https://www.dagstuhl.de/07022

### January 9 – 13 , 2007, Dagstuhl Seminar 07022

# Visualization and Processing of Tensor Fields

## Organizers

David H. Laidlaw (Brown University – Providence, US)

Joachim Weickert (Universität des Saarlandes, DE)

## For support, please contact

## Documents

List of Participants

Dagstuhl's Impact: Documents available

## Motivation

While scalar- and vector-valued data sets are omnipresent as grayscale
and color images in the fields of scientific visualization and image
processing, also matrix-valued data sets (so-called **tensor fields**)
have gained significant importance recently. This has been triggered by
the following developments:

- Medical imaging techniques such as
**diffusion tensor magnetic resonance imaging (DT-MRI)**are becoming more and more widespread. DT-MRI is a 3-D imaging method that yields a diffusion tensor in each voxel. This diffusion tensor describes the diffusive behavior of water molecules in the tissue. It can be represented by a positive semidefinite 3 x 3 matrix in each voxel. - Tensors have shown their use as
**feature descriptors**in image analysis, segmentation and grouping. This includes numerous applications of the structure tensor in fields ranging from motion analysis to texture segmentation, but also the tensor voting framework. -
**Tensor factorizations**have been proposed as compact multilinear models for multidimensional visual datasets. They successfully exploit spatial redundancy. - A number of scientific applications require to visualize tensor
fields. The tensor concept is a common
**physical description of anisotropic behaviour**, for instance in geomechanics / earthquake simulation, satellite gradiometry, liquid crystals, and material science.

## Problems

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

- How should one visualize these high-dimensional data in an appropriate way?
- How can user interaction be coupled with visualization to serve scientific users' problems?
- How can structure of tensor fields be addressed? Topological methods have been used in visualization, but more research in this area is needed.
- 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 tensor in their eigenvalues and eigenvectors and process these entities?
- 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 should one adapt the processing to a task at hand, e.g. the tractography of fiber-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 obvious if and how they can be extended to tensor fields. Often this requires to introduce sophisticated novel metrics in the space of tensor data.
- 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.
- Is it possible to derive a generic visualization and processing paradigm for tensor fields that originate from different application areas?
- What are the scientific application areas that can be served by tensor field visualization and analysis? What are the fundamental relevant problems from those application areas?

Since tensor fields have been investigated in different application domains and the field is fairly young, not many systematic investigations have been carried out so far. It is thus not surprising that many results are scattered throughout the scientific literature, and often people are only aware of a small fraction of the relevant papers. In April 2004, a Dagstuhl Perspective Workshop organised by Hans Hagen and Joachim Weickert was the first international forum where leading experts on visualization and processing of tensor fields had the opportunity to meet, sometimes for the first time. This workshop has identified several key issues and has triggered fruitful collaborations that have also led to the first book in its area. It contains a number of survey chapters that have been written in collaboration and that should enable also nonexperts to get access to the core ideas of this rapidly emerging field. Participants of the 2004 Perspective Workshop were very enthousiastic about the interdisciplinarity and the interaction possibilities, and they were very interested in pursuing this concept further in a second workshop. This is the goal of the current follow-up Dagstuhl seminar.

## Goals

Similar to the first meeting, we want to gather people from scientific visualization, image processing, medical imaging and other tensor-oriented application fields in a real interdisciplinary atmosphere, and we intend to publish the scientific output of this meeting in a postproceedings volume. This time, however, the following innovations are planned:

- Since a number of fundamental issues has already been identified at the 2004 Perspective Workshop - and quite some progress has been achieved - we would also like to encourage younger participants to present their more recent findings. Many researchers have entered this field fairly recently. To give room for the latest advances, about 50 per cent new people who have not attended the first tensor meeting have been invited. Moreover, in order to gain better insights into the foundations of tensor fields, also more experts from applied mathematics have been invited.
- Directions that have not or hardly been addressed in the first
workshop and will play a role in our current proposal include
- tensor voting ideas
- tensor approximations of high dimensional visual data
- a more consequent use of differential geometry in image analysis of tensor fields
- methods based on wavelets
- clustering, labeling of clusters, and calculating quantitative measures from regions
- a number of alternative applications beyond DT-MRI, such as tagged MRI for deformation analysis of the heart muscle, liquid crystals, geomechanical / earthquake data, satellite gradiometry, and material science.

- We explicitly encourage all participants to give stimulating, provocative and even controversal presentations that trigger discussions rather than polished, but less exciting technical talks.

Since this area is rather young, representatives of most relevant groups can meet within the framework of a relatively small seminar. We are confident that the unique atmosphere of Schloss Dagstuhl provides an ideal location to initiate a closer interaction within this emerging scientific field.

## Dagstuhl Seminar Series

- 18442: "Visualization and Processing of Anisotropy in Imaging, Geometry, and Astronomy" (2018)
- 16142: "Multidisciplinary Approaches to Multivalued Data: Modeling, Visualization, Analysis" (2016)
- 14082: "Visualization and Processing of Higher Order Descriptors for Multi-Valued Data" (2014)
- 11501: "Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data" (2011)
- 09302: "New Developments in the Visualization and Processing of Tensor Fields" (2009)
- 04172: "Visualization and Image Processing of Tensor Fields" (2004)

## Classification

- Computer Graphics / Computer Vision

## Keywords

- Visualization
- Image processing
- Tensor fields
- Diffusion tensor imaging (DT-MRI)
- Topology
- Partial differential equations (PDEs)
- Feature extraction
- Segmentation
- Registration
- Fiber tracking