A Comparison Among Distances Based on Neighborhood Sequences in Regular Grids

  • Benedek Nagy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3540)


The theory of neighborhood sequences is applicable in many image-processing algorithms. The theory is well developed for the square grid. Recently there are some results for the hexagonal grid as well. In this paper, we are considering all the three regular grids in the plane. We show that there are some very essential differences occurring. On the triangular plane the distance has metric properties. The distances on the square and the hexagonal case may not meet the triangular inequality. There are non-symmetric distances on the hexagonal case. In addition, contrary to the other two grids,the distance can depend on the order of the initial elements of the neighborhood sequence.Moreover in the hexagonal grid it is possible that circles with different radii are the same (using different neighborhood sequences). On the square grid the circles with the same radius are in a well ordered set, but in the hexagonal case there can be non-comparable circles.


Digital geometry Neighborhood sequences Square grid Hexagonal grid Triangular grid Distance 


  1. 1.
    Danielsson, P.E.: 3D Octagonal Metrics. In: Proceedings of Eighth Scandinavian Conference on Image Processing, pp. 727–736 (1993)Google Scholar
  2. 2.
    Das, P.P., Chakrabarti, P.P., Chatterji, B.N.: Distance functions in digital geometry. Information Science 42, 113–136 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Deutsch, E.S.: Thinning algorithms on rectangular, hexagonal and triangular arrays. Communications of the ACM 15(3), 827–837 (1972)CrossRefGoogle Scholar
  4. 4.
    Fazekas, A., Hajdu, A., Hajdu, L.: Lattice of generalized neighborhood sequences in nD and ∞D. Publicationes Mathematicae Debrecen 60, 405–427 (2002)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Hajdu, A., Nagy, B.: Approximating the Euclidean circle using neighbourhood sequences. In: Proceedings of the third KEPAF conference, Domaszék, Hungary, January 2002, pp. 260–271 (2002)Google Scholar
  6. 6.
    Hajdu, A., Nagy, B., Zörgő, Z.: Indexing and segmenting colour images using neighborhood sequences. In: Proceedings of IEEE International Conference on Image Processing, Barcelona, Spain, September 2003, vol. 1, pp. 957–960 (2003)Google Scholar
  7. 7.
    Hajdu, A., Kormos, J., Nagy, B., Zörgő, Z.: Choosing appropriate distance measurement in digital image segmentation. Ann. Univ. Sci. Budapest. Sect. Comput. 24, 193–208 (2004)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Nagy, B.: Distance functions based on neighbourhood sequences. Publicationes Mathematicae Debrecen 63, 483–493 (2003)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Nagy, B.: Shortest Path in Triangular Grids with Neighborhood Sequences. Journal of Computing and Information Technology 11, 111–122 (2003)CrossRefGoogle Scholar
  10. 10.
    Nagy, B.: Metrics based on neighborhood sequences in triangular grids. Pure Mathematics and Applications 13, 259–274 (2002)MathSciNetGoogle Scholar
  11. 11.
    Nagy, B.: A symmetric coordinate system for the hexagonal networks. In: Proceedings of Information Society 2004 – Theoretical Computer Science (ACM Slovenija conference), Ljubljana, Slovenia, October 2004, vol. D, pp. 181–184 (2004)Google Scholar
  12. 12.
    Nagy, B.: Non-metrical distances on the hexagonal plane. In: Proceedings of the 7th International Conference on Pattern Recognition and Image Analysis: New Information Technologies, St. Petersburg, Russian Federation, October 2004, pp. 335–338 (2004); Accepted for publication in Pattern Recognition and Image AnalysisGoogle Scholar
  13. 13.
    Rosenfeld, A., Pfaltz, J.L.: Distance functions on digital pictures. Pattern Recognition 1, 33–61 (1968)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Yamashita, M., Ibaraki, T.: Distances defined by neighborhood sequences. Pattern Recognition 19, 237–246 (1986)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Benedek Nagy
    • 1
    • 2
  1. 1.Department of Computer Science, Faculty of InformaticsUniversity of DebrecenDebrecenHungary
  2. 2.Research Group on Mathematical LinguisticsRovira i Virgili UniversityTarragonaSpain

Personalised recommendations