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  • Volume 3 (2013)
  • no. 1
  • p. 195-201
no. 1
Digital shapes, digital boundaries and rigid transformations: A topological discussion
Yukiko Kenmochi1; Phuc Ngo2; Nicolas Passat3; Hugues Talbot1
1 LIGM, Université Paris-Est, France
2 CEA LIST - DIGITEO Labs, France
3 CReSTIC, Université de Reims, France
Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 195-201.
  • Abstract

Curvature is a continuous and infinitesimal notion. These properties induce geometrical difficulties in digital frameworks, and the following question is naturally asked: “How to define and compute curvatures of digital shapes?” In fact, not only geometrical but also topological difficulties are also induced in digital frameworks. The – deeper – question thus arises: “Can we still define and compute curvatures?” This latter question, that is relevant in the context of digitization, i.e., when passing from ℝ n to ℤ n , can also be stated in ℤ n itself, when applying geometric transformations on digital shapes. This paper proposes a preliminary discussion on this topic.

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Published online: 2014-11-12
Zbl: 06938616
DOI: 10.5802/acirm.68
Classification: 00X99
Keywords: topology, digitization, geometric transformations
Author's affiliations:
Yukiko Kenmochi 1; Phuc Ngo 2; Nicolas Passat 3; Hugues Talbot 1

1 LIGM, Université Paris-Est, France
2 CEA LIST - DIGITEO Labs, France
3 CReSTIC, Université de Reims, France
  • BibTeX
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     author = {Yukiko Kenmochi and Phuc Ngo and Nicolas Passat and Hugues Talbot},
     title = {Digital shapes, digital boundaries and rigid transformations: {A} topological discussion},
     journal = {Actes des rencontres du CIRM},
     pages = {195--201},
     publisher = {CIRM},
     volume = {3},
     number = {1},
     year = {2013},
     doi = {10.5802/acirm.68},
     zbl = {06938616},
     language = {en},
     url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.68/}
}
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Yukiko Kenmochi; Phuc Ngo; Nicolas Passat; Hugues Talbot. Digital shapes, digital boundaries and rigid transformations: A topological discussion. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 195-201. doi : 10.5802/acirm.68. https://acirm.centre-mersenne.org/articles/10.5802/acirm.68/
  • References
  • Cited by

[1] A. Gross; L. Latecki Digitizations preserving topological and differential geometric properties, Computer Vision and Image Understanding, Volume 62 (1995) no. 3, pp. 370-381 | Article

[2] R. Klette; A. Rosenfeld Digital Geometry: Geometric Methods for Digital Picture Analysis, Morgan Kaufmann, 2004 | Zbl: 1064.68090

[3] T. Y. Kong; A. Rosenfeld Digital topology: Introduction and survey, Computer Vision, Graphics, and Image Processing, Volume 48 (1989) no. 3, pp. 357-393 | Article

[4] L. J. Latecki; C. Conrad; A. D. Gross Preserving topology by a digitization process, Journal of Mathematical Imaging and Vision, Volume 8 (1998) no. 2, pp. 131-159 | Article | MR: 1614921 | Zbl: 0895.68138

[5] L. J. Latecki; U. Eckhardt; A. Rosenfeld Well-composed sets, Computer Vision and Image Understanding, Volume 61 (1995) no. 1, pp. 70-83 | Article

[6] Mathematical Morphology: From Theory to Applications (L. Najman; H. Talbot, eds.), ISTE/J. Wiley & Sons, 2010 | Zbl: 1198.94012

[7] P. Ngo; N. Passat; Y. Kenmochi; H. Talbot Well-composed images and rigid transformations, ICIP 2013, 20th International Conference on Image Processing, Proceedings (2013), pp. 3035-3039 | Article

[8] P. Ngo; N. Passat; Y. Kenmochi; H. Talbot Topology-preserving rigid transformation of 2D digital images, IEEE Transactions on Image Processing, Volume 23 (2014) no. 2, pp. 885-897 | Article | MR: 3159547 | Zbl: 1374.94278

[9] T. Pavlidis Algorithms for Graphics and Image Processing, Springer-Verlag, 1982 | Article | MR: 643798 | Zbl: 0482.68086

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