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Higher-Order Tensors in Diffusion Imaging

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Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathematical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging. This survey gives a careful introduction to the foundations of higher-order tensor algebra, and explains how some concepts from linear algebra generalize to the higher-order case. From the application side, it reviews a variety of distinct higher-order tensor models that arise in the context of diffusion imaging, such as higher-order diffusion tensors, q-ball or fiber Orientation Distribution Functions (ODFs), and fourth-order covariance and kurtosis tensors. By bridging the gap between mathematical foundations and application, it provides an introduction that is suitable for practitioners and applied mathematicians alike, and propels the field by stimulating further exchange between the two.

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Acknowledgements

A. Ghosh and R. Deriche are partially supported by the NucleiPark research project (ANR Program “Maladies Neurologique et maladies Psychiatriques”) and the France Parkinson Association. L.-H. Lim is partially supported by an AFOSR Young Investigator Award (FA9550-13-1-0133), an NSF CAREER Award (DMS-1057064), and an NSF Collaborative Research Grant (DMS-1209136).

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Authors and Affiliations

  1. University of Bonn, Bonn, Germany

    Thomas Schultz

  2. Max Planck Institute for Intelligent Systems, Tübingen, Germany

    Thomas Schultz

  3. Eindhoven University of Technology, Eindhoven, The Netherlands

    Andrea Fuster & Luc Florack

  4. INRIA Sophia Antipolis-Méditerranée, Sophia Antipolis Cedex, France

    Aurobrata Ghosh & Rachid Deriche

  5. University of Chicago, Chicago, IL, USA

    Lek-Heng Lim

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  1. Thomas Schultz
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  2. Andrea Fuster
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  3. Aurobrata Ghosh
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  4. Rachid Deriche
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  5. Luc Florack
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  6. Lek-Heng Lim
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Correspondence to Thomas Schultz , Andrea Fuster or Aurobrata Ghosh .

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Editors and Affiliations

  1. Dept. of Radiology, Harvard Medical School, Brigham, and Women's Hospital, Boston, Massachusetts, USA

    Carl-Fredrik Westin

  2. Technical University of Delft, Dept of Computer Graphics and Visualization, Delft, The Netherlands

    Anna Vilanova

  3. Universität des Saarlandes, Dept. of Mathematics, Saarbrücken, Germany

    Bernhard Burgeth

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Schultz, T., Fuster, A., Ghosh, A., Deriche, R., Florack, L., Lim, LH. (2014). Higher-Order Tensors in Diffusion Imaging. In: Westin, CF., Vilanova, A., Burgeth, B. (eds) Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54301-2_6

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