Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere
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This paper constructs a semi-discrete tight frame of tensor needlets associated with a quadrature rule for tangent vector fields on the unit sphere S^2 of R^3 --- tensor needlets. The proposed tight tensor needlets provide a multiscale representation of any square integrable tangent vector field on S^2, which leads to a multiresolution analysis (MRA) for the field. From the MRA, we develop fast algorithms for tensor needlet transforms, including the decomposition and reconstruction of the needlet coefficients between levels, via a set of filter banks and scalar FFTs. The fast tensor needlet transforms have near linear computational cost proportional to N√(N) for N evaluation points or coefficients. Numerical examples for the simulated and real data demonstrate the efficiency of the proposed algorithm.
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