DOI QR코드

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DIGITAL GEOMETRY AND ITS APPLICATIONS

Han, Sang-Eon

  • Received : 2008.05.21
  • Accepted : 2008.05.29
  • Published : 2008.06.25

Abstract

Digital geometry has strongly contributed to the study of a discrete topological space $X{\subset}{\mathbf{Z}}^n$ with k-adjacency of ${\mathbf{Z}}^n$. As a survey-type article, we review various utilities of digital geometry.

Keywords

k-adjacency relations of ${\mathbf{Z}}^n$;Digital continuity;Geometric realization;Relative k-homotopy;Strong k-deformation retract;k-homotopic thinning;($k_0,k_1$)-isomorphism;Digital ($k_0,k_1$)-covering;Discrete Deck's transformation group;Universal ($k_0,k_1$)-covering

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 (1994) 1003-1011. https://doi.org/10.1016/0167-8655(94)90032-9
  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, 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
  5. 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
  6. 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
  7. S. Fourey and R. Malgouyres, A digital linking number for discrete curves, International Journal of Pattern Recognition and Artificial Intelligence 15 (2001) 1053-1074. https://doi.org/10.1142/S0218001401001295
  8. S.E. Han, Computer topology and its applications, Honam Math. Jour. 25(1)(2003) 153-162.
  9. S.E. Han, Minimal digital pseudotorus with k-adjacency, Honam Mathematical Journal 26(2)(2004) 237-246.
  10. 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.
  11. S.E. Han, Digital coverings and their applications, Jour. of Applied Mathematics and Computing 18(1-2)(2005) 487-495.
  12. 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
  13. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27 (1) (2005) 115-129.
  14. 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
  15. S.E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag Berlin, pp.214-225 (2006).
  16. S.E. Han, Erratum to "Non-product property of the digital fundamental group", Information Sciences 176(1)(2006) 215-216. https://doi.org/10.1016/j.ins.2005.03.014
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. S.E. Han, Map preserving local properties of a digital image, Acta Applicanta Mathematicae (2008), to appear. https://doi.org/10.1007/s10440-008-9250-2
  26. 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).
  27. 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).
  28. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  29. T.Y. Kong, A digital fundamental group Computers and Graphics 13 (1989) 159-166. https://doi.org/10.1016/0097-8493(89)90058-7
  30. T.Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, (1996).
  31. 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
  32. W.S. Massey, Algebraic Topology, Springer-Verlag, New York, 1977.
  33. A. Rosenfeld, Arcs and curves in digital pictures, Jour. of the ACM 20 (1973) 81-87. https://doi.org/10.1145/321738.321745
  34. A. Rosenfeld and R. Klette, Digital geometry, Information Sciences 148 (2003) 123-127.
  35. E.H. Spanier, Algebraic Topology, McGraw-Hill Inc., New York, 1966.