Journal of Visualization

, Volume 22, Issue 3, pp 641–660 | Cite as

A survey on visualization of tensor field

  • Chongke BiEmail author
  • Lu Yang
  • Yulin Duan
  • Yun Shi
Regular Paper


Tensor field has been widely used in various applications, such as medical imaging, industrial manufacturing, high-dimensional data analysis, and so forth. However, it is a challenging task to understand tensor field intuitively. Therefore, tensor field visualization has become an important research topic. In this survey, we present a comprehensive survey for two kinds of visualization methods for tensor fields: glyphs and streamlines. For glyphs, the eigenvalues of tensor fields will be used to classify existing visualization methods. There are mainly three types of eigenvalues: diffusion tensor fields with all positive real eigenvalues; the tensor field with negative real eigenvalues; the tensor field with imaginary eigenvalues. The methods showing the difference between two tensors (glyphs) are also introduced. For streamlines, there are mainly three important issues: the selection of seed points (streamlines), interpolation of tensor fields, the singularity problem around isotropic tensors. Finally, we discuss challenges and open questions for future studies.

Graphical abstract


Tensor field Visualization Glyph Streamline Anisotropy 



This work was partly supported by the National Natural Science Foundation of China under Grant Nos. 61702360, 61572057, 61836001, partly by the Tianjin Natural Science Foundation of China under Granted No. 16JCQNJC04100.


