Digital shapes, digital boundaries and rigid transformations: A topological discussion

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

Actes des rencontres du CIRM (2013)

  • Volume: 3, Issue: 1, page 195-201
  • ISSN: 2105-0597

Abstract

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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.

How to cite

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Kenmochi, Yukiko, et al. "Digital shapes, digital boundaries and rigid transformations: A topological discussion." Actes des rencontres du CIRM 3.1 (2013): 195-201. <http://eudml.org/doc/275281>.

@article{Kenmochi2013,
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 $\mathbb\{R\}^n$ to $\mathbb\{Z\}^n$, can also be stated in $\mathbb\{Z\}^n$ itself, when applying geometric transformations on digital shapes. This paper proposes a preliminary discussion on this topic.},
affiliation = {LIGM, Université Paris-Est, France; CEA LIST - DIGITEO Labs, France; CReSTIC, Université de Reims, France; LIGM, Université Paris-Est, France},
author = {Kenmochi, Yukiko, Ngo, Phuc, Passat, Nicolas, Talbot, Hugues},
journal = {Actes des rencontres du CIRM},
keywords = {topology; digitization; geometric transformations; 2D digital image; discrete rigid transformation; simple point; DRT graph; Eulerian model},
language = {eng},
month = {11},
number = {1},
pages = {195-201},
publisher = {CIRM},
title = {Digital shapes, digital boundaries and rigid transformations: A topological discussion},
url = {http://eudml.org/doc/275281},
volume = {3},
year = {2013},
}

TY - JOUR
AU - Kenmochi, Yukiko
AU - Ngo, Phuc
AU - Passat, Nicolas
AU - Talbot, Hugues
TI - Digital shapes, digital boundaries and rigid transformations: A topological discussion
JO - Actes des rencontres du CIRM
DA - 2013/11//
PB - CIRM
VL - 3
IS - 1
SP - 195
EP - 201
AB - 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 $\mathbb{R}^n$ to $\mathbb{Z}^n$, can also be stated in $\mathbb{Z}^n$ itself, when applying geometric transformations on digital shapes. This paper proposes a preliminary discussion on this topic.
LA - eng
KW - topology; digitization; geometric transformations; 2D digital image; discrete rigid transformation; simple point; DRT graph; Eulerian model
UR - http://eudml.org/doc/275281
ER -

References

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  1. A. Gross, L. Latecki, Digitizations preserving topological and differential geometric properties, Computer Vision and Image Understanding 62 (1995), 370-381 
  2. R. Klette, A. Rosenfeld, Digital Geometry: Geometric Methods for Digital Picture Analysis, (2004), Morgan Kaufmann Zbl1064.68090
  3. T. Y. Kong, A. Rosenfeld, Digital topology: Introduction and survey, Computer Vision, Graphics, and Image Processing 48 (1989), 357-393 
  4. L. J. Latecki, C. Conrad, A. D. Gross, Preserving topology by a digitization process, Journal of Mathematical Imaging and Vision 8 (1998), 131-159 Zbl0895.68138
  5. L. J. Latecki, U. Eckhardt, A. Rosenfeld, Well-composed sets, Computer Vision and Image Understanding 61 (1995), 70-83 
  6. Mathematical Morphology: From Theory to Applications, (2010), NajmanL.L. Zbl1198.94012
  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), 3035-3039, IEEE Signal Processing Society, Melbourne, Australia 
  8. P. Ngo, N. Passat, Y. Kenmochi, H. Talbot, Topology-preserving rigid transformation of 2D digital images, IEEE Transactions on Image Processing 23 (2014), 885-897 Zbl1291.68421
  9. T. Pavlidis, Algorithms for Graphics and Image Processing, (1982), Springer-Verlag Zbl0482.68087

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