Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON DIGITAL HOMOTOPY EQUIVALENCE

  • Han, Sang-Eon (Department of Computer and Applied Mathematics Honam University)
  • Received : 2007.02.21
  • Accepted : 2007.03.31
  • Published : 2007.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

The notions of digital k-homotopy equivalence and digital ($k_0,k_1$)-homotopy equivalence were developed in [13, 16]. By the use of the digital k-homotopy equivalence, we can investigate digital k-homotopy equivalent properties of Cartesian products constructed by the minimal simple closed 4- and 8-curves in $\mathbf{Z}^2$.

Keywords

References

  1. G. Bertrand, Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern Recognition Letters, 15 (1994) 1003-1011. https://doi.org/10.1016/0167-8655(94)90032-9
  2. G. Bertrand and R. Malgouyres, Some topological properties of discrete surfaces, Jour, of Mathematical Imaging and Vision, 20 (1999) 207-221.
  3. L. Boxer, A classical construction for the digital fundamental group, Jour, of Mathematical Imaging and Vision, 10 (1999) 51-62. https://doi.org/10.1023/A:1008370600456
  4. L. Boxer, Properties of digital homotopy, Jour, of Mathematical Imaging and Vision 22 (2005) 19-26. https://doi.org/10.1007/s10851-005-4780-y
  5. A.I. Bykov, L.G. Zerkalov, M.A. Rodriguez Pineda, Index of a point of 3-D digital binary image and algorithm of computing its Euler characteristic, Pattern Recognition 32 (1999) 845-850. https://doi.org/10.1016/S0031-3203(98)00023-5
  6. S.E. Han, Algorithm for Discriminating Digital Images w.r.t. a Digital ($k_0,k_1$)-homeomorphism, Jour, of Applied Mathematics and Computing 18(1-2)(2005) 505-512.
  7. S.E.Han, Connected sum of digital closed surfaces, Information Sciences 176(3)(2006) 332-348. https://doi.org/10.1016/j.ins.2004.11.003
  8. S.E. Han, Digital coverings and their applications, Jour, of Applied Mathematics and Computing 18(1-2)(2005) 487-495.
  9. S.E. Han, Digital fundamental group and Euler characteristic of acon­nected sum of digital closed surfaces, Information Sciences (Available online at www.sciencedirect.com) (Article in press) (2007).
  10. S.E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag Berlin, pp.214-225, 2006.
  11. S.E. Han, Equivalent ($k_0, k_1$ )-covering and generalized digital lifting, Information Sciences (Available online at www. sciencedirect.com) (Articles in press)(2007).
  12. S.E.Han, Extension problem of a Khalimsky continuous map with digital ($k_0, k_1$)-continuity, to appear, Information Sciences.
  13. S.E. Han, Minimal digital pseudotorus with k-adjacency, Honam Mathematical Journal 26 (2) (2004) 237-246.
  14. S.E. Han, Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces, Information Sciences 176(2)(2006) 120-134. https://doi.org/10.1016/j.ins.2005.01.002
  15. S.E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (1-3) (2005) 73-91. https://doi.org/10.1016/j.ins.2004.03.018
  16. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27 (1) (2005) 115-129.
  17. S.E. Han, Strong k-deformation retract and its applications, to appear, Journal of Korean Mathematical Society. https://doi.org/10.4134/JKMS.2007.44.6.1479
  18. S.E. Han, The fundamental group of a closed k-surface, Information Sciences (Available online at www.sciencedirect.com)(Articles in press).
  19. S.E. Han and B.G. Park, Digital graph ($k_0, k_1$)-homotopy equivalence and its applications, http://atlas-conferences.com/c/a/k/b/35.htm(2003).
  20. S.E. Han and B.G. Park, Digital graph ($k_0, k_1$)-isomorphism and its applications, http://atlas-conferences.com/c/a/k/b/35.htm(2003).
  21. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  22. T.Y. Kong, A. Rosenfeld, (Digital topology - A brief introduction and bibliogra­phy) Topological Algorithms for the Digital Image Processing, (Elsevier Science, Amsterdam, 1996).
  23. R. Kopperman, R. Meyer and R. G. Wilson, A Jordan surface theorem for three-dimensional digital spaces, Discrete and computational Geometry, 6 (1991) 155-161. https://doi.org/10.1007/BF02574681
  24. R. Malgouyres, Homotopy in 2-dimensional digital images, Theoretical Computer Science, 230 (2000) 221-233. https://doi.org/10.1016/S0304-3975(98)00347-8
  25. D.G. Morgenthaler, A. Rosenfeld, Surfaces in three dimensional digital images, Information and Control, 36 (1981) 227-247. https://doi.org/10.1016/S0019-9958(81)90290-4

Cited by

  1. DIGITAL COVERING THEORY AND ITS APPLICATIONS vol.30, pp.4, 2008, https://doi.org/10.5831/HMJ.2008.30.4.589
  2. Homotopy equivalence which is suitable for studying Khalimsky nD spaces vol.159, pp.7, 2012, https://doi.org/10.1016/j.topol.2011.07.029
  3. Comparison among digital fundamental groups and its applications vol.178, pp.8, 2008, https://doi.org/10.1016/j.ins.2007.11.030
  4. The k-Homotopic Thinning and a Torus-Like Digital Image in Z n vol.31, pp.1, 2008, https://doi.org/10.1007/s10851-007-0061-2
  5. DIGITAL GEOMETRY AND ITS APPLICATIONS vol.30, pp.2, 2008, https://doi.org/10.5831/HMJ.2008.30.2.207