Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion Imaging

  • Laura Astola
  • Luc Florack
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5567)

Abstract

We study three dimensional volumes of higher order tensors, using Finsler geometry. The application considered here is in medical image analysis, specifically High Angular Resolution Diffusion Imaging (HARDI) [1] of the brain. We want to find robust ways to reveal the architecture of the neural fibers in brain white matter. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled with a symmetric positive definite second order tensor, based on the assumption that there exists one dominant direction of fibers restricting the thermal motion of water molecules, leading naturally to a Riemannian framework. HARDI may potentially overcome the shortcomings of DTI by allowing multiple relevant directions, but invalidates the Riemannian approach. Instead Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide the exact criterion to determine whether a field of spherical functions has a Finsler structure. We also show a fiber tracking method in Finsler setting. Our model also incorporates a scale parameter, which is beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as real HARDI data.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tuch, D., Reese, T., Wiegell, M., Makris, N., Belliveau, J., van Wedeen, J.: High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magnetic Resonance in Medicine 48(6), 1358–1372 (2002)Google Scholar
  2. 2.
    Stejskal, E., Tanner, J.: Spin diffusion measurements: Spin echoes ion the presence of a time-dependent field gradient. The Journal of Chemical Physics 42(1), 288–292 (1965)CrossRefGoogle Scholar
  3. 3.
    Cohen de Lara, M.: Geometric and symmetry properties of a nondegenerate diffusion process. The Annals of Probability 23(4), 1557–1604 (1995)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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: Dohi, T., Kikinis, R. (eds.) MICCAI 2002. LNCS, vol. 2488, pp. 459–466. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Lenglet, C., Deriche, R., Faugeras, O.: Inferring white matter geometry from diffusion tensor MRI: Application to connectivity mapping. In: Pajdla, T., Matas, J.G. (eds.) ECCV 2004. LNCS, vol. 3024, pp. 127–140. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Astola, L., Florack, L., ter Haar Romeny, B.: Measures for pathway analysis in brain white matter using diffusion tensor images. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI 2007. LNCS, vol. 4584, pp. 642–649. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Astola, L., Florack, L.: Sticky vector fields and other geometric measures on diffusion tensor images. In: MMBIA 2008, IEEE Computer Society Workshop on Mathematical Methods in Biomedical Image Analysis, held in conjunction with CVPR 2008, Anchorage, Alaska, The United States. CVPR, vol. 20, pp. 1–7. Springer, Heidelberg (2008)Google Scholar
  8. 8.
    Özarslan, E., Mareci, T.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magnetic resonance in Medicine 50, 955–965 (2003)CrossRefGoogle Scholar
  9. 9.
    Barmpoutis, A., Jian, B., Vemuri, B., Shepherd, T.: Symmetric positive 4th order tensors and their estimation from diffusion weighted MRI. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI 2007. LNCS, vol. 4584, pp. 308–319. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Florack, L., Balmashnova, E.: Decomposition of high angular resolution diffusion images into a sum of self-similar polynomials on the sphere. In: Proceedings of the Eighteenth International Conference on Computer Graphics and Vision, GraphiCon 2008, Moscow, Russian Federation, June 2008, pp. 26–31 (2008) (invited paper)Google Scholar
  11. 11.
    Florack, L., Balmashnova, E.: Two canonical representations for regularized high angular resolution diffusion imaging. In: MICCAI Workshop on Computational Diffusion MRI, New York, USA, September 10, 2008, pp. 94–105 (2008)Google Scholar
  12. 12.
    Melonakos, J., Pichon, E., Angenent, S., Tannenbaum, A.: Finsler active contours. IEEE Transactions on Pattern Analysis and Machine Intelligence 30(3), 412–423 (2008)CrossRefGoogle Scholar
  13. 13.
    Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar
  14. 14.
    Shen, Z.: Lectures on Finsler Geometry. World Scientific, Singapore (2001)CrossRefMATHGoogle Scholar
  15. 15.
    Tuch, D.: Q-ball imaging. Magnetic Resonance in Medicine 52(4), 577–582 (2002)CrossRefGoogle Scholar
  16. 16.
    Jansons, K., Alexander, D.: Persistent angular structure: New insights from diffusion magnetic resonance imaging data. Inverse Problems 19, 1031–1046 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Özarslan, E., Shepherd, T., Vemuri, B., Blackband, S., Mareci, T.: Resolution of complex tissue microarchitecture using the diffusion orientation transform. NeuroImage 31, 1086–1103 (2006)CrossRefGoogle Scholar
  18. 18.
    Jian, B., Vemuri, B., Özarslan, E., Carney, P., Mareci, T.: A novel tensor distribution model for the diffusion-weighted MR signal. NeuroImage 37, 164–176 (2007)CrossRefGoogle Scholar
  19. 19.
    Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast and robust analytical q-ball imaging. Magnetic Resonance in Medicine 58(3), 497–510 (2006)CrossRefGoogle Scholar
  20. 20.
    Müller, C. (ed.): Analysis of Spherical Symmetries in Euclidean Spaces. Applied Mathematical Sciences, vol. 129. Springer, New York (1998)MATHGoogle Scholar
  21. 21.
    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, USA, vol. 1, pp. 1076–1083. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  22. 22.
    Hamani, C., Saint-Cyr, J., Fraser, J., Kaplitt, M., Lozano, A.: The subthalamic nucleus in the context of movement disorders. Brain, a Journal of Neurology 127, 4–20 (2004)CrossRefGoogle Scholar
  23. 23.
    Paxinos, G., Watson, C.: The Rat Brain In Stereotaxic Coordinates. Academic Press, San Diego (1998)Google Scholar
  24. 24.
    Brunenberg, E., Prckovska, V., Platel, B., Strijkers, G., ter Haar Romeny, B.M.: Untangling a fiber bundle knot: Preliminary results on STN connectivity using DTI and HARDI on rat brains. In: Proceedings of the 17th Meeting of the International Society for Magnetic Resonance in Medicine (ISMRM), Honolulu, Hawaii (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Laura Astola
    • 1
  • Luc Florack
    • 1
  1. 1.Department of mathematics and computer scienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations