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Digital shapes, digital boundaries and rigid transformations: A topological discussion
Actes des rencontres du CIRM, Volume 3 (2013) no. 1, p. 195-201

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.

Published online : 2014-11-13
DOI : https://doi.org/10.5802/acirm.68
Classification:  00X99
Keywords: topology, digitization, geometric transformations 
@article{ACIRM_2013__3_1_195_0,
     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},
     publisher = {CIRM},
     volume = {3},
     number = {1},
     year = {2013},
     pages = {195-201},
     doi = {10.5802/acirm.68},
     language = {en},
     url = {https://acirm.centre-mersenne.org/item/ACIRM_2013__3_1_195_0}
}

Kenmochi, Yukiko; Ngo, Phuc; Passat, Nicolas; Talbot, Hugues. 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/item/ACIRM_2013__3_1_195_0/

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

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

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

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

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

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

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

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

[9] T. Pavlidis Algorithms for Graphics and Image Processing, Springer-Verlag (1982)