Fiber Orientation Distribution Functions and Orientation Tensors for Different Material Symmetries

  • Maher MoakherEmail author
  • Peter J. Basser
Part of the Mathematics and Visualization book series (MATHVISUAL)


In this paper we give closed-form expressions of the orientation tensors up to the order four associated with some axially-symmetric orientation distribution functions (ODF), including the well-known von Mises-Fisher, Watson, and de la Vallée Poussin ODFs. Each is characterized by a mean direction and a concentration parameter. Then, we use these elementary ODFs as building blocks to construct new ones with a specified material symmetry and derive the corresponding orientation tensors. For a general ODF we present a systematic way of calculating the corresponding orientation tensors from certain coefficients of the expansion of the ODF in spherical harmonics.


Orientation Distribution Function Concentration Parameter Modal Vector Fabric Tensor Orientation Tensor 
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  1. 1.
    Advani, S.G., Tucker III, C.L.: The use of tensors to describe and predict fiber orientation in short fiber composites. J. Rheol. 31(8), 751–784 (1987)CrossRefzbMATHGoogle Scholar
  2. 2.
    Advani, S.G., Tucker III, C.L.: Closure approximations for three-dimensional structure tensors. J. Rheol. 34(3), 367–386 (1990)CrossRefGoogle Scholar
  3. 3.
    Backus, G.: A geometric picture of anisotropic elastic tensors. Rev. Geophys. Space Phys. 8(3), 633–671 (1970)CrossRefGoogle Scholar
  4. 4.
    Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E.P.: Symmetric tensor decomposition. Linear Algebra Appl. 433(11–12), 1851–1872 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chebbi, Z.: Study of brain white matter fiber crossings using fourth-order diffusion tensors estimated from HARDI data. Master’s thesis, National Engineering School at Tunis (2009)Google Scholar
  6. 6.
    Cieslinski, M.M., Steel, P.J., Lincoln, S.F., Easton, C.J.: Centrosymmetric and non-centrosymmetric packing of aligned molecular fibers in the solid state self assemblies of cyclodextrin-based rotaxanes. Supramol. Chem. 18, 529–536 (2006)CrossRefGoogle Scholar
  7. 7.
    Eberlea, A.P.R., Vélez-Garcíab, G.M., Bairda, D.G., Wapperomc P.: Fiber orientation kinetics of a concentrated short glass fiber suspension in startup of simple shear flow. J. Non-Newtonian Fluid Mech. 165, 110–119 (2010)CrossRefGoogle Scholar
  8. 8.
    Fisher, N.I., Lewis, T., Embleton, B.J.J.: Statistical Analysis of Spherical Data. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  9. 9.
    Florack, L., Balmashnova, E.: Two canonical representations for regularized high angular resolution diffusion imaging. In: Alexander, D., Gee, J., Whitaker, R. (eds.) MICCAI Workshop on Computational Diffusion MRI, New York, pp. 85–96 (2008)Google Scholar
  10. 10.
    Florack, L., Balmashnova, E., Astola, L., Brunenberg, E.: A new tensorial framework for single-shell high angular resolution diffusion imaging. J. Math. Imaging Vision 38(3), 171–181 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ghosh, A., Megherbi, T., Oulebsir-Boumghar, F., Deriche, R.: Fiber orientation distribution from non-negative sparse recovery. In: IEEE 10th International Symposium on Biomedical Imaging (ISBI), 2013, pp. 254–257 (2013)Google Scholar
  12. 12.
    Goldacker, T., Abetz, V., Stadler, R., Erukhimovich, I., Leibler, L.: Non-centrosymmetric superlattices in block copolymer blends. Nature 398, 137–139 (1999)CrossRefGoogle Scholar
  13. 13.
    Gur, Y., Jiao, F., Zhu, S.X., Johnson, C.R.: White matter structure assessment from reduced HARDI data using low-rank polynomial approximations. In: Panagiotaki, E., O’Donnell, L., Schultz, T., Zhang, G.H. (eds.) Proceedings of the Computational Diffusion MRI, pp. 186–197 (2012)Google Scholar
  14. 14.
    Jack, D.A., Smith, D.E.: Elastic properties of short-fiber polymer composites, derivation and demonstration of analytical forms for expectation and variance from orientation tensors. J. Compos. Mater. 42(3), 277–308 (2008)CrossRefGoogle Scholar
  15. 15.
