Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON HOMOTOPIES ASSOCIATED WITH KHALIMSKY TOPOLOGY

  • HAN, SANG-EON (Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • LEE, SIK (Department of Mathematics Education, Chonnam National University)
  • Received : 2015.09.29
  • Accepted : 2015.12.03
  • Published : 2015.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

Several kinds of homotopies have been substantially used to study topological properties of digital spaces. The present paper, as a survey article, studies some recent results in the field of homotopy theory associated with Khalimsky topology. In particular, Khalimsky topological properties of digital products related to the establishment of the homotopies are mainly treated.

Keywords

References

  1. P. Alexandroff, Diskrete Raume, Mat. Sb. 2 (1937) 501-518.
  2. R. Ayala, E. Dominguez, A.R. Frances, and A. Quintero, Homotopy in digital spaces, Discrete Applied Math, 125(1) (2003) 3-24. https://doi.org/10.1016/S0166-218X(02)00221-4
  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. S.E. Han, On the classification of the digital images up to digital homotopy equivalence, Jour. Comput. Commun. Res. 10(2000) 207-216.
  5. 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
  6. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27 (1)(2005) 115-129.
  7. S.E. Han, S.E. Han, Equivalent ($k_{0},\;k_{1}$)-covering and generalized digital lifting, Information Sciences 178(2) (2008) 550-561. https://doi.org/10.1016/j.ins.2007.02.004
  8. S.E. Han, Continuities and homeomorphisms in computer topology and their applications, J. Korean Math. Soc. 45 (2008) 923-952. https://doi.org/10.4134/JKMS.2008.45.4.923
  9. S.E. Han, The k-homotopic thinning and a torus-like digital image in $Z^n$, Journal of Mathematical Imaging and Vision 31 (1) (2008) 1-16. https://doi.org/10.1007/s10851-007-0061-2
  10. S.E. Han, Cartesian product of the universal covering property Acta Applicandae Mathematicae 108 (2009) 363-383. https://doi.org/10.1007/s10440-008-9316-1
  11. S.E. Han, KD-($k_{0},\;k_{1}$)-homotopy equivalence and its applications Journal of Korean Mathematical Society 47(5) (2010) 1031-1054. https://doi.org/10.4134/JKMS.2010.47.5.1031
  12. S.E. Han, Multiplicative property of the digital fundamental group, Acta Appli-candae Mathematicae 110(2) (2010) 921-944. https://doi.org/10.1007/s10440-009-9486-5
  13. S.E. Han, Homotopy equivalence which is suitable for studying Khalimsky nD spaces, Topology Appl. 159 (2012) 1705-1714. https://doi.org/10.1016/j.topol.2011.07.029
  14. S.E. Han, Study on topological spaces with the semi-$T_{1/2}$ separation axiom Honam Mathematical Journal, (2013) 35(4) 707-716. https://doi.org/10.5831/HMJ.2013.35.4.707
  15. S.E. Han, An equivalent property of a normal adjacency of a digita product Honam Mathematical Journal, (2014) 36(3) 199-215. https://doi.org/10.5831/HMJ.2014.36.1.199
  16. S.E. Han, Generalizations of continuity of maps and homeomorphisms for studying 2D digital topological spaces and their applications, Topology Appl., online first press (2015).
  17. S.E. Han, Contractibility and fixed point property: The case of Khalimsky topo-logical spaces, submitted (2015).
  18. S.E. Han and Sik Lee, Remarks on digital products with normal adjacency rela-tions Honam Mathematical Journal, (2013) 35(3) 515-424. https://doi.org/10.5831/HMJ.2013.35.3.515
  19. 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/36.htm(2003).
  20. 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).
  21. S.E. Han, Wei Yao, Homotopies based on Marcus Wyse topology and their applications, Topology and its applications, in press.
  22. G. T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993) 381-396. https://doi.org/10.1006/cgip.1993.1029
  23. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  24. E. Khalimksy, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl., 36(1) (1990) 1-17. https://doi.org/10.1016/0166-8641(90)90031-V
  25. T.Y. Kong, A digital fundamental group Computers and Graphics 13 (1989) 159-166. https://doi.org/10.1016/0097-8493(89)90058-7
  26. T.Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
  27. 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
  28. A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, 4 (1986) 177-184. https://doi.org/10.1016/0167-8655(86)90017-6
  29. F. Wyse and D. Marcus et al., Solution to problem 5712, Amer. Math. Monthly 77(1970) 1119. https://doi.org/10.2307/2316121