Skip to main content
SpringerLink
Log in
Book cover

Mathematical Methods for Signal and Image Analysis and Representation pp 49–63Cite as

3D-Coherence-Enhancing Diffusion Filtering for Matrix Fields

  • Chapter
  • First Online:

Part of the book series: Computational Imaging and Vision ((CIVI,volume 41))

Abstract

Coherence-enhancing diffusion filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flow-like features in images. The completion of line-like structures 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 visualize, for example, the nerve fibers in brain tissue. As any physical measurement DT-MRI is subjected to errors causing faulty representations of the tissue corrupted by noise and with visually interrupted lines or fibers.

In this paper we address that problem by proposing a coherence-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 gap-closing and flow-enhancing qualities of the technique presented.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

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

    MATH  Google Scholar 

  2. Bjornemo, M., Brun, A.: White matter fiber tracking diffusion tensor MRI. Master’s thesis, Linkøping University, Linkøping (2002)

    Google Scholar 

  3. Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Technical report 113, Department of Mathematics, Saarland University, Saarbrücken, Germany (2004)

    Google Scholar 

  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)

    Article  Google Scholar 

  5. Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach for singular PDEs for the processing of matrix fields. In: Sgallari, F., Murli, A., Paragios, N. (eds.) Scale Space and Variational Methods in Computer Vision: Proceedings of the 1st International Conference, SSVM 2007, Ischia, Italy, May–June 2007. Lecture Notes in Computer Science, vol. 4485, pp. 556–567. Springer, Berlin (2007)

    Chapter  Google Scholar 

  6. Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81(2–3), 179–197 (2007)

    Article  MathSciNet  Google Scholar 

  7. Burgeth, B., Didas, S., Weickert, J.: A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. In: Laidlaw, D.H., Weickert, J. (eds.) Visualization and Processing of Tensor Fields. Mathematics and Visualization, pp. 305–323. Springer, Berlin (2009)

    Chapter  Google Scholar 

  8. Campbell, J.S.W., Siddiqi, K., Vemuri, B.C., Pike, G.B.: A geometric flow for white matter fiber tract reconstruction. In: Proceedings IEEE International Symposium on Biomedical Imaging: Macro to Nano, pp. 505–508. IEEE Computer Society Press, Los Alamitos (2002)

    Chapter  Google Scholar 

  9. Chefd’Hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Constrained flows on matrix-valued functions: application to diffusion tensor regularization. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) Proceedings of the 7th European Conference on Computer Vision, Copenhagen, Denmark, May–June 2002. Lecture Notes in Computer Science, vol. 2350–2353, pp. 251–265. Springer, Berlin (2002)

    MATH  Google Scholar 

  10. Coulon, O., Alexander, D.C., Arridge, S.R.: A regularization scheme for diffusion tensor magnetic resonance images. In: Insana, M.F., Leahy, R.M. (eds.) Proceedings of the 17th International Conference on Information Processing in Medical Imaging, IPMI 2001, Davis, USA, 2001. Lecture Notes in Computer Science, vol. 2082, pp. 92–105. Springer, Berlin (2001)

    Google Scholar 

  11. Duits, R., Franken, E.M.: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images. Int. J. Comput. Vis. 92(3), 231–264 (2011)

    Article  MathSciNet  Google Scholar 

  12. Fillard, P., Pennec, X., Arsigny, V., Ayache, N.: Clinical DT-MRI estimation, smoothing, and fiber tracking with log-Euclidean metrics. IEEE Trans. Med. Imaging 26(11) (2007)

    Google Scholar 

  13. Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proceedings of the ISPRS Intercomission Conference on Fast Processing of Photogrammetric Data, pp. 281–305 (1987)

    Google Scholar 

  14. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990). Corrected reprint of the 1985 original

    MATH  Google Scholar 

  15. McGraw, T., Vemuri, B.C., Chen, Y., Rao, M., Mareci, T.: DT-MRI denoising and neuronal fiber tracking. Med. Image Anal. 8(2), 95–111 (2004)

    Article  Google Scholar 

  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. Proceedings of the 2nd International Symposium, ISVC 2006, Lake Tahoe, NV, USA, November 6–8, 2006. Lecture Notes in Computer Science, vols. 4291–4292, pp. 454–464. Springer, Berlin (2006)

    Google Scholar 

  17. Tschumperlé, D., Deriche, R.: Regularization of orthonormal vector sets using coupled PDE’s. In: Proceedings 1st IEEE Workshop on Variational and Level Set Methods, pp. 3–10. IEEE Computer Society, Los Alamitos (2001)

    Chapter  Google Scholar 

  18. Weickert, J.A.: Anisotropic Diffusion in Image Processing. ECMI Series. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  19. Weickert, J.A.: Coherence-enhancing diffusion filtering. Int. J. Comput. Vis. 31(2–3), 111–127 (1999)

    Google Scholar 

  20. Weickert, J.: Design of Nonlinear Diffusion Filters. In: Jähne, B.J., Haußecker, H. (eds.) Computer Vision and Applications, pp. 439–458. Academic Press, San Diego (2000)

    Chapter  Google Scholar 

  21. 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 

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

    MATH  Google Scholar 

  23. Zhang, S., Demiralp, C., Laidlaw, D.H.: Visualizing diffusion tensor MR images using streamtubes and streamsurfaces. IEEE Trans. Vis. Comput. Graph. 9(4), 454–462 (2003)

    Article  Google Scholar 

  24. Zhukov, L., Barr, A.H.: Oriented tensor reconstruction: tracing neural pathways from diffusion tensor MRI. In: Proceedings IEEE Visualization 2002, pp. 387–394. IEEE Computer Society, Los Alamitos (2002)

    Google Scholar 

Download references

Acknowledgements

We are grateful to Anna Vilanova i Bartrolí (Eindhoven University 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 is by courtesy of Gordon Kindlmann (University of Chicago).

Author information

Authors and Affiliations

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

    Bernhard Burgeth & Joachim Weickert

  2. Department of Computing, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK

    Luis Pizarro

  3. Abteilung Bildverarbeitung, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    Stephan Didas

Authors
  1. Bernhard Burgeth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis Pizarro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephan Didas
    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

Corresponding author

Correspondence to Bernhard Burgeth .

Editor information

Editors and Affiliations

  1. Dept. Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands

    Luc Florack

  2. Dept. Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, Eindhoven, 5600 MB, The Netherlands

    Remco Duits

  3. Dept. Applied Mathematics, Delft University of Technology, Delft, 2628 CD, The Netherlands

    Geurt Jongbloed

  4. Probability & Stochastic Networks (PNA), Centrum Wiskunde & Informatica, Amsterdam, 1098 XG, The Netherlands

    Marie-Colette van Lieshout

  5. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, 45141, Germany

    Laurie Davies

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag London Limited

About this chapter

Cite this chapter

Burgeth, B., Pizarro, L., Didas, S., Weickert, J. (2012). 3D-Coherence-Enhancing Diffusion Filtering for Matrix Fields. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation