Digital pseudomanifolds, digital weakmanifolds and Jordan–Brouwer separation theorem

AbstractIn this paper we introduce the new notion of n-pseudomanifold and n-weakmanifold in an (n+1)-digital image using (2(n+1),3(n+1)−1)-adjacency. For these classes, we prove the digital version of the Jordan–Brouwer separation theorem. To accomplish this objective, we construct a polyhedral representation of the (n+1)-digital image based on a cubical complex decomposition which enables us to translate some results from polyhedral topology into the digital space. Our main result extends the class of “thin” objects that are defined locally and verifying the Jordan–Brouwer separation theorem

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 digital index theorem

Proc. of the 7th Int. Workshop on Combinatorial Image Analysis. IWCIA00. Caen. France. July 2000.This paper is devoted to 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, 6),(18, 26),(26, 26), as well as for the strong 26-surfaces of Bertrand-Malgouyres.Dirección General de Investigación Científica y TécnicaDirección General de Enseñanza Superio

A boundary representation for extracting sharp surfaces from regularly-gridded 3d objects

Geometry extraction from volume data is important in many applications. On a regular 3d grid, current approaches do not consistently preserve object details such as sharp corners and edges of 26-connected objects. We describe a boundary representation in which we geometrically constrain the connectivity, so that such details can be maintained. Application of our model for object surfacing compares favorable to current surfacing methods

Digital homotopy with obstacles

As a sequel of [4] Ayala, R., E. Dom´ıguez, A. R. Franc´es and A. Quintero, Homotopy in Digital Spaces, Discrete and Applied Mathematics, To Appear, this paper is devoted to the computation of the digital fundamental group π d 1 (O/S; σ) defined by loops in the digital object O for which the digital object S acts as an “obstacle”. We prove that for arbitrary digital spaces the group π d 1 (O/S; σ) maps onto the usual fundamental group of the difference of continuous analogues |AO∪S | − |AS |. Moreover, we show that this epimorphism turns to be an isomorphism for a large class of digital spaces including most of the examples in digital topology.Dirección General de Enseñanza Superio

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

Boundaries in digital spaces

[EN] Intuitively, a boundary in an N-dimensional digital space is a connected component of the (N − 1)-dimensional surface of a connected object. In this paper we make these concepts precise, and show that the boundaries so specified have properties that are intuitively desirable. We provide some efficient algorithms for tracking such boundaries. We illustrate that the algorithms can be used, in particular, for computer graphic display of internal structures (such as the skull and the spine) in the human body based on the output of medical imaging devices (such as CT scanners). In the process some interesting mathematical results are proven regarding “digital Jordan boundaries,” such as a specification of a local condition that guarantees the global condition of “Jordanness.”The research of the author is currently supported by NIH grant HL070472 and NSF grant DMS0306215.Herman, GT. (2007). Boundaries in digital spaces. Applied General Topology. 8(1):93-149. doi:10.4995/agt.2007.1918.SWORD931498