Journal of Mathematical Imaging and Vision

, Volume 44, Issue 1, pp 19–37 | Cite as

Digital Imaging: A Unified Topological Framework

  • Loïc Mazo
  • Nicolas Passat
  • Michel Couprie
  • Christian Ronse
Article

Abstract

In this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on ℤn 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,3n−1) and (3n−1,2n)-adjacencies in ℤn) can then be correctly modelled in \(\mathbb{F}^{n}\). Moreover, it is established that the digital fundamental group of a digital image in ℤn 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 (ℤn) or cubical complexes (\(\mathbb{F}^{n}\)), and to potentially unify some of them.

Keywords

Digital imaging Digital topology Cubical complexes Homotopy Fundamental group 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Loïc Mazo
    • 1
    • 2
  • Nicolas Passat
    • 1
  • Michel Couprie
    • 2
  • Christian Ronse
    • 1
  1. 1.Université de Strasbourg, LSIITUMR CNRS 7005Illkirch CedexFrance
  2. 2.Université Paris-Est, Laboratoire d’Informatique Gaspard-Monge, Équipe A3SIESIEE ParisNoisy le Grand CedexFrance

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