A Novel Riemannian Metric for Geodesic Tractography in DTI

  • Andrea Fuster
  • Antonio Tristan-Vega
  • Tom Dela Haije
  • Carl-Fredrik Westin
  • Luc Florack
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor . Such a metric is used for white matter tractography and connectivity analysis. We propose a modified metric tensor given by the adjugate rather than the inverse diffusion tensor. Tractography experiments on real brain diffusion data show improvement in the vicinity of isotropic diffusion regions compared to results for inverse (sharpened) diffusion tensors.



Tom Dela Haije gratefully acknowledges The Netherlands Organisation for Scientific Research (NWO) for financial support. Andrea Fuster would like to thank Lauren O’Donnell for feedback on brain white matter anatomy and Ana Achúcarro.


  1. 1.
    Astola, L., Florack, L., ter Haar Romeny, B.M.: Measures for pathway analysis in brain white matter using diffusion tensor images. In: Karssemeijer, N., Lelieveldt, B.P.F. (eds.) Proceedings of the IPMI 2007, Kerkrade. Lecture Notes in Computer Science, vol. 4584, pp. 642–649. Springer (2007)Google Scholar
  2. 2.
    Astola, L., Fuster, A., Florack, L.: A Riemannian scalar measure for diffusion tensor images. Pattern Recognit. 44(9), 1885–1891 (2011)CrossRefGoogle Scholar
  3. 3.
    Basser, P.J., Mattiello, J., Le Bihan, D.: Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. 103, 247–254 (1994)CrossRefGoogle Scholar
  4. 4.
    de Lara, M.C.: Geometric and symmetry properties of a nondegenerate diffusion process. Ann. Probab. 23(4), 1557–1604 (1995). doi:10.1214/aop/1176987794. http://projecteuclid.org/euclid.aop/1176987794 Google Scholar
  5. 5.
    Descoteaux, M., Deriche, R., Lenglet, C.: Diffusion tensor sharpening improves white matter tractography. In: SPIE Image Processing: Medical Imaging, San Diego, pp. 1084–1087 (2007)Google Scholar
  6. 6.
    Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)CrossRefMATHGoogle Scholar
  7. 7.
    Fletcher, P.T., Tao, R., Jeong, K.W., Whitaker, R.T.: A volumetric approach to quantifying region-to-region white matter connectivity in diffusion tensor MRI. In: Karssemeijer, N., Lelieveldt, B.P.F. (eds.) Proceedings of the IPMI 2007, Kerkrade. Lecture Notes in Computer Science, vol. 4584, pp. 346–358. Springer (2007)Google Scholar
  8. 8.
    Hao, X., Whitaker, R.T., Fletcher, P.T.: Adaptive Riemannian metrics for improved geodesic tracking of white matter. In: Székely, G., Hahn, H.K. (eds.) Proceedings of the IPMI 2011, Kloster Irsee. Lecture Notes in Computer Science, vol. 6801, pp. 13–24. Springer, Berlin (2011)Google Scholar
  9. 9.
    Jbabdi, S., Bellec, P., Toro, R., Daunizeau, J., Pélégrini-Issac, M., Benali, H.: Accurate anisotropic fast marching for diffusion-based geodesic tractography. Int. J. Biomed. Imaging 2008, 1–12 (2008). doi:10.1155/2008/320195. http://www.hindawi.com/journals/ijbi/2008/320195/
  10. 10.
    Lazar, M., Weinstein, D.M., Tsuruda, J.S., Hasan, K.M., Arfanakis, K., Meyerand, M.E., Badie, B., Rowley, H.A., Haughton, V., Field, A., Alexander, A.L.: White matter tractography using diffusion tensor deflection. Hum. Brain Mapp. 18(4), 306–321 (2003). doi:10.1002/hbm.10102. http://dx.doi.org/10.1002/hbm.10102 Google Scholar
  11. 11.
    Lenglet, C., Deriche, R., Faugeras, O.: Inferring white matter geometry from diffusion tensor MRI: application to connectivity mapping. In: Pajdla, T., Matas, J. (eds.) Proceedings of the 8th European Conference on Computer Vision, Prague, May 2004. Lecture Notes in Computer Science, vol. 3021–3024, pp. 127–140. Springer, Berlin (2004)Google Scholar
  12. 12.
    Melonakos, J., Pichon, E., Angenent, S., Tannenbaum, A.: Finsler active contours. IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 412–423 (2008)CrossRefGoogle Scholar
  13. 13.
    O’Donnell, L., Haker, S., Westin, C.F.: New approaches to estimation of white matter connectivity in diffusion tensor MRI: elliptic PDEs and geodesics in a tensor-warped space. In: Proceedings of Medical Imaging, Computing and Computer Assisted Intervention, Tokyo. Lecture Notes in Computer Science, vol. 2488, pp. 459–466. Springer (2002)Google Scholar
  14. 14.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Pieper, S., Halle, M., Kikinis, R.: 3D Slicer. In: IEEE International Symposium on Biomedical Imaging ISBI 2004, Arlington, pp. 632–635 (2004)Google Scholar
  16. 16.
    Prados, E., Soatto, S., Lenglet, C., Pons, J.P., Wotawa, N., Deriche, R., Faugeras, O.: Control theory and fast marching techniques for brain connectivity mapping. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, June 2006, vol. 1, pp. 1076–1083. IEEE Computer Society (2006)Google Scholar
  17. 17.
    Tournier, J.D., Calamante, F., Gadian, D.G., Connelly, A.: Diffusion-weighted magnetic resonance imaging fibre tracking using a front evolution algorithm. NeuroImage 20(1), 276–288 (2003). doi:10.1016/S1053-8119(03)00236-2. http://www.sciencedirect.com/science/article/pii/S1053811903002362

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrea Fuster
    • 1
  • Antonio Tristan-Vega
    • 2
  • Tom Dela Haije
    • 1
  • Carl-Fredrik Westin
    • 3
  • Luc Florack
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.University of ValladolidValladolidSpain
  3. 3.Harvard Medical SchoolBostonUSA

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