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Issue No.06 - November/December (2010 vol.16)
pp: 1595-1604
Thomas Schultz , Computer Science Department and the Computation Institute, University of Chicago
Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.
Tensor Glyphs, Stress Tensors, Rate-of-Deformation Tensors, Geometry Tensors, Glyph Design
Thomas Schultz, "Superquadric Glyphs for Symmetric Second-Order Tensors", IEEE Transactions on Visualization & Computer Graphics, vol.16, no. 6, pp. 1595-1604, November/December 2010, doi:10.1109/TVCG.2010.199
[1] A. Barr, Superquadrics and angle-preserving transformations. IEEE Computer Graphics and Applications, 18 (1): 11–23, 1981.
[2] P. Basser, J. Mattiello, and D. LeBihan, MR diffusion tensor spectroscopy and imaging. Biophysics Journal, 66 (1): 259–267, 1994.
[3] W. Benger and H.-C. Hege, Tensor splats. In Visualization and Data Analysis (Proc. SPIE), volume 5295, pages 151–162, 2004.
[4] B. Cabral and L. C. Leedom, Imaging vector fields using line integral convolution. In Computer Graphics (Proc. ACM SIGGRAPH), pages 263–270, 1993.
[5] P. Danielsson, Q. Lin, and Q. Ye, Efficient detection of Second-Degree variations in 2D and 3D images. Journal of Visual Communication and Image Representation, 12 (3): 255–305, 2001.
[6] W. C. de Leeuw and J. J. van Wijk, A probe for local flow field visualization. In Proc. IEEE Visualization, pages 39–45, 1993.
[7] T. Delmarcelle and L. Hesselink, Visualizing second-order tensor fields with hyperstreamlines. IEEE Computer Graphics and Applications, 13 (4): 25–33, 1993.
[8] C. Dick, J. Georgii, R. Burgkart, and R. Westermann, Stress tensor field visualization for implant planning in orthopedics. IEEE Trans. on Visualization and Computer Graphics, 15 (6): 1399–1406, 2009.
[9] Z. Ding, J. C. Gore, and A. W. Anderson, Classification and quantification of neuronal fiber pathways using diffusion tensor MRI. Magnetic Resonance in Medicine, 49 (4): 716–721, 2003.
[10] R. K. Dodd, A new approach to the visualization of tensor fields. Graphical Models and Image Processing, 60 (4): 286–303, 1998.
[11] G. Elber, Line art illustrations of parametric and implicit forms. IEEE Trans. on Visualization and Computer Graphics, 4 (1): 71–81, 1998.
[12] C. Feddern, J. Weickert, and B. Burgeth, Level-set methods for tensor-valued images. In O. D. Faugeras, and N. Paragios editors, , Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision, pages 65–72, 2003.
[13] L. Feng, I. Hotz, B. Hamann, and K. Joy, Anisotropic noise samples. IEEE Trans. on Visualization and Computer Graphics, 14 (2): 342–354, 2008.
[14] A. Girshick, V. Interrante, S. Haker, and T. Lemoine, Line direction matters: An argument for the use of principal directions in 3D line drawings. In Proc. 1st Intl. Symp. Non-Photorealistic Animation and Rendering, pages 43–52. ACM, 2000.
[15] A. Globus, C. Levit, and T. Lasinski, A tool for visualizing the topology of three-dimensional vector fields. In Proc. IEEE Visualization, pages 33–40, 1991.
[16] R. Haber, Visualization techniques for engineering mechanics. Computing Systems in Engineering, 1 (1): 37–50, 1990.
[17] H. Hagen, S. Hahmann, T. Schreiber, Y. Nakajima, B. Wordenweber, and P. Hollemann-Grundstedt, Surface interrogation algorithms. IEEE Computer Graphics and Applications, 12 (5): 53–60, 1992.
[18] Y. M.A. Hashash, J. I.-C. Yao, and D. C. Wotring, Glyph and hyperstreamline representation of stress and strain tensors and material constitutive response. Intl. Journal for Numerical and Analytical Methods in Geomechanics, 27: 603–626, 2003.
[19] A. Hertzmann and D. Zorin, Illustrating smooth surfaces. In Computer Graphics (Proc. ACM SIGGRAPH), pages 517–526, 2000.
[20] L. Hesselink, Y. Levy, and Y. Lavin, The topology of symmetric, second-order 3D tensor fields. IEEE Trans. on Visualization and Computer Graphics, 3 (1): 1–11, 1997.
