A Directional Rouy-Tourin Scheme for Adaptive Matrix-Valued Morphology

  • Luis Pizarro
  • Bernhard Burgeth
  • Michael Breuß
  • Joachim Weickert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5720)

Abstract

In order to describe anisotropy in image processing models or physical measurements, matrix fields are a suitable choice. In diffusion tensor magnetic resonance imaging (DT-MRI), for example, information about the diffusive properties of water molecules is captured in symmetric positive definite matrices. The corresponding matrix field reflects the structure of the tissue under examination. Recently, morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images have been extended to matrix-valued data.

In this article we consider an adaptive, PDE-driven dilation process for matrix fields. The anisotropic morphological evolution is steered with a matrix constructed from a structure tensor for matrix valued data. An important novel ingredient is a directional variant of the matrix-valued Rouy-Tourin scheme that enables our method to complete or enhance anisotropic structures effectively. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method.

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References

  1. 1.
    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  2. 2.
    Serra, J.: Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France (1967)Google Scholar
  3. 3.
    Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)MATHGoogle Scholar
  4. 4.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)MATHGoogle Scholar
  5. 5.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic Press, London (1988)Google Scholar
  6. 6.
    Soille, P.: Morphological Image Analysis, 2nd edn. Springer, Berlin (2003)MATHGoogle Scholar
  7. 7.
    Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis 123, 199–257 (1993)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Arehart, A.B., Vincent, L., Kimia, B.B.: Mathematical morphology: The Hamilton–Jacobi connection. In: Proc. Fourth International Conference on Computer Vision, Berlin, pp. 215–219. IEEE Computer Society Press, Los Alamitos (1993)Google Scholar
  9. 9.
    Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing 42, 3377–3386 (1994)CrossRefGoogle Scholar
  10. 10.
    Sapiro, G., Kimmel, R., Shaked, D., Kimia, B.B., Bruckstein, A.M.: Implementing continuous-scale morphology via curve evolution. Pattern Recognition 26, 1363–1372 (1993)CrossRefGoogle Scholar
  11. 11.
    van den Boomgaard, R.: Mathematical Morphology: Extensions Towards Computer Vision. PhD thesis, University of Amsterdam, The Netherlands (1992)Google Scholar
  12. 12.
    Breuß, M., Weickert, J.: A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision 25(2), 187–201 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Verdú-Monedero, R., Angulo, J.: Spatially-variant directional mathematical morphology operators based on a diffused average squared gradient field. In: Blanc-Talon, J., Bourennane, S., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2008. LNCS, vol. 5259, pp. 542–553. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Breuß, M., Burgeth, B., Weickert, J.: Anisotropic continuous-scale morphology. In: Martí, J., Benedí, J.M., Mendonça, A., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4478, pp. 515–522. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Burgeth, B., Breuß, M., Pizarro, L., Weickert, J.: PDE-driven adaptive morphology for matrix fields. In: Tai, X.-C., et al. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 247–258. Springer, Heidelberg (2009)Google Scholar
  16. 16.
    Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81, 179–197 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis 29, 867–884 (1992)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proc. ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, Interlaken, Switzerland, June 1987, pp. 281–305 (1987)Google Scholar
  19. 19.
    Bigün, J., Granlund, G.H., Wiklund, J.: Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(8), 775–790 (1991)CrossRefGoogle Scholar
  20. 20.
    Bigün, J.: Vision with Direction. Springer, Berlin (2006)MATHGoogle Scholar
  21. 21.
    Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)CrossRefGoogle Scholar
  22. 22.
    Feddern, C., Weickert, J., Burgeth, B., Welk, M.: Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision 69(1), 91–103 (2006)CrossRefMATHGoogle Scholar
  23. 23.
    Di Zenzo, S.: A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing 33, 116–125 (1986)CrossRefMATHGoogle Scholar
  24. 24.
    Burgeth, B., Didas, S., Weickert, J.: A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. In: Laidlaw, D., Weickert, J. (eds.): Visualization and Processing of Tensor Fields. Mathematics and Visualization, pp. 305–323. Springer, Heidelberg (2009)Google Scholar
  25. 25.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  26. 26.
    Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing 25(4), 496–511 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Luis Pizarro
    • 1
  • Bernhard Burgeth
    • 1
  • Michael Breuß
    • 1
  • Joachim Weickert
    • 1
  1. 1.Mathematical Image Analysis Group Faculty for Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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