Advertisement

Discrete Topological Transformations for Image Processing

  • Michel Couprie
  • Gilles Bertrand
Part of the Lecture Notes in Computational Vision and Biomechanics book series (LNCVB, volume 2)

Abstract

Topology-based image processing operators usually aim at transforming an image while preserving its topological characteristics. This chapter reviews some approaches which lead to efficient and exact algorithms for topological transformations in 2D, 3D and grayscale images. Some transformations that modify topology in a controlled manner are also described. Finally, based on the framework of critical kernels, we show how to design a topologically sound parallel thinning algorithm guided by a priority function.

Keywords

Topological Characteristic Grayscale Image Medial Axis Simple Point Cubical Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work has been partially supported by the “ANR BLAN07-2_184378 MicroFiss” project and the “ANR-2010-BLAN-0205 Kidico” project.

References

  1. 1.
    Aktouf, Z., Bertrand, G., Perroton, L.: A three-dimensional holes closing algorithm. Pattern Recognit. Lett. 23(5), 523–531 (2002). zbMATHCrossRefGoogle Scholar
  2. 2.
    Attali, D., Boissonnat, J.D., Edelsbrunner, H.: Stability and computation of the medial axis—a state-of-the-art report. In: Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, pp. 109–125. Springer, Berlin (2009) CrossRefGoogle Scholar
  3. 3.
    Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognit. Lett. 15, 1003–1011 (1994) CrossRefGoogle Scholar
  4. 4.
    Bertrand, G.: On P-simple points. C. R. Acad. Sci., Sér. 1 Math. 321, 1077–1084 (1995) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bertrand, G.: On critical kernels. C. R. Acad. Sci., Sér. 1 Math. 345, 363–367 (2007) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bertrand, G., Couprie, M.: New 3D parallel thinning algorithms based on critical kernels. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4245, pp. 580–591. Springer, Berlin (2006) CrossRefGoogle Scholar
  7. 7.
    Bertrand, G., Couprie, M.: Two-dimensional thinning algorithms based on critical kernels. J. Math. Imaging Vis. 31(1), 35–56 (2008) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bertrand, G., Couprie, M.: On parallel thinning algorithms: minimal non-simple sets, P-simple points and critical kernels. J. Math. Imaging Vis. 35(1), 23–35 (2009) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bertrand, G., Everat, J.C., Couprie, M.: Image segmentation through operators based upon topology. J. Electron. Imaging 6(4), 395–405 (1997) CrossRefGoogle Scholar
  10. 10.
    Bertrand, G., Malandain, G.: A new characterization of three-dimensional simple points. Pattern Recognit. Lett. 15(2), 169–175 (1994) zbMATHCrossRefGoogle Scholar
  11. 11.
    Blum, H.: An associative machine for dealing with the visual field and some of its biological implications. Biol. Prototypes Synthetic Syst. 1, 244–260 (1961) Google Scholar
  12. 12.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Models for the Perception of Speech and Visual Form, pp. 362–380. MIT Press, Cambridge (1967) Google Scholar
  13. 13.
    Borgefors, G.: Distance transformations in digital images. Comput. Vis. Graph. Image Process. 34, 344–371 (1986) CrossRefGoogle Scholar
  14. 14.
    Chaussard, J.: Topological tools for discrete shape analysis. Ph.D. thesis, Université Paris-Est (December 2010) Google Scholar
  15. 15.
    Chaussard, J., Couprie, M., Talbot, H.: Robust skeletonization using the discrete λ-medial axis. Pattern Recognit. Lett. 32(9), 1384–1394 (2011) CrossRefGoogle Scholar
  16. 16.
    Chazal, F., Lieutier, A.: The λ-medial axis. Graph. Models 67(4), 304–331 (2005) zbMATHCrossRefGoogle Scholar
  17. 17.
    Couprie, M.: Note on fifteen 2D parallel thinning algorithms. Tech. Rep. IGM2006-01, Université de Marne-la-Vallée (2006) Google Scholar
