Advertisement

How to Make n-D Plain Maps Defined on Discrete Surfaces Alexandrov-Well-Composed in a Self-Dual Way

  • Nicolas Boutry
  • Thierry GéraudEmail author
  • Laurent Najman
Article

Abstract

In 2013, Najman and Géraud proved that by working on a well-composed discrete representation of a gray-level image, we can compute what is called its tree of shapes, a hierarchical representation of the shapes in this image. This way, we can proceed to morphological filtering and to image segmentation. However, the authors did not provide such a representation for the non-cubical case. We propose in this paper a way to compute a well-composed representation of any gray-level image defined on a discrete surface, which is a more general framework than the usual cubical grid. Furthermore, the proposed representation is self-dual in the sense that it treats bright and dark components in the image the same way. This paper can be seen as an extension to gray-level images of the works of Daragon et al. on discrete surfaces.

Keywords

Well-composedness Discrete surfaces Digital topology Alexandrov spaces Frontier orders Cross section topology Tree of shapes Mathematical morphology 

Notes

Acknowledgements

We would like to acknowledge the time and effort devoted by the reviewers, which greatly improved the quality of our paper.

References

  1. 1.
    Alexander, J.W.: A proof and extension of the Jordan–Brouwer separation theorem. Trans. Am. Math. Soc. 23(4), 333–349 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alexandrov, P.S.: Diskrete Räume. Matematicheskii Sbornik 2(3), 501–519 (1937)zbMATHGoogle Scholar
  3. 3.
    Alexandrov, P.S.: Combinatorial Topology, vol. 1-3. Dover Publications, New York (2011)Google Scholar
  4. 4.
    Alexandrov, P.S., Hopf, H., Topologie, I.: Die grundlehren der mathematischen wissenschaften in einzeldarstellungen, vol. 45. Springer, Berlin (1945)Google Scholar
  5. 5.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bertrand, G.: New notions for discrete topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science Series, vol. 1568, pp. 218–228. Springer (1999)Google Scholar
  7. 7.
    Bertrand, G., Everat, J.-C., Couprie, M.: Topological approach to image segmentation. In: SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation. Vision Geometry V, vol. 2826, pp. 65–76. International Society for Optics and Photonics (1996)Google Scholar
  8. 8.
    Bertrand, G., Everat, J.-C., Couprie, M.: Image segmentation through operators based on topology. J. Electron. Imaging 6(4), 395–405 (1997)CrossRefGoogle Scholar
  9. 9.
    Beucher, S., Meyer, F.: The morphological approach to segmentation: the watershed transformation. Opt. Eng. 34, 433–433 (1992)Google Scholar
  10. 10.
    Boutry, N.: A study of well-composedness in \(n\)-D. Ph.D. thesis, Université Paris-Est, Noisy-Le-Grand, France (2016)Google Scholar
  11. 11.
    Boutry, N., Géraud, T., Najman, L.: How to make \(n\)-D functions digitally well-composed in a self-dual way. In: International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing. Lecture Notes in Computer Science Series, vol. 9082, pp. 561–572. Springer (2015)Google Scholar
  12. 12.
    Boutry, N., Géraud, T., Najman, L.: How to make \(n\)-D images well-composed without interpolation. In: International Conference on Image Processing. IEEE (2015)Google Scholar
  13. 13.
    Boutry, N., Géraud, T., Najman, L.: A tutorial on well-composedness. J. Math. Imaging Vis. 60, 443–478 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Boutry, N., González-Díaz, R., Jiménez, M.J.: Weakly well-composed cell complexes over \(n\)-D pictures. Inf. Sci. (2018).  https://doi.org/10.1016/j.ins.2018.06.005 Google Scholar
  15. 15.
    Boutry, N., Najman, L., Géraud, T.: About the equivalence between AWCness and DWCness. Research report, LIGM: Laboratoire d’Informatique Gaspard-Monge ; LRDE: Laboratoire de Recherche et de Développement de l’EPITA (HAL Id: hal-01375621) (2016)Google Scholar
  16. 16.
    Boutry, N., Najman, L., Géraud, T.: Well-composedness in Alexandrov spaces implies digital well-composedness in \({\mathbb{Z}}^n\). In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science Series, vol. 10502, pp. 225–237. Springer (2017)Google Scholar
  17. 17.
    Carlinet, E., Géraud, T.: A color tree of shapes with illustrations on filtering, simplification, and segmentation. In: International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing. Lecture Notes in Computer Science Series, vol. 9082, pp. 363–374. Springer (2015)Google Scholar
  18. 18.
    Caselles, V., Monasse, P.: Grain filters. J. Math. Imaging Vis. 17(3), 249–270 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Caselles, V., Monasse, P.: Geometric Description of Images as Topographic Maps. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  20. 20.
