Advertisement

The Algebra of Fourth-Order Tensors with Application to Diffusion MRI

  • Maher Moakher
Part of the Mathematics and Visualization book series (MATHVISUAL)

Summary

In this chapter we give different descriptions of fourth-order tensors. We show that under certain symmetries it is possible to describe a fourth-order tensor in three-dimensional space by a second-order tensor in a six-dimensional space. Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors. We discuss the algebra of the space of fourth-order symmetric tensors and describe different metrics on this space. Special emphasis is placed on totally symmetric tensors and on orientation tensors. Applications to high angular resolution diffusion imaging are discussed.

Keywords

Symmetric Tensor Orientation Distribution Function Tensor Model High Angular Resolution Orientation Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. G. Advani and C. L. Tucker III. The use of tensors to describe and predict fiber orientation in short fiber composites. J. Rheol., 31(8):751–784, 1987.CrossRefGoogle Scholar
  2. [2]
    J. Applequist. Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media. J. Phys. A Math. Gen., 22:4303–4330, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    G. Backus. A geometric picture of anisotropic elastic tensors. Reviews of Geophysics and Space Physics, 8(3):633–671, 1970.CrossRefGoogle Scholar
  4. A. Barmpoutis, B. Jian, and B. C. Vemuri. Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI. Technical Report REP-2007–312, University of Florida, 2007.Google Scholar
  5. [5]
    P. J. Basser, J. Mattiello, and D. Le Bihan. Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. B, 123:247–254, 1994.CrossRefGoogle Scholar
  6. [6]
    A. Campanella and M. L. Tonon. A note on the Cauchy relations. Meccanica, 29:105–108, 1994.zbMATHCrossRefGoogle Scholar
  7. [7]
    M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche. Apparent diffusion coefficients from high angular resolution diffusion imaging: estimation and applications. Magn. Reson. Med., 56:395–410, 2006.CrossRefGoogle Scholar
  8. [8]
    S. Forte and M. Vianello. Symmetry classes for elasticity tensors. J. Elasticity, 43:81–108, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. E. Gurtin. An Introduction to Continuum Mechanics, volume 158 of Mathematics in Science and Engineering. Academic Press, New York, 1981.Google Scholar
  10. [10]
    A. E. H. Love. A Treatise on the Mathematical Theory of Elasticity, 4th edition. Dover, New York, 1944.zbMATHGoogle Scholar
  11. [11]
    K. V. Mardia. Statistics of directional data. J. Royal Stat. Soc. Ser B, 37(3):349–393, 1975.zbMATHMathSciNetGoogle Scholar
  12. [12]
    A. J. McConnell. Applications of the Absolute Differential Calculus. Blackie and Son, London, 1931.zbMATHGoogle Scholar
  13. [13]
    M. Moakher. Fourth-order Cartesian tensors: Old and new facts, notions and applications. Q. J. Mech. Appl. Math., 61(2):181–203, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    M. Moakher and A. N. Norris. The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry. J. Elasticity, 85:215–263, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    E. Özarslan and T. H. Mareci. Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn. Reson. Med., 50:955–965, 2003.CrossRefGoogle Scholar
  16. [16]
    K. Reich. Die Entwicklung des Tensorkalküls: Vom absoluten Differentialkalkül zur Relativitätstheorie. Number 11 in Science Networks - Historical Studies. Birkhäuser Verlag, Basel/Boston, 1994.Google Scholar
  17. [17]
    [17] D. S. Tuch, R. M. Weisskoff, J. W. Belliveau, and V. J. Wedeen. High angular resolution diffusion imaging of the human brain. In Proceedings of the 7th Annual Meeting of ISMRM, Philadelphia, 1999, page 321.Google Scholar
  18. [18]
    L. J. Walpole. Fourth rank tensors of the thirty-two crystal classes: multiplication tables. Proc. R. Soc. Lond., A391:149–179, 1984.Google Scholar
  19. [20]
    G. S. Watson. Distributions on the circle and sphere. J. Appl. Probab., 19:265–280, 1982.CrossRefGoogle Scholar
  20. [21]
    J. Weickert and H. Hagen, editors. Visualization and Processing of Tensor Fields. Springer, Berlin, 2006.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Maher Moakher
    • 1
  1. 1.Laboratory for Mathematical and Numerical Modeling in Engineering ScienceNational Engineering School at TunisTunis-BelvédèreTunisia

Personalised recommendations