  18. 18.
    Couprie, M., Bertrand, G.: Topology preserving alternating sequential filter for smoothing 2D and 3D objects. J. Electron. Imaging 13(4), 720–730 (2004) CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Couprie, M., Bezerra, F.N., Bertrand, G.: Topological operators for grayscale image processing. J. Electron. Imaging 10(4), 1003–1015 (2001) CrossRefGoogle Scholar
  21. 21.
    Daragon, X., Couprie, M.: Segmentation topologique du neo-cortex cérébral depuis des données IRM. In: Proc. Congrès RFIA, vol. 3, pp. 809–818 (2002) Google Scholar
  22. 22.
    Dokládal, P., Bloch, I., Couprie, M., Ruijters, D., Urtasun, R., Garnero, L.: Segmentation of 3D head MR images using morphological reconstruction under constraints and automatic selection of markers. Pattern Recognit. 36, 2463–2478 (2003) CrossRefGoogle Scholar
  23. 23.
    Fabbri, R., Costa, L.D.F., Torelli, J.C., Bruno, O.M.: 2D Euclidean distance transform algorithms: a comparative study. ACM Comput. Surv. 40(1), 1–44 (2008) CrossRefGoogle Scholar
  24. 24.
    Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Appl. Math. 125(1), 59–80 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hall, R.W.: Tests for connectivity preservation for parallel reduction operators. Topol. Appl. 46(3), 199–217 (1992) zbMATHCrossRefGoogle Scholar
  26. 26.
    Kong, T.Y.: On the problem of determining whether a parallel reduction operator for n-dimensional binary images always preserves topology. In: Vision Geometry II. Proc. SPIE, vol. 2060, pp. 69–77 (1993) Google Scholar
  27. 27.
    Kong, T.Y.: On topology preservation in 2D and 3D thinning. Int. J. Pattern Recognit. Artif. Intell. 9, 813–844 (1995) CrossRefGoogle Scholar
  28. 28.
    Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: Proc. DGCI. Lecture Notes in Computer Science, vol. 1347, pp. 3–18 (1997) Google Scholar
  29. 29.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48, 357–393 (1989) CrossRefGoogle Scholar
  30. 30.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46, 141–161 (1989) CrossRefGoogle Scholar
  31. 31.
    Ma, C.M.: On topology preservation in 3D thinning. Comput. Vis. Graph. Image Process. 59(3), 328–339 (1994) Google Scholar
  32. 32.
    Malandain, G., Bertrand, G., Ayache, N.: Topological segmentation of discrete surfaces. Int. J. Comput. Vis. 10(2), 183–197 (1993) CrossRefGoogle Scholar
  33. 33.
    Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Appl. Math. 21(1), 67–79 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Rosenfeld, A.: Digital topology. Am. Math. Mon. 86, 621–630 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Rosenfeld, A., Pfaltz, J.L.: Distance functions on digital pictures. Pattern Recognit. 1, 33–61 (1968) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rueda, A., Acosta, O., Couprie, M., Bourgeat, P., Fripp, J., Dowson, N., Romero, E., Salvado, O.: Topology-corrected segmentation and local intensity estimates for improved partial volume classification of brain cortex in MRI. J. Neurosci. Methods 188(2), 305–315 (2010) CrossRefGoogle Scholar
  37. 37.
    Rutovitz, D.: Pattern recognition. J. R. Stat. Soc. 129, 504–530 (1966) Google Scholar
  38. 38.
    Stefanelli, S., Rosenfeld, A.: Some parallel thinning algorithms for digital pictures. J. Assoc. Comput. Mach. 18(2), 255–264 (1971) zbMATHCrossRefGoogle Scholar
  39. 39.
    Talbot, H., Vincent, L.: Euclidean skeletons and conditional bisectors. In: Proc. VCIP’92. Proc. SPIE, vol. 1818, pp. 862–876 (1992) Google Scholar
  40. 40.
    Whitehead, J.H.C.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 45(2), 243–327 (1939) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Laboratoire d’Informatique Gaspard-Monge, Équipe A3SIUniversité Paris-Est, ESIEE ParisMarne-la-ValléeFrance

Personalised recommendations