Journal of Mathematical Imaging and Vision

, Volume 44, Issue 3, pp 254–281 | Cite as

Topology on Digital Label Images

  • Loïc Mazo
  • Nicolas Passat
  • Michel Couprie
  • Christian Ronse
Article

Abstract

In digital imaging, after several decades devoted to the study of topological properties of binary images, there is an increasing need of new methods enabling to take into (topological) consideration n-ary images (also called label images). Indeed, while binary images enable to handle one object of interest, label images authorise to simultaneously deal with a plurality of objects, which is a frequent requirement in several application fields. In this context, one of the main purposes is to propose topology-preserving transformation procedures for such label images, thus extending the ones (e.g., growing, reduction, skeletonisation) existing for binary images. In this article, we propose, for a wide range of digital images, a new approach that permits to locally modify a label image, while preserving not only the topology of each label set, but also the topology of any arrangement of the labels understood as the topology of any union of label sets. This approach enables in particular to unify and extend some previous attempts devoted to the same purpose.

Keywords

Digital imaging Topology Label images Homotopy Simple points 

References

  1. 1.
    Buneman, O.P.: A grammar for the topological analysis of plane figures. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 5, pp. 383–393 (1969) Google Scholar
  2. 2.
    Kong, T.Y.: A digital fundamental group. Comput. Graph. 13(2), 159–166 (1989) CrossRefGoogle Scholar
  3. 3.
    González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147, 245–263 (2005). Advances in Discrete Geometry and Topology MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Hilditch, C.J.: Linear skeletons from square cupboards. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 4, pp. 403–420 (1969) Google Scholar
  5. 5.
    Bertrand, G.: On P-simple points. C. R. Acad. Sci., Sér. Math. 1(321), 1077–1084 (1995) MathSciNetGoogle Scholar
  6. 6.
    Ronse, C.: A topological characterization of thinning. Theor. Comput. Sci. 43(0), 31–41 (1986) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Passat, N., Mazo, L.: An introduction to simple sets. Pattern Recognit. Lett. 30(15), 1366–1377 (2009) CrossRefGoogle Scholar
  8. 8.
    Mangin, J.-F., Frouin, V., Bloch, I., Régis, J., López-Krahe, J.: From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations. J. Math. Imaging Vis. 5(4), 297–318 (1995) CrossRefGoogle Scholar
  9. 9.
    Miri, S., Passat, N., Armspach, J.-P.: Topology-preserving discrete deformable model: application to multi-segmentation of brain MRI. In: International Conference on Image and Signal Processing—ICISP 2008. Lecture Notes in Computer Science, vol. 5099, pp. 67–75. Springer, Berlin (2008) Google Scholar
  10. 10.
    Poupon, F., Mangin, J.-F., Hasboun, D., Poupon, C., Magnin, I., Frouin, V.: Multi-object deformable templates dedicated to the segmentation of brain deep structures. In: MICCAI’98: Proceedings of the First International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 1134–1143. Springer, Berlin (1998) Google Scholar
  11. 11.
    Bazin, P.-L., Pham, D.: Topology-preserving tissue classification of magnetic resonance brain images. IEEE Trans. Med. Imaging 26(4), 487–496 (2007) CrossRefGoogle Scholar
  12. 12.
    Liu, J., Huang, S., Nowinski, W.: Registration of brain atlas to MR images using topology preserving front propagation. J. Signal Process. Syst. 55(1), 209–216 (2009) CrossRefGoogle Scholar
  13. 13.
    Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17(1), 146–160 (1970) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Duda, O., Hart, P.E., Munson, J.H.: Graphical data processing research study and experimental investigation. Tech. Rep. AD650926. Stanford Research Institute (1967) Google Scholar
  15. 15.
    Damiand, G., Dupas, A., Lachaud, J.-O.: Fully deformable 3D digital partition model with topological control. Pattern Recognit. Lett. 32, 1374–1383 (2011) CrossRefGoogle Scholar
  16. 16.
    Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognit. Lett. 15, 1003–1011 (1994) CrossRefGoogle Scholar
  17. 17.
    Latecki, L.J.: Multicolor well-composed pictures. Pattern Recognit. Lett. 16(4), 425–431 (1995) CrossRefGoogle Scholar
  18. 18.
    Siqueira, S., Latecki, L., Tustison, N., Gallier, J., Gee, J.: Topological repairing of 3D digital images. J. Math. Imaging Vis. 30(3), 249–274 (2008) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cointepas, Y., Bloch, I., Garnero, L.: A cellular model for multi-objects multi-dimensional homotopic deformations. Pattern Recognit. 34, 1785–1798 (2001) MATHCrossRefGoogle Scholar
  20. 20.
    Bazin, P.-L., Ellingsen, L., Pham, D.: Digital homeomorphisms in deformable registration. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI. Lecture Notes in Computer Science, vol. 4584, pp. 211–222. Springer, Berlin (2007) Google Scholar
  21. 21.
