# Bijective Rigid Motions of the 2D Cartesian Grid

Published in DGCI, 2016

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. (2016) Bijective Rigid Motions of the 2D Cartesian Grid. In: Normand N., Guédon J., Autrusseau F. (eds) Discrete Geometry for Computer Imagery. DGCI 2016. Lecture Notes in Computer Science, vol 9647. Springer, pp 359-371, doi:10.1007/978-3-319-32360-2_28

Author(s): K. Pluta, P. Romon, Y. Kenmochi, N. Passat

Abstract: 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.

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Errata is not provided but several typos and mistakes were corrected in the journal version of this paper (see Bijective Digitized Rigid Motions on Subsets of the Plane)