PDE-Driven Adaptive Morphology for Matrix Fields

  • Bernhard Burgeth
  • Michael Breuß
  • Luis Pizarro
  • Joachim Weickert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5567)

Abstract

Matrix fields are important in many applications since they are the adequate means to describe anisotropic behaviour in image processing models and physical measurements. A prominent example is diffusion tensor magnetic resonance imaging (DT-MRI) which is a medical imaging technique useful for analysing the fibre structure in the brain. Recently, morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images have been extended to three dimensional fields of symmetric positive definite matrices.

In this article we propose a novel method to incorporate adaptivity into the matrix-valued, PDE-driven dilation process. The approach uses a structure tensor concept for matrix data to steer anisotropic morphological evolution in a way that enhances and completes line-like structures in matrix fields. Numerical experiments performed on synthetic and real-world data confirm the gap-closing and line-completing qualities of the proposed method.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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
  2. 2.
    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
  3. 3.
    Bigün, J.: Vision with Direction. Springer, Berlin (2006)MATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Breuß, M., Burgeth, B., Weickert, J.: Anisotropic continuous-scale morphology. In: Martí, J., Benedí, J.M., Mendonça, A.M., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4478, pp. 515–522. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing 42, 3377–3386 (1994)CrossRefGoogle Scholar
  7. 7.
    Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)CrossRefGoogle Scholar
  8. 8.
    Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for tensor data: Ordering versus PDE-based approach. Image and Vision Computing 25(4), 496–511 (2007)CrossRefGoogle Scholar
  9. 9.
    Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81, 179–197 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Burgeth, B., Didas, S., Weickert, J.: A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. Technical Report 197, Department of Mathematics, Saarland University, Saarbrücken, Germany (July 2007); to appear in: Laidlaw, D., Weickert, J. (eds.): Visualization and Processing of Tensor Fields. Springer, Heidelberg (2009)Google Scholar
  11. 11.
    Chefd’Hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 251–265. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Di Zenzo, S.: A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing 33, 116–125 (1986)CrossRefMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    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
  15. 15.
    Goutsias, J., Heijmans, H.J.A.M., Sivakumar, K.: Morphological operators for image sequences. Computer Vision and Image Understanding 62, 326–346 (1995)CrossRefGoogle Scholar
  16. 16.
    Goutsias, J., Vincent, L., Bloomberg, D.S. (eds.): Mathematical Morphology and its Applications to Image and Signal Processing. Computational Imaging and Vision, vol. 18. Kluwer, Dordrecht (2000)MATHGoogle Scholar
  17. 17.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)MATHGoogle Scholar
  18. 18.
    Heijmans, H.J.A.M., Roerdink, J.B.T.M. (eds.): Mathematical Morphology and its Applications to Image and Signal Processing. Computational Imaging and Vision, vol. 12. Kluwer, Dordrecht (1998)MATHGoogle Scholar
  19. 19.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  20. 20.
    Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition 7, 53–58 (1975)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 25(4), 395–404 (2007)CrossRefGoogle Scholar
  22. 22.
    Louverdis, G., Vardavoulia, M.I., Andreadis, I., Tsalides, P.: A new approach to morphological color image processing. Pattern Recognition 35, 1733–1741 (2002)CrossRefMATHGoogle Scholar
  23. 23.
    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  24. 24.
    Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)MATHGoogle Scholar
  25. 25.
    Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2002)MATHGoogle Scholar
  26. 26.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics 79, 12–49 (1988)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rao, A.R., Schunck, B.G.: Computing oriented texture fields. CVGIP: Graphical Models and Image Processing 53, 157–185 (1991)Google Scholar
  28. 28.
    Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis 29, 867–884 (1992)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    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
  30. 30.
    Schultz, T., Burgeth, B., Weickert, J.: Flexible segmentation and smoothing of DT-MRI fields through a customizable structure tensor. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Nefian, A., Meenakshisundaram, G., Pascucci, V., Zara, J., Molineros, J., Theisel, H., Malzbender, T. (eds.) ISVC 2006. LNCS, vol. 4291, pp. 455–464. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  31. 31.
    Serra, J.: Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France (1967)Google Scholar
  32. 32.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)MATHGoogle Scholar
  33. 33.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic Press, London (1988)Google Scholar
  34. 34.
    Soille, P.: Morphological Image Analysis, 2nd edn. Springer, Berlin (2003)MATHGoogle Scholar
  35. 35.
    van den Boomgaard, R.: Mathematical Morphology: Extensions Towards Computer Vision. PhD thesis, University of Amsterdam, The Netherlands (1992)Google Scholar
  36. 36.
    Weickert, J.: Coherence-enhancing diffusion of colour images. In: Sanfeliu, A., Villanueva, J.J., Vitrià, J. (eds.) Proc. Seventh National Symposium on Pattern Recognition and Image Analysis, Barcelona, Spain, April 1997, vol. 1, pp. 239–244 (1997)Google Scholar
  37. 37.
    Weickert, J.: Coherence-enhancing diffusion filtering. International Journal of Computer Vision 31(2/3), 111–127 (1999)CrossRefGoogle Scholar
  38. 38.
    Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 251–268. AMS, Providence (2002)CrossRefGoogle Scholar
  39. 39.
    Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Springer, Berlin (2006)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

Personalised recommendations