Digital Imaging: A Unified Topological Framework

Abstract

In 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 ( $\mathbb{F}^{n}$ ) of cubical complexes. In particular, it is shown that all the standard pairs of adjacencies (namely the (4,8) and (8,4)-adjacencies in ℤ2, the (6,18), (18,6), (6,26), and (26,6)-adjacencies in ℤ3, and more generally the (2n,3 n −1) and (3 n −1,2n)-adjacencies in ℤ n ) can then be correctly modelled in $\mathbb{F}^{n}$ . 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 $\mathbb{F}^{n}$ , 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 ( $\mathbb{F}^{n}$ ), and to potentially unify some of them.