© 2014

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

  • Carl-Fredrik Westin
  • Anna Vilanova
  • Bernhard Burgeth


  • Discusses challenges in both theoretical and practical issues related to analyzing and visualizing fields of tensors and higher order descriptors

  • Provides examples of tensors as applied to research in engineering and brain connectivity

  • Multidisciplinary approach is intended to inspire and advance new research on analyzing and visualizing fields of tensors and higher order descriptors

Conference proceedings

Part of the Mathematics and Visualization book series (MATHVISUAL)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Tensor Data Visualization

    1. Front Matter
      Pages 1-1
    2. Mario Hlawitschka, Ingrid Hotz, Andrea Kratz, G. Elisabeta Marai, Rodrigo Moreno, Gerik Scheuermann et al.
      Pages 3-15
    3. Andrea Kratz, Cornelia Auer, Ingrid Hotz
      Pages 17-34
  3. Representation and Processing of Higher-Order Descriptors

    1. Front Matter
      Pages 35-35
    2. Hans Knutsson, Carl-Fredrik Westin
      Pages 37-73
    3. Bernhard Burgeth, Andreas Kleefeld
      Pages 75-95
    4. Thomas C. J. Dela Haije, Remco Duits, Chantal M. W. Tax
      Pages 97-126
  4. Higher Order Tensors and Riemannian-Finsler Geometry

    1. Front Matter
      Pages 127-127
    2. Thomas Schultz, Andrea Fuster, Aurobrata Ghosh, Rachid Deriche, Luc Florack, Lek-Heng Lim
      Pages 129-161
    3. Aurobrata Ghosh, Rachid Deriche
      Pages 163-188
    4. Neda Sepasian, Jan H. M. ten Thije Boonkkamp, Luc M. J. Florack, Bart M. Ter Haar Romeny, Anna Vilanova
      Pages 209-225
  5. Tensor Signal Processing

    1. Front Matter
      Pages 227-227
    2. Madhura Ingalhalikar, Parmeshwar Khurd, Ragini Verma
      Pages 229-247
    3. Ofer Pasternak, Klaus Maier-Hein, Christian Baumgartner, Martha E. Shenton, Yogesh Rathi, Carl-Fredrik Westin
      Pages 249-270
    4. Rodrigo Moreno, Magnus Borga, Örjan Smedby
      Pages 271-292
  6. Applications of Tensor Processing

    1. Front Matter
      Pages 293-293
    2. Eugene Zhang
      Pages 295-311
    3. Yogesh Rathi, Martha E. Shenton, Carl-Fredrik Westin
      Pages 313-324

About these proceedings


Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties.

Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs.

The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI.

Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI).

A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research.

The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.


Diffusion tensor imaging (DT-MRI) High angular resolution diffusion imaging (HARDI) Higher-order descriptors Higher-order tensors Image processing Visualization

Editors and affiliations

  • Carl-Fredrik Westin
    • 1
  • Anna Vilanova
    • 2
  • Bernhard Burgeth
    • 3
  1. 1.Dept. of RadiologyHarvard Medical School, Brigham, and Women's HospitalBostonUSA
  2. 2.Technical University of Delft, Dept of Computer Graphics and VisualizationDelftThe Netherlands
  3. 3.Universität des Saarlandes, Dept. of MathematicsSaarbrückenGermany

Bibliographic information