Visualization and Analysis of Second‐Order Tensors: Moving Beyond the Symmetric Positive‐Definite Case

Tensors provide a powerful language to describe physical phenomena. Consequently, they have a long tradition in physics and appear in various application areas, either as the final result of simulations or as intermediate product. Due to their complexity, tensors are hard to interpret. This motivates the development of well‐conceived visualization methods. As a sub‐branch of scientific visualization, tensor field visualization has been especially pushed forward by diffusion tensor imaging. In this review, we focus on second‐order tensors that are not diffusion tensors. Until now, these tensors, which might be neither positive‐definite nor symmetric, are under‐represented in visualization and existing visualization tools are often not appropriate for these tensors. Hence, we discuss the strengths and limitations of existing methods when dealing with such tensors as well as challenges introduced by them. The goal of this paper is to reveal the importance of the field and to encourage the development of new visualization methods for tensors from various application fields.

[1]  Hans Hagen,et al.  Tensor Field Reconstruction Based on Eigenvector and Eigenvalue Interpolation , 2010, Scientific Visualization: Advanced Concepts.

[2]  Robert S. Laramee,et al.  Asymmetric Tensor Field Visualization for Surfaces , 2011, IEEE Transactions on Visualization and Computer Graphics.

[3]  Hans Knutsson,et al.  A Review of Tensors and Tensor Signal Processing , 2009, Tensors in Image Processing and Computer Vision.

[4]  Ingrid Hotz,et al.  Glyph- and Texture-based Visualization of Segmented Tensor Fields , 2012, GRAPP/IVAPP.

[5]  David F. Tate,et al.  A Novel Interface for Interactive Exploration of DTI Fibers , 2009, IEEE Transactions on Visualization and Computer Graphics.

[6]  Bernd Hamann,et al.  Anisotropic Noise Samples , 2008, IEEE Transactions on Visualization and Computer Graphics.

[7]  D. Le Bihan,et al.  Diffusion tensor imaging: Concepts and applications , 2001, Journal of magnetic resonance imaging : JMRI.

[8]  HesselinkLambertus,et al.  The Topology of Symmetric, Second-Order 3D Tensor Fields , 1997 .

[9]  Gunther H. Weber,et al.  Interactive Volume Rendering of Diffusion Tensor Data , 2009 .

[10]  Thomas Schultz,et al.  A Maximum Enhancing Higher‐Order Tensor Glyph , 2010, Comput. Graph. Forum.

[11]  Simon R. Arridge,et al.  Tensor Field Regularisation for DT-MR Images , 2001 .

[12]  N. Ayache,et al.  Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics , 2007 .

[13]  Rebecca M. Brannon,et al.  Exploring 2D tensor fields using stress nets , 2005, VIS 05. IEEE Visualization, 2005..

[14]  G. Kindlmann,et al.  Superquadric Glyphs for Symmetric Second-Order Tensors , 2010, IEEE Transactions on Visualization and Computer Graphics.

[15]  Eugene Zhang,et al.  Interactive procedural street modeling , 2008, ACM Trans. Graph..

[16]  Martin Rumpf,et al.  Visualization of Principal Curvature Directions by Anisotropic Diffusion , 2000, VMV.

[17]  Rebecca M. Brannon,et al.  Visualization of salt-induced stress perturbations , 2004, IEEE Visualization 2004.

[18]  Xavier Tricoche,et al.  Degenerate 3D Tensors , 2006, Visualization and Processing of Tensor Fields.

[19]  Björn Meyer,et al.  A Visual Approach to Analysis of Stress Tensor Fields , 2011, Scientific Visualization: Interactions, Features, Metaphors.

[20]  Zhizhou Wang,et al.  An affine invariant tensor dissimilarity measure and its applications to tensor-valued image segmentation , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[21]  David H. Laidlaw,et al.  Visualizing diffusion tensor images of the mouse spinal cord , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[22]  Atilla Baskurt,et al.  A new CAD mesh segmentation method, based on curvature tensor analysis , 2005, Comput. Aided Des..

