Abstract
The characteristics of a digitization of a Euclidean planar shape depends on the digitization process but also on the shape border regularity. The notion of Local Turn Boundedness (LTB) was introduced by the authors in Le Quentrec, É. et al.: Local Turn-Boundedness: A curvature control for a good digitization, DGCI 2019 so as to have multigrid convergent perimeter estimation on Euclidean shapes. If it was proved that the par-regular curves are locally turn bounded, the relation with the quasi-regularity introduced in Ngo, P.et al.: Convexity-Preserving Rigid Motions of 2D Digital Objects, DGCI 2017 had not yet been explored. Our paper is dedicated to prove that for planar shapes, local turn-boundedness implies quasi-regularity.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Actually, the equivalence does not perfectly hold as seen taking \(S=\bar{B}(0,r)\).
References
Alexandrov, A.D., Reshetnyak, Y.G.: General Theory of Irregular Curves, Mathematics and Its Applications, vol. 29. Springer, Dordrecht (1989). https://doi.org/10.1007/978-94-009-2591-5
Chazal, F., Cohen-Steiner, D., Lieutier, A.: A sampling theory for compact sets in Euclidean space. Discr. Comput. Geom. 41(3), 461–479 (2009). https://doi.org/10.1007/s00454-009-9144-8
Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93(3), 418–491 (1959). http://www.jstor.org/stable/1993504
Lachaud, J., Thibert, B.: Properties of Gauss digitized shapes and digital surface integration. J. Math Imaging Vis. 54(2), 162–180 (2016). https://doi.org/10.1007/s10851-015-0595-7
Latecki, L., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vision 8, 131–159 (1998). https://doi.org/10.1023/A:1008273227913
Le Quentrec, É., Mazo, L., Baudrier, É., Tajine, M.: Local Turn-Boundedness: A curvature control for a good digitization. In: Couprie, M., Cousty, J., Kenmochi, Y., Mustafa, N. (eds.) Discrete Geometry for Computer Imagery, pp. 51–61. Springer, Cham (2019). https://rd.springer.com/chapter/10.1007/978-3-030-14085-4_5
Le Quentrec, É., Mazo, L., Baudrier, É., Tajine, M.: Local Turn-Boundedness: A Curvature Control for Continuous Curves with Application to Digitization. J. Math. Imaging Vis. 62(5), 673–692 (2020). https://doi.org/10.1007/s10851-020-00952-x
Le Quentrec, É., Mazo, L., Baudrier, É., Tajine, M.: Monotonic sampling of a continuous closed curve from its Gauss digitization. Application to length estimation. Technical report, Icube Laboratory, University of Strasbourg, CNRS (2020). https://hal.archives-ouvertes.fr/hal-02987858, submitted
Meine, H., Köthe, U., Stelldinger, P.: A topological sampling theorem for robust boundary reconstruction and image segmentation. Discrete Appl. Math. 157(3), 524–541 (2009). https://doi.org/10.1016/j.dam.2008.05.031, http://www.sciencedirect.com/science/article/pii/S0166218X08002643
Ngo, P., Kenmochi, Y., Debled-Rennesson, I., Passat, N.: Convexity-preserving rigid motions of 2D digital objects. In: Kropatsch, W.G., Artner, N.M., Janusch, I. (eds.) DGCI 2017. LNCS, vol. 10502, pp. 69–81. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66272-5_7
Ngo, P., Passat, N., Kenmochi, Y., Debled-Rennesson, I.: Geometric Preservation of 2D Digital Objects Under Rigid Motions. J. Math. Imaging Vis. 61(2), 204–223 (2019). https://doi.org/10.1007/s10851-018-0842-9, http://link.springer.com/10.1007/s10851-018-0842-9
Pavlidis, T.: Algorithms for Graphics and Image Processing. Springer, Heidelberg (1982)
Stelldinger, P., Terzic, K.: Digitization of non-regular shapes in arbitrary dimensions. Image Vis. Comput. 26(10), 1338–1346 (2008). https://doi.org/10.1016/j.imavis.2007.07.013, http://www.sciencedirect.com/science/article/pii/S0262885607001370
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Le Quentrec, É., Mazo, L., Baudrier, É., Tajine, M. (2021). Locally Turn-Bounded Curves Are Quasi-Regular. In: Lindblad, J., Malmberg, F., Sladoje, N. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2021. Lecture Notes in Computer Science(), vol 12708. Springer, Cham. https://doi.org/10.1007/978-3-030-76657-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-76657-3_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-76656-6
Online ISBN: 978-3-030-76657-3
eBook Packages: Computer ScienceComputer Science (R0)