Skip to main content

Surface Thinning in 3D Cubical Complexes

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5852))

Abstract

We introduce a parallel thinning algorithm with directional substeps based on the collapse operation, which is guaranteed to preserve topology and to provide a thin result. Then, we propose two variants of a surface-preserving thinning scheme, based on this parallel directional thinning algorithm. Finally, we propose a methodology to produce filtered surface skeletons, based on the above thinning methods and the recently introduced discrete λ-medial axis.

This work has been partially supported by the “ANR BLAN07–2_184378 MicroFiss” project.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Attali, D., Boissonnat, J.-D., Edelsbrunner, H.: Stability and computation of the medial axis — a state-of-the-art report. In: Möller, T., Hamann, B., Russell, B. (eds.) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. LNCS, pp. 1–19. Springer, Heidelberg (to appear, 2009)

    Google Scholar 

  2. Attali, D., Lachaud, J.O.: Delaunay conforming iso-surface, skeleton extraction and noise removal. Computational Geometry: Theory and Applications 19, 175–189 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Attali, D., Montanvert, A.: Modelling noise for a better simplification of skeletons. In: Proc. International Conference on Image Processing (ICIP), vol. 3, pp. 13–16 (1996)

    Google Scholar 

  4. Bertrand, G.: On critical kernels. Comptes Rendus de l’Académie des Sciences, Série Math.  I(345), 363–367 (2007)

    Google Scholar 

  5. Bertrand, G., Couprie, M.: Two-dimensional parallel thinning algorithms based on critical kernels. Journal of Mathematical Imaging and Vision 31(1), 35–56 (2008)

    Article  MathSciNet  Google Scholar 

  6. Bertrand, G., Couprie, M.: A new 3D parallel thinning scheme based on critical kernels. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 580–591. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  7. Bertrand, G., Couprie, M.: On parallel thinning algorithms: minimal non-simple sets, P-simple points and critical kernels. Journal of Mathematical Imaging and Vision 35(1), 23–35 (2009)

    Article  MathSciNet  Google Scholar 

  8. Borgefors, G., Ragnemalm, I., Sanniti di Baja, G.: The Euclidean distance transform: finding the local maxima and reconstructing the shape. In: Proc. of the 7th Scandinavian Conference on Image Analysis, vol. 2, pp. 974–981 (1991)

    Google Scholar 

  9. Chaussard, J., Couprie, M., Talbot, H.: A discrete lambda-medial axis. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 421–433. Springer, Heidelberg (2009)

    Google Scholar 

  10. Chazal, F., Lieutier, A.: The lambda medial axis. Graphical Models 67(4), 304–331 (2005)

    Article  MATH  Google Scholar 

  11. Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D and 4D discrete spaces. IEEE Trans. on Pattern Analysis and Machine Intelligence 31(4), 637–648 (2009)

    Article  Google Scholar 

  12. Davies, E.R., Plummer, A.P.N.: Thinning algorithms: a critique and a new methodology. Pattern Recognition 14, 53–63 (1981)

    Article  MathSciNet  Google Scholar 

  13. Ge, Y., Fitzpatrick, J.M.: On the generation of skeletons from discrete Euclidean distance maps. IEEE Trans. on Pattern Analysis and Machine Intelligence 18(11), 1055–1066 (1996)

    Article  Google Scholar 

  14. Hesselink, W.H., Roerdink, J.B.T.M.: Euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform. IEEE Trans. on Pattern Analysis and Machine Intelligence 30(12), 2204–2217 (2008)

    Article  Google Scholar 

  15. Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Computer Vision, Graphics and Image Processing 48, 357–393 (1989)

    Article  Google Scholar 

  16. Kong, T.Y., Litherland, R., Rosenfeld, A.: Problems in the topology of binary digital images. In: Open problems in topology, pp. 376–385. Elsevier, Amsterdam (1990)

    Google Scholar 

  17. Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46, 141–161 (1989)

    Article  Google Scholar 

  18. Liu, L.: 3d thinning on cell complexes for computing curve and surface skeletons. Master’s thesis, Washington University in Saint Louis (May 2009)

    Google Scholar 

  19. Malandain, G., Fernández-Vidal, S.: Euclidean skeletons. Image and Vision Computing 16, 317–327 (1998)

    Article  Google Scholar 

  20. Ogniewicz, R.L., Kübler, O.: Hierarchic Voronoi skeletons. Pattern Recognition 28(33), 343–359 (1995)

    Article  Google Scholar 

  21. Pudney, C.: Distance-ordered homotopic thinning: a skeletonization algorithm for 3D digital images. Computer Vision and Image Understanding 72(3), 404–413 (1998)

    Article  Google Scholar 

  22. Rémy, E., Thiel, E.: Exact medial axis with Euclidean distance. Image and Vision Computing 23(2), 167–175 (2005)

    Article  Google Scholar 

  23. Rosenfeld, A.: A characterization of parallel thinning algorithms. Information and Control 29, 286–291 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  24. Serra, J.: Image analysis and mathematical morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  25. Siddiqi, K., Bouix, S., Tannenbaum, A., Zucker, S.: The Hamilton-Jacobi skeleton. In: International Conference on Computer Vision (ICCV), pp. 828–834 (1999)

    Google Scholar 

  26. Soille, P.: Morphological image analysis. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  27. Talbot, H., Vincent, L.: Euclidean skeletons and conditional bisectors. In: Proc. VCIP 1992, SPIE, vol. 1818, pp. 862–876 (1992)

    Google Scholar 

  28. Vincent, L.: Efficient computation of various types of skeletons. In: Proc. Medical Imaging V, SPIE, vol. 1445, pp. 297–311 (1991)

    Google Scholar 

  29. Whitehead, J.H.C.: Simplicial spaces, nuclei and m-groups. Proceedings of the London Mathematical Society 45(2), 243–327 (1939)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chaussard, J., Couprie, M. (2009). Surface Thinning in 3D Cubical Complexes. In: Wiederhold, P., Barneva, R.P. (eds) Combinatorial Image Analysis. IWCIA 2009. Lecture Notes in Computer Science, vol 5852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10210-3_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10210-3_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10208-0

  • Online ISBN: 978-3-642-10210-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics