Skip to main content
SpringerLink
Log in
Book cover

Mathematical Methods for Signal and Image Analysis and Representation pp 83–100Cite as

SPD Tensors Regularization via Iwasawa Decomposition

  • Chapter
  • First Online:

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

Abstract

In this chapter we propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on a local differential geometric approach where the manifold of symmetric positive-definite (SPD) matrices, P n, is parameterized via the Iwasawa coordinate system. The distances on P n are measured here in terms of a natural GL(n)-invariant metric. Via the mathematical concept of fibre bundles, we describe the tensor-valued image as a section where the metric over the section is induced by the metric over P n. Then, a functional over the sections accompanied by a suitable data fitting term is defined. The variation of this functional with respect to the Iwasawa coordinates leads to a set of \(\frac{1}{2}n(n+1)\) coupled equations of motion for these coordinates. Then, by means of the gradient descent method, these equations of motion define a Beltrami flow over P n. It turns out that the local coordinate approach via the Iwasawa coordinate system results in very simple numerics. Regularization results of structure tensors and diffusion tensors as well as results of fibres tractography for DTI are 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. Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. Reson. 111(3), 209–219 (1996)

    Article  Google Scholar 

  2. Basser, P.J., Mattiello, J., Le Bihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66(1), 259–267 (1994)

    Google Scholar 

  3. Batchelor, P.G., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Magn. Reson. Med. 53(1), 221–225 (2005)

    Article  Google Scholar 

  4. Bigun, J., Granlund, G.H., Wiklund, J.: Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Trans. Pattern Anal. Mach. Intell. 13(8), 775–790 (1991)

    Article  Google Scholar 

  5. Blomgren, P., Chan, T.F.: Color TV: total variation methods for restoration of vector valued images. IEEE Trans. Image Process. 7(3), 304–309 (1998)

    Article  Google Scholar 

  6. Chefd’hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Regularizing flows for constrained matrix-valued images. J. Math. Imaging Vis. 20(1–2), 147–162 (2004)

    Article  MathSciNet  Google Scholar 

  7. 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 

  8. Feddern, C., Weickert, J., Burgeth, B., Welk, M.: Curvature-driven PDE methods for matrix-valued images. Int. J. Comput. Vis. 69(1), 91–103 (2006)

    Article  Google Scholar 

  9. 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 

  10. Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)

    Article  Google Scholar 

  11. 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 

  12. Gur, Y., Sochen, N.: Coordinate-free diffusion over compact Lie groups. 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. 580–591. Springer, Berlin (2007)

    Chapter  Google Scholar 

  13. Jorgenson, J., Lang, S.: Pos n(R) and Eisenstein Series. Lecture Notes in Mathematics, vol. 1868. Springer, Berlin (2005)

    Book  Google Scholar 

  14. Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, New York (2001)

    MATH  Google Scholar 

  15. Kimmel, R., Sochen, N., Malladi, R.: From high energy physics to low level vision. In: Scale-Space Theory in Computer Vision. Lecture Notes in Computer Science, vol. 1252, pp. 236–247. Springer, Berlin (1997)

    Chapter  Google Scholar 

  16. Lang, S.: Fundamentals of Differential Geometry. Springer, New York (1999)

    Book  Google Scholar 

  17. Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26(3), 735–747 (2005)

    Article  MathSciNet  Google Scholar 

  18. Moakher, M., Zerai, M.: Riemannian curvature-driven flows for tensor-valued data. 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. 592–602. Springer, Berlin (2007)

    Chapter  Google Scholar 

  19. Mori, S., Wakana, S., van Zijl, P.C.M., Nagae-Poetscher, L.M.: MRI Atlas of Human White Matter. Elsevier, Amsterdam (2005)

    Google Scholar 

  20. Pajevic, S., Pierpaoli, C.: Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: application to white matter fiber tract mapping in the human brain. Magn. Reson. Med. 42, 526–540 (1999)

    Article  Google Scholar 

  21. Pasternak, O., Sochen, N., Intrator, N., Assaf, Y.: Variational multiple-tensors fitting of fiber-ambiguous DW-MRI voxels. J. Magn. Reson. Imaging 26(8), 1133–1144 (2008)

    Article  Google Scholar 

  22. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006)

    MATH  Google Scholar 

  23. Pierpaoli, C., Jezzard, P., Basser, P.J., Barnett, A., Chiro, G.D.: Diffusion tensor MRI of the human brain. Radiology 201, 637–648 (1996)

    Article  Google Scholar 

  24. Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981)

    Article  MathSciNet  Google Scholar 

  25. Rao, A.R., Schunck, B.G.: Computing oriented texture fields. CVGIP, Graph. Models Image Process. 53(2), 157–185 (1991)

    Article  Google Scholar 

  26. Rosenfeld, B.: The Geometry of Lie Groups. Kluwer Academic, Dordrecht (1997)

    Book  Google Scholar 

  27. Shafrir, D., Sochen, N., Deriche, R.: Regularization of mappings between implicit manifolds of arbitrary dimension and codimension. In: Paragios, N., Faugeras, O., Chan, T., Schnör, C. (eds.) Proceedings of the 3rd IEEE Workshop on Variational, Geometric and Level-Set Methods in Computer Vision, VLSM 2005, Beijing, China, October 16, 2005. Lecture Notes in Computer Science, vol. 3752, pp. 344–355. Springer, Berlin (2005)

    MATH  Google Scholar 

  28. Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7(3), 310–318 (1998)

    Article  MathSciNet  Google Scholar 

  29. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications, vol. 2. Springer, Berlin (1988)

    Book  Google Scholar 

  30. Tschumperlé, D., Deriche, R.: Orthonormal vector sets regularization with PDE’s and applications. Int. J. Comput. Vis. 50(3), 237–252 (2002)

    Article  Google Scholar 

  31. Wang, Z., Vemuri, B.C., Chen, Y., Mareci, T.H.: A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans. Med. Imaging 23(8), 930–939 (2004)

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SCI Institute, University of Utah, 72 S. Central Campus Dr., Salt Lake City, UT, 84112, USA

    Yaniv Gur

  2. Department of Psychiatry, Brigham and Women’s Hospital Harvard Medical School, 1249 Boylston Street, Boston, MA, 02215, USA

    Ofer Pasternak

  3. Department of Applied Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Nir Sochen

Authors
  1. Yaniv Gur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ofer Pasternak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nir Sochen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaniv Gur .

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

Gur, Y., Pasternak, O., Sochen, N. (2012). SPD Tensors Regularization via Iwasawa Decomposition. 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_5

Download citation

Publish with us

Policies and ethics

Search

Navigation