    Čomić, L., Magillo, P.: Repairing 3D binary images using the BCC grid with a 4-valued combinatorial coordinate system. Inf. Sci. (2018).  https://doi.org/10.1016/j.ins.2018.02.049 Google Scholar
  21. 21.
    Daragon, X.: Surfaces discrètes et frontières d’objets dans les ordres. Ph.D. thesis, Université de Marne-la-Vallée (2005)Google Scholar
  22. 22.
    Daragon, X., Couprie, M., Bertrand, G.: Discrete surfaces and frontier orders. J. Math. Imaging Vis. 23(3), 379–399 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Eckhardt, U., Latecki, L.J.: Digital topology. Institut für Angewandte Mathematik (1994)Google Scholar
  24. 24.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Foundations of Computer Science, pp. 454–463. IEEE (2000)Google Scholar
  25. 25.
    Evako, A.V., Kopperman, R., Mukhin, Y.V.: Dimensional properties of graphs and digital spaces. J. Math. Imaging Vis. 6(2–3), 109–119 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Géraud, T., Carlinet, E., Crozet, S., Najman, L.: A quasi-linear algorithm to compute the tree of shapes of \(n\)-D images. In: International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing. Lecture Notes in Computer Science Series, vol. 7883, pp. 98–110. Springer (2013)Google Scholar
  27. 27.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Well-composed cell complexes. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science Series, vol. 6607, pp. 153–162. Springer (2011)Google Scholar
  28. 28.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: 3D well-composed polyhedral complexes. Discrete Appl. Math. 183, 59–77 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Encoding specific 3D polyhedral complexes using 3D binary images. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science Series, vol. 9647, pp. 268–281. Springer (2016)Google Scholar
  30. 30.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Efficiently storing well-composed polyhedral complexes computed over 3D binary images. J. Math. Imaging Vis. 59(1), 106–122 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Greenberg, M.J.: Lectures on Algebraic Topology. Mathematics Lecture Note, vol. 33556. W.A. Benjamin, New York (1977)Google Scholar
  32. 32.
    Hudson, J.F.: Piecewise Linear Topology, vol. 1. Benjamin, New York (1969)Google Scholar
  33. 33.
    Kelley, J.L.: General Topology Graduate. Texts in Mathematics, vol. 27. Springer, New York (1975)Google Scholar
  34. 34.
    Kopperman, R., Meyer, P.R., Wilson, R.G.: A Jordan surface theorem for three-dimensional digital spaces. Discrete Comput. Geom. 6(2), 155–161 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lachaud, J.-O., Montanvert, A.: Continuous analogs of digital boundaries: a topological approach to ISO-surfaces. Graph. Models Image Process. 62(3), 129–164 (2000)CrossRefGoogle Scholar
  36. 36.
    Latecki, L.J.: Well-composed sets. Adv. Electron. Electron Phys. 112, 95–163 (2000)Google Scholar
  37. 37.
    Levillain, R., Géraud, T., Najman, L.: Writing reusable digital topology algorithms in a generic image processing framework. In: Applications of Discrete Geometry and Mathematical Morphology. Lecture Notes in Computer Science Series, vol. 7346, pp. 140–153. Springer (2012)Google Scholar
  38. 38.
    Lima, E.L.: The Jordan–Brouwer separation theorem for smooth hypersurfaces. Am. Math. Mon. 95(1), 39–42 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lorensen, W.E., Cline, H.E.: Marching cubes: a high resolution 3D surface construction algorithm. In: ACM SIGGRAPH Computer Graphics, vol. 21, pp. 163–169. ACM (1987)Google Scholar
  40. 40.
    Meyer, F.: Skeletons and perceptual graphs. Signal Process. 16(4), 335–363 (1989)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Najman, L., Géraud, T.: Discrete set-valued continuity and interpolation. In: Mathematical Morphology and Its Applications to Signal and Image Processing. Lecture Notes in Computer Science Series, vol. 7883, pp. 37–48. Springer (2013)Google Scholar
  42. 42.
    Najman, L., Talbot, H.: Mathematical Morphology: From Theory to Applications. Wiley, New York (2013)CrossRefzbMATHGoogle Scholar
  43. 43.
    Serra, J., Soille, P.: Mathematical Morphology and Its Applications to Image Processing, vol. 2. Springer, New York (2012)zbMATHGoogle Scholar
  44. 44.
    Whitehead, G.W.: Elements of Homotopy Theory. Graduate Texts in Mathematics, vol. 61. Springer, New York (1978)CrossRefGoogle Scholar
  45. 45.
    Xu, Y., Géraud, T., Najman, L.: Morphological filtering in shape spaces: applications using tree-based image representations. In: International Conference on Pattern Recognition, pp. 485–488. IEEE (2012)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.EPITA Research and Development Laboratory (LRDE)Le Kremlin-BicêtreFrance
  2. 2.Université Paris-Est, LIGM, Équipe A3SI, ESIEEChamps-sur-MarneFrance

Personalised recommendations