Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

DIGITAL COVERING THEORY AND ITS APPLICATIONS

  • Kim, In-Soo (Department of Mathematics, Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • Han, Sang-Eon (Faculty of Liberal Education Center, Institute of Pure and Applied Mathematics, Chonbuk National University)
  • Received : 2008.11.11
  • Accepted : 2008.12.15
  • Published : 2008.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

As a survey-type article, the paper reviews various digital topological utilities from digital covering theory. Digital covering theory has strongly contributed to the calculation of the digital k-fundamental group of both a digital space(a set with k-adjacency or digital k-graph) and a digital product. Furthermore, it has been used in classifying digital spaces, establishing almost Van Kampen theory which is the digital version of van Kampen theorem in algebrate topology, developing the generalized universal covering property, and so forth. Finally, we remark on the digital k-surface structure of a Cartesian product of two simple closed $k_i$-curves in ${\mathbf{Z}}^n$, $i{\in}{1,2}$.

Keywords

References

  1. 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
  2. G. Bertrand, Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern Recognition Letters 15 (1991) 1003-1011.
  3. G. Bertrand and M. Malgouyres, Some topological properties of discrete surfaces. Jour. of Mathematical Imaging and Vision 20 (1999) 207-221.
  4. L. Boxer, Digitally continuous functions. Pattern Recognition Letters 15 (1991), 833-839.
  5. 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
  6. L. Boxer, Digital Products, Wedge; and Covering Spaces, Jour. of Mathematical Imaging and Vision 25 (2006) 159-171. https://doi.org/10.1007/s10851-006-9698-5
  7. 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
  8. S. Fourey, and R. Malgouyres, A digital linking number for discrete curves, International Journal of Pattern Recognition and Artificial lntelligence 15 (2001) 1053-1074. https://doi.org/10.1142/S0218001401001295
  9. N.D. Georgiou, S.E. Han, On computer topological function space, Journal of the Korean Mathematical Society 46(2009) in press.
  10. S.E. Han, Computer topology and its applications, Honam Math. Jour. 25(1)(2003) 153-162.
  11. S.E. Han, Minimal digital pseudotorus with k-adjacency, Honam Mathematical Journal 26(2)(2004) 237-246.
  12. 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.
  13. S.E. Han, Digital coverings and their applications, Jour. of Applied Mathematics and Computing 18(1-2)(2005) 487-495.
  14. 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
  15. S.E. Han. On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27 (1) (2005) 115-129.
  16. 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
  17. S.E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag Berlin, pp.214-225 (2006).
  18. S.E. Han, Erratum to "Non-product property of the digital fundamental group", Information Sciences 176 (1)(2000) 215-216.
  19. S.E. Han, Minimal simple closed k-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
  20. S.E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences 177(16)(2007) 3314-3326. https://doi.org/10.1016/j.ins.2006.12.013
  21. S.E. Han, Remarks on digital k-homotopy equivalence, Honam Mathematical Journal 29(1) (2007) 101-118. https://doi.org/10.5831/HMJ.2007.29.1.101
  22. S.E. Han, Strong k-deformation retract and its applications, Journal of the Korean Mathematical Society 44(6)(2007) 1479-1503. https://doi.org/10.4134/JKMS.2007.44.6.1479
  23. S.E. Han, The fundamental group of a closed k-surface, Information Sciences 177(18)(2007) 3731-3748. https://doi.org/10.1016/j.ins.2007.02.031
  24. S.E. Han, Comparison among digital fundamental groups and its applications, Information Sciences 178(2008) 2091-2104. https://doi.org/10.1016/j.ins.2007.11.030
  25. S.E. Han, Continuities and homeomorphisms in computer topology and their applications, Journal of the Korean Mathematical Society 45(4)(2008) 923-952. https://doi.org/10.4134/JKMS.2008.45.4.923
  26. 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
  27. S.E. Han, The k-homotopic thinning and 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
  28. S.E. Han, Map preserving local properties of a digital image, Acta Applicandae Mathematicae, 104(2) (2008) 177-190. https://doi.org/10.1007/s10440-008-9250-2
  29. S.E. Han, Cartesian product of the universal covering property, Acta Applicandae Mathematicae. (2009). to appear.
  30. S.E. Han, Existence problem or a generalized univeral covering space, Acta Applicandae Mathematicae, (2009), to appear.
  31. S.E. Han, 32-pseudo-surface in $Z_4$ and their comparison by digital 32-homotopy equivalence, Journal of the Korean Mathematical Society, (2009), to appear.
  32. 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).
  33. 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).
  34. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  35. T.Y. Kong, A digital fundamental group Computer& and Graphics 13 (1989) 159-166. https://doi.org/10.1016/0097-8493(89)90058-7
  36. T.Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, (1996).
  37. 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
  38. W.S. Massey, Algebraic Topology, Springer-Verlag, New York, 1977.
  39. A. Rosenfeld, Arcs and curves in digital pictures, Jour. of the ACM 20 (1973) 81-87. https://doi.org/10.1145/321738.321745
  40. A. Rosenfeld and R. Klette, Digital geometry , Information Sciences 148 (2003) 123-127.
  41. E.H. Spanier, Algebraic Topology, McCraw-Hill Inc., New York, 1966.

Cited by

  1. REGULAR COVERING SPACE IN DIGITAL COVERING THEORY AND ITS APPLICATIONS vol.31, pp.3, 2009, https://doi.org/10.5831/HMJ.2009.31.3.279
  2. COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3 vol.32, pp.1, 2010, https://doi.org/10.5831/HMJ.2010.32.1.141
  3. UTILITY OF DIGITAL COVERING THEORY vol.36, pp.3, 2014, https://doi.org/10.5831/HMJ.2014.36.3.695