[21] I. Hotz, L. Feng, H. Hagen, B. Hamann, and K. Joy, Tensor field visualization using a metric interpretation. In J. Weickert, and H. Hagen editors, , Visualization and Processing of Tensor Fields, chapter 16, pages 269–280. Springer, 2006.
[22] I. Hotz, L. Feng, H. Hagen, B. Hamann, K. Joy, and B. Jeremic, Physically based methods for tensor field visualization. In Proc. IEEE Visualization, pages 123–130, 2004.
[23] E. Hsu, Generalized line integral convolution rendering of diffusion tensor fields. In Proc. International Society of Magnetic Resonance in Medicine (ISMRM), page 790, 2001.
[24] T. J. Jankun-Kelly, Y S. Lanka, and J. E. Swan II, An evaluation of glyph perception for real symmetric traceless tensor properties. Computer Graphics Forum (Special Issue on EuroVis 2010), 29 (3): 1133–1142, 2010.
[25] T. J. Jankun-Kelly and K. Mehta, Superellipsoid-based, real symmetric traceless tensor glyphs motivated by nematic liquid crystal alignment visualization. IEEE Trans. on Visualization and Computer Graphics (Proc. IEEE Visualization), 12 (5): 1197–1204, 2006.
[26] B. Jeremić, G. Scheuermann, J. Frey, Z. Yang, B. Hamann, K. I. Joy, and H. Hagen, Tensor visualizations in computational geomechanics. Intl. Journal for Numerical and Analytical Methods in Geomechanics, 26 (10): 925–944, 2002.
[27] D. Keefe, D. Karelitz, E. Vote, and D. H. Laidlaw, Artistic collaboration in designing VR visualizations. IEEE Computer Graphics and Applications, 25 (2): 18–23, 2005.
[28] G. Kindlmann, Superquadric tensor glyphs. In Proc. EG/IEEE TCVG Symposium on Visualization, pages 147–154, May 2004.
[29] G. Kindlmann, X. Tricoche, and C.-F. Westin, Delineating white matter structure in diffusion tensor MRI with anisotropy creases. Medical Image Analysis, 11 (5): 492–502, 2007.
[30] G. Kindlmann, D. Weinstein, and D. Hart, Strategies for direct volume rendering of diffusion tensor fields. IEEE Trans. on Visualization and Computer Graphics, 6 (2): 124–138, 2000.
[31] G. Kindlmann and C.-F. Westin, Diffusion tensor visualization with glyph packing. IEEE Trans. on Visualization and Computer Graphics, 12 (5): 1329–1335, 2006.
[32] G. Kindlmann, R. Whitaker, T. Tasdizen, and T. Möller, Curvature-based transfer functions for direct volume rendering: Methods and applications. In Proc. IEEE Visualization 2003, pages 513–520, 2003.
[33] G. L. Kindlmann, R. S.J. Estépar, S. M. Smith, and C.-F. Westin, Sampling and visualizing creases with scale-space particles. IEEE Trans. on Visualization and Computer Graphics, 15 (6): 1415–1424, 2009.
[34] R. M. Kirby, H. Marmanis, and D. H. Laidlaw, Visualizing multivalued data from 2D incompressible flows using concepts from painting. In Proc. IEEE Visualization, pages 333–340, 1999.
[35] D. H. Laidlaw, E. T. Ahrens, D. Kremers, M. J. Avalos, R. E. Jacobs, and C. Readhead, Visualizing diffusion tensor images of the mouse spinal cord. In Proc. IEEE Visualization, pages 127–134, 1998.
[36] M. A. Livingston, Visualization of rotation fields. In Proc. IEEE Visualization, pages 491–494, 1997.
[37] T. Luft, C. Colditz, and O. Deussen, Image enhancement by unsharp masking the depth buffer. ACM Transactions on Graphics (Proc. ACM SIGGRAPH), 25 (3): 1206–1213, 2006.
[38] J. E. Marsden and A. J. Tromba, Vector Calculus. W.H. Freeman and Company, New York, New York, 1996.
[39] M. D. Meyer, P. Georgel, and R. T. Whitaker, Robust particle systems for curvature dependent sampling of implicit surfaces. In Proc. Shape Modeling and Applications (SMI), pages 124–133, June 2005.
