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

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

A Digital Index Theorem

Abstract This paper is devoted to prove a Digital Index Theorem for digital (n − 1)-manifolds in a digital space (R n, f), where f belongs to a large family of lighting functions on the standard cubical decomposition R n 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