## Spherical Fast Multiscale Approximation by Locally Compact Orthogonal Wavelets

• Using a stereographical projection to the plane we construct an O(N log(N)) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.

Author: Frank Bauer, Martin Gutting urn:nbn:de:hbz:386-kluedo-16127 Schriften zur Funktionalanalysis und Geomathematik (44) Preprint English 2009 2009 Technische Universität Kaiserslautern 2009/09/17 Scattered-Data-Interpolation ; Wavelet ; Wavelet-Analyse ; Wavelet-Transformation Final paper available at http://10.1007/s13137-011-0015-0 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 42-XX FOURIER ANALYSIS / 42Cxx Nontrigonometric harmonic analysis / 42C40 Wavelets and other special systems 43-XX ABSTRACT HARMONIC ANALYSIS (For other analysis on topological and Lie groups, see 22Exx) / 43Axx Abstract harmonic analysis / 43A90 Spherical functions [See also 22E45, 22E46, 33C55] 65-XX NUMERICAL ANALYSIS / 65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx) / 65D10 Smoothing, curve fitting 65-XX NUMERICAL ANALYSIS / 65Txx Numerical methods in Fourier analysis / 65T60 Wavelets 86-XX GEOPHYSICS [See also 76U05, 76V05] / 86-08 Computational methods Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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