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Geometric Preservation of 2D Digital Objects Under Rigid Motions

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Abstract

Rigid motions (i.e. transformations based on translations and rotations) are simple, yet important, transformations in image processing. In \(\mathbb {R}^n\), they are both topology and geometry preserving. Unfortunately, these properties are generally lost in \(\mathbb {Z}^n\). In particular, when applying a rigid motion on a digital object, one generally alters its structure but also the global shape of its boundary. These alterations are mainly caused by digitization during the transformation process. In this specific context, some solutions for the handling of topological issues were proposed in \(\mathbb {Z}^2\). In this article, we also focus on geometric issues in \(\mathbb {Z}^2\). Indeed, we propose a rigid motion scheme that preserves geometry and topology properties of the transformed digital object: a connected object will remain connected, and some geometric properties (e.g. convexity, area and perimeter) will be preserved. To reach that goal, our main contributions are twofold. First, from an algorithmic point of view, our scheme relies on (1) a polygonization of the digital object, (2) the transformation of the intermediate piecewise affine object of \(\mathbb {R}^2\) and (3) a digitization step for recovering a result within \(\mathbb {Z}^2\). The intermediate modeling of a digital object of \(\mathbb {Z}^2\) as a piecewise affine object of \(\mathbb {R}^2\) allows us to avoid the geometric alterations generally induced by standard pointwise rigid motions. However, the final digitization of the polygon back to \(\mathbb {Z}^2\) has to be carried out cautiously. In particular, our second, theoretical contribution is a notion of quasi-regularity that provides sufficient conditions to be fulfilled by a continuous object for guaranteeing both topology and geometry preservation during its digitization.

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Notes

  1. This definition of r-regularity can be equivalently expressed as the invariance of \({ X }\) with respect to both opening and closing by a structuring element defined as a close ball of radius r. This mathematical morphology analogy will be given in Sect. 3.

  2. The boundary of \(\mathsf {X}\) is defined here as the boundary of the continuous object obtained as the union of the closed Voronoi cells associated to the points of \(\mathsf {X}\), in \(\mathbb {R}{}^2\).

  3. Kim introduced in [25] the definition of cellular convexity and proved the equivalence to the one using the convex hull [25, Lemma 10]. In [26], Eckhardt reformulated and renamed this notion H-convexity.

  4. http://ipol-geometry.loria.fr/~phuc/ipol_demo/RigidMotion2D.

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments. In particular, we are especially grateful to referee 3, who pointed out an improved definition of quasi-r-regularity.

This work was partly funded by the French Agence Nationale de la Recherche, grant agreement ANR-15-CE40-0006 (CoMeDiC, https://lama.univ-savoie.fr/comedic), by the French Programmed’Investissements d’Avenir (LabEx Bézout, ANR-10-LABX-58) and by a mobility grant from the French Groupe de Recherche IGRV (CNRS).

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Authors and Affiliations

  1. Université de Lorraine, LORIA, UMR 7503, Villers-lés-Nancy, France

    Phuc Ngo & Isabelle Debled-Rennesson

  2. Université de Reims Champagne-Ardenne, CReSTIC, Reims, France

    Nicolas Passat

  3. Université Paris-Est, LIGM, CNRS, Marne-la-Vallée, France

    Yukiko Kenmochi

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  1. Phuc Ngo
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  4. Isabelle Debled-Rennesson
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Ngo, P., Passat, N., Kenmochi, Y. et al. Geometric Preservation of 2D Digital Objects Under Rigid Motions. J Math Imaging Vis 61, 204–223 (2019). https://doi.org/10.1007/s10851-018-0842-9

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