Curves, Hypersurfaces, and Good Pairs of Adjacency Relations

  • Valentin E. Brimkov
  • Reinhard Klette
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)


In this paper we propose several equivalent definitions of digital curves and hypersurfaces in arbitrary dimension. The definitions involve properties such as one-dimensionality of curves and (n – 1)-dimensionality of hypersurfaces that make them discrete analogs of corresponding notions in topology. Thus this work appears to be the first one on digital manifolds where the definitions involve the notion of dimension. In particular, a digital hypersurface in nD is an (n – 1)-dimensional object, as it is in the case of continuous hypersurfaces. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a clssification of good pairs in arbitrary dimension.


digital geometry digital topology digital curve digital hypersurface good pair 


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  1. 1.
    Alexander, J.C., Thaler, A.I.: The boundary count of digital pictures. J. ACM 18, 105–112 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Andres, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. Graphical Models Image Processing 59, 302–309 (1997)CrossRefGoogle Scholar
  3. 3.
    Bertrand, G., Malgouyres, R.: Some topological properties of surfaces in Z 3. J. Mathematical Imaging Vision 11, 207–221 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brimkov, V.E., Andres, E., Barneva, R.P.: Object discretizations in higher dimensions. Pattern Recognition Letters 23, 623–636 (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, L., Cooley, D.H., Zhang, J.: The equivalence between two definitions of digital surfaces. Information Sciences 115, 201–220 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cohen-Or, D., Kaufman, A., Kong, T.Y.: On the soundness of surface voxelizations. In: Kong, T.Y., Rosenfeld, A. (eds.) Topological Algorithms for Digital Image Processing, pp. 181–204. Elsevier, Amsterdam (1996)CrossRefGoogle Scholar
  7. 7.
    Couprie, M., Bertrand, G.: Tessellations by connection. Pattern Recognition Letters 23, 637–647 (2002)zbMATHCrossRefGoogle Scholar
  8. 8.
    Duda, R.O., Hart, P.E., Munson, J.H.: Graphical-data-processing research study and experimental investigation. In: TR ECOM-01901-26, March 1967, Stanford Research Institute, Menlo Park (1967)Google Scholar
  9. 9.
    Eckhardt, U., Latecki, L.: Topologies for the digital spaces Z 2 and Z 3. Computer Vision Image Understanding 90, 295–312 (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    Françon, J.: Discrete combinatorial surfaces. Graphical Models Image Processing 57, 20–26 (1995)CrossRefGoogle Scholar
  11. 11.
    Herman, G.T.: Boundaries in digital spaces: Basic theory. In: Kong, T.Y., Rosenfeld, A. (eds.) Topological Algorithms for Digital Image Processing, pp. 233–261. Elsevier, Amsterdam (1996)CrossRefGoogle Scholar
  12. 12.
    Kenmochi, Y., Imiya, A., Ichikawa, A.: Discrete combinatorial geometry. Pattern Recognition 30, 1719–1728 (1997)zbMATHCrossRefGoogle Scholar
  13. 13.
    Kim, C.E.: Three-dimensional digital line segments. IEEE Trans. Pattern Analysis Machine Intelligence 5, 231–234 (1983)zbMATHCrossRefGoogle Scholar
  14. 14.
    Klette, R., Rosenfeld, A.: Digital Geometry - Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  15. 15.
    Kong, T.Y.: Digital topology. In: Davis, L.S. (ed.) Foundations of Image Understanding, Massachusetts, pp. 33–71. Kluwer, Boston (2001)Google Scholar
  16. 16.
    Kong, T.Y.: Topological adjacency relations on Z n. Theoretical Computer Science 283, 3–28 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kong, T.Y., Roscoe, A.W., Rosenfeld, A.: Concepts of digital topology. Topology and its Applications 46, 219–262 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kopperman, R., Meyer, P.R., Wilson, R.: A Jordan surface theorem for three-dimensional digital spaces. Discrete Computational Geometry 6, 155–161 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kovalevsky, V.: Multidimensional cell lists for investigating 3-manifolds. Discrete Applied Mathematics 125, 25–44 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lachaud, J.-O., Montanvert, A.: Continuous analogs of digital boundaries: A topological approach to isosurfaces. Graphical Models 62, 129–164 (2000)CrossRefGoogle Scholar
  21. 21.
    Malgouyres, R.: A definition of surfaces of Z 3: A new 3D discrete Jordan theorem. Theoretical Computer Science 186, 1–41 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Morgenthaler, D.G., Rosenfeld, A.: Surfaces in three-dimensional digital images. Information Control 51, 227–247 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mylopoulos, J.P., Pavlidis, T.: On the topological properties of quantized spaces. I. The notion of dimension. J. ACM 18, 239–246 (1971)MathSciNetGoogle Scholar
  24. 24.
    Reveillès, J.-P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  25. 25.
    Rosenfeld, A.: Adjacency in digital pictures. Information and Control 26, 24–33 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Rosenfeld, A.: Compact figures in digital pictures. IEEE Trans. Systems, Man, Cybernetics 4, 221–223 (1974)zbMATHGoogle Scholar
  27. 27.
    Rosenfeld, A., Kong, T.Y., Wu, A.Y.: Digital surfaces. CVGIP: Graphical Models Image Processing 53, 305–312 (1991)zbMATHCrossRefGoogle Scholar
  28. 28.
    Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. J. ACM 13, 471–494 (1966)zbMATHCrossRefGoogle Scholar
  29. 29.
    Tourlakis, G.: Homological methods for the classification of discrete Euclidean structures. SIAM J. Applied Mathematics 33, 51–54 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tourlakis, G., Mylopoulos, J.: Some results in computational topology. J. ACM 20, 430–455 (1973)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Udupa, J.K.: Connected, oriented, closed boundaries in digital spaces: Theory and algorithms. In: Kong, T.Y., Rosenfeld, A. (eds.) Topological Algorithms for Digital Image Processing, pp. 205–231. Elsevier, Amsterdam (1996)CrossRefGoogle Scholar
  32. 32.
    Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Valentin E. Brimkov
    • 1
  • Reinhard Klette
    • 2
  1. 1.Fairmont State UniversityFairmontUSA
  2. 2.CITR TamakiUniversity of AucklandAucklandNew Zealand

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