  1. Abbasloo A, Wiens V, Hermann M, Schultz T (2016) Visualizing tensor normal distributions at multiple levels of detail. IEEE Trans Vis Comput Graph 22:975–984Google Scholar
  2. Alexander AL, Kindlmann GL, Parker DL, Tsurada JS (2000) A geometric analysis of diffusion tensor measurements of the human brain. IMagn Reson Med 44:283–291Google Scholar
  3. Ankele M, Schultz T (2019) DT-MRI streamsurfaces revisited. IEEE Trans Vis Comput Graph 25(1):1112–1121Google Scholar
  4. Arsigny V, Fillard P, Pennec X, Ayache N (2006) Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn Reson Med 56(2):411–421Google Scholar
  5. Assaf Y, Ben-Bashat D, Chapman J, Peled S, Biton I, Kafri M, Segev Y, Hendler T, Korczyn A, Gralf M, Cohen Y (2002) High b-value q-space analyzed diffusion-weighted MRI: application to multiple sclerosis. Magn Reson Med 47(1):115–126Google Scholar
  6. Auer C, Hotz I (2011) Complete tensor field topology on 2D triangulated manifolds embedded in 3D. Comput Graph Forum 30(3):831–840Google Scholar
  7. Barrick TR, Clark CA (2004) Singularities in diffusion tensor fields and their relevance in white matter fiber tractography. Neuro Image 22(2):481–491Google Scholar
  8. Bashat D, Sira L, Gralf M, Planka P, Hendler T, Cohen Y, Assaf Y (2005) Normal white matter development from infancy to adulthood: comparing diffusion tensor and high b value diffusion weighted MR images. J Magn Reson Imaging 21(5):503–511Google Scholar
  9. Basser PJ (1995) Inferring microstructural features and the physiological state of tissues from diffusion-weighted images. NMR Biomed 8(7):333–344Google Scholar
  10. Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A (2000) In vivo fiber tractography using DT-MRI data. Magn Reson Med 44(4):625–632Google Scholar
  11. Batchelor PG, Moakher M, Atkinson D, Calamante F, Connelly A (2005) A rigorous framework for diffusion tensor calculus. Magn Reson Med 53(1):221–225Google Scholar
  12. Belhumeur PN, Kriegman DJ, Yuille AL (1999) The bas-relief ambiguity. Int J Comput Vis 35(1):33–44Google Scholar
  13. Bergmann Ø, Kindlmann G, Lundervold A, Westin C (2006) Diffusion k-tensor estimation from Q-ball imaging using discretized principal axes. In: Proceedings of medical image computing and computer-assisted intervention, pp 268–275Google Scholar
  14. Bergmann Ø, Kindlmann GL, Peled S, Westin C (2007) Two-tensor fiber tractography. In: Proceedings of the 4th IEEE international symposium on biomedical imaging, pp 796–799Google Scholar
  15. Bi C (2012) Degeneracy-aware interpolation of diffusion tensor fields and its applications. Ph.D. thesisGoogle Scholar
  16. Bi C, Takahashi S, Fujishiro I (2010) Interpolating 3D diffusion tensors in 2D planar do- main by locating degenerate lines. In: Proceedings of the 6th international conference on advances in visual computing, springer lecture notes in computer science, vol 6453, pp 328–337Google Scholar
  17. Bi C, Takahashi S, Ishida H, Fujishiro I (2010) Interpolating 3D diffusion tensors through optimizing rotational transfomations of anisotropic features. In: Poster proceedings of IEEE pacific visualization 2010. IEEE Computer Society, pp 3–4Google Scholar
  18. Bi C, Takahashi S, Fujishiro I (2011) Interpolation of 3D diffusion tensor fields by locating degenerate lines. In: Poster proceedings of IEEE pacific visualization 2011. IEEE Computer Society, pp 9–10Google Scholar
  19. Bi C, Sakurai D, Takahashi S, Ono K (2012) Interactive control of mesh topology in quadrilateral mesh generation based on 2D tensor fields. In: Proceedings of the 8th international conference on advances in visual computing, springer lecture notes in computer science, vol 7432, pp 726–735Google Scholar
  20. Bi C, Takahashi S, Fujishiro I (2012) Degeneracy-aware interpolation of 3D diffusion tensor fields. In: SPIE visualization and data analysis 2012, vol 8294, pp 1–8Google Scholar
  21. Bi C, Yuan Y, Zhang R, Xiang Y, Wang Y, Zhang J (2017) A dynamic mode decom- position based edge detection method for art images. IEEE Photon J 9(6):1–13Google Scholar
  22. Bi C, Yuan Y, Zhang J, Shi Y, Xiang Y, Wang Y, Zhang R (2018) Dynamic mode decomposition based video shot detection. IEEE Access 6:21397–21407Google Scholar
