Paths, Homotopy and Reduction in Digital Images

Authors:
Loïc Mazo;Nicolas Passat;Michel Couprie;Christian Ronse
Affiliations:
LSIIT, UMR CNRS 7005, Université de Strasbourg, Illkirch Cedex, France 67412 and Laboratoire d'Informatique Gaspard-Monge, Université Paris-Est, ESIEE Paris, France;LSIIT, UMR CNRS 7005, Université de Strasbourg, Illkirch Cedex, France 67412;Laboratoire d'Informatique Gaspard-Monge, Université Paris-Est, ESIEE Paris, France;LSIIT, UMR CNRS 7005, Université de Strasbourg, Illkirch Cedex, France 67412
Venue:
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Year:
2011

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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.