    Jones, M.N.: Spherical Harmonics and Tensors for Classical Field Theory. Wiley, New York (1985)Google Scholar
  16. 16.
    Kanatani, K.-I.: Distribution of directional data and fabric tensors. Int. J. Eng. Sci. 22(2), 149–164 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lakes, R.: Elastic and viscoelastic behavior of chiral materials. Int. J. Mech. Sci. 43(7), 1579–1589 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, C., Bammer, R., Acar, B., Moseley, M.E.: Characterizing non-gaussian diffusion by using generalized diffusion tensors. Magn. Reson. Med. 51, 924–937 (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mardia, K.V.: Statistics of directional data. J. R. Stat. Soc. Ser. B 37(3), 349–393 (1975)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Megherbi, T., Kachouane, M., Oulebsir-Boumghar, F., Deriche, R.: Crossing fibers detection with an analytical high order tensor decomposition. Comput. Math. Methods Med. 2014, 18 pp. (2014) [Article ID 476837]Google Scholar
  21. 21.
    Moakher, M.: Fourth-order Cartesian tensors: old and new facts, notions and applications. Q. J. Mech. Appl. Math. 61(2), 181–203 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ostroverkhov, V., Ostroverkhova, O., Petschek, R.G., Singer, K.D., Sukhomlinova, L., Twieg, R.J., Wang, S.-X., Chien, L.C.: Optimization of the molecular hyperpolarizability for second harmonic generation in chiral media. Chem. Phys. 257, 263–274 (2000)CrossRefGoogle Scholar
  23. 23.
    Özarslan, E., Mareci, T.H.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn. Reson. Med. 50(5), 955–965 (2003)CrossRefGoogle Scholar
  24. 24.
    Özarslan, E., Shepherd, T.M., Vemuri, B.C., Blackband, S.J., Mareci, T.H.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). NeuroImage 31(3), 1086–1103 (2006)CrossRefGoogle Scholar
  25. 25.
    Papenfuss, C., Ván, P.: Scalar, vectorial, and tensorial damage parameters from the mesoscopic background. Proc. Est. Acad. Sci. 57(3), 132–141 (2008)CrossRefGoogle Scholar
  26. 26.
    Schaeben, H.: A simple standard orientation density function: the hyperspherical de la Vallée Poussin kernel. Phys. Status Solidi (B) 200(2), 367–376 (1997)Google Scholar
  27. 27.
    Schultz, T., Seidel, H.-P.: Estimating crossing fibers: a tensor decomposition approach. IEEE Trans. Vis. Comput. Graph. 14(6), 1635–1642 (2008)CrossRefGoogle Scholar
  28. 28.
    Schultz, T., Fuster, A., Ghosh, A., Deriche, R., Florack, L., Lim, L.-H.: Higher-order tensors in diffusion imaging. In: Westin, C.F., Vilanova, A., Burgeth, B. (eds.) Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization, pp. 129–161. Springer, Berlin/Heidelberg (2014)Google Scholar
  29. 29.
    Snieder, R.: A Guided Tour of Mathematical Methods for the Physical Sciences, 2nd edn. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  30. 30.
    Tournier, J.-D., Calamante, F., Gadian, D.G., Connelly, A.: Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23(3), 1176–1185 (2004)CrossRefGoogle Scholar
  31. 31.
    Tuch, D.S.: Q-ball imaging. Magn. Reson. Med. 52(6), 1358–1372 (2004)CrossRefGoogle Scholar
  32. 32.
    van der Boogaart, K.G., Hielscher, R., Prestin, J., Schaeben, H.: Kernel-based methods for inversion of the Radon transform on SO(3) and their applications to texture analysis. J. Comput. Appl. Math. 199, 122–140 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Voyiadjis, G.Z., Kattan, P.I.: Evolution of fabric tensors in damage mechanics of solids with micro-cracks: Part I - Theory and fundamental concepts. Mech. Res. Commun. 34(2), 145–154 (2007)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, B., Zhou, J., Koschny, T., Kafesaki, M., Soukoulis, C.M.: Chiral metamaterials: simulations and experiments. J. Opt. A Pure Appl. Opt. 11, 114003 (2009)CrossRefGoogle Scholar
  35. 35.
    Watson, G.S.: Distributions on the circle and sphere. J. Appl. Probab. 19, 265–280 (1982)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at TunisUniversity of Tunis El ManarTunis-BelvédèreTunisia
  2. 2.Section on Tissue Biophysics & Biomimetics, PPITS, NICHDNational Institutes of HealthBethesdaUSA

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