paper is to prove a Digital Jordan-Brouwer Theorem and an Index Theorem for simplicity 26-surfaces. For this, we follow the approach to Digital Topology introduced in [2], and find a digital space such that the continuous analogue of each simplicity 26-surface is a combinatorial 2-manifold. Thus, the separation theorems quoted above turn out to be an immediate consequence of the general results obtained in [2] and [3] for arbitrary digital n-manifolds.

A maximum set of (26,6)-connected digital surfaces

In the class

An axiomatic approach to digital topology

\mathcal{H}

Local characterization of a maximum set of digital (26,6)-surfaces

of (26,6)–connected homogeneous digital spaces on R3 we find a digital space EU with the largest set of digital surfaces in that class. That is, if a digital objet S is a digital surface in any space

A Digital Lighting Function for Strong 26-Surfaces

E \epsilon \mathcal{H}

A plate-based definition of discrete surfaces

then S is a digital surface in EU too.