[23]  Alex T. Pang,et al.  2D asymmetric tensor analysis , 2005, VIS 05. IEEE Visualization, 2005..

[24]  Alisa Neeman,et al.  Visualizing tensor fields in geomechanics , 2005, VIS 05. IEEE Visualization, 2005..

[25]  P. Basser,et al.  Estimation of the effective self-diffusion tensor from the NMR spin echo. , 1994, Journal of magnetic resonance. Series B.

[26]  Carl-Fredrik Westin,et al.  A tensor-like representation for averaging, filtering and interpolation of 3-D object orientation data , 2005, IEEE International Conference on Image Processing 2005.

[27]  Carl-Fredrik Westin,et al.  Geodesic-Loxodromes for Diffusion Tensor Interpolation and Difference Measurement , 2007, MICCAI.

[28]  Thierry Delmarcelle,et al.  The Visualization of Second-Order Tensor Fields , 2013 .

[29]  G. Kindlmann,et al.  Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images , 2006, Magnetic resonance in medicine.

[30]  Lambertus Hesselink,et al.  The topology of symmetric, second-order tensor fields , 1994, Proceedings Visualization '94.

[31]  Hans-Peter Seidel,et al.  Estimating Crossing Fibers: A Tensor Decomposition Approach , 2008, IEEE Transactions on Visualization and Computer Graphics.

[32]  Carl-Fredrik Westin,et al.  Diffusion Tensor Visualization with Glyph Packing , 2006, IEEE Transactions on Visualization and Computer Graphics.

[33]  Pierre Alliez,et al.  Anisotropic polygonal remeshing , 2003, ACM Trans. Graph..

[34]  Rebecca M. Brannon,et al.  Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors , 2008, VG/PBG@SIGGRAPH.

[35]  Gordon L. Kindlmann,et al.  Tensorlines: advection-diffusion based propagation through diffusion tensor fields , 1999, Proceedings Visualization '99 (Cat. No.99CB37067).

[36]  Ronald D. Kriz,et al.  Visualization of Zeroth, Second, Fourth, Higher Order Tensors, and Invariance of Tensor Equations " , 2005 .

[37]  K. Shimada,et al.  Anisotropic Triangular Meshing of Parametric Surfaces via Close Packing of Ellipsoidal Bubbles , 2007 .

[38]  J. Weickert,et al.  Visualization and Processing of Tensor Fields (Mathematics and Visualization) , 2005 .

[39]  Markus Rütten,et al.  Analyzing Vortex Breakdown Flow Structures by Assignment of Colors to Tensor Invariants , 2006, IEEE Transactions on Visualization and Computer Graphics.

[40]  Carl-Fredrik Westin,et al.  Invariant Crease Lines for Topological and Structural Analysis of Tensor Fields , 2008, IEEE Transactions on Visualization and Computer Graphics.

[41]  Thomas Schultz,et al.  Topological Features in 2D Symmetric Higher‐Order Tensor Fields , 2011, Comput. Graph. Forum.

[42]  Alex T. Pang,et al.  Topological lines in 3D tensor fields and discriminant Hessian factorization , 2005, IEEE Transactions on Visualization and Computer Graphics.

[43]  Rebecca M. Brannon,et al.  Visualization of geologic stress perturbations using Mohr diagrams , 2005, IEEE Transactions on Visualization and Computer Graphics.

[44]  Gerik Scheuermann,et al.  Fast and Memory Efficient GPU-Based Rendering of Tensor Data , 2011 .

[45]  Vivek Verma,et al.  A flow-guided streamline seeding strategy , 2000, Proceedings Visualization 2000. VIS 2000 (Cat. No.00CH37145).

[46]  B. Vemuri,et al.  Diffusion tensor field restoration and segmentation , 2004 .

