# 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

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topKenmochi, 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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- 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
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- Mathematical Morphology: From Theory to Applications, (2010), NajmanL.L. Zbl1198.94012
- 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
- 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
- T. Pavlidis, Algorithms for Graphics and Image Processing, (1982), Springer-Verlag Zbl0482.68087

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