  23. Bi C, Fu B, Chen J, Zhao Y, Yang L, Duan Y, Shi Y (2019) Machine learning based fast multi-layer liquefaction disaster assessment. World Wide Web: internet and web information systems. pp 1–16.
  24. Burgkart R, Westermann R, Georgii J, Dick C (2009) Stress tensor field visualization for implant planning in orthopedics. IEEE Trans Vis Comput Graph 15(6):1399–1406Google Scholar
  25. Chefd’hotel C, Tschumperlé D, Deriche R, Faugeras O (2004) Regularizing flows for constrained matrix-valued images. J Math Imaging Vis 20(1–2):147–162MathSciNetzbMATHGoogle Scholar
  26. Chen Y, Cohen J, Krolik J (2007) Similarity-guided streamline placement with error evaluation. IEEE Trans Vis Comput Graph 13(6):1448–1455Google Scholar
  27. De Leeuw WC, Van Wijk JJ (1993)A probe for local flow field visualization. In: Proceedings of the 4th conference on visualization, pp 39–45Google Scholar
  28. Ebert D, Rheingans P (2000) Volume illustration: non-photorealistic rendering of volume data. In: Proceedings of IEEE visualization 2000, pp 195–202Google Scholar
  29. Ebert D, Shaw C (2001) Minimally immersive flow visualization. IEEE Trans Vis Comput Graph 7(4):343–350Google Scholar
  30. Engelke W, Lawonn K, Preim B, Hotz I (2019) Autonomous particles for interactive flow visualization. Comput Graph Forum 0(0), 1–12Google Scholar
  31. Ennis D, Kindlmann G, Heim P, Rodriguez I, Wen H, McVeigh E (2004) Visualization of high-resolution myocardial strain and diffusion tensors using superquadric glyphs. In: Proceedings of the 12th annual meeting of international society for magnetic resonance in medicine (ISMRM), p. 1Google Scholar
  32. Feng L, Hotz I, Hamann B, Joy K (2008) Anisotropic noise samples. IEEE Trans Vis Comput Graph 14(2):342–354Google Scholar
  33. Fillard P, Arsigny V, Pennec X, Ayache N (2006) Clinical DT-MRI estimation, smooth- ing and fiber tracking with Log-Euclidean metrics. In: Proceedings of the 3rd IEEE international symposium on biomedical imaging, pp. 786–789Google Scholar
  34. Fletcher PT, Joshi SC (2004) Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors. In: Proceedings of computer vision and mathematical methods in medical and biomedical image analysis, springer lecture notes in computer science, vol 3117, pp 87–98Google Scholar
  35. Fletcher PT, Joshi S (2007) Riemannian geometry for the statistical analysis of diffusion tensor data. Sig Process 87(2):250–262zbMATHGoogle Scholar
  36. Fu F, Abukhdeir N (2015) A topologically-informed hyperstreamline seeding method for alignment tensor fields. IEEE Trans Vis Comput Graph 21(3):413–419Google Scholar
  37. Fujishiro I, Chen L, Takeshima Y, Nakamura H, Suzuki Y (2002) Parallel visualization of gigabyte datasets in GeoFEM. Concurr Comput Pract Exp 14(6–7):521–530zbMATHGoogle Scholar
  38. Gerrits T, Rssl C, Theisel H (2017) Glyphs for general second-order 2D and 3D tensors. IEEE Trans Vis Comput Graph 23(1):980–989Google Scholar
  39. Gleicher M, Albers D, Walker R, Jusrfi I, Hansen C, Roberts J (2011) Visual comparison for information visualization. Inf Vis 10(4):289–309Google Scholar
  40. Golub GH, Loan CFV (eds) (1996) Matrix computations. Johns Hopkins University Press, MarylandzbMATHGoogle Scholar
  41. Haber R (1990) Visualization techniques for engineering mechanics. Comput Syst Eng 1(1):37–50Google Scholar
  42. Hagen H, Hahmann S, Schreiber T, Nakajima Y, Wordenweber B, Hollemann- Grundstedt P (1992) Surface interrogation algorithms. IEEE Comput Graph Appl 12(5):53–60Google Scholar
  43. Hasan K, Parker D, Alexander A (2001) Comparison of gradient encoding schemes for diffusion-tensor MRI. J Magn Reson Imaging 13(5):769–780Google Scholar
  44. Hashash Y, Yao J, Wotring D (2003) Glyph and hyperstreamline representation of stress and strain tensors and material constitutive response. Int J Numer Anal Methods Geomech 27:603–626zbMATHGoogle Scholar
  45. Hesselink L, Levy Y, Lavin Y (1997) The topology of symmetric, second-order 3D tensor fields. IEEE Trans Vis Comput Graph 3(1):1–11Google Scholar
