Journal of Mathematical Imaging and Vision

, Volume 59, Issue 1, pp 84–105 | Cite as

Bijective Digitized Rigid Motions on Subsets of the Plane

  • Kacper PlutaEmail author
  • Pascal Romon
  • Yukiko Kenmochi
  • Nicolas Passat


Rigid motions in \(\mathbb {R}^2\) are fundamental operations in 2D image processing. They satisfy many properties: in particular, they are isometric and therefore bijective. Digitized rigid motions, however, lose these two properties. To investigate the lack of injectivity or surjectivity and more generally their local behavior, we extend the framework initially proposed by Nouvel and Rémila to the case of digitized rigid motions. Yet, for practical applications, the relevant information is not global bijectivity, which is seldom achieved, but bijectivity of the motion restricted to a given finite subset of \(\mathbb {Z}^2\). We propose two algorithms testing that condition. Finally, because rotation angles are rarely given with infinite precision, we propose a third algorithm providing optimal angle intervals that preserve this restricted bijectivity.


Digital geometry Rigid motions Combinatorial analysis Local characterization Bijective transformations 


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.LIGM (UMR 8049), LAMA (UMR 8050), UPEM, UPEC, CNRSUniversité Paris-EstMarne-la-ValléeFrance
  2. 2.LAMA (UMR 8050), UPEM, UPEC, CNRSUniversité Paris-EstMarne-la-ValléeFrance
  3. 3.LIGM (UMR 8049), UPEM, CNRS, ESIEE Paris, ENPCUniversité Paris-EstMarne-la-ValléeFrance
  4. 4.CReSTICUniversité de Reims Champagne-ArdenneReimsFrance

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