December 11 – 16 , 2011, Dagstuhl Seminar 11501

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


Bernhard Burgeth (Universität des Saarlandes, DE)
Anna Vilanova (TU Eindhoven, NL)
Carl-Fredrik Westin (Harvard Medical School – Boston, US)

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This seminar is the 4th in a series of Dagstuhl Seminars devoted to the visualization and processing of tensor fields and higher order descriptors. Tensor fields play an important role in many different scientific disciplines and application domains such as medical imaging, image processing, fluid dynamics, and structural mechanics. Analysis and visualization of multi-valued data have gained significant importance in scientific visualization and image processing due to rapid advances in medical imaging and and in the engineering industry.

In medical imaging, multi-valued data include diffusion weighted magnetic resonance imaging (dMRI), a medical imaging modality that allows the measurement of water diffusion in tissue in vivo. These measurements allow the description of diffusion in living fibrous tissue (e.g., white matter or muscle). The diffusion can be described by a diffusion tensor (i.e., a positive semidefinite 3 X 3 matrix). It is customary to acquire more complex data than can be described by the tensor model and recently the analysis has been extended to higher-order descriptors (i.e., higher-order tensors or spherical harmonics). There are several open questions how to best analyze and visualize such data.

Seminar Topics and Breakout Sessions

The emphasis of the seminar was on presenting the recent developments in the multidisciplinary field as well as identifying new challenges. We discussed a broad set of topics and challenges that cover both theoretical and practical issues related to analyzing and visualizing fields of tensors and higher order descriptors. During the workshop we discussed

  • Higher-order models in dMRI beyond the diffusion tensor
  • Higher-order tensors in image processing
  • Computational analysis and visualization of airflow dynamics
  • Novel differential geometric approaches to brain connectivity from dMRI
  • Connectivity concepts in mathematical morphology for tensor fields
  • Tensor concepts in structural mechanics and material science
  • Visualization of uncertainty
  • dMRI in brain studies for clinical applications

This year we scheduled time for breakout sessions that would foster focused discussions in smaller groups. During the first day of the meeting the group defined a list of important topics and open questions. Three of those were chosen and defined the breakout sessions:

  • How do we define a suitable Finsler metric from diffusion data?
  • How do we define biologically meaningful metrics from diffusion tensor and higher order model diffusion data?
  • What are important questions in engineering that can be answered with visualization?

These breakout sessions turned out to be very successful and the groups scheduled extra time in the evenings for continued discussions. The format of the new breakout sessions fits very well in the Dagstuhl environment promoting discussions and interactions. If we get the chance to organize another meeting at Dagstuhl, the breakout sessions will definitely be a part of the schedule.


The participants all agreed that the meeting was successful and stimulating. Seminar participants are already collaborating on a Springer book summarizing the results of the meeting. The Springer book will have about twenty chapters authored by the meeting participants, and we expect the book to be published in early 2013. The participants expressed interest in documenting the discussions in the breakout session in book chapters, in addition to the science described in their regular presentation.

The environment at Schloss Dagstuhl has generated several new scientific collaborations. The work in the engineering breakout session has resulted in a new project of four participants (Stommel, Burgeth, Scheuermann, Hotz) and a submission of a grant proposal to the Landesforschungsförderprogramm (LFFP) des Saarlandes. The proposal is currently under review. Three seminar participants who met at the meeting (O'Donnell, Hui Zhang, Schultz), from the USA, Great Britain, and Germany, are collaboratively organizing a workshop on computational diffusion MRI at the conference for Medical Image Computing and Computer-Assisted Intervention 2012.

It was voted that the group will apply for another meeting in this series, and that in addition to the current organizers (Carl-Fredrik Westin, Bernhard Burgeth, Anna Vilanova Bartroli), add Dr. Ingrid Hotz (ZIB - Berlin) as an organizer of the next event.

The organizers thank all the attendees for their contributions and extend special thanks to the team of Schloss Dagstuhl for helping to make this workshop a success. As always, we enjoyed the warm atmosphere of the Schloss, which supports both formal presentations as well as informal exchanges of ideas.

Dagstuhl Seminar Series


  • Computer Graphics / Computer Vision


  • Visualization
  • Image processing
  • Tensor fields
  • Diffusion tensor imaging (DT-MRI)
  • High angular resolution diffusion imaging (HARDI)
  • Partial differential equations (PDEs)
  • Feature extraction
  • Fiber tracking
  • Elastography
  • Structural mechanics
  • Fluid dynamics


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