Abstract
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.
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References
Bishop, R., Goldberg, S.: Tensor Analysis on Manifolds. Dover Publications, New York (1968)
Demmel, J.W., Gilbert, J., Li, X.S.: SuperLU users’ guide. Tech. rep. CSD-97-944, University of California (1997)
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)
Dong, S., timo Bremer, P., Garl, M.: Spectral surface quadrangulation. ACM Trans. Graph. 25, 1057–1066 (2006)
Dragomir, S., Perrone, D.: Harmonic Vector Fields: Variational Principles and Differential Geometry. Elsevier Science Ltd, Oxford (2011)
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)
Ebling, J., Scheuermann, G.: Clifford Fourier transform on vector fields. IEEE Trans. Vis. Comput. Graph. 11(4), 469–479 (2005)
Elcott, S., Schröder, P.: Building your own DEC at home. In: SIGGRAPH ’05: ACM SIGGRAPH 2005 Courses, p. 8. ACM, New York (2005)
Elcott, S., Tong, Y., Kanso, E., Schröder, P., Desbrun, M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 4-es (2007). doi:http://doi.acm.org/10.1145/1189762.1189766
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. Oxford University Press, Oxford (1998)
Fisher, M., Schröder, P.: Design of tangent vector fields. ACM Trans. Graph. 26, 56 (2007)
Flanders, H.: Differential forms with applications to the physical sciences. Dover Publications, Mineola (1989)
Fletcher, C.: Computational Galerkin Methods. Springer Series in Computational Physics. Springer, New York (1984)
Galerkin, B.G.: On electrical circuits for the approximate solution of the laplace equation. Vestnik Inzh. 19, 897–908 (1915)
Heath, M.: Scientific Computing. McGraw-Hill, Boston (2002)
Hirani, A.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology (2003)
Huebner, K., Dewhirst, D., Smith, D., Byrom, T.: The Finite Element Method for Engineers. Wiley India Pvt. Ltd., New York (2008)
Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge/New York (2004)
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)
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. citeseer.ist.psu.edu/article/lehoucq97arpack.html
Loan, G., Golub, G.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)
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)
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)
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)
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)
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 http://dl.acm.org/citation.cfm?id=1281957.1281983
Vallet, B., Lévy, B.: Spectral geometry processing with manifold harmonics. Comput. Graph. Forum (Proceedings Eurographics) 27, 251–260 (2008)
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)
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Wagner, C., Garth, C., Hagen, H. (2012). Harmonic Field Analysis. In: Laidlaw, D., Vilanova, A. (eds) New Developments in the Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27343-8_19
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DOI: https://doi.org/10.1007/978-3-642-27343-8_19
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