Concepts of Binary Morphological Operations Dilation and Erosion on the Triangular Grid

  • Mohsen Abdalla
  • Benedek NagyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10149)


In this paper, basic concepts of digital binary morphological operations, i.e., dilation and erosion are investigated on a triangular grid. Every triangle pixel is addressed by a unique coordinate triplet with sum zero (even pixels) or one (odd pixels). Even and odd pixels have different orientations. The triangular grid is not a lattice, that is, not every translation with a grid vector maps the grid to itself. Therefore, to extend the morphological operations to the triangular grid is not straightforward. We introduce three types of definition for both of dilation and erosion. Various examples and properties of the considered dilation and erosion are analyzed on the triangular grid.


Digital image processing Mathematical morphology Binary morphology Dilation Erosion Triangular grid Non-traditional grids 


  1. 1.
    Deutsch, E.S.: Thinning algorithms on rectangular, hexagonal, and triangular arrays. Commun. ACM 15(9), 827–837 (1972)CrossRefGoogle Scholar
  2. 2.
    Dineen, G.P.: Programming pattern recognition. In: Proceedings of Western Joint Computer Conference, Los Angeles, CA, pp. 94–100, 1–3 March 1955Google Scholar
  3. 3.
    Ghosh, P.K., Deguchi, K.: Mathematics of Shape Description. Wiley, New York (2009)Google Scholar
  4. 4.
    Golay, M.J.E.: Hexagonal parallel pattern transformations. IEEE Trans. Comput. 18(8), 733–740 (1969)CrossRefGoogle Scholar
  5. 5.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Prentice-Hall Inc., Upper Saddle River (2006)Google Scholar
  6. 6.
    Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. Image Process. 4(9), 1213–1222 (1995)CrossRefGoogle Scholar
  7. 7.
    Kardos, P., Palágyi, K.: Topology preservation on the triangular grid. Ann. Math. Artif. Intell. 75, 53–68 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kirsch, R.A.: Experiments in processing life motion with a digital computer. In: Proceedings of Eastern Joint Computer Conference, pp. 221–229 (1957)Google Scholar
  9. 9.
    Klette, R., Rosenfeld, A.: Digital Geometry. Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  10. 10.
    Luczak, E., Rosenfeld, A.: Distance on a hexagonal grid. IEEE Trans. Comput. C-25(5), 532–533 (1976)CrossRefzbMATHGoogle Scholar
  11. 11.
    Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)zbMATHGoogle Scholar
  12. 12.
    Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nagy, B.: Finding shortest path with neighborhood sequences in triangular grids. In: ITI-ISPA 2001, 2nd IEEE R8-EURASIP International Symposium on Image and Signal Processing and Analysis, Pula, Croatia, pp. 55–60 (2001)Google Scholar
  14. 14.
    Nagy, B.: A family of triangular grids in digital geometry. In: ISPA 2003, 3rd International Symposium on Image and Signal Processing and Analysis, Rome, Italy, pp. 101–106 (2003)Google Scholar
  15. 15.
    Nagy, B.: Generalized triangular grids in digital geometry. Acta Math. Acad. Paedagogicae Nyíregyháziensis 20, 63–78 (2004)zbMATHGoogle Scholar
  16. 16.
    Nagy, B.: Isometric transformations of the dual of the hexagonal lattice. In: ISPA 2009 – 6th International Symposium on Image and Signal Processing and Analysis, Salzburg, Austria, pp. 432–437 (2009)Google Scholar
  17. 17.
    Nagy, B.: Weighted distances on a triangular grid. In: Barneva, R.P., Brimkov, V.E., Šlapal, J. (eds.) IWCIA 2014. LNCS, vol. 8466, pp. 37–50. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-07148-0_5 CrossRefGoogle Scholar
  18. 18.
    Nagy, B.: Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids. Ann. Math. Artif. Intell. 75, 117–134 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nagy, B., Lukić, T.: Dense projection tomography on the triangular tiling. Fundam. Informaticae 145, 125–141 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New York (1982)zbMATHGoogle Scholar
  21. 21.
    Shih, F.: Binary Morphology. Image Processing and Mathematical Morphology. CRC Press, Boca Raton (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Soille, P., Rivest, J.F.: Principles and Applications of Morphological Image Analysis. Springer, Berlin (1992)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Arts and Sciences, Department of MathematicsEastern Mediterranean UniversityFamagustaTurkey

Personalised recommendations