    Whitehead, J.H.C.: Simplicial Spaces, Nuclei and m-Groups. Proc. Lond. Math. Soc. s2-45, 243–327 (1939) CrossRefGoogle Scholar
  22. 22.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Digital imaging: a unified topological framework. Journal of Mathematical Imaging and Vision. doi:10.1007/s10851-011-0308-9
  23. 23.
    Bertrand, G., Couprie, M.: Two-dimensional thinning algorithms based on critical kernels. J. Math. Imaging Vis. 31(1), 35–56 (2008) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Paths, homotopy and reduction in digital images. Acta Appl. Math. 113(2), 167–193 (2011) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bing, R.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. Lectures on Modern Mathematics II, 93–128 (1964) MathSciNetGoogle Scholar
  26. 26.
    Zeeman, E.: On the dunce hat. Topology 2, 341–358 (1964) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Whitehead, J.H.C.: Combinatorial homotopy. I. Bull. Am. Math. Soc. 55, 213–245 (1949) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Appl. Math. 125(1), 59–80 (2003) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Maunder, C.R.F.: Algebraic Topology. Dover, New York (1996) Google Scholar
  30. 30.
    Munkres, J.: Elements of Algebraic Topology. Westview Press, Boulder (1996) Google Scholar
  31. 31.
    May, A.: A Concise Course in Algebraic Topology. University Chicago Press, Chicago (1999) MATHGoogle Scholar
  32. 32.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar
  33. 33.
    Giblin, P.: Graphs, Surfaces and Homology. Cambridge University Press, Cambridge (2010) MATHCrossRefGoogle Scholar
  34. 34.
    Alexandroff, P.: Diskrete Räume, Rec. Math. [Mat. Sbornik] N.S. 501–519 (1937) Google Scholar
  35. 35.
    Birkhoff, G.: Rings of sets. Duke Math. J. 3(3), 443–454 (1937) MathSciNetCrossRefGoogle Scholar
  36. 36.
    McCord, M.: Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33(3), 465–474 (1966) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Bertrand, G.: New notions for discrete topology. In: DCGI’99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, pp. 218–228 (1999) CrossRefGoogle Scholar
  38. 38.
    Stong, R.E.: Finite topological spaces. Trans. Am. Math. Soc. 123(25), 325–340 (1966) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    May, J.P.: Finite topological spaces (lecture notes). url: www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf (2008)
  40. 40.
    Barmak, J.A., Minian, E.G.: Simple homotopy types and finite spaces. Adv. Math. 218, 87–104 (2008) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Barmak, J.A., Minian, E.G.: One-point reductions of finite spaces, h-regular CW-complexes and collapsibility, Algebraic & Geometric. Topology 8(3), 1763–1780 (2008) MathSciNetMATHGoogle Scholar
  42. 42.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48, 357–393 (1989) CrossRefGoogle Scholar
  43. 43.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989) CrossRefGoogle Scholar
  44. 44.
    Kovalevsky, V.: Axiomatic digital topology. J. Math. Imaging Vis. 26(1), 41–58 (2006) MathSciNetCrossRefGoogle Scholar
  45. 45.
    Kovalesky, V.: Geometry of Locally Finite Spaces. Publishing House Dr. Baerbel Kovalevski (2008) Google Scholar
  46. 46.
    Herman, G.T.: Geometry of Digital Spaces. Birkhäuser, Basel (1998) MATHGoogle Scholar
  47. 47.
    Kronheimer, E.: The topology of digital images. Topol. Appl. 46, 279–303 (1992) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Ronse, C., Agnus, V.: Morphology on label images: flat-type operators and connections. J. Math. Imaging Vis. 22(2), 283–307 (2005) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Ronse, C., Agnus, V.: Geodesy on label images, and applications to video sequence processing. J. Vis. Commun. Image Represent. 19, 392–408 (2008) CrossRefGoogle Scholar
  50. 50.
    Ayala, R., Domínguez, E., Francés, A., Quintero, A.: Digital lighting functions. In: Procs. Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1347, pp. 139–150. Springer, Berlin (1997) Google Scholar
  51. 51.
    Ronse, C.: An isomorphism for digital images, Journal of Combinatorial Theory, Series A 39(2) Google Scholar
  52. 52.
    Spanier, E.H.: Algebraic Topology. Springer, Berlin (1994) Google Scholar
  53. 53.
    Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: DGCI’97: Proceedings of the 7th International Workshop on Discrete Geometry for Computer Imagery, pp. 3–18. Springer, Berlin (1997) Google Scholar
  54. 54.
    Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D and 4D discrete spaces. IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 637–648 (2009) CrossRefGoogle Scholar
  55. 55.
    Blyth, T.: Lattices and Ordered Algebraic Structures. Springer, London (2005) MATHGoogle Scholar
  56. 56.
    Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Loïc Mazo
    • 1
    • 2
  • Nicolas Passat
    • 1
  • Michel Couprie
    • 2
  • Christian Ronse
    • 1
  1. 1.LSIIT, UMR CNRS 7005Université de StrasbourgStrasbourgFrance
  2. 2.Laboratoire d’Informatique Gaspard-Monge, Équipe A3SI, ESIEEUniversité Paris-EstParisFrance

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