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Top Challenges in the Visualization of Engineering Tensor Fields Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Tensor Invariants and Glyph Design

Authors : Andrea Kratz, Cornelia Auer, Ingrid Hotz

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

Publisher: Springer Berlin Heidelberg

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Abstract

Tensors provide a mathematical language for the description of many physical phenomena. They appear everywhere where the dependence of multiple vector fields is approximated as linear. Due to this generality they occur in various application areas, either as result or intermediate product of simulations. As different as these applications, is the physical meaning and relevance of particular mathematical properties. In this context, domain specific tensor invariants that describe the entities of interest play a crucial role. Due to their importance, we propose to build any tensor visualization upon a set of carefully chosen tensor invariants. In this chapter we focus on glyph-based representations, which still belong to the most frequently used tensor visualization methods. For the effectiveness of such visualizations the right choice of glyphs is essential. This chapter summarizes some common glyphs, mostly with origin in mechanical engineering, and link their interpretation to specific tensor invariants.

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previous chapter Top Challenges in the Visualization of Engineering Tensor Fields
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Literature
1.
Bahn, M.M.: Invariant and orthonormal scalar measures derived from magnetic resonance diffusion tensor imaging. J. Magn. Reson. 141(1), 68–77 (1999)CrossRefMathSciNet
2.
Barr, A.H.: Superquadrics and angle-preserving transformations. IEEE Comput. Graph. Appl. 1(1), 11–23 (1981)CrossRef
3.
Benger, W., Hege, H.-C.: Tensor splats. In: Proceedings of Visualization and Data Analysis, San Jose, pp. 151–162 (2004)
4.
Bhalerao, A., Westin, C.-F.: Tensor splats: visualising tensor fields by texture mapped volume rendering. In: Proceedings of MICCAI’03, Montréal, pp. 294–901 (2003)
5.
Bilgen, M., Narayana, P.A.: Mohr diagram interpretation of anisotropic diffusion indices in MRI. Magn. Reson. Imaging 21(5), 567–572 (2003)CrossRef
6.
7.
Criscione, J.C., Humphrey, J.D., Douglas, A.S., Hunter, W.C.: An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. J. Mech. Phys. Solids 48, 2445–2465 (2000)CrossRefMATH
8.
Crossno, P., Rogers, D.H., Brannon, R.M., Coblentz, D., Fredrich, J.T.: Visualization of geologic stress perturbations using Mohr diagrams. IEEE Trans. Vis. Comput. Graph. 11(5), 508–518 (2005)CrossRef
9.
DoCarmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, Upper Saddle River (1976)
10.
Ennis, D.B., Kindlmann, G.: Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images. Magn. Reson. Med. 55(1), 136–146 (2006)CrossRef
11.
Feng, L., Hotz, I., Hamann, B., Joy, K.: Anisotropic noise samples. IEEE Trans. Vis. Comput. Graph. 14(2), 342–354 (2008)CrossRef
12.
13.
Hashash, Y.M.A., Yao, J.I.-C., Wotring, D.C.: Glyph and hyperstreamline representation of stress and strain tensors and material constitutive response. Int. J. Numer. Anal. Methods Geomech. 27(7), 603–626 (2003)CrossRefMATH
14.
Hlawitschka, M., Scheuermann, G.: Hot-lines – tracking lines in higher order tensor fields. In: Proceedings of IEEE Visualization (Vis’05), Minneapolis, pp. 27–34 (2005)
15.
Hotz, I., Feng, L., Hagen, H., Hamann, B., Jeremic, B., Joy, K.I.: Physically based methods for tensor field visualization. In: Proceedings of IEEE Visualization (Vis’04), Austin, pp. 123–130 (2004)
16.
Jankun-Kelly, T., Mehta, K.: Superellipsoid-based, real symmetric traceless tensor glyphs motivated by nematic liquid crystal alignment visualizations. IEEE Trans. Vis. Comput. Graph. (Proceedings Visualization ’06) 12, 1197–1204 (2006)
17.
Kindlmann, G.: Superquadric tensor glyphs. In: Proceeding of the Joint Eurographics – IEEE TCVG Symposium on Visualization, Konstanz, pp. 147–154 (2004)
18.
Kindlmann, G., Ennis, D.B., Whitaker, R.T., Westin, C.-F.: Diffusion tensor analysis with invariant gradients and rotation tangents. IEEE Trans. Med. Imaging 26, 1483–1499 (2007)CrossRef
19.
Kindlmann, G., Westin, C.-F.: Diffusion tensor visualization with glyph packing. IEEE Trans. Vis. Comput. Graph. (Vis ’06) 12(5), 1329–1336 (2006)
20.
Kratz, A., Auer, C., Stommel, M., Hotz, I.: Visualization and analysis of second-order tensors: moving beyond the symmetric positive-definite case. Comput. Graph. Forum – State of the Art Reports 32(1), 49–74 (2013)
21.
Kratz, A., Baum, D., Hotz, I.: Anisotropic sampling of planar and two-manifold domains for texture generation and glyph distribution. Trans. Vis. Comput. Graph. (TVCG) 19, 1782–1794 (2013)
22.
Kriz, R., Yaman, M., Harting, M., Ray, A.: Visualization of zeroth, second, fourth, higher order tensors, and invariance of tensor equations. Summer Research Project 2003, 2005. University Visualization and Animation Group, Virginia Tech, Blacksburg
23.
Lie, A.E., Kehrer, J., Hauser, H.: Critical design and realization aspects of glyph-based 3d data visualization. In: Proceedings of the Spring Conference on Computer Graphics (SCCG ’09), Budmerice, pp. 27–34 (2009)
24.
Lund, B.: Crustal stress studies using microearthquakes and boreholes. PhD thesis, Department of Earth Sciences, Uppsala University (2000)
25.
Moore, J.G., Schorn, S.A., Moore, J.: Methods of classical mechanics applied to turbulence stresses in a tip leakage vortex. J. Turbonmachinery 118(4), 622–630 (1996)CrossRef
26.
Schultz, T., Kindlmann, G.: A maximum enhancing higher-order tensor glyph. Comput. Graph. Forum (EuroVis’10) 29(3), 1143–1152 (2010)
27.
Schultz, T., Kindlmann, G.L.: Superquadric glyphs for symmetric second-order tensors. IEEE Trans. Vis. Comput. Graph. (Vis’10) 16(6), 1595–1604 (2010)
28.
Westin, C.-F., Maier, S.E., Khidhir, B., Everett, P., Jolesz, F.A., Kikinis, R.: Image processing for diffusion tensor magnetic resonance imaging. In: Proceedings of MICCAI’99, Cambridge, pp. 441–452 (1999)
29.
Westin, C.-F., Peled, S., Gudbjartsson, H., Kikinis, R., Jolesz, F.A.: Geometrical diffusion measures for MRI from tensor basis analysis. In: Proceedings of the International Society for Magnetic Resonance Medicine (ISMRM), Vancouver, pp. 1742 (1997)
Metadata
Title
Tensor Invariants and Glyph Design
Authors
Andrea Kratz
Cornelia Auer
Ingrid Hotz
Copyright Year
2014
Publisher
Springer Berlin Heidelberg
Book
Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data

Print ISBN: 978-3-642-54300-5

Electronic ISBN: 978-3-642-54301-2

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-642-54301-2_2

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