[47]  P. Thomas Fletcher,et al.  Riemannian geometry for the statistical analysis of diffusion tensor data , 2007, Signal Process..

[48]  Ingrid Hotz,et al.  Particle-Based Anisotropic Sampling for Two-Dimensional Tensor Field Visualization , 2011, VMV.

[49]  Robert S. Laramee,et al.  Asymmetric Tensor Analysis for Flow Visualization , 2009, IEEE Transactions on Visualization and Computer Graphics.

[50]  Barr,et al.  Superquadrics and Angle-Preserving Transformations , 1981, IEEE Computer Graphics and Applications.

[51]  Joan G. Moore,et al.  Education Committee Best Paper of 1995 Award: Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex , 1996 .

[52]  Carl-Fredrik Westin,et al.  Tensor Splats: Visualising Tensor Fields by Texture Mapped Volume Rendering , 2003, MICCAI.

[53]  Jaya Sreevalsan-Nair,et al.  2D Tensor Field Segmentation , 2011, Scientific Visualization: Interactions, Features, Metaphors.

[54]  T. McGraw,et al.  A Subdivision Approach to Tensor Field Interpolation , 2008 .

[55]  Rüdiger Westermann,et al.  Stress Tensor Field Visualization for Implant Planning in Orthopedics , 2009, IEEE Transactions on Visualization and Computer Graphics.

[56]  David H. Laidlaw,et al.  Exploring 3D DTI Fiber Tracts with Linked 2D Representations , 2009, IEEE Transactions on Visualization and Computer Graphics.

[57]  Gordon Kindlmann,et al.  Superquadric tensor glyphs , 2004, VISSYM'04.

[58]  Donald Danielson.  Vectors And Tensors In Engineering And Physics: Second Edition , 1992 .

[59]  Ofer Pasternak,et al.  The effect of metric selection on the analysis of diffusion tensor MRI data , 2010, NeuroImage.

[60]  Lambertus Hesselink,et al.  Singularities in nonuniform tensor fields , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[61]  Ghassan Hamarneh,et al.  DT-MRI segmentation using graph cuts , 2007, SPIE Medical Imaging.

[62]  J. Weickert,et al.  Level-Set Methods for Tensor-Valued Images , 2003 .

[63]  Lambertus Hesselink,et al.  The Topology of Symmetric Tensor Fields , 1997 .

[64]  Ketan Mehta,et al.  Superellipsoid-based, Real Symmetric Traceless Tensor Glyphs Motivated by Nematic Liquid Crystal Alignment Visualization , 2006, IEEE Transactions on Visualization and Computer Graphics.

[65]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[66]  A. Pang,et al.  Degenerate 3 D Tensors , 2022 .

[67]  Hans Hagen,et al.  Vector and Tensor Field Topology Simplification on Irregular Grids , 2001, VisSym.

[68]  Eugene Zhang,et al.  Interactive Tensor Field Design and Visualization on Surfaces , 2007, IEEE Transactions on Visualization and Computer Graphics.

[69]  Xuelong Li,et al.  Tensors in Image Processing and Computer Vision , 2009, Advances in Pattern Recognition.

[70]  Brian Cabral,et al.  Imaging vector fields using line integral convolution , 1993, SIGGRAPH.

[71]  Robert B. Haber,et al.  Visualization techniques for engineering mechanics , 1990 .

[72]  Bernd Hamann,et al.  Eigenvector-based Interpolation and Segmentation of 2D Tensor Fields , 2011, Topological Methods in Data Analysis and Visualization.

[73]  David H. Laidlaw,et al.  An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications , 2006, Visualization and Processing of Tensor Fields.

[74]  John Moore,et al.  Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex , 1995 .

[75]  Tim Weyrich,et al.  Eurographics Symposium on Point-based Graphics (2006) Gpu-based Ray-casting of Quadratic Surfaces , 2022 .

[76]  Werner Benger,et al.  Tensor splats , 2004, Visualization and Data Analysis.