[40] A. Neeman, B. Jeremic, and A. Pang, Visualizing tensor fields in geomechanics. In Proc. IEEE Visualization, pages 35–42, 2005.
[41] E. Özarslan and T. Mareci, Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magnetic Resonance in Medicine, 50 (5): 955–965, 2003.
[42] S. Pajevic and C. Pierpaoli, Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: Application to white matter fiber tract mapping in the human brain. Magnetic Resonance in Medicine, 42 (3): 526–540, 1999.
[43] F. H. Post, B. Vrolijk, H. Hauser, R. S. Laramee, and H. Doleisch, The state of the art in flow visualisation: Feature extraction and tracking. Computer Graphics Forum, 22 (4): 775–792, 2003.
[44] T. Ropinski, J. Meyer-Spradow, M. Specht, K. H. Hinrichs, and B. Preim, Surface glyphs for visualizing multimodal volume data. In Proc. Vision, Modeling, and Visualization, pages 3–12, 2007.
[45] R. J. Rost, OpenGL shading manual. Addison-Wesley, 2nd edition, 2006.
[46] W. Schroeder, K. Martin, and B. Lorensen, The Visualization Toolkit: An Object Oriented Approach to Graphics, chapter 6. Kitware, 2003.
[47] W. J. Schroeder and K. M. Martin, The Visualization Handbook, chapter 1: Overview of Visualization, pages 3–38. Academic Press, 2004.
[48] T. Schultz and G. Kindlmann, A maximum enhancing higher-order tensor glyph. Computer Graphics Forum (Proc. EuroVis), 29 (3): 1143–1152, 2010.
[49] T. Schultz, H. Theisel, and H.-P. Seidel, Crease surfaces: From theory to extraction and application to diffusion tensor MRI. IEEE Trans. on Visualization and Computer Graphics, 16 (1): 109–119, 2010.
[50] C. D. Shaw, D. S. Ebert, J. M. Kukla, A. Zwa, I. Soboroff, and D. A. Roberts, Data visualization using automatic, perceptually-motivated shapes. In Visual Data Exploration and Analysis (Proc. SPIE), volume 3298, pages 208–213, 1998.
[51] A. Sigfridsson, T. Ebbers, E. Heiberg, and L. Wigström, Tensor field visualisation using adaptive filtering of noise fields combined with glyph rendering. In Proc. IEEE Visualization, pages 371–378, 2002.
[52] V. Slavin, R. Pelcovits, G. Loriot, A. Callan-Jones, and D. Laidlaw, Techniques for the visualization of topological defect behavior in nematic liquid crystals. IEEE Trans. on Visualization and Computer Graphics, 12 (5): 1323–1328, 2006.
[53] M. Tarini, P. Cignoni, and C. Montani, Ambient occlusion and edge cueing to enhance real time molecular visualization. IEEE Transactions on Visualization and Computer Graphics, 12 (5): 1237–1244, 2006.
[54] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, Saddle connectors - an approach to visualizing the topological skeleton of complex 3D vector fields. In Proc. IEEE Visualization, pages 225–232, 2003.
[55] W.-S. Tong, C.-K. Tang, P. Mordohai, and G. Medioni, First order augmentation to tensor voting for boundary inference and multiscale analysis in 3D. IEEE Trans. on Pattern Analysis and Machine Intelligence, 26 (5): 594–611, 2004.
[56] A. Vilanova, S. Zhang, G. Kindlmann, and D. H. Laidlaw, An introduction to visualization of diffusion tensor imaging and its applications. In J. Weickert, and H. Hagen editors, , Visualization and Processing of Tensor Fields, pages 121–153. Springer, 2006.
[57] J. Weickert and H. Hagen editors. , Visualization and Processing of Tensor Fields. Springer, 2006.
[58] E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, page 1894. CRC Press, 2003.
[59] C.-F. Westin, S. E. Maier, B. Khidhir, P. Everett, F. A. Jolesz, and R. Kikinis, Image processing for diffusion tensor magnetic resonance imaging. In Proceedings MICCAI, volume 1679 of LNCS, pages 441–452, 1999.
[60] M. R. Wiegell, H. B.W. Larsson, and V. J. Wedeen, Fiber crossing in human brain depicted with diffusion tensor MR imaging. Radiology, 217 (3): 897–903, 2000.
[61] X. Zheng and A. Pang, HyperLIC. In Proc. IEEE Visualization, pages 249–256, 2003.
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