  46. Hotz I, Feng L, Hagen H, Hamann B, Joy KI, Jeremic B (2004) Physically based methods for tensor field visualization. IEEE Vis 2004:123–130Google Scholar
  47. Hotz I, Feng L, Hagen H, Hamann B, Joy K (2006) Tensor field visualization using a metric interpretation. In: Visualization and processing of tensor fields, pp 269–280Google Scholar
  48. Hotz I, Sreevalsan-Nair J, Hamann B (2010) Tensor field reconstruction based on eigen- vector and eigenvalue interpolation. In: Scientific visualization: advanced concepts, pp 110–123Google Scholar
  49. Hsu E (2001) Generalized line integral convolution rendering of diffusion tensor fields. In: Proceedings of international society of magnetic resonance in medicine (ISMRM), p 790Google Scholar
  50. Ikits M, Brederson JD, Hansen CD, Johnson CR (2003) A constraint-based technique for haptic volume exploration. IEEE Vis 2003:263–269Google Scholar
  51. Ito S, Okuda H: HPC-MW (2007) A problem solving environment for developing parallel FEM applications. In: Proceedings of the 8th international conference on applied parallel computing: state of the art in scientific computing, springer lecture notes in computer science, vol 4699, pp 694–702Google Scholar
  52. Jeremic B, Scheuermann G, Frey J, Yang Z, Hamann B, Joy K, Hagen H (2002) Tensor visualizations in computational geomechanics. Int J Numer Anal Methods Geomech 26(10), 925–944Google Scholar
  53. Jobard B, Lefer W (1997) Creating evenly-spaced streamlines of arbitrary density. In: Visualization in scientific computing, pp 43–56Google Scholar
  54. Jones D (2004) The effect of gradient sampling schemes on measures derived from diffusion tensor MRI: a monte carlo study. Magn Reson Med 51(4):807–815Google Scholar
  55. Jones D, Griffin L, Alexander D, Catani M, Horsfield M, Howard R, Williams S (2002) Spatial normalization and averaging of diffusion tensor MRI data sets. Neuroimage 17(2):592–617Google Scholar
  56. Kindlmann GL (2004) Superquadric tensor glyphs. In: Proceedings of IEEE TCVG symposium on visualization 2004, pp 147–154Google Scholar
  57. Kindlmann GL (2004) Visualization and analysis of diffusion tensor fields. Ph.D. thesisGoogle Scholar
  58. Kindlmann GL, Weinstein DM (1999) Hue-balls and lit-tensors for direct volume rendering of diffusion tensor fields. IEEE Visualization 1999:183–189Google Scholar
  59. Kindlmann GL, Westin C (2006) Diffusion tensor visualization with glyph packing. IEEE Trans Vis Comput Graph 12(5):1329–1336Google Scholar
  60. Kindlmann GL, Weinstein D, Hart D (2000) Strategies for direct volume rendering of diffusion tensor fields. IEEE Trans Vis Comput Graph 6(2):124–138Google Scholar
  61. Kindlmann GL, Tricoche X, Westin C (2006) Anisotropy creases delineate white matter structure in diffusion tensor MRI. In: Proceedings of 9th international conference on medical image computing and computer-assisted intervention, springer lecture notes in computer science, vol 4190, pp 126–133Google Scholar
  62. Kindlmann GL, Estepar RSJ, Niethammer M, Haker S, Westin CF (2007) Geodesic-Loxodromes for diffusion tensor interpolation and difference measurement. In: Proceedings of medical image computing and computer-assisted intervention, springer lecture notes in computer science, vol 4791, pp 1–9Google Scholar
  63. Kindlmann GL, Tricoche X, Westin C (2007b) Delineating white matter structure in diffusion tensor MRI with anisotropy creases. Med Image Anal 11(5):492–502Google Scholar
  64. Kirby R, Marmanis H, Laidlaw D (1999) Visualizing multivalued data from 2D incompressible flows using concepts from painting. In: Proceedings of IEEE visualization, pp 333–340Google Scholar
  65. Kratz A, Meyer B, Hotz I (2011) A Visual approach to analysis of stress tensor fields. Sci Vis Interact Featur Metaphors Dagstuhl Follow-Ups 2:188–211Google Scholar
  66. Kratz A, Auer C, Stommel M, Hotz I (2013) Visualization and analysis of second-order tensors: moving beyond the symmetric positive-definite case. Comput Graph Forum 32(1):49–74Google Scholar
  67. Kratz A, Schöneich M, Zobel V, Burgeth B, Scheuermann G, Hotz I, Stommel M (2014) Tensor visualization driven mechanical component design. Proc IEEE Pac Vis Symp 2014:145–152Google Scholar
