Fiber Enhancement in Diffusion-Weighted MRI

  • Remco Duits
  • Tom C. J. Dela Haije
  • Arpan Ghosh
  • Eric Creusen
  • Anna Vilanova
  • Bart ter Haar Romeny
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)


Diffusion-Weighted MRI (DW-MRI) measures local water diffusion in biological tissue, which reflects the underlying fiber structure. In order to enhance the fiber structure in the DW-MRI data we consider both (convection-)diffusions and Hamilton-Jacobi equations (erosions) on the space \(\mathbb{R}^3 \rtimes S^2\) of 3D-positions and orientations, embedded as a quotient in the group SE(3) of 3D-rigid body movements. These left-invariant evolutions are expressed in the frame of left-invariant vector fields on SE(3), which serves as a moving frame of reference attached to fiber fragments. The linear (convection-)diffusions are solved by a convolution with the corresponding Green’s function, whereas the Hamilton-Jacobi equations are solved by a morphological convolution with the corresponding Green’s function. Furthermore, we combine dilation and diffusion in pseudo-linear scale spaces on \(\mathbb{R}^{3}\rtimes S^2\). All methods are tested on DTI-images of the brain. These experiments indicate that our techniques are useful to deal with both the problem of limited angular resolution of DTI and the problem of spurious, non-aligned crossings in HARDI.


DTI HARDI DW-MRI sub-Riemannian geometry scale spaces Lie groups Hamilton-Jacobi equations erosion 


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  1. 1.
    Akian, M., Quadrat, J., Viot, M.: Bellman processes. Lecture Notes in Control and Information Science 199, 302–311 (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alexander, D.C., Barker, G.J., Arridge, S.R.: Detection and modeling of non-gaussian apparent diffusion coefficient profiles in human brain data. Magnetic Rosonance in Medicine 48, 331–340 (2002)CrossRefGoogle Scholar
  3. 3.
    Basser, P.J., Mattiello, J., Lebihan, D.: MR diffusion tensor spectroscopy and imaging. Biophysical Journal 66, 259–267 (1994)CrossRefGoogle Scholar
  4. 4.
    Boscain, U., Duplaix, J., Gauthier, J.P., Rossi, F.: Anthropomorphic image reconstruction via hypoelliptic diffusion. (accepted for publication in JMIV, to appear)Google Scholar
  5. 5.
    Burgeth, B., Pizarro, L., Didas, S., Weickert, J.: 3d-coherence-enhancing diffusion filtering for matrix fields. In: Florack, Duits, Jongbloed, van Lieshout, Davies (eds.) Locally Adaptive Filters in Signal and Image Processing (to appear, 2011)Google Scholar
  6. 6.
    Burgeth, B., Weickert, J.: An explanation for the logarithmic connection between linear and morphological systems. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 325–339. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Creusen, E.J., Duits, R., Dela Haije, T.C.J.: Numerical schemes for linear and non-linear enhancement of DW-MRI. In: Bruckstein, A.M., et al. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 14–25. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Descoteaux, M.: High Angular Resolution Diffusion MRI: From Local Estimation to Segmentation and Tractography. PhD thesis, Universite de Nice (2008)Google Scholar
  9. 9.
    Duits, R., Creusen, E.J., Ghosh, A., Dela Haije, T.C.J.: Diffusion, convection and erosion on SE(3)/(0×SO(2)) and their application to the enhancement of crossing fibers. Published on Arxiv, nr. arXiv:1103.0656v4 also available as CASA-report (2011),,
  10. 10.
    Duits, R., Franken, E.M.: Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, part I: Linear left-invariant diffusion equations on SE(2). Quarterly of Appl. Math., A.M.S. 68, 255–292 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 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. International Journal of Computer Vision, IJCV 92(3), 231–264 (2011), CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Duits, R., van Almsick, M.A.: The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2d-Euclidean motion group (April 2008)Google Scholar
  13. 13.
    Florack, L.: Codomain scale space and regularization for high angular resolution diffusion imaging. In: CVPR Workshop on Tensors in Image Processing and Computer Vision, Anchorage, Alaska, The United States, vol. 20 (June 2008)Google Scholar
  14. 14.
    Florack, L.M.J., Maas, R., Niessen, W.J.: Pseudo-linear scale-space theory. International Journal of Computer Vision 31(2/3), 247–259 (1999)CrossRefGoogle Scholar
  15. 15.
    Gur, Y., Sochen, N.: Regularizing Flows over Lie groups. Journal of Mathematical Imaging and Vision 33(2), 195–208 (2009)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Mumford, D.: Elastica and computer vision. In: Algebraic Geometry and Its Applications, pp. 491–506. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  17. 17.
    Petitot, J.: Neurogéomètrie de la vision–Modèles mathématiques et physiques des architectures fonctionelles. Les Éditions de l’École Polytechnique (2008)Google Scholar
  18. 18.
    Prckovska, V., Rodrigues, P., Duits, R., Vilanova, A., ter Haar Romeny, B.M.: Extrapolating fiber crossings from DTI data. Can we infer similar fiber crossings as in HARDI? In: CDMRI 2010 Proc. MICCAI Workshop Computational Diffusion MRI, China (2010)Google Scholar
  19. 19.
    Rodrigues, P., Duits, R., Vilanova, A., ter Haar Romeny, B.M.: Accelerated Diffusion Operators for Enhancing DW-MRI. In: Eurographics Workshop on Visual Computing for Biology and Medicine, Leipzig, Germany, pp. 49–56 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Remco Duits
    • 1
    • 2
  • Tom C. J. Dela Haije
    • 2
  • Arpan Ghosh
    • 1
  • Eric Creusen
    • 1
    • 2
  • Anna Vilanova
    • 2
  • Bart ter Haar Romeny
    • 2
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyThe Netherlands
  2. 2.Department of Biomedical EngineeringEindhoven University of TechnologyThe Netherlands

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