Differential Methods for Multi-Dimensional Visual Data Analysis

  • Werner Benger
  • René Heinzl
  • Dietmar Hildenbrand
  • Tino Weinkauf
  • Holger Theisel
  • David Tschumperlé
Reference work entry


Images in scientific visualization are the end-product of data processing. Starting from higher-dimensional datasets, such as scalar-, vector-, tensor- fields given on 2D, 3D, 4D domains, the objective is to reduce this complexity to two-dimensional images comprehensible to the human visual system. Various mathematical fields such as in particular differential geometry, topology (theory of discretized manifolds), differential topology, linear algebra, Geometric Algebra, vectorfield and tensor analysis, and partial differential equations contribute to the data filtering and transformation algorithms used in scientific visualization. The application of differential methods is core to all these fields. The following chapter will provide examples from current research on the application of these mathematical domains to scientific visualization and ultimately generating of images for analysis of multi-dimensional datasets.


Vector Field Fiber Bundle Base Space Tensor Field Geometric Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References and Further Reading

  1. 1.
    The homepage of geomerics ltd.
  2. 2.
    Abłamowicz R, Fauser B (2009) Clifford/ bigebra, a maple package for Clifford (co)algebra computations. Available at\(\text{ \copyright }\) 1996–2009, RA&BF
  3. 3.
    Bayro-Corrochano E, Vallejo R, Arana-Daniel N (2005) Geometric preprocessing, geometric feedforward neural networks and Clifford support vector machines for visual learning. Special issue of Journal Neurocomputing 67:54–105Google Scholar
  4. 4.
    Benger W (2004) Visualization of general relativistic tensor fields via a fiber bundle data model. PhD thesis, FU BerlinMATHGoogle Scholar
  5. 5.
    Benger W (2008) Colliding galaxies, rotating neutron stars and merging black holes – visualising high dimensional data sets on arbitrary meshes. N J Phys 10.
  6. 6.
    Benger W, Ritter, M, Acharya S, Roy S, Jijao F (2009) Fiberbundle-based visualization of a stir tank fluid. In WSCG 2009, PlzenGoogle Scholar
  7. 7.
    Bochev P, Hyman M (2006) Principles of compatible discretizations. In: Proceedings of IMA Hot Topics Workshop on Compatible Discretizations. Springer, vol IMA 142, pp 89–120Google Scholar
  8. 8.
    Brouwer L (1912) Zur Invarianz des n-dimensionalen Gebiets. Mathematische Annalen, 1Google Scholar
  9. 9.
    Buchholz S, Hitzer EMS, Tachibana K (2007) Optimal learning rates for Clifford neurons. In: International Conference on Artificial Neural Networks, Porto, Portugal. vol 1, pp 864–873, 9–13Google Scholar
  10. 10.
    Butler DM, Bryson S (1992) Vector bundle classes form a powerful tool for scientific visualization. Comput Phys 6:576–584Google Scholar
  11. 11.
    Butler DM, Pendley MH (1989) A visualization model based on the mathematics of fiber bundles. Comput Phys 3(5):45–51Google Scholar
  12. 12.
    Clifford WK (1882a) Applications of grassmann’s extensive algebra. In: Tucker R (ed) Mathematical Papers. Macmillian, London, pp 266–276Google Scholar
  13. 13.
    Clifford WK (1882b) On the classification of geometric algebras. In: Tucker R (ed) Mathematical Papers. Macmillian, London, pp 397–401Google Scholar
  14. 14.
    Dorst L, Fontijne D, Mann S (2007) Geometric algebra for computer science, an object-oriented approach to geometry. Morgan Kaufman, San MateoGoogle Scholar
  15. 15.
    Ebling J (2005) Clifford fourier transform on vector fields. IEEE Trans Visual Comput Gr 11(4):469–479. IEEE member Scheuermann, GerikCrossRefGoogle Scholar
  16. 16.
    Firby P, Gardiner C (1982) Surface topology, Chap 7. Ellis Horwood, Vector Fields on Surfaces, pp 115–135Google Scholar
  17. 17.
    Fontijne D (2007) Efficient implementation of geometric algebra. PhD thesis, University of AmsterdamGoogle Scholar
  18. 18.
    Fontijne D, Bouma T, Dorst L (2005) Gaigen: a geometric algebra implementation generator.
  19. 19.
    Fontijne D, Dorst L (2003) Modeling 3D euclidean geometry. IEEE Comput Graph Appl 23(2):68–78CrossRefGoogle Scholar
  20. 20.
    Garth C, Tricoche X, Scheuermann G (2004) Tracking of vector field singularities in unstructured 3D time-dependent datasets. In: Proceedings of the IEEE Visualization, pp 329–336Google Scholar
  21. 21.
    Globus A, Levit C, Lasinski T (1991) A tool for visualizing the topology of threedimensional vector fields. In: Proceedings of the IEEE Visualization ’91, pp 33–40Google Scholar
  22. 22.
    Gottlieb DH (1990) Vector fields and classical theorems of topology. Rendiconti del Seminario Matematico e Fisico, Milano, 60Google Scholar
  23. 23.
    Gottlieb DH (1996) All the way with gauss-bonnet and the sociology of mathematics. The American Mathematical Monthly 103(6): 457–469MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Gross P, Kotiuga PR (2004) Electromagnetic theory and computation: a topological approach. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  25. 25.
    Hart J (1999) Using the cw-complex to represent the topological structure of implicit surfaces and solids. In: Implicit Surfaces ’99, Eurographics/SIGGRAPH, pp 107–112. ∼ jch/papers/cw.pdf
  26. 26.
    Hatcher A (2002) Algebraic topology. Cambridge University Press, CambridgeMATHGoogle Scholar
  27. 27.
    Hauser H, Gröller E (2000) Thorough insights by enhanced visualization of flow topology. In: 9th International Symposium on Flow Visualization,
  28. 28.
    Helman J, Hesselink L (1989) Representation and display of vector field topology in fluid flow data sets. IEEE Computer 22(8):27–36CrossRefGoogle Scholar
  29. 29.
    Helman J, Hesselink L (1991) Visualizing vector field topology in fluid flows. IEEE Comput Graph Appl 11:36–46CrossRefGoogle Scholar
  30. 30.
    Hestenes D (1986) New foundations for classical mechanics. Reidel, DordrechtMATHCrossRefGoogle Scholar
  31. 31.
    Hestenes D, Sobczyk G (1984) Clifford algebra to geometric calculus: a unified language for mathematics and physics. Dordrecht, ReidelMATHCrossRefGoogle Scholar
  32. 32.
    Hildenbrand D, Fontijne D, Perwass C, Dorst L (2004) Tutorial geometric algebra and its application to computer graphics. In: Eurographics Conference GrenobleGoogle Scholar
  33. 33.
    Hildenbrand D, Fontijne D, Wang Y, Alexa M, Dorst L (2006) Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra. In Eurographics conference Vienna.Google Scholar
  34. 34.
    Hildenbrand D, Lange H, Stock F, Koch A (2008) Efficient inverse kinematics algorithm based on conformal geometric algebra using reconfigurable hardware. In GRAPP conference MadeiraGoogle Scholar
  35. 35.
    Hildenbrand D, Pitt J (2008) The Gaalop home page.
  36. 36.
    Hocking J, Young G (1961) Topology. Addison-Wesley, Dover, New YorkMATHGoogle Scholar
  37. 37.
    Hunt J (1987) Vorticity and vortex dynamics in complex turbulent flows. Proceedings of CANCAM, Transactions of the Canadian Society for Mechanical Engineering, 11:21Google Scholar
  38. 38.
    Jeong J, Hussain F (1995) On the identification of a vortex. J Fluid Mech 285:69–94MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Kenwright D, Henze C, Levit C (1999) Feature extraction of separation and attachment lines. IEEE Trans Vis Comput Graph 5(2):135–144CrossRefGoogle Scholar
  40. 40.
    Löffelmann H, Doleisch H, Gröller E (1998) Visualizing dynamical systems near critical points. In: Spring Conference on Computer Graphics and its Applications. Budmerice, Slovakia, pp 175–184Google Scholar
  41. 41.
    Mann S, Rockwood A (2002) Computing singularities of 3D vector fields with geometric algebra. In: Proceedings of the IEEE Visualization, pp 283–289Google Scholar
  42. 42.
    Mattiussi C (2001) The geometry of time-stepping. In: Teixeira FL (ed) Geometric methods in computational electromagnetics, PIER 32. EMW, Cambridge, pp 123–149Google Scholar
  43. 43.
    McCormick B, DeFanti T, Brown M (1987) Visualization in scientific computing. Comput Gr 21(6)Google Scholar
  44. 44.
    Naeve A, Rockwood A (2001) Course 53 geometric algebra. In: Siggraph conference Los Angeles.Google Scholar
  45. 45.
    Perwass C (2005) The CLU home page.
  46. 46.
    Perwass C (2009) Geometric algebra with applications in engineering. Springer, BerlinMATHGoogle Scholar
  47. 47.
    Petsche H-J (2009) The Grassmann Bicentennial Conference home page.
  48. 48.
    Pham MT, Tachibana K, Hitzer EMS, Yoshikawa T, Furuhashi T (2008) Classification and clustering of spatial patterns with geometric algebra. In: AGACSE conference Leipzig.Google Scholar
  49. 49.
    Reyes-Lozano L, Medioni G, Bayro-Corrochano E (2007) Registration of 3d points using geometric algebra and tensor voting. J Comput Vis 75(3):351–369CrossRefGoogle Scholar
  50. 50.
    Rosenhahn B, Sommer G (2005) Pose estimation in conformal geometric algebra. J Math Imaging Vis 22:27–70MathSciNetCrossRefGoogle Scholar
  51. 51.
    Stalling D, Steinke T (1996) Visualization of vector fields in quantum chemistry. Technical Report, ZIB Preprint SC-96-01Google Scholar
  52. 52.
    Theisel H, Weinkauf T, Hege H-C, Seidel H-P (2003) Saddle connectors – an approach to visualizing the topological skeleton of complex 3D vector fields. In Proceedings of the IEEE Visualization, pp 225–232Google Scholar
  53. 53.
    Theisel H, Weinkauf T, Hege H-C, Seidel H-P (2004) Grid-independent detection of closed stream lines in 2D vector fields. In Proceedings of the Vision, Modeling and Visualization 2004, November 16–78, USA, pp 421–428, ∼ weinkauf/publications/bibtex/theisel04b.bib
  54. 54.
    Tonti E (1976/1977) The Reason for Analogies between Physical Theories. Appl Math Model 1(1):37–50MathSciNetCrossRefGoogle Scholar
  55. 55.
    Treinish LA (1997) Data explorer data model.
  56. 56.
    Veldhuizen T (1995) Using \(\mathrm{C}++\) template metaprograms. \(\mathrm{C}++\) Report 7(4):36–43. Reprinted in \(\mathrm{C}++\) Gems, ed. Stanley LippmanGoogle Scholar
  57. 57.
    Venkataraman S, Benger W, Long A, Byungil Jeong LR (2006) Visualizing hurricane katrina – large data management, rendering and display challenges. In: GRAPHITE 2006, 29 November–2 December, Kuala Lumpur, MalaysiaGoogle Scholar
  58. 58.
    Weinkauf T (2008) Extraction of topological structures in 2D and 3D vector fields. PhD thesis, University Magdeburg.
  59. 59.
    Weinkauf T, Theisel H, Hege H-C, Seidel H-P (2004) Boundary switch connectors for topological visualization of complex 3D vector fields. In: Data Visualization 2004. Proceedings of the VisSym 2004, May 19–21, Konstanz, Germany, pp 183–192, ∼ weinkauf/publications/bibtex/weinkauf04a.bib
  60. 60.
    Zomorodian AJ (2005) Topology for computing. In: Cambridge Monographs on Applied and Computational MathematicsGoogle Scholar
  61. 61.
    Alvarez L, Guichard F, Lions PL, Morel JM (1993) Axioms and fundamental equations of image processing. Arch Ration Mech Anal 123(3): 199–257MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Aubert G, Kornprobst P (2002) Mathematical problems in image processing: partial differential equations and the calculus of variations, applied mathematical sciences, vol 147. Springer, JanuaryGoogle Scholar
  63. 63.
    Barash D (2002) A fundamental relationship between bilateral filtering, adaptive smoothing and the nonlinear diffusion equation. IEEE Trans Pattern Anal Mach Intell 24(6):844CrossRefGoogle Scholar
  64. 64.
    Becker J, Preusser T, Rumpf M (2000) PDE methods in flow simulation post processing. Comput Vis Sci 3(3):159–167MATHCrossRefGoogle Scholar
  65. 65.
    Bertalmio M, Sapiro G, Caselles V, Ballester C (2000) Image inpainting. ACM SIGGRAPH, Int Conf Comp Gr Interact Tech pp 417–424Google Scholar
  66. 66.
    Black MJ, Sapiro G, Marimont DH, Heeger D (1998) Robust anisotropic diffusion. IEEE Trans Image Process 7(3):421–432CrossRefGoogle Scholar
  67. 67.
    Cabral B, Leedom LC (1993) Imaging vector fields using line integral convolution. SIGGRAPH’93, in Computer Graphics Vol.27, pp 263–272Google Scholar
  68. 68.
