Neighborhood Decomposition of Convex Structuring Elements for Mathematical Morphology on Hexagonal Grid

  • Syng-Yup Ohn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4263)


In this paper, we present a new technique to find the optimal neighborhood decomposition for convex structuring elements used in morphological image processing on hexagonal grid. In neighborhood decomposition, a structuring element is decomposed into a set of neighborhood structuring elements, each of which consists of the combination of the origin pixel and its six neighbor pixels. Generally, neighborhood decomposition reduces the amount of computation required to perform morphological operations such as dilation and erosion. Firstly, we define a convex structuring element on a hexagonal grid and formulate the necessary and sufficient condition to decompose a convex structuring element into the set of basis convex structuring elements. Secondly, decomposability of a convex structuring element into the set of primal bases is also proved. Furthermore, cost function is used to represent the amount of computation or execution time required for performing dilations on different computing environments and by different implementation methods. The decomposition condition and the cost function are applied to find the optimal neighborhood decomposition of a convex structuring element, which guarantees the minimal amount of computation for morphological operations. Example decompositions show that the decomposition results in great reduction in the amount of computation for morphological operations.


Cost Function Mathematical Morphology Morphological Operation Primal Basis Hexagonal Grid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Serra, J.: Introduction to Mathematical Morphology. Computer Vision, Graphics and Image Processing 35, 285–305 (1986)CrossRefGoogle Scholar
  2. 2.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)MATHGoogle Scholar
  3. 3.
    Haralick, R.M., Sternberg, S.R., Zhuang, X.: Image Analysis Using Mathematical Morphology. IEEE Trans. on PAMI 9, 532–550 (1987)Google Scholar
  4. 4.
    Aykac, D., Hoffman, E.A., McLennan, G., Reinhardt, J.M.: Segmentation and analysis of the human airway tree from three-dimensional X-ray CT images. IEEE Transactions on Medical Imaging 22, 940–950 (2003)CrossRefGoogle Scholar
  5. 5.
    Zana, F., Klein, J.-C.: Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation. IEEE Trans. on Image Processing 10, 1010–1019 (2001)MATHCrossRefGoogle Scholar
  6. 6.
    Soille, P., Pesaresi, M.: Advances in mathematical morphology applied to geoscience and remote sensing. IEEE Trans.on Geoscience and Remote Sensing 40, 2042–2055 (2002)CrossRefGoogle Scholar
  7. 7.
    Zhuang, X., Haralick, R.M.: Morphological Structuring Element Decomposition. Computer Vision, Graphics and Image Processing 35, 370–382 (1986)CrossRefGoogle Scholar
  8. 8.
    Dadda, L.: Parallel algorithms and architectures for CPUs and dedicated processors: development and trends. In: Algorithms and Architectures for Parallel Processing ICAPP 1995, vol. 2, pp. 939–948 (1995)Google Scholar
  9. 9.
    Svolos, A.I., Konstantopoulos, C.G., Kaklamanis, C.: Efficient Binary Morphological Algorithms on a Massively Parallel Processor. In: International Parallel and Distributed Processing Symposium 2000 Proceedings, pp. 281–286 (2000)Google Scholar
  10. 10.
    York, G., Managuli, R., Kim, Y.: Fast Binary and Grayscale Mathematical Morphology on VLIW. In: Proc. Of SPIE: Real-Time Imaging IV, pp. 45–55 (1999)Google Scholar
  11. 11.
    Levialdi, S.: Computer Architectures for Image Analysis. In: 9th International Conference on Pattern Recognition, vol. 2, pp. 1148–1158 (1988)Google Scholar
  12. 12.
    Xu, J.: Decomposition of Convex Polygonal Morphological Structuring Elements into Neighborhood Subsets. IEEE Trans. on PAMI 13, 153–162 (1991)Google Scholar
  13. 13.
    Park, H., Chin, R.T.: Optimal Decomposition of Convex Morphological Structuring Elements for 4-connected Parallel Array Processors. IEEE Trans. on PAMI 16, 304–313 (1994)MATHGoogle Scholar
  14. 14.
    Wuthrich, C.A., Stucki, P.: An algorithmic comparison between square- and hexagonal grids. CVGIP:Graphical Models Image Processing 53, 324–339 (1991)CrossRefGoogle Scholar
  15. 15.
    Her, I., Yuan, C.-T.: Resampling on a pseudo-hexagonal grid. CVGIP: Graphical Models Image Processing 56, 336–347 (1994)CrossRefMATHGoogle Scholar
  16. 16.
    Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. on Image Processing 4, 1213–1222 (1995)CrossRefGoogle Scholar
  17. 17.
    Ohn, S.-Y.: Decomposition of 3D Convex Structuring Element in Morphological Operation for Parallel Processing Architectures. In: Kamel, M., Campilho, A.C. (eds.) ICIAR 2005. LNCS, vol. 3656, pp. 644–651. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Syslo, M.M., Deo, N., Kowalik, J.S.: Discrete Optimization Algorithms. Prentice-Hall, Englewood Cliffs (1983)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Syng-Yup Ohn
    • 1
  1. 1.Department of Computer and Information EngineeringHankuk Aviation UniversitySeoulKorea

Personalised recommendations