Acta Applicandae Mathematicae

, Volume 113, Issue 2, pp 167–193 | Cite as

Paths, Homotopy and Reduction in Digital Images

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

Abstract

The development of digital imaging (and its subsequent applications) has led to consideration and investigation of topological notions that are 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.

Keywords

Topology Digital imaging Paths Fundamental group Homotopy-type preservation 

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Loïc Mazo
    • 1
    • 2
  • Nicolas Passat
    • 1
  • Michel Couprie
    • 2
  • Christian Ronse
    • 1
  1. 1.LSIIT, UMR CNRS 7005Université de StrasbourgIllkirch CedexFrance
  2. 2.Laboratoire d’Informatique Gaspard-MongeUniversité Paris-EstESIEE ParisFrance

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