Digital imaging: A unified topological framework

Loïc Mazo (1, 2) , Nicolas Passat (1) , Michel Couprie (2) , Christian Ronse (1)

1 LSIIT - Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

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 of cubical complexes (F^n). 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, 3n−1) and (3n−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.

Mots clés: homotopy; cubical complexes; digital imaging; fundamental group; digital topology

Domaines: Topologie géométrique [math.GT]; Vision par ordinateur et reconnaissance de formes [cs.CV]

Loïc Mazo, Nicolas Passat, Michel Couprie, Christian Ronse. Digital imaging: A unified topological framework. Journal of Mathematical Imaging and Vision, 2012, 44 (1), pp.19-37. ⟨10.1007/s10851-011-0308-9⟩. ⟨hal-00622529⟩