Skip to main content
Springer Nature Link
Log in

PDE-Driven Adaptive Morphology for Matrix Fields

  • Conference paper
Scale Space and Variational Methods in Computer Vision (SSVM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5567))

  • 3030 Accesses

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.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Bigün, J.: Vision with Direction. Springer, Berlin (2006)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  6. Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing 42, 3377–3386 (1994)

    Article  Google Scholar 

  7. Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  9. Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81, 179–197 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  12. Di Zenzo, S.: A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing 33, 116–125 (1986)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Goutsias, J., Heijmans, H.J.A.M., Sivakumar, K.: Morphological operators for image sequences. Computer Vision and Image Understanding 62, 326–346 (1995)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  17. Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  19. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  20. Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition 7, 53–58 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 25(4), 395–404 (2007)

    Article  Google Scholar 

  22. Louverdis, G., Vardavoulia, M.I., Andreadis, I., Tsalides, P.: A new approach to morphological color image processing. Pattern Recognition 35, 1733–1741 (2002)

    Article  MATH  Google Scholar 

  23. Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)

    Google Scholar 

  24. Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)

    MATH  Google Scholar 

  25. Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2002)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rao, A.R., Schunck, B.G.: Computing oriented texture fields. CVGIP: Graphical Models and Image Processing 53, 157–185 (1991)

    Google Scholar 

  28. Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis 29, 867–884 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  31. Serra, J.: Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France (1967)

    Google Scholar 

  32. Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)

    MATH  Google Scholar 

  33. Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic Press, London (1988)

    Google Scholar 

  34. Soille, P.: Morphological Image Analysis, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  35. van den Boomgaard, R.: Mathematical Morphology: Extensions Towards Computer Vision. PhD thesis, University of Amsterdam, The Netherlands (1992)

    Google Scholar 

  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. Weickert, J.: Coherence-enhancing diffusion filtering. International Journal of Computer Vision 31(2/3), 111–127 (1999)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  39. Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Springer, Berlin (2006)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, 66041, Saarbrücken, Germany

    Bernhard Burgeth, Michael Breuß, Luis Pizarro & Joachim Weickert

Authors
  1. Bernhard Burgeth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Breuß
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luis Pizarro
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joachim Weickert
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Xue-Cheng Tai

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  3. Simula Research Laboratory, M. Linges v 17, Fornebu, P.O.Box 134, N-1325, Lysaker, Norway

    Marius Lysaker

  4. SINTEF ICT, Oslo, Norway

    Knut-Andreas Lie

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Burgeth, B., Breuß, M., Pizarro, L., Weickert, J. (2009). PDE-Driven Adaptive Morphology for Matrix Fields. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02256-2_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02255-5

  • Online ISBN: 978-3-642-02256-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics