Higher-Order Tensors in Diffusion Imaging

Bernhard Burgeth, Anna Vilanova, and Carl-Fredrik Westin (Editors)
Thomas Schultz, Andrea Fuster, Aurobrata Ghosh, Rachid Deriche, Luc Florack, and Lek-Heng Lim
In: Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (accepted for publication), Springer, 2013
 

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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Bibtex

@INCOLLECTION{SchultzTenDag2013,
     author = {Schultz, Thomas and Fuster, Andrea and Ghosh, Aurobrata and Deriche, Rachid and Florack, Luc and
               Lim, Lek-Heng},
     editor = {Burgeth, Bernhard and Vilanova, Anna and Westin, Carl-Fredrik},
      title = {Higher-Order Tensors in Diffusion Imaging},
  booktitle = {Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (accepted
               for publication)},
     series = {Mathematics + Visualization},
       year = {2013},
  publisher = {Springer},
   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.}
}