Edge-Enhancing Diffusion Filtering for Matrix Fields

  • Bernhard Burgeth
  • Luis Pizarro
  • Stephan Didas
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


The elimination of noise and small details from an image while simultaneously preserving or enhancing the edge structures in an image is a ever-lasting task in image processing. Edge-enhancing anisotropic diffusion is known to tackle this problem successfully. The problem of noise removal and edge enhancement is also a major concern in diffusion tensor magnetic resonance imaging (DT-MRI). This medical image acquisition technique outputs a 3D matrix field of symmetric 3 ×3-matrices, and it helps to visualise, for example, the nerve fibres in brain tissue. As any physical measurement DT-MRI is subjected to errors causing faulty representations of the tissue structure corrupted by noise. In this paper we address that problem by proposing a edge-enhancing diffusion filtering methodology for matrix fields. The approach is based on a generic structure tensor concept for matrix fields that relies on the operator-algebraic properties of symmetric matrices, rather than their channel-wise treatment of earlier proposals. Numerical experiments with artificial and real DT-MRI data confirm the noise-removing and edge-enhancing qualities of the technique presented.


Fractional Anisotropy Structure Tensor Diffusion Tensor Magnetic Resonance Imaging Matrix Field Block Operator Matrix 
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.



We are grateful to Anna Vilanova i Bartrolí (Eindhoven Institute of Technology) and Carola van Pul (Maxima Medical Center, Eindhoven) for providing us with the DT-MRI data set and for discussing questions concerning data conversion. The original helix data set is by courtesy of Gordon Kindlmann (University of Chicago). The thank Thomas Schultz (University of Chicago) for providing the tractography results.


  1. 1.
    Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. Reson. 11, 209–219 (1996)Google Scholar
  2. 2.
    Bigun, J.: Vision with Direction. Springer, Berlin (2006)Google Scholar
  3. 3.
    Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Technical report 113, Department of Mathematics, Saarland University, Saarbrücken, October 2004Google Scholar
  4. 4.
    Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for matrix-data: ordering versus PDE-based approach. Image Vis. Comput. 25(4), 496–511 (2007)Google Scholar
  5. 5.
    Burgeth, B., Didas, S., Florack, L., Weickert, J.: A Generic Approach for Singular PDEs for and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 4485, pp. 556–567. Springer, Berlin (2007)Google Scholar
  6. 6.
    Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81, 179–197 (2007)Google Scholar
  7. 7.
    Burgeth, B., Didas, S., Weickert, J.: A general structure tensor concept and coherenceenhancing diffusion filtering for matrix fields. In: Laidlaw, D., Weickert, J. (eds.) Visualization and Processing of Tensor Fields, pp. 305–323. Springer, Berlin (2009)Google Scholar
  8. 8.
    Burgeth, B., Pizarro, L., Didas, S., Weickert, J.: 3D-coherence-enhancing diffusion filtering for matrix fields. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, M.-C., Davies, L. (eds.) Locally Adaptive Filters in Signal and Image Processing, pp. 55–70. Springer (2011)Google Scholar
  9. 9.
    Chefd’Hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Constrained flows of matrixvalued functions: Application to diffusion tensor regularization. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) Computer Vision – ECCV 2002. Lecture Notes in Computer Science, vol. 2350, pp. 251–265. Springer, Berlin (2002)Google Scholar
  10. 10.
    Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proceedings of ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, Interlaken, pp. 281–305 (1987)Google Scholar
  11. 11.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)Google Scholar
  12. 12.
    Mori, S., van Zijl, P.C.: Fiber tracking: principles and strategies – a technical review. NRM Biomed 15, 468–480 (2002)Google Scholar
  13. 13.
    Nucifora, P.G., Verma, R., Lee, S.-K., Melhem, E.R.: Diffusion-tensor MR imaging and tractography: exploring brain microstructure and connectivity. Radiology 245, 367–384 (2007)Google Scholar
  14. 14.
    Pajevic, S., Basser, P.J.: Parametric and non-parametric statistical analysis of DT-MRI data. J. Magn. Reson. 161(1), 1–14 (2003)Google Scholar
  15. 15.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)Google Scholar
  16. 16.
    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.) Advances in Visual Computing. Lecture Notes in Computer Science, vol. 4291, pp. 454–464. Springer, Berlin (2006)Google Scholar
  17. 17.
    Tschumperlé, D., Deriche, R.: Regularization of orthonormal vector sets using coupled PDE’s. In: Proceedings of First IEEE Workshop on Variational and Level Set Methods in Computer Vision, Vancouver, pp. 3–10. IEEE Computer Society Press (2001)Google Scholar
  18. 18.
    Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. Comput. Suppl. 11, 221–236 (1996)Google Scholar
  19. 19.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)Google Scholar
  20. 20.
    Weickert, J.: Coherence-enhancing diffusion filtering. Int. J. Comput. Vis. 31(2/3), 111–127 (1999)Google Scholar
  21. 21.
    Weickert, J.: Design of nonlinear diffusion filters. In: Jähne, B., Haußecker, H. (eds.) Computer Vision and Applications, pp. 439–458. Academic, San Diego (2000)Google Scholar
  22. 22.
    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)Google Scholar

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© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Department of ComputingImperial College LondonLondonUK
  3. 3.Abteilung BildverarbeitungFraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWMKaiserslauternGermany

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