Encyclopedia of Computer Graphics and Games

Living Edition
| Editors: Newton Lee

Tensor Field Visualization

  • Tim McGraw
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-08234-9_96-1


A tensor, specifically a second order tensor, is a linear mapping from vectors to vectors and is represented by a multidimensional array of values called its “components.”

A tensor field is a mapping from each point in some spatial domain (usually 2D or 3D) to a tensor.

Tensor field visualization is the process of visually representing tensor fields so that features of interest in the field become apparent to the viewer.


Some physical phenomena can be represented by a single number, or scalar value. Temperature and density are well-known examples. Other quantities characterized by a magnitude and direction, like force and velocity, are represented as a vector. Yet other phenomena, like mechanical stress and diffusion, are represented by a matrix. This progression, from scalar to vector to matrix, is generalized by the concept of tensor order. Tensors of order 0 are represented by scalars, tensors of order 1 are represented by vectors, and tensors of order 2 are...

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  1. Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66(1), 259–267 (1994)Google Scholar
  2. Basser, P.J., Pajevic, S., Pierpaoli, C., Duda, J., Aldroubi, A.: In vivo fiber tractography using DT-MRI data. Magn. Reson. Med. 44(4), 625–632 (2000)Google Scholar
  3. Cabral, B., Leedom, L.C.: Imaging vector fields using line integral convolution. In: Proceedings of the 20th annual conference on computer graphics and interactive techniques, pp. 263–270. ACM, Anaheim, CA (1993)Google Scholar
  4. Delmarcelle, T., Hesselink, L.: Visualizing second-order tensor fields with hyperstreamlines. IEEE Comput. Graph. Appl. 13(4), 25–33 (1993)Google Scholar
  5. Hsu, E.: Generalized line integral convolution rendering of diffusion tensor fields. In: Proceedings of the International Society for Magnetic Resonance in Medicine, 9th Scientific Meeting and Exhibition, Glasgow, vol. 790 (2001)Google Scholar
  6. Kindlmann, G.: Superquadric tensor glyphs. In: Proceedings of the Sixth Joint Eurographics-IEEE TCVG conference on Visualization, pp. 147–154. Eurographics Association, Konstanz, Germany (2004)Google Scholar
  7. Kindlmann, G., Westin, C.-F.: Diffusion tensor visualization with glyph packing. IEEE Trans. Vis. Comput. Graph. 12(5), 1329–1336 (2006)Google Scholar
  8. Kindlmann, G., Tricoche, X., Westin, C.-F.: Delineating white matter structure in diffusion tensor MRI with anisotropy creases. Med. Image Anal. 11(5), 492–502 (2007)Google Scholar
  9. Kondratieva, P., Kruger, J., Westermann, R.: The application of GPU particle tracing to diffusion tensor field visualization. In: IEEE Visualization 2005, pp. 73–78. IEEE, Minneapolis, MN (2005)Google Scholar
  10. Laidlaw, D.H., Ahrens, E.T., Kremers, D., Avalos, M.J., Jacobs, R.E., Readhead, C.: Visualizing diffusion tensor images of the mouse spinal cord. In: Proceedings of Visualization’98, pp. 127–134. IEEE, Research Triangle Park, NC (1998)Google Scholar
  11. McGraw, T., Vemuri, B.C., Wang, Z., Chen, Y., Rao, M., Mareci, T.: Line integral convolution for visualization of fiber tract maps from DTI. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 615–622. Springer, Tokyo, Japan (2002)Google Scholar
  12. Özarslan, E., Mareci, T.H.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn. Reson. Med. 50(5), 955–965 (2003)Google Scholar
  13. Özarslan, E., Vemuri, B.C., Mareci, T.H.: Generalized scalar measures for diffusion MRI using trace, variance, and entropy. Magn. Reson. Med. 53(4), 866–876 (2005)Google Scholar
  14. Schultz, T., Theisel, H., Seidel, H.-P.: Topological visualization of brain diffusion MRI data. IEEE Trans. Vis. Comput. Graph. 13(6), 1496–1503 (2007)Google Scholar
  15. Tricoche, X., Kindlmann, G., Westin, C.-F.: Invariant crease lines for topological and structural analysis of tensor fields. IEEE Trans. Vis. Comput. Graph. 14(6), 1627–1634 (2008)Google Scholar
  16. Weldeselassie, Y.T., Barmpoutis, A., Atkins, M.S.: Symmetric positive semidefinite Cartesian tensor fiber orientation distributions (CT-FOD). Med. Image Anal. 16(6), 1121–1129 (2012)Google Scholar
  17. Westin, C.-F., Maier, S.E., Mamata, H., Nabavi, A., Jolesz, F.A., Kikinis, R.: Processing and visualization for diffusion tensor MRI. Med. Image Anal. 6(2), 93–108 (2002)Google Scholar
  18. Zheng, X., Pang, A.: HyperLIC. In: Proceedings of the 14th IEEE Visualization 2003 (VIS’03), pp. 249–256. IEEE Computer Society, Seattle, WA (2003)Google Scholar
  19. Zheng, X., Pang, A.: Topological lines in 3D tensor fields. In: Proceedings of the Conference on Visualization’04, pp. 313–320. IEEE Computer Society, Austin, TX (2004)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Purdue UniversityWest LafayetteUSA