Abstract
Using a stereographical projection to the plane we construct an \({\mathcal{O}(N\log(N))}\) 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 min 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.
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Bauer, F., Gutting, M. Spherical fast multiscale approximation by locally compact orthogonal wavelets. Int J Geomath 2, 69–85 (2011). https://doi.org/10.1007/s13137-011-0015-0
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DOI: https://doi.org/10.1007/s13137-011-0015-0
Keywords
- Orthogonal wavelets on the sphere
- Spherical multiresolution analysis
- Fast wavelet transform
- Scattered data
- Approximation of spherical functions