    Chan T, Shen J (2001) Non-texture inpaintings by curvature-driven diffusions. J Vis Commun Image Represent 12(4):436–449CrossRefGoogle Scholar
  69. 69.
    Charbonnier P, Blanc-Féraud L, Aubert G, Barlaud M (1997) Deterministic edge-preserving regularization in computed imaging. IEEE Trans Image Process 6(2):298–311CrossRefGoogle Scholar
  70. 70.
    Di Zenzo S (1986) A note on the gradient of a multi-image. Comput Vision Gr Image Process 33:116–125CrossRefGoogle Scholar
  71. 71.
    Kimmel R, Malladi R, Sochen N (2000) Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. Int J Comput Vision 39(2):111–129MATHCrossRefGoogle Scholar
  72. 72.
    Koenderink JJ (1984) The structure of images. Biol Cybern 50:363–370MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Kornprobst P, Deriche R, Aubert G (1997) Non-linear operators in image restoration. In Proceedings of the 1997 Conference on Computer Vision and Pattern Recognition (CVPR ’97) (June 17–19, 1997). CVPR. IEEE Computer Society, Washington, DC, pp 325Google Scholar
  74. 74.
    Lindeberg T (1994) Scale-space theory in computer vision. Kluwer Academic, DordrechtGoogle Scholar
  75. 75.
    Nielsen M, Florack L, Deriche R (1997) Regularization, scale-space and edge detection filters. J Math Imaging Vis 7(4):291–308MathSciNetCrossRefGoogle Scholar
  76. 76.
    Perona P, Malik J (1990) Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 12(7):629–639CrossRefGoogle Scholar
  77. 77.
    Preußer T, Rumpf M (1999) Anisotropic nonlinear diffusion in flow visualization. In Proceedings of the Conference on Visualization ’99: Celebrating Ten Years (San Francisco, California, United States). IEEE Visualization. IEEE Computer Society Press, Los Alamitos, CA, pp 325–332Google Scholar
  78. 78.
    Rudin L, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60:259–268MATHCrossRefGoogle Scholar
  79. 79.
    Sapiro G (2001) Geometric partial differential equations and image analysis. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  80. 80.
    Sapiro G, Ringach DL (1996) Anisotropic diffusion of multi-valued images with applications to color filtering. IEEE Trans Image Process 5(11):1582–1585CrossRefGoogle Scholar
  81. 81.
    Tomasi C, Manduchi R (1998) Bilateral Filtering for Gray and Color Images. In Proceedings of the Sixth international Conference onComputer Vision (January 04–07, 1998). ICCV. IEEE Computer Society, Washington, DC, pp 839Google Scholar
  82. 82.
    Tschumperlé D, Deriche R (2005) Vector-valued image regularization with PDE’s: a common framework for different applications. IEEE Trans Pattern Anal Mach Intell 27(4)Google Scholar
  83. 83.
    Tschumperlé D (2006) Fast anisotropic smoothing of multi-valued images using curvature- reserving PDE’s. Int J Comput Vis 68(1):65–82, ISSN: 0920-5691CrossRefGoogle Scholar
  84. 84.
    Vemuri BC, Chen Y, Rao M, McGraw T, Wang Z, Mareci T (2001) Fiber Tract Mapping from Diffusion Tensor MRI. In Proceedings of the IEEE Workshop on Variational and Level Set Methods (Vlsm’01) (July 13–13, 2001). VLSM. IEEE Computer Society, Washington, DC, pp 81Google Scholar
  85. 85.
    Weickert J (1998) Anisotropic diffusion in image processing. Teubner-Verlag, StuttgartMATHGoogle Scholar
  86. 86.
    Weickert J (1999) Coherence-enhancing diffusion of colour images. Image Vis Comput 17: 199–210CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Werner Benger
    • 1
  • René Heinzl
    • 2
  • Dietmar Hildenbrand
    • 3
  • Tino Weinkauf
    • 4
  • Holger Theisel
    • 5
  • David Tschumperlé
    • 6
  1. 1.Center for Computation and Technology at Lousiana State UniversityBaton RougeUSA
  2. 2.Shenteq s.r.oBratislavaSlovak Republic
  3. 3.University of Technology DarmstadtDarmstadtGermany
  4. 4.New York UniversityNew YorkUSA
  5. 5.Institut fur Simulation und Graphik AG Visual ComputingMagdeburgGermany
  6. 6.GREYC (UMR-CNRS 6072)CAEN CedexFrance

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