A survey of algebraic methods in digital topology

Journal of Mathematical Imaging and Vision

Volumn 6, Issue 2-3, 1996, Pages 139-159

  McAndrew, Alasdair ^a   Osborne, Charles ^b  

^a VICTORIA UNIVERSITY   (Australia)

^b MONASH UNIVERSITY   (Australia)

Author keywords

Algebraic topology; Digital topology; Homology; Homotopy

Indexed keywords

ALGEBRAIC TOPOLOGY; DIGITAL TOPOLOGY; HOMOLOGY; HOMOTOPY;

ALGEBRA; BOOLEAN ALGEBRA; IMAGE ANALYSIS; SET THEORY; SURFACES; TOPOLOGY;

IMAGE PROCESSING;

EID: 0030169392     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00119836     Document Type: Article

References (51)

1. M.K. Agoston, Algebraic Topology, A First Course, Marcel Dekker Inc.: New York, NY, 1976.
2. J.C. Alexander and A.I. Thaler, "The boundary count of digital pictures," J. Assoc. Computing Machinery, Vol. 18, No. 1, pp. 105-112, 1971.
3. G. Burde and W. Zieschang, Knots, W. de Gruyter, New York, NY, 1985.
4. R.H. Crowell and R.H. Fox, Introduction to Knot Theory, Blaisdell Publishing Company: Boston, 1963.
5. P.J. Giblin, Graphs, Surfaces and Homology, 2nd Ed., Chapman and Hall: London, 1987.
6. C. McA. Gordon and J. Luecke, "Knots are determined by their complements," J. Amer. Math. Soc., Vol. 2, pp. 371-415, 1989.
7. M.J. Greenberg and J.R. Harper, Algebraic Topology, A First Course, Benjamin/Cummings Publishing Company: Reading, Mass., 1981.
8. Z. Guo and R.W. Hall, "Parallel thinning with two-subiteration algorithms," Communications of the ACM, Vol. 32, No. 3, pp. 359-373, 1989.
9. K.J. Halford and K. Preston, "3D Skeletonization of elongated solids," Computer Vision, Graphics and Image Processing, Vol. 27, pp. 78-91, 1984.
10. G.T. Herman, "Discrete multidimensional Jordan surfaces," Contemporary Mathematics, Vol. 119, pp. 85-94, 1991.
11. J.P.P. Hudson, Piecewise Linear Topology, W.A. Benjamin Inc.: Menlo Park, Calif., 1969.
12. J.G. Hocking and G.S. Young, Topology, Addison Wesley: Reading, Mass., 1961.
13. B.-K. Jang and R.T. Chin, "Analysis of thinning algorithms using mathematical morphology," IEEE Transactions in Pattern Analysis and Machine Intelligence, Vol. PAMI-12, pp. 541-551, 1990.
14. L. Janos and A. Rosenfeld, "Digital connectedness: An algebraic approach," Pattern Recognition Letters, Vol. 1, pp. 135-139, 1983.
15. L.H. Kauffman, Formal Knot Theory, Princeton University Press: Princeton, NJ, 1983.
16. L.H. Kauffman, On Knots, Princeton University Press: Princeton, NJ, 1987.
17. L.H. Kauffman, "New invariants in the theory of knots," Amer. Math. Monthhly, Vol. 95, pp. 195-242, 1988.
18. L.H. Kauffman, Knots and Physics, World Scientific: Singapore, 1991.
19. E. Khalimsky, "Motion, deformation and homotopy in finite spaces," in Proceedings, 1987 IEEE Conference on Systems, Man and Cybernetics, 1987, pp. 221-234,
20. T.Y. Kong, "A digital fundamental group," Computers and Graphics, Vol. 13, No. 2, pp. 159-166, 1989.
21. T.Y. Kong, "Problem of determining whether a parallel reduction operator for n-dimensional binary images always preserves topology," in Proceedings, Vision Geometry II, R.A. Melter and A.Y. Wu (Eds.) SPIE, 1993, Vol. 2060, pp. 69-77.
22. T.Y. Kong, A.W. Roscoe, and A. Rosenfeld, "Concepts of Algebraic Topology," Topology and Its Applications, Vol. 46, No. 3, pp. 219-262, 1992.
23. T.Y. Kong and A. Rosenfeld, "Digital topology: Introduction and survey," Computer Vision, Graphics, and Image Processing, Vol. 48, pp. 357-393, 1989.
24. M.A. Krasnow, A. Stasiak, S.J. Spengler, F. Dean, T. Koller, and N.R. Cozzarelli, "Determination of the absolute handedness of knots and catenanes of DNA," Nature, Vol. 304, August 11, 1983.
