Advertisement

Digital Two-Dimensional Bijective Reflection and Associated Rotation

  • Eric AndresEmail author
  • Mousumi Dutt
  • Arindam Biswas
  • Gaelle Largeteau-Skapin
  • Rita Zrour
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)

Abstract

In this paper, a new bijective reflection algorithm in two dimensions is proposed along with an associated rotation. The reflection line is defined by an arbitrary Euclidean point and a straight line passing through this point. The reflection line is digitized and the 2D space is paved by digital perpendicular (to the reflection line) straight lines. For each perpendicular line, integer points are reflected by central symmetry with respect to the reflection line. Two consecutive digital reflections are combined to define a digital bijective rotation about arbitrary center, i.e. bijective digital rigid motion.

Keywords

Digital reflection Bijective digital rotation Digital rotation Bijective digital rigid motion 

References

  1. 1.
    Goodman, R.: Alice through looking glass after looking glass: the mathematics of mirrors and kaleidoscopes. Am. Math. Mon. 111(4), 281–298 (2004). http://www.jstor.org/stable/4145238MathSciNetCrossRefGoogle Scholar
  2. 2.
    Richard, A., Fuchs, L., Largeteau-Skapin, G., Andres, E.: Decomposition of nD-rotations: classification, properties and algorithm. Graph. Model. 73(6), 346–353 (2011)CrossRefGoogle Scholar
  3. 3.
    Fredriksson, K., Mäkinen, V., Navarro, G.: Rotation and lighting invariant template matching. Inf. Comput. 205(7), 1096–1113 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yilmaz, A., Javed, O., Shah, M.: Object tracking: a survey. ACM Comput. Surv. 38(4), 13 (2006)CrossRefGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Andres, E.: The quasi-shear rotation. In: Miguet, S., Montanvert, A., Ubéda, S. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 307–314. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-62005-2_26CrossRefGoogle Scholar
  7. 7.
    Carstens, H.-G., Deuber, W., Thumser, W., Koppenrade, E.: Geometrical bijections in discrete lattices. Comb. Probab. Comput. 8(1–2), 109–129 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Nouvel, B., Rémila, E.: Characterization of bijective discretized rotations. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 248–259. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-30503-3_19CrossRefGoogle Scholar
  9. 9.
    Nouvel, B., Rémila, É.: Incremental and transitive discrete rotations. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.) IWCIA 2006. LNCS, vol. 4040, pp. 199–213. Springer, Heidelberg (2006).  https://doi.org/10.1007/11774938_16CrossRefGoogle Scholar
  10. 10.
    Roussillon, T., Coeurjolly, D.: Characterization of bijective discretized rotations by gaussian integers, Technical report, LIRIS UMR CNRS 5205, January 2016Google Scholar
  11. 11.
    Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijective rigid motions of the 2D cartesian grid. In: Normand, N., Guédon, J., Autrusseau, F. (eds.) DGCI 2016. LNCS, vol. 9647, pp. 359–371. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-32360-2_28CrossRefGoogle Scholar
  12. 12.
    Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijectivity certification of 3D digitized rotations. In: Bac, A., Mari, J.-L. (eds.) CTIC 2016. LNCS, vol. 9667, pp. 30–41. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-39441-1_4CrossRefGoogle Scholar
  13. 13.
    Pluta, K., Roussillon, T., Coeurjolly, D., Romon, P., Kenmochi, Y., Ostromoukhov, V.: Characterization of bijective digitized rotations on the hexagonal grid, Technical report, HAL, submitted to Journal of Mathematical Imaging and Vision, June 2017. https://hal.archives-ouvertes.fr/hal-01540772
  14. 14.
    Andres, E.: Shear based bijective digital rotation in hexagonal grids. Submitted to Pattern recognition Letters 2018. https://hal.archives-ouvertes.fr/hal-01900148v1
  15. 15.
    Andres, E., Largeteau-Skapin, G., Zrour, R.: Shear based bijective digital rotation in triangular grids. Submitted to Pattern recognition Letters (2018). https://hal.archives-ouvertes.fr/hal-01900149v1
  16. 16.
    Reveillès, J.-P.: Calcul en nombres entiers et algorithmique, Ph.D. thesis. Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  17. 17.
    Jacob, M.-A., Andres, E.: On discrete rotations. In: Discrete Geometry for Computer Imagery, p. 161 (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Eric Andres
    • 1
    Email author
  • Mousumi Dutt
    • 2
  • Arindam Biswas
    • 3
  • Gaelle Largeteau-Skapin
    • 1
  • Rita Zrour
    • 1
  1. 1.University of Poitiers, Laboratory XLIM, ASALI, UMR CNRS 7252, BP 30179Futuroscope ChasseneuilFrance
  2. 2.Department of Computer Science and EngineeringSt. Thomas’ College of Engineering and TechnologyKolkataIndia
  3. 3.Department of Information TechnologyIndian of Engineering Science and Technology, ShibpurHowrahIndia

Personalised recommendations