Harmonic Field Analysis

  • Christian Wagner
  • Christoph Garth
  • Hans Hagen
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Harmonic analysis techniques are established and successful tools in a variety of application areas, with the Fourier decomposition as one well-known example. In this chapter, we describe recent work on possible approaches to use Harmonic Analysis on fields of arbitrary type to facilitate global feature extraction and visualization. We find that a global approach is hampered by significant computational costs, and thus describe a local framework for harmonic vector field analysis to address this concern. In addition to a description of our approach, we provide a high-level overview of mathematical concepts underlying it and address practical modeling and calculation issues. As a potential application, we demonstrate the definition of empirical features based on local harmonic analysis of vector fields that reduce field data to low dimensional feature sets and offers possibilities for visualization and analysis.


Vector Field Simplicial Complex Global Approach Fourier Decomposition Basis Coefficient 
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.


  1. 1.
    Bishop, R., Goldberg, S.: Tensor Analysis on Manifolds. Dover Publications, New York (1968)Google Scholar
  2. 2.
    Demmel, J.W., Gilbert, J., Li, X.S.: SuperLU users’ guide. Tech. rep. CSD-97-944, University of California (1997)Google Scholar
  3. 3.
    Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: SIGGRAPH ’06: ACM SIGGRAPH 2006 Courses, pp. 39–54. ACM, New York (2006)Google Scholar
  4. 4.
    Dong, S., timo Bremer, P., Garl, M.: Spectral surface quadrangulation. ACM Trans. Graph. 25, 1057–1066 (2006)Google Scholar
  5. 5.
    Dragomir, S., Perrone, D.: Harmonic Vector Fields: Variational Principles and Differential Geometry. Elsevier Science Ltd, Oxford (2011)Google Scholar
  6. 6.
    Ebling, J., Scheuermann, G.: Clifford convolution and pattern matching on vector fields. In: Proceedings of the 14th IEEE Visualization 2003 (VIS’03), p. 26. IEEE Computer Society, Piscataway (2003)Google Scholar
  7. 7.
    Ebling, J., Scheuermann, G.: Clifford Fourier transform on vector fields. IEEE Trans. Vis. Comput. Graph. 11(4), 469–479 (2005)Google Scholar
  8. 8.
    Elcott, S., Schröder, P.: Building your own DEC at home. In: SIGGRAPH ’05: ACM SIGGRAPH 2005 Courses, p. 8. ACM, New York (2005)Google Scholar
  9. 9.
    Elcott, S., Tong, Y., Kanso, E., Schröder, P., Desbrun, M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 4-es (2007). doi: Google Scholar
  10. 10.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. Oxford University Press, Oxford (1998)Google Scholar
  11. 11.
    Fisher, M., Schröder, P.: Design of tangent vector fields. ACM Trans. Graph. 26, 56 (2007)Google Scholar
  12. 12.
    Flanders, H.: Differential forms with applications to the physical sciences. Dover Publications, Mineola (1989)Google Scholar
  13. 13.
    Fletcher, C.: Computational Galerkin Methods. Springer Series in Computational Physics. Springer, New York (1984)Google Scholar
  14. 14.
    Galerkin, B.G.: On electrical circuits for the approximate solution of the laplace equation. Vestnik Inzh. 19, 897–908 (1915)Google Scholar
  15. 15.
    Heath, M.: Scientific Computing. McGraw-Hill, Boston (2002)Google Scholar
  16. 16.
    Hirani, A.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology (2003)Google Scholar
  17. 17.
    Huebner, K., Dewhirst, D., Smith, D., Byrom, T.: The Finite Element Method for Engineers. Wiley India Pvt. Ltd., New York (2008)Google Scholar
  18. 18.
    Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge/New York (2004)Google Scholar
  19. 19.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)Google Scholar
  20. 20.
    Lehoucq, R., Sorensen, D.C., Yang, C.: Arpack users guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. Communication 6(3), 147 (1998). Tech. Rep., SIAM, Philadelphia. Google Scholar
  21. 21.
    Loan, G., Golub, G.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)Google Scholar
  22. 22.
    Reuter, M., Wolter, F., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: Proceedings of the 2005 ACM Symposium on Solid and Physical Modeling, Cambridge, p. 106. ACM, New York (2005)Google Scholar
  23. 23.
    Schlemmer, M., Heringer, M., Morr, F., Hotz, I., Hering-Bertram, M., Garth, C., Kollmann, W., Hamann, B., Hagen, H.: Moment invariants for the analysis of 2D flow fields. IEEE Trans. Vis. Comput. Graph. 13(6), 1743 (2007)Google Scholar
  24. 24.
    Sorensen, D.: Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations. Institute for Computer Applications in Science and Engineering, Hampton. In: Keyes, D.E., Sameh, A., Venkatakrishnan, V. (eds.) Contractor, pp. 1–34. Kluwer, New York (1996)Google Scholar
  25. 25.
    Taubin, G.: A signal processing approach to fair surface design. In: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, pp. 351–358, ACM, New York (1995)Google Scholar
  26. 26.
    Tong, Y., Alliez, P., Cohen-Steiner, D., Desbrun, M.: Designing quadrangulations with discrete harmonic forms. In: Proceedings of the Fourth Eurographics Symposium on Geometry Processing, SGP ’06, pp. 201–210. Eurographics Association, Aire-la-Ville (2006). URL
  27. 27.
    Vallet, B., Lévy, B.: Spectral geometry processing with manifold harmonics. Comput. Graph. Forum (Proceedings Eurographics) 27, 251–260 (2008)Google Scholar
  28. 28.
    Wardetzky, M., Mathur, S., Kälberer, F., Grinspun, E.: Discrete laplace operators: no free lunch. In: SIGGRAPH Asia ’08: ACM SIGGRAPH ASIA 2008 Courses, pp. 1–5. ACM, New York (2008)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.University of KaiserslauternKaiserslauternGermany

Personalised recommendations