Fundamental groups and Euler characteristics of sphere-like digital images

Authors

  • Laurence Boxer Niagara University
  • P. Christopher Staecker Fairfield University

DOI:

https://doi.org/10.4995/agt.2016.4624

Keywords:

digital topology, digital image, fundamental group, Euler characteristic

Abstract

The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but dierent notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.

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Author Biographies

Laurence Boxer, Niagara University

Laurence Boxer received his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign, in 1976. He is Professor and past Chair of Computer and Information Sciences at Niagara University, and Research Professor of Computer Science and Engineering at SUNY-Bualo.

P. Christopher Staecker, Fairfield University

P. Christopher Staecker received his Ph.D. in Mathematics from the University of California, Los Angeles, in 2005. He is now an Associate Professor in the Department of Mathematics at Faireld University in Faireld, Connecticut, USA.

References

L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4

L. Boxer, A classical construction for the digital fundamental group, Pattern Recognition Letters 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456

L. Boxer, Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22 (2005), 19-26. https://doi.org/10.1007/s10851-005-4780-y

L. Boxer, Homotopy properties of sphere-like digital images, Journal of Mathematical Imaging and Vision 24 (2006), 167-175. https://doi.org/10.1007/s10851-005-3619-x

L. Boxer, Digital products, wedges and covering spaces, Journal of Mathematical Imaging and Vision 25 (2006), 159-171. https://doi.org/10.1007/s10851-006-9698-5

L. Boxer, Continuous maps on digital simple closed curves, Applied Mathematics 1 (2010), 377-386. https://doi.org/10.4236/am.2010.15050

L. Boxer, I. Karaca and A. Oztel, Topological invariants in digital images, Journal of Mathematical Sciences: Advances and Applications 11, no. 2 (2011), 109-140.

L. Boxer and P. C. Staecker, Connectivity preserving multivalued functions in digital topology, Journal of Mathematical Imaging and Vision 55, no. 3 (2016), 370-377. https://doi.org/10.1007/s10851-015-0625-5

L. Boxer and P. C. Staecker, Remarks on pointed digital homotopy, submitted (http://arxiv.org/abs/1503.03016).

L. Boxer and P. C. Staecker, Homotopy relations for digital images, submitted (http://arxiv.org/abs/1509.06576).

L. Chen, Gradually varied surfaces and its optimal uniform approximation, SPIE Proceedings

L. Chen, Discrete surfaces and manifolds, Scientific Practical Computing, Rockville, MD, 2004 J. Haarman, M. P. Murphy, C. S. Peters and P. C. Staecker, Homotopy equivalence in finite digital images, Journal of Mathematical Imaging and Vision 53 (2015), 288-302. https://doi.org/10.1007/s10851-014-0551-y

S.-E. Han, Connected sum of digital closed surfaces, Information Sciences 176, no. 3 (2006), 332-348. https://doi.org/10.1016/j.ins.2004.11.003

S.-E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences 177 (2007), 3314-3326. https://doi.org/10.1016/j.ins.2006.12.013

S.-E. Han, Equivalent $(k_0,k_1)$-covering and generalized digital lifting, Information Sciences 178 (2008), 550-561. https://doi.org/10.1016/j.ins.2007.02.004

E. Khalimsky, Motion, deformation, and homotopy in finite spaces, in Proceedings IEEE International Conference on Systems, Man, and Cybernetics, 1987, 227-234.

T. Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), 159-166. https://doi.org/10.1016/0097-8493(89)90058-7

T. Y. Kong and A. Rosenfeld, eds., Topological algorithms for digital image processing, Elsevier, 1996.

A. Rosenfeld, Digital topology, American Mathematical Monthly 86 (1979), 621-630. https://doi.org/10.1080/00029890.1979.11994873

A. Rosenfeld, 'Continuous' functions on digital images, Pattern Recognition Letters 4 (1987), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6

Q. F. Stout, Topological matching, Proceedings 15th Annual Symposium on Theory of Computing, 1983, 24-31. https://doi.org/10.1145/800061.808729

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Published

2016-10-03

How to Cite

[1]
L. Boxer and P. C. Staecker, “Fundamental groups and Euler characteristics of sphere-like digital images”, Appl. Gen. Topol., vol. 17, no. 2, pp. 139–158, Oct. 2016.

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