  68. Kubicki M, Park H, Westin C, Nestor P, Mulkern R, Maier S, Niznikiewicz M, Connor E, Levitt J, Frumin M (2005) DTI and MTR abnormalities in schizophrenia: analysis of white matter integrity. Neuroimage 26(4):1109–1118Google Scholar
  69. Lei N, Zheng X, Jiang J, Lin YY, Gu DX (2017) Quadrilateral and hexahedral mesh generation based on surface foliation theory. Comput Methods Appl Mech Eng 316:758–781MathSciNetGoogle Scholar
  70. Lenglet C, Rousson M, Deriche R, Faugeras O (2006) Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing. J Math Imaging Vis 25(3):423–444MathSciNetGoogle Scholar
  71. Max N, Crawfis R, Grant C (1994) Visualizing 3D velocity fields near contour surfaces. In: Proceedings of IEEE visualization, pp 248–255Google Scholar
  72. McGraw T, Vemuri BC, Chen Y, Rao M, Mareci T (2004) DT-MRI denoising and neuronal fiber tracking. Med Image Anal 8(2):95–111Google Scholar
  73. McGraw T, Vemuri BC, Ozarslan E, Chen Y, Mareci T (2009) Variational denoising of diffusion-weighted MRI. Inverse Probl Imaging 3(4):625–648MathSciNetzbMATHGoogle Scholar
  74. McGraw T, Kawai T, Yassine I, Zhu L (2011) Visualizing high-order symmetric tensor field structure with differential operators. J Appl Math 2011:1–27MathSciNetzbMATHGoogle Scholar
  75. Mebarki A, Alliez P, Devillers O (2005) Farthest point seeding for efficient placement of streamlines. In: Proceedings of IEEE visualization, pp 479–486Google Scholar
  76. Merino-Caviedes S, Martin-Fernandez M (2008) A general interpolation method for symmetric second-rank tensors in two dimensions. In: Proceedings of the 5th IEEE international symposium on biomedical imaging, pp 931–934Google Scholar
  77. Meuschke M, Voß S, Beuing O, Preim B, Lawonn K (2017) Glyph-based comparative stress tensor visualization in cerebral aneurysms. Comput Graph Forum 36(3):99–108Google Scholar
  78. Mishra A, Lu Y, Meng J, Anderson AW, Ding Z (2006) Unified framework for anisotropic interpolation and smoothing of diffusion tensor images. Neuro Image 31(4):1525–1535Google Scholar
  79. Moore JG, Schorn SA, Moore J (1995) Methods of classical mechanics applied to turbulence stresses in a tip leakage vortex. In: Proceedings of international gas turbine and aeroengine congress & exposition, pp 1–11Google Scholar
  80. Mori S, Crain BJ, Chacko VP, Zijl PCV (1999) Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Ann Neurol 45(2):265–269Google Scholar
  81. Muraki S, Fujishiro I, Suzuki Y, Takeshima Y (2006) Diffusion-based tractography: visualizing dense white matter connectivity from 3D tensor fields. Proc Vol Graph 2006:119–126Google Scholar
  82. Neeman A, Jeremic B, Pang A (2005) Visualizing tensor fields in geomechanics. In: Proceedings of IEEE visualization, pp 35–42Google Scholar
  83. Obermaier H, Billen MI, Hagen H, Hering-Bertram M (2011) Interactive visualization of scattered moment tensor data. In: Proceedings of SPIE visualization and data analysis 2011, vol 7868, 78680IGoogle Scholar
  84. Ogawa Y, Fujishiro I, Suzuki Y, Takeshima Y (2009) Designing 6DOF haptic transfer functions for effective exploration of 3D diffusion tensor fields. In: Proceedings of world haptics conference, pp 470–475Google Scholar
  85. Oster T, Rössl C, Theisel H (2018) Core lines in 3d second-order tensor fields. Comput Graph Forum 37(3):327–337Google Scholar
  86. Ozarslan E, Marecl T (2003) Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn Reson Med 50(5):955–965Google Scholar
  87. Pajevic S, Aldroubi A, Basser PJ (2002) A continuous tensor field approximation of dis- crete DT-MRI data for extracting microstructural and architectural features of tissue. J Magn Reson 154(1):85–100Google Scholar
  88. Palacios J, Yeh H, Wang W, Zhang Y, Laramee RS, Sharma R, Schultz T, Zhang E (2016) Feature surfaces in symmetric tensor fields based on eigenvalue manifold. IEEE Trans Vis Comput Graph 22(3):1248–1260Google Scholar
  89. Palke D, Lin Z, Chen G, Yeh H, Vincent P, Laramee R, Zhang E (2011) Asym- metric tensor field visualization for surfaces. IEEE Trans Vis Comput Graph 17(12):1979–1988Google Scholar
  90. Pasternak O, Sochen N, Basser PJ (2010) The effect of metric selection on the anal- ysis of diffusion tensor MRI data. NeuroImage 49(3):2190–2204Google Scholar
