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

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


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.


Symmetric Tensor Orientation Distribution Function Tensor Model High Angular Resolution Orientation Tensor 
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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

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