18. – 23. April 2004, Dagstuhl Seminar 04172

Visualization and Image Processing of Tensor Fields


Hans Hagen (DFKI – Kaiserslautern, DE)
Joachim Weickert (Universität des Saarlandes, DE)

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Recently, matrix-valued data sets (so-called tensor fields) have gained significant importance in the fields of scientific visualization and image processing. This has been triggered by the following developments:

  • Novel medical imaging techniques such as diffusion tensor magnetic resonance imaging (DT-MRI) have been introduced. DT-MRI is a 3-D imaging method that yields a diffusion tensor in each voxel. This diffusion tensor describes the diffusive behaviour of water moluecules in the tissue. It can be represented by a positive semidefinite 3 x 3 matrix in each voxel.
  • Tensors have shown their use as a general tool in image analysis, segmentation and grouping. This also includes widespread applications of the so-called structure tensor in fields ranging from motion analysis to texture segmentation.
  • A number of scientific applications require to visualize tensor fields. The tensor concept is a common physical description of anisotropic behaviour, especially in solid mechanics and civil engineering (e.g. stress-strain relationships, inertia tensors, diffusion tensors, permittivity tensors).


This has led to a number of challenging scientific questions, e.g.

  • How can one visualize these high-dimensional data in an appropriate way?
  • What are the relevant features to be processed? Is it better to have component-wise processing, to introduce some coupling between the tensor channels or to decompose the tensors in their eigenvalues and eigenvectors and process these entities separately?
  • How should one process these data such that essential properties of the tensor fields are not sacrificed? For instance, often one knows that the tensor field is positive semidefinite. In this case it would be very problematic if an image processing method would create matrices with negative eigenvalues.
  • How can structure of tensor fields be addressed? Topological methods have been used in visualization, but more research in this area is needed.
  • How should one adapt the processing to a task at hand, e.g. the enhancement of fibre-like structures in brain imaging? This may be very important for a number of medical applications such as connectivity studies.
  • How can one perform higher-level operations on these data, e.g. segment tensor fields? Current segmentation methods have been designed for scalar- or vector-valued data, and it is not clear if and how they can be extended to tensor fields.
  • How can one perform operations on tensor fields in an algorithmically efficient manner? Many tensor fields use 3 x 3 matrices as functions on a three-dimensional image domain. This may involve a very large amount of data such that a clear need for highly efficient algorithms arises.

Since this research area is very young, these fundamental questions have not been solved yet. In image processing, e.g., the filtering of tensor fields has not been investigated before 2000. Moreover, a lack of interdisciplinary interaction is another reason for the numerous unsolved problems in this area: Many medical imaging people are unaware of recent progress in the the tensor-based image analysis area, while image processing specialists do not know much about recent medical imaging techniques such as DT-MRI. Their research would also benefit significantly if they had the possibility to visualize the results of their tensor-based filters in a suitable way. On the other hand, most computer graphics specialists do not yet use advanced image processing methods to smooth and enhance their tensor fields.


In this Dagstuhl seminar, we want to bring together experts from scientific visualization, image processing and medical imaging in a real interdisciplinary workshop. Each invited participant has contributed to specific aspects in the area of tensor field imaging. The interaction of these participants will clarify the needs that every group has and the scientific perpectives it can contribute to the field of tensor imaging. We expect that identifying these mid-term perspectives should lead to a significant boost of the scientific output on tensor-based imaging.

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