25. L. Lam, S.-W. Lee and C.Y. Suen, "Thinning methodologies - A comprehensive survey," IEEE Transactions in Pattern Analysis and Machine Intelligence, Vol. PAMI-14, No. 9, pp. 869-885, 1992.
26. C.-N. Lee and A. Rosenfeld, "Connectivity issues in 2D and 3D images," in Proceedings, IEEE Conference on Computer Vision and Pattern Recognition, 1986, pp. 278-285.
27. C.-N. Lee, T. Poston and A. Rosenfeld, "Holes and genus of 2D and 3D images," CVGIP: Graphical Models and Image Processing, Vol. 55, No. 1, pp. 20-47, 1993.
28. S. Lobregt, P.W. Verbeek, and F.G. Groen, "Three-dimensional skeletonization: Principle and algorithm," IEEE Transactions in Pattern Analysis and Machine Intelligence, Vol. PAMI-2, pp. 75-77, 1982.
29. A. McAndrew and C.F. Osborne, "Algebraic methods for multidimensional digital topology," in Proceedings, Vision Geometry II, R.A. Melter and A.Y. Wu (Eds.) SPIE, 1993, Vol. 2060, pp. 14-25.
30. A. McAndrew and C.F. Osborne, "An investigation into digitized knot structures," in Proceedings, DICTA'93, K.K. Fung and A. Ginige (Eds.) 1993, pp. 55-62.
31. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Dover Publications Inc.: New York, NY, 1976.
32. W.S. Massey, A Basic Course in Algebraic Topology, SpringerVerlag: New York, NY, 1991.
33. C.R.F. Maunder, Algebraic Topology, Cambridge University Press: Cambridge, U.K., 1980.
34. J. Mayer, Algebraic Topology, Prentice-Hall Inc.: Englewood Cliffs, NJ, 1972.
35. D.G. Morgenthaler and A. Rosenfeld, "Surfaces in three-dimensional digital images," Information and Control, Vol. 51, pp. 227-247, 1981.
36. L. Neuwirth, "The theory of knots," Scientific American, pp. 84-96, June 1979.
37. K. Preston, "Multidimensional Logical Transforms," IEEE Transactions in Pattern Analysis and Machine Intelligence, Vol. PAMI-5, No. 5, pp. 539-554, 1983.
38. G.M. Reed, "On the characterization of simple closed surfaces in three-dimensional digital images," Computer Vision, Graphics, and Image Processing, Vol. 25, pp. 226-235, 1984.
39. G.M. Reed and A. Rosenfeld, "Recognition of surfaces in three-dimensional digital images," Information and Control, Vol. 53, pp. 108-120, 1982.
40. K. Reidemeister, Knot Theory, Tr. L.F. Brown, C.O. Christensen, and B.A. Smith, BCS Associates: Moscow, Idaho, 1983.
41. D. Rolfsen, Knots and Links, Publish or Perish: Wilmington, DE, 1976.
42. A. Rosenfeld, "Connectivity in digital pictures," J. Assoc. Computing Machinery, Vol. 17, No. 1, pp. 146-160, 1970.
43. J.J. Rotman, An Introduction to Algebraic Topology, SpringerVerlag: New York, NY, 1988.
44. J.P. Serra, Image Analysis and Mathematical Morphology, Academic Press: London, 1982.
45. E.H. Spanier, Algebraic Topology, Springer-Verlag: New York, NY, 1966.
46. R. Stefanelli and A. Rosenfeld, "Some parallel algorithms for digital pictures," J. Assoc. Computing Machinery, Vol. 18, No. 2, pp. 235-254, 1971.
47. J. Stillwell, Classical Topology and Combinatorial Group Theory, 2nd Ed., Springer-Verlag: New York, NY, 1993.
48. L.N. Stout, "Two discrete forms of the Jordan curve theorem," American Mathematical Monthly, Vol. 95, pp. 332-336, 1988.
49. G. Tourlakis and J. Mylopoulos, "Some results in computational topology," J. Assoc. Computing Machinery, Vol. 20, No. 3, pp. 439-455, 1973.
50. Y.F. Tsao and K.S. Fu, "A parallel thinning algorithm for 3-D pictures," Computer Vision, Graphics, and Image Processing, Vol. 17, pp. 315-331, 1981.
51. R.-Y. Wu and W.-H. Tsai, "A new one-pass parallel thinning algorithm for binary images," Pattern Recognition Letters, Vol. 13, No. 10, pp. 715-723, 1992.