Skip to main content
SpringerLink
Log in

Topology preservation on the triangular grid

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

There are exactly three regular planar grids, which are formed by tiling the 2-dimensional Euclidean space with regular triangles, squares, and hexagons. The topology of the square grid is well-understood, but it cannot be said of the remaining two regular sampling schemes. This work deals with the topological properties of digital binary pictures sampled on the triangular grid. Some characterizations of simple pixels and sufficient conditions for topology preserving operators are reported. These results provide the theoretical background to various topological algorithms including thinning, shrinking, generating discrete Voronoi diagrams, and contour smoothing on the triangular grid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertrand, G., Couprie, M.: Transformations topologiques discrètes. In: Coeurjolly, D., Montanvert, A., Chassery, J. (eds.) Géométrie discrète et images numériques, pp. 187–209. Hermès Science Publications (2007)

  2. Bertrand, G., Couprie, M.: Two-dimensional thinning algorithms based on critical kernels. J. Math. Imaging Vision 31(1), 35–56 (2008)

    Article  MathSciNet  Google Scholar 

  3. Bloch, I., Pescatore, J., Garnero, L.: A new characterization of simple elements in a tetrahedral mesh. Graph. Model. 67(4), 260–284 (2005)

    Article  MATH  Google Scholar 

  4. Brimkov, V.E., Barneva, R.P.: Analytical Honeycomb Geometry for Raster and Volume Graphics. Comput. J. 48, 180–199 (2005)

    Article  Google Scholar 

  5. Deutsch, E.S.: On parallel operations on hexagonal arrays. IEEE Trans. Comp. C-19(10), 982–983 (1970)

    Article  Google Scholar 

  6. Deutsch, E.S.: Thinning algorithms on rectangular, hexagonal, and triangular arrays. Commun. ACM 15, 827–837 (1972)

    Article  Google Scholar 

  7. Gaspar, F.J., Gracia, J.L., Lisbona, F.J., Rodrigo, C.: On geometric multigrid methods for triangular grids using three-coarsening strategy. Appl. Numer. Math. 59(7), 1693–1708 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hall, R.W. In: Kong, T.Y., Rosenfeld, A. (eds.) : Parallel connectivity-preserving thinning algorithms. In: Topological algorithms for digital image processing, pp. 145–179. Elsevier Science (1996)

  9. Hall, R.W., Kong, T.Y., Rosenfeld, A. In: Kong, T.Y., Rosenfeld, A. (eds.) : Shrinking binary images. In: Topological algorithms for digital image processing, pp. 31–98. Elsevier Science (1996)

  10. Hartmann, E.: A marching method for the triangulation of surfaces. Vis. Comput. 14, 95–108 (1998)

    Article  MATH  Google Scholar 

  11. Kardos, P., Palágyi, K.: On topology preservation for hexagonal parallel thinning algorithms. In: Proceedings 14th International Workshop on Combinatorial Image Analysis, IWCIA 2011, Madrid, Spain, Lecture Notes in Computer Science, Vol. 6636, pp. 31–42. Springer (2011)

  12. Kardos, P., Palágyi, K.: On topology preservation for triangular thinning algorithms. In: Proceedings 15th International Workshop on Combinatorial Image Analaysis, IWCIA 2012, Austin, TX, USA, Lecture Notes in Computer Science, Vol. 7655, pp. 128–142. Springer (2012)

  13. Kardos, P., Palágyi, K.: Topology preserving hexagonal thinning. Int. J. Comput. Math. 90, 1607–1617 (2013)

    Article  MATH  Google Scholar 

  14. Kardos, P., Palágyi, K.: Sufficient conditions for topology preserving additions and general operators. In: Proceddings 14th IASTED International Conference Computer Graphics and Imaging, CGIM 2013, pp. 107–114. IASTED ACTA Press (2013)

  15. Kong, T.Y.: On topology preservation in 2-d and 3-d thinning. Int. J. Pattern Recognit. Artif. Intell. 9, 813–844 (1995)

    Article  Google Scholar 

  16. Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Comp. Vision Graph. Image Process. 48, 357–393 (1989)

    Article  Google Scholar 

  17. Lam, L., Lee, S.-W., Suen, C.Y.: Thinning methodologies — A comprehensive survey. IEEE Trans. Pattern Anal. Mach. Intell. 14, 869–885 (1992)

    Article  Google Scholar 

  18. Lee, M., Jayanthi, S.: Hexagonal Image Processing: A Practical Approach (Advances in Pattern Recognition). Springer-Verlag (2005)

  19. Ma, C.M.: On topology preservation in 3D thinning. CVGIP: Image Underst. 59, 328–339 (1994)

    Article  Google Scholar 

  20. Marchand-Maillet, S., Sharaiha, Y.M.: Binary Digital Image Processing – A Discrete Approach. Academic Press (2000)

  21. Nagy, B.: Characterization of digital circles in triangular grid. Pattern Recogn. Lett. 25(11), 1231–1242 (2004)

    Article  Google Scholar 

  22. Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discret. Appl. Math. 21, 67–79 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Sahr, K., White, D., Kimerling, A.J.: Geodesic discrete global grid systems. Cartogr. Geogr. Inf. Sci. 30(2), 121–134 (2003)

    Article  Google Scholar 

  24. Serra, J.: Image analysis and mathematical morphology. Academic Press (1982)

  25. Staunton, R.C.: An analysis of hexagonal thinning algorithms and skeletal shape representation. Pattern Recog. 29, 1131–1146 (1996)

    Article  Google Scholar 

  26. Staunton, R.C.: One-pass parallel hexagonal thinning algorithm. In: Proceedings IEE 7th International Conference Image Processing and Its Applications, pp. 841–845 (1999)

  27. In: Suen, C.Y., Wang, P.S.P. (eds.) : Thinning methodologies for pattern recognition. Series in Machine Perception and Artificial Intelligence 8. World Scientific, Singapore (1994)

  28. Wiederhold, P., Morales, S.: Thinning on quadratic, triangular, and hexagonal cell complexes. In: Proceedings 13th International Workshop on Combinatorial Image Analysis, IWCIA 2009, Lecture Notes in Computer Science, Vol. 5852, pp. 162–175. Springer (2009)

  29. Wuthrich, C., Stucki, P.: An algorithm comparison between square- and hexagonal-based grids. Graph. Model Image Process. 53, 324–339 (1991)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Image Processing and Computer Graphics, University of Szeged, Szeged, Hungary

    Péter Kardos & Kálmán Palágyi

Authors
  1. Péter Kardos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kálmán Palágyi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kálmán Palágyi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kardos, P., Palágyi, K. Topology preservation on the triangular grid. Ann Math Artif Intell 75, 53–68 (2015). https://doi.org/10.1007/s10472-014-9426-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-014-9426-6

Keywords

Search

Navigation