[77]  Alex T. Pang,et al.  Topological lines in 3D tensor fields , 2004, IEEE Visualization 2004.

[78]  John F. Hughes,et al.  Smooth interpolation of orientations with angular velocity constraints using quaternions , 1992, SIGGRAPH.

[79]  Carl-Fredrik Westin,et al.  Segmentation of Tensor Fields: Recent Advances and Perspectives , 2009, Tensors in Image Processing and Computer Vision.

[80]  Nicholas Ayache,et al.  Clinical DT-MRI estimation, smoothing and fiber tracking with Log-Euclidean metrics , 2006, 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006..

[81]  Robert Michael Kirby,et al.  Advanced Reaction-Diffusion Models for Texture Synthesis , 2006, J. Graph. Tools.

[82]  M. Bilgen,et al.  Mohr diagram interpretation of anisotropic diffusion indices in MRI. , 2003, Magnetic resonance imaging.

[83]  Gordon L. Kindlmann,et al.  Hue-balls and lit-tensors for direct volume rendering of diffusion tensor fields , 1999, Proceedings Visualization '99 (Cat. No.99CB37067).

[84]  David E. Breen,et al.  Level Set Modeling and Segmentation of DT-MRI Brain Data , 2001 .

[85]  David S. Ebert,et al.  Using shape to visualize multivariate data , 1999, NPIVM '99.

[86]  P. Basser,et al.  Diffusion tensor MR imaging of the human brain. , 1996, Radiology.

[87]  Lambertus Hesselink,et al.  Visualizing second-order tensor fields with hyperstreamlines , 1993, IEEE Computer Graphics and Applications.

[88]  Carl-Fredrik Westin,et al.  Segmentation of Thalamic Nuclei from DTI Using Spectral Clustering , 2006, MICCAI.

[89]  David H. Laidlaw,et al.  Visualizing Diffusion Tensor MR Images Using Streamtubes and Streamsurfaces , 2003, IEEE Trans. Vis. Comput. Graph..

[90]  Werner Benger,et al.  Visualization of General Relativistic Tensor Fields via a Fiber Bundle Data Model , 2004 .

[91]  N. Ayache,et al.  Log‐Euclidean metrics for fast and simple calculus on diffusion tensors , 2006, Magnetic resonance in medicine.

[92]  Gordon L. Kindlmann,et al.  Strategies for Direct Volume Rendering of Diffusion Tensor Fields , 2000, IEEE Trans. Vis. Comput. Graph..

[93]  Hans Hagen,et al.  Physically based methods for tensor field visualization , 2004, IEEE Visualization 2004.

[94]  Y. Hashash,et al.  Glyph and hyperstreamline representation of stress and strain tensors and material constitutive response , 2003 .

[95]  Hans Hagen,et al.  An Introduction to Tensors , 2006, Visualization and Processing of Tensor Fields.

[96]  Bernd Hamann,et al.  Interactive Glyph Placement for Tensor Fields , 2007, ISVC.

[97]  David E. Breen,et al.  Level set modeling and segmentation of diffusion tensor magnetic resonance imaging brain data , 2003, J. Electronic Imaging.

[98]  Carl-Fredrik Westin,et al.  3D Bayesian Regularization of Diffusion Tensor MRI Using Multivariate Gaussian Markov Random Fields , 2004, MICCAI.

[99]  Penny Rheingans,et al.  NIH-NSF visualization research challenges report summary , 2006, IEEE Computer Graphics and Applications.

[100]  Gerik Scheuermann,et al.  HOT-lines: tracking lines in higher order tensor fields , 2005, VIS 05. IEEE Visualization, 2005..

[101]  Maher Moakher,et al.  Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization , 2006, Visualization and Processing of Tensor Fields.

[102]  Rachid Deriche,et al.  Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing , 2006, Journal of Mathematical Imaging and Vision.

[103]  Hans Hagen,et al.  Visualization of Seismic Soils Structure Interaction Simulations , 2001, VIIP.