Journal of Mathematical Imaging and Vision

, Volume 61, Issue 2, pp 224–236 | Cite as

Combinatorics of the Gauss Digitization Under Translation in 2D

  • Étienne BaudrierEmail author
  • Loïc Mazo


The action of a translation on a continuous object before its digitization generates several digital objects. This paper focuses on the combinatorics of the generated digital objects up to integer translations. In the general case, a worst-case upper bound is exhibited and proved to be reached on an example. Another upper bound is also proposed by making a link between the number of the digital objects and the boundary curve, through its self-intersections on the torus. An upper bound, quadratic in digital perimeter, is then derived in the convex case and eventually an asymptotic upper bound, quadratic in the grid resolution, is exhibited in the polygonal case. A few significant examples finish the paper.


Discrete geometry Gauss digitization Translation Combinatorics 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.ICube, University of Strasbourg, CNRSIllkirchFrance

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