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Locally supported rational spline wavelets on a sphere

Author: Daniela Rosca
Journal: Math. Comp. 74 (2005), 1803-1829
MSC (2000): Primary 42C40, 41A63; Secondary 41A15, 65D07, 41A17
Published electronically: March 14, 2005
MathSciNet review: 2164098
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Abstract: In this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations, giving explicit expressions of these wavelets. Our wavelets have small support, a fact which is very important in working with large amounts of data, since the algorithms for decomposition, compression and reconstruction deal with sparse matrices. We also give a quasi-interpolant associated to a given triangulation and study the approximation error. Some numerical examples are given to illustrate the efficiency of our wavelets.

References [Enhancements On Off] (What's this?)

  • 1. G. P. Bonneau, Optimal Triangular Haar Bases for Spherical Data, in IEEE Visualization '99, San Francisco, CA, 1999.
  • 2. A. Cohen, L. M. Echeverry and Q. Sun, Finite Elements Wavelets, Report, University Pierre et Marie Curie, Paris, 2000.
  • 3. S. Dahlke, W. Dahmen and I. Weinreich, Multiresolution Analysis and Wavelets on $\mathbb{S}^{2}$ and $\mathbb{S}^{3},$ Numer. Funct. Anal. Optim. 16 (1995), 19-41. MR 1322896 (96a:42044)
  • 4. M. Floater and E. Quak, Piecewise Linear Prewavelets on Arbitrary Triangulations, Numer. Math. 82 (1999), 221-252.MR 1685460 (2000a:42053)
  • 5. M. Floater and E. Quak, A Semi-prewavelet Approach to Piecewise Linear Prewavelets on Triangulation, Approximation Theory IX (C.K. Chui and L.L. Schumaker, eds.), vol. 2, Vanderbilt University Press, 1998, pp. 63-70. MR 1743034
  • 6. M. Floater and E. Quak, Linear Independence and Stability of Piecewise Linear Prewavelets on Arbitrary Triangulations, SIAM J. Numer. Anal. 38 2000, no. 1, 58-79. MR 1770342 (2001e:65237)
  • 7. M. Floater, E. Quak and M. Reimers, Filter Bank Algorithms for Piecewise Linear Prewavelets on Arbitrary Triangulation, J. Comp. Appl. Math. 119 (2000), 185-207. MR 1774217 (2001i:65145)
  • 8. W. Freeden and U. Windheuser, Combined Spherical Harmonics and Wavelet Expansions - A Future Concept in Earth's Gravitational Potential Determination, Appl. Comput. Harm. Anal. 4 (1997), 1-37. MR 1429676 (97j:86004)
  • 9. J. Göttelmann, Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere, Appl. Comput. Harm. Anal. 7 (1999), 1-33.MR 1699606 (2000j:42051)
  • 10. M. Lounsbery, T. DeRose and J. Warren, Multiresolution Analysis for Surfaces of Arbitrary Topological Type, ACM Transactions on Graphics 16 (1997), no. 1, 34-73.
  • 11. F. Narcowich and J. D. Ward, Wavelets Associated with Periodic Basis Functions, Appl. Comput. Harm. Anal. 3 (1996), 40-56. MR 1374394 (96j:42022)
  • 12. F. Narcowich and J. D. Ward, Nonstationary Wavelets on the $m$-Sphere for scattered Data, Appl. Comput. Harm. Anal. 3 (1996), 324-336.MR 1420501 (97h:42020)
  • 13. G. Nielson, I. Jung and J. Sung, Haar Wavelets over Triangular Domains with Applications to Multiresolution Models for Flow over a Sphere, IEEE Visualization '97, IEEE 1997, pp. 143-150.
  • 14. P. Oswald, Multilevel Finite Element Approximation: Theory and Applications, B. G. Teubner, Stuttgart, 1994. MR 1312165 (95k:65110)
  • 15. D. Potts and M. Tasche, Interpolatory Wavelets on the Sphere, Approximation Theory VIII (C. K. Chui and L. L. Schumaker, eds.), vol. 2, World Scientific, Singapore, 1995, pp. 335-342.MR 1471800 (98e:42040)
  • 16. D. Potts, G. Steidl and M. Tasche, Kernels of Spherical Harmonics and Spherical Frames, Advanced Topics in Multivariate Approximations (F. Fontanella, K. Jetter, P-J. Laurent, eds.), World Scientific, Singapore, 1996. MR 1661417 (99g:42042)
  • 17. D. Rosca, Haar wavelets on spherical triangulations, in Advances in Multiresolution for Geometric Modelling (N. A. Dodgson, M. S. Floater, M. A. Sabin, eds.), Springer Verlag, 2005, pp. 405-417.
  • 18. D. Rosca, Optimal Haar wavelets on spherical triangulations, Pure Mathematics and Applications, Budapest (to appear).
  • 19. D. Rosca, Piecewise Constant Wavelets Defined on Closed Surfaces, J. Comput. Anal. Appl. (to appear).
  • 20. H. Schaeben, D. Potts and J. Prestin, Spherical Wavelets with Application in Preferred Crystallographic Orientation, IAMG' 2001, Cancun, 2001.
  • 21. P. Schröder and W. Sweldens, Spherical Wavelets: Efficiently Representing Functions on the Sphere, Computer Graphics Proceedings (SIGGRAPH 95), 1995, pp. 161-172.
  • 22. P. Schröder and W. Sweldens, Spherical Wavelets: Texture Processing, preprint.
  • 23. L. Schumaker and C. Traas, Fitting Scattered Data on Sphere-like Surface Using Tensor Products of Trigonometric and Polynomial Splines, Numer. Math. 60 (1991), 133-144. MR 1131503 (92j:65012)
  • 24. I. Weinreich, A Construction of $C^{1}$-Wavelets on the Two-Dimensional Sphere, Appl. Comput. Harm. Anal. 10 (2001), 1-26.MR 1808197 (2001k:42048)

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Additional Information

Daniela Rosca
Affiliation: Institute of Mathematics, University of Lübeck, Wallstrasse 40, Lübeck 23560, Germany
Address at time of publication: Department of Mathematics, Technical University of Cluj-Napoca, str. Daicoviciu 15, Cluj-Napoca 400020, Romania

Keywords: Wavelets, multivariate approximation, interpolation
Received by editor(s): October 3, 2003
Received by editor(s) in revised form: April 12, 2004
Published electronically: March 14, 2005
Additional Notes: Research supported by the EU Research Training Network MINGLE, HPRN-CT-1999-00117.
Article copyright: © Copyright 2005 American Mathematical Society

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