Advertisement

Journal of Mathematical Imaging and Vision

, Volume 60, Issue 5, pp 707–716 | Cite as

Characterization of Bijective Digitized Rotations on the Hexagonal Grid

  • Kacper Pluta
  • Tristan Roussillon
  • David Cœurjolly
  • Pascal Romon
  • Yukiko Kenmochi
  • Victor Ostromoukhov
Article
  • 43 Downloads

Abstract

Digitized rotations on discrete spaces are usually defined as the composition of a Euclidean rotation and a rounding operator; they are in general not bijective. Nevertheless, it is well known that digitized rotations defined on the square grid are bijective for some specific angles. This infinite family of angles has been characterized by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In this article, we characterize bijective digitized rotations on the hexagonal grid using arithmetic properties of the Eisenstein integers.

Keywords

Hexagonal grid Digital geometry Digital topology Honeycomb geometry Digitized rotations Bijective transformations Geometric transformations 

Notes

Acknowledgements

This work received funding from the project CoMeDiC (ANR–15–CE40–0006).

References

  1. 1.
    Anglin, W.S.: Using Pythagorean triangles to approximate angles. Am. Math. Mon. 95(6), 540–541 (1988)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Condat, L., Van De Ville, D., Blu, T.: Hexagonal versus orthogonal lattices: a new comparison using approximation theory. In: ICIP 2005, vol. 3, pp. III–1116. IEEE (2005)Google Scholar
  3. 3.
    Condat, L., Ville, D.V.D.: Quasi-interpolating spline models for hexagonally-sampled data. IEEE Trans. Image Process. 16(5), 1195–1206 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Conway, J., Smith, D.: On Quaternions and Octonions. Ak Peters Series. Taylor & Francis, Boca Raton (2003)MATHGoogle Scholar
  5. 5.
    Fredriksson, K.: Rotation Invariant Template Matching. Ph.D. thesis, University of Helsinki (2001)Google Scholar
  6. 6.
    Gilder, J.: Integer-Sided Triangles with an angle of 60\(^\circ \). Math. Gaz. 66(438), 261–266 (1982)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gordon, R.A.: Properties of Eisenstein triples. Math. Mag. 85(1), 12–25 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hales, T.C.: The honeycomb conjecture. Discrete Computat. Geom. 25(1), 1–22 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1979)MATHGoogle Scholar
  10. 10.
    Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. Image Process. 4(9), 1213–1222 (1995)CrossRefGoogle Scholar
  11. 11.
    Jacob, M.A., Andres, E.: On discrete rotations. In: 5th International Workshop on Discrete Geometry for Computer Imagery, pp. 161–174 (1995)Google Scholar
  12. 12.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  13. 13.
    Kong, T., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Gr. Image Process. 48(3), 357–393 (1989)CrossRefGoogle Scholar
  14. 14.
    Middleton, L., Sivaswamy, J.: Edge detection in a hexagonal-image processing framework. Image Vis. Comput. 19(14), 1071–1081 (2001)CrossRefGoogle Scholar
  15. 15.
    Middleton, L., Sivaswamy, J.: Hexagonal Image Processing: A Practical Approach. Advances in Pattern Recognition. Springer, Berlin (2005)MATHGoogle Scholar
  16. 16.
    Nouvel, B., Rémila, E.: On colorations induced by discrete rotations. In: DGCI, Proceedings, Lecture Notes in Computer Science, vol. 2886, pp. 174–183. Springer (2003)Google Scholar
  17. 17.
    Nouvel, B., Rémila, E.: Characterization of bijective discretized rotations. In: Klette, R., Žunić, J. (eds.) Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 3322, pp. 248–259. Springer, Berlin (2005)CrossRefGoogle Scholar
  18. 18.
    Nouvel, B., Rémila, E.: Configurations induced by discrete rotations: periodicity and quasi-periodicity properties. Discrete Appl. Math. 147(2–3), 325–343 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ostromoukhov, V., Hersch, R.D.: Halftoning by rotating non-bayer dispersed dither arrays. SPIE Milest. Ser. 154, 238–255 (1999)Google Scholar
  20. 20.
    Ostromoukhov, V., Hersch, R.D., Amidror, I.: Rotated dispersed dither: a new technique for digital halftoning. In: Proceedings of the 21st annual conference on Computer graphics and interactive techniques, pp. 123–130. ACM (1994)Google Scholar
  21. 21.
    Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijective digitized rigid motions on subsets of the plane. J. Math. Imaging Vis. 59(1), 84–105 (2017)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Honeycomb Geometry: Rigid Motions on the Hexagonal Grid. In: DGCI 2017, pp. 33–45. Springer International Publishing, Cham (2017)Google Scholar
  23. 23.
    Roussillon, T., Cœurjolly, D.: Characterization of bijective discretized rotations by Gaussian integers. Research report, LIRIS UMR CNRS 5205 (2016). https://hal.archives-ouvertes.fr/hal-01259826
  24. 24.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)MATHGoogle Scholar
  25. 25.
    Yilmaz, A., Javed, O., Shah, M.: Object tracking: a survey. ACM Comput. Surv. 38(4), 13 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LAMA (UMR 8050), UPEM, UPEC, CNRSUniversité Paris-EstMarne-la-ValléeFrance
  2. 2.LIRIS (UMR 5205), CNRSUniversité de LyonLyonFrance
  3. 3.LIGM (UMR 8049), UPEM, CNRS, ESIEE Paris, ENPCUniversité Paris-EstMarne-la-ValléeFrance

Personalised recommendations