  91. Peikert R, Sadlo F (2008) Height ridge computation and filtering for visualization. Proc IEEE Pac Vis Symp 2008:119–126zbMATHGoogle Scholar
  92. Pierpaoli C, Basser PJ (2000) Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med 36(6):893–906Google Scholar
  93. Raith F, Blecha C, Nagel T, Parisio F, Kolditz O, Gunther F, Stommel M, Scheuermann G (2019) Tensor field visualization using fiber surfaces of invariant space. IEEE Trans Vis Comput Graph 25(1):1122–1131Google Scholar
  94. Ren L, Zhang L, Wang L, Tao F, Chai X (2017) Cloud manufacturing: key characteristics and applications. Int J Comput Integr Manuf 30(6):501–515Google Scholar
  95. Ren L, Cheng X, Wang X, Cui J, Zhang L (2019) Multi-scale dense gate recurrent unit networks for bearing remaining useful life prediction. Fut Gener Comput Syst 94:601–609. Google Scholar
  96. Savadjiev P, Kindlmann GL, Bouix S, Shenton ME, Westin C (2009) Local white matter geometry indices from diffusion tensor gradients. In: Proceedings of medical image computing and computer-assisted intervention, springer lecture notes in computer science, vol 5761, pp 345–352Google Scholar
  97. Schultz T (2011) Topological features in 2D symmetric higher-order tensor fields. Comput Graph Forum 30(3):841–850Google Scholar
  98. Schultz T, Kindlmann GL (2010a) A maximum enhancing higher-order tensor glyph. Comput Graph Forum 29(3):1143–1152Google Scholar
  99. Schultz T, Kindlmann GL (2010b) Superquadric glyphs for symmetric second-order tensors. IEEE Trans Vis Comput Graph 16(6):1595–1604Google Scholar
  100. Schultz T, Seidel HP (2008) Estimating crossing fibers: a tensor decomposition approach. IEEE Trans Vis Comput Graph 14(6):1635–1642Google Scholar
  101. Schultz T, Schlaffke L, Schölkopf B, Schmidt-Wilcke T (2013) HiFiVE: a Hilbert space embedding of fiber variability estimates for uncertainty modeling and visualization. Comput Graph Forum 32(3):121–130Google Scholar
  102. Schultz T, Vilanova A (2018) Diffusion MRI visualization. NMR Biomed 0(0): e3902Google Scholar
  103. Schultz T, Theisel H, Seidel HP (2007) Topological visualization of brain diffusion MRI data. IEEE Trans Vis Comput Graph 13(6):1496–1503Google Scholar
  104. Schultz T, Theisel H, Seidel HP (2010) Crease surfaces: from theory to extraction and application to diffusion tensor MRI. IEEE Trans Vis Comput Graph 16(1):109–119Google Scholar
  105. Seltzer N, Kindlmann G (2016) Glyphs for asymmetric second-order 2D tensors. In: Proceedings of IEEE VGTC conference on visualization, pp 141–150Google Scholar
  106. Sepasian N, Ten Thije Boonkkamp J, Vilanova A (2015) Diffusion tensor imaging: brain pathway reconstruction. Nieuw Archief voor Wiskunde 5/16(4): 259–265Google Scholar
  107. Sigfridsson A, Ebbers T, Heiberg E, Wigström L (2002) Tensor field visualization using adaptive filtering of noise fields combined with glyph rendering. In: Proceedings of IEEE visualization, pp 371–378Google Scholar
  108. Slavin V, Pelcovits R, Loriot G, Callan-Jones A, Laidlaw D (2006) Techniques for the visualization of topological defect behavior in nematic liquid crystals. IEEE Trans Vis Comput Graph 12(5):1323–1328Google Scholar
  109. Sreevalsan-Nair J, Auer C, Hamann B, Hotz I (2011) Eigenvector-based interpolation and segmentation of 2D tensor fields. In: Topological methods in data analysis and visualization, mathematics and visualization, pp 139–150Google Scholar
  110. Theisei H, Weinkauf T, Hege H, Seidei H (2003) Saddle connectors? An approach to visualizing the topological skeleton of complex 3D vector fields. In: Proceedings of IEEE visualization, pp 325–232Google Scholar
  111. Tricoche X, Kindlmann GL, Westin C (2008) Invariant crease lines for topological and structural analysis of tensor fields. IEEE Trans Vis Comput Graph 14(6):1627–1634Google Scholar
  112. Turk G, Banks D (1996) Image-guided streamline placement. In: Proceedings of the 23rd annual conference on computer graphics and interactive techniques, pp 453–460Google Scholar
  113. Verma V, Kao D, Pang A (2000) Flow-guided streamline seeding strategy. In: Proceedings of IEEE visualization, pp 163–170Google Scholar
