Bijective Rigid Motions of the 2D Cartesian Grid

  • Kacper PlutaEmail author
  • Pascal Romon
  • Yukiko Kenmochi
  • Nicolas Passat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9647)


Rigid motions are fundamental operations in image processing. While they are bijective and isometric in \(\mathbb {R}^2\), they lose these properties when digitized in \(\mathbb {Z}^2\). To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on \(\mathbb {Z}^2\), initially proposed by Nouvel and Rémila for rotations on \(\mathbb {Z}^2\). This allows us to study bijective rigid motions on \(\mathbb {Z}^2\), and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of \(\mathbb {Z}^2\) is bijective.


Finite Subset Critical Line Rigid Motion Integer Point Forward Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors express their thanks to Laurent Najman of ESIEE Paris for his remarks concerning the backward algorithm, and to Mariusz Jędrzejczyk of Norbert Barlicki Memorial Teaching Hospital No. 1, Department of Radiology and Diagnostic Imaging, for providing the computer tomography image of a human chest.

The research leading to these results has received funding from the Programme d’Investissements d’Avenir (LabEx Bézout, ANR-10-LABX-58).


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Kacper Pluta
    • 1
    • 2
    Email author
  • Pascal Romon
    • 2
  • Yukiko Kenmochi
    • 1
  • Nicolas Passat
    • 3
  1. 1.Université Paris-Est, LIGM, CNRS-ESIEEParisFrance
  2. 2.Université Paris-Est, LAMAParisFrance
  3. 3.Université de Reims Champagne-Ardenne, CReSTICReimsFrance

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