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 (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 Z 2 , the (6, 18), (18, 6), (6, 26), and (26, 6)-adjacencies in Z 3 , and more generally the (2n, 3 n − 1) and (3 n − 1, 2n)-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 ), and to potentially unify some of them.

The development of digital imaging (and its subsequent applications) has led to consider and investigate topological notions, well-defined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particular that the standard definition of path in algebraic topology is coherent w.r.t. the ones (often empirically) used in digital imaging. From this statement, we retrieve, and actually extend, an important result related to homotopy-type preservation, namely the equivalence between the fundamental group of a digital space and the group induced by digital paths. Based on this sound definition of paths, we also (re)explore various (and sometimes equivalent) ways to reduce a digital image in a homotopy-type preserving fashion.