Tensors in Visualization: Tutorial at VisWeek 2010

This is the webpage of our tutorial on

  • the visualization of second-order and higher-order tensor fields, with the main applications being diffusion MRI and fluid flows;
  • the use of higher-order tensors as mathematical models in the visualization of other types of data (e.g., volume data).

which was held at VisWeek, Salt Lake City, UT, USA, on Oct 26, 2010.


Level: Intermediate to advanced. We will expect basic knowledge of linear algebra and of standard techniques for tensor visualization (like streamlines or ellipsoid glyphs).


Introduction Gordon / Thomas 10 minutes
Tensor Topology Xavier 25 minutes
Tensor Field Features Gordon 25 minutes
Asymmetric Tensors Eugene 30 minutes
Coffee - 30 minutes
HARDI Thomas / Anna 40 minutes
Data Tensor Models Alex 40 minutes
Discussion All 10 minutes


Tensor fields arise in several scientific applications in which visualization research has traditionally been interested: Two prominent examples are diffusion-weighted magnetic resonance imaging and fluid flows. Organization of data into data tensors can provide a useful mathematical tool even for processing and visualizing simple scalar volume datasets. This tutorial will present a coherent and coordinated explanation of these topics, which minimizes redundancy while describing the flow of ideas between the various approaches. Particular emphasis is put on topics on which additional research is needed.

The first half of the tutorial will focus on second-order tensors, which can be represented as matrices with respect to a fixed coordinate system. An indispensible step in the visualization of tensor fields is to select a part of the data for display, since trying to show all available information at once would lead to visual clutter. The first strategy to address this problem has been to generalize notions from vector field topology to second-order tensor fields. We will introduce the basic concepts and the resulting methods, discuss their potential and limitations in practice. We then consider several more general features in second-order tensor fields, which build on topological considerations, as well as borrowing from other areas such as computer vision. Finally, in the analysis of fluid flows, asymmetric tensors arise, for which only few dedicated visualization tools exist. We will explain the challenges of asymmetric tensor visualization with respect to the more widely understood symmetric case, present the state of the art, and point out open problems.

The second half will start with treating tensors of orders higher than two in the context of High Angular Resolution Diffusion Imaging (HARDI). HARDI is an evolving research topic that includes some diverging strategies on how to best represent and process the resulting data, including models whose parameterizations fall outside tensor models. We will put these methods in context while showing how higher-order tensors can be used to represent real-valued and antipodally symmetric functions and thus provide a useful framework for much of the current work in HARDI. We will describe the main threads of HARDI research and point out differences in their interpretation, which are crucial for meaningful research on their visualization.

Finally, we turn to a larger class of volume datasets: Data tensors or multi-way arrays are often encountered when we have a collection of multivariate data which can be organized into a data tensor based on their causal factors. For example, a vectorized facial image is the result of multiple causal factors, such as the underlying geometry that defines the person's identity, the expression, the age, etc. We will introduce linear and multilinear factorization models that apply to higher order data tensors, and demonstrate practical examples in which they have proven beneficial for compression and automated analysis.


Xavier Tricoche Tensor Topology [pdf]
Gordon L. Kindlmann Tensor Field Features [pdf]
Eugene Zhang Asymmetric Tensors [pdf]
Thomas Schultz HARDI Models [pdf]
Anna Vilanova HARDI Visualization [pdf] [pptx (includes animations)]

Last update of slides: Oct 31, 2010