  114. Wang B, Hotz I (2017) Robustness for 2D symmetric tensor field topology. In: Proceedings of modeling, analysis, and visualization of anisotropy, pp 3–27Google Scholar
  115. Weickert J, Welk M (2006) Tensor field interpolation with PDEs. In: Visualization and processing of tensor fields, mathematics and visualization, Springer, pp 315–325Google Scholar
  116. Weiss K, Lindstrom P (2016) Adaptive multilinear tensor product wavelets. IEEE Trans Vis Comput Graph 22(1):985–994Google Scholar
  117. Weisstein EW (2003) CRC Concise encyclopedia of mathematics. CRC Press, Boca Raton, p 1894zbMATHGoogle Scholar
  118. Westin CF, Maler SE, Khidhir B, Everett P, Jolesz FA, Kikinis R (1999) Image processing for diffusion tensor magnetic resonance imaging. In: Proceedings of the 2nd conference on medical image computing and computer-assisted intervention (MICCAI), pp 441–452Google Scholar
  119. Westin CF, Peled S, Gudbjartsson H, Kikinis R, Jolesz FA (1997) Geometrical diffusion measures for MRI from tensor basis analysis. In: Proceedings of international society for magnetic resonance in medicine, p 1742Google Scholar
  120. Westin CF, Maier SE, Mamata H, Nabavi A, Jolesz FA, Kikinis R (2002) Processing and visualization for diffusion tensor MRI. Med Image Anal 6(2):93–108Google Scholar
  121. Wu K, Liu Z, Zhang S, Moorhead R (2010) Topology-aware evenly spaced streamline placement. IEEE Trans Vis Comput Graph 16(5):791–801Google Scholar
  122. Xu K, Gao X, Chen G (2018) Hexahedral mesh quality improvement via edge-angle optimization. Comput Graph 70:17–27Google Scholar
  123. Yang L, Wang B, Zhang R, Zhou H, Wang R (2018) Analysis on location accuracy for the binocular stereo vision system. IEEE Photon J 10(1): 1–13Google Scholar
  124. Yassine I, McGraw T (2009) 4th order diffusion tensor interpolation with divergence and curl constrained bézier patches. In: Proceedings of the 6th IEEE international symposium on biomedical imaging, pp 634–637Google Scholar
  125. Yassine I, McGraw T (2008) A subdivision approach to tensor field interpolation. In: Proceedings of workshop on computational diffusion MRI, pp 117–124Google Scholar
  126. Ye X, Kao D, Pang A (2005) Strategy for seeding 3D streamlines. In: Proceedings of IEEE visualization, pp 471–478Google Scholar
  127. Yusoff YA, Mohamad F, Sunar MS, Selamat A (2016) Flow visualization techniques: a review. In: Proceedings of international conference on industrial, engineering and other applications of applied intelligent systems, pp 527–538Google Scholar
  128. Zhang C, Caan MWA, Höllt T, Eisemann E, Vilanova A (2017) Overview + detail visualization for ensembles of diffusion tensors. Comput Graph Forum 36(3):121–132Google Scholar
  129. Zhang E, Yeh H, Lin Z, Laramee R (2009) Asymmetric tensor analysis for flow visualization. IEEE Trans Vis Comput Graph 15(1):106–122Google Scholar
  130. Zhang C, Schultz T, Lawonn K, Eisemann E, Vilanova A (2016) Glyph-based comparative visualization for diffusion tensor fields. IEEE Trans Vis Comput Graph 22(1):797–806Google Scholar
  131. Zheng, X., Pang, A.: HyperLIC. In: IEEE Visualization 2003, pp. 249 - 256 (2003)Google Scholar
  132. Zheng X, Pang A (2004) Topological lines in 3D tensor fields. IEEE Vis 2004:313–320Google Scholar
  133. Zheng X, Parlett B, Pang A (2005a) Topological structures of 3D tensor fields. IEEE Vis 2005:551–558Google Scholar
  134. Zheng X, Parlett BN, Pang A (2005b) Topological lines in 3D tensor fields and discriminant Hessian factorization. IEEE Trans Vis Comput Graph 11(4):395–407Google Scholar
  135. Zobel V, Scheuermann G (2018) Extremal curves and surfaces in symmetric tensor fields. Vis Comput 34(10):1427–1442Google Scholar

Copyright information

© The Visualization Society of Japan 2019

Authors and Affiliations

  1. 1.College of Intelligence and ComputingTianjin UniversityTianjinChina
  2. 2.School of Mechanical EngineeringTianjin University of TechnologyTianjinChina
  3. 3.Institute of Agricultural Resources and Regional PlanningChinese Academy of Agricultural SciencesBeijingChina

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