Constructing Wavelet Frames and Orthogonal Wavelet Bases on the Sphere

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Constructing orthogonal wavelet bases on the sphere

The stereographic projection determines a bijection between the two-sphere, minus the North Pole, and the tangent plane at the South Pole. This correspondence induces a unitary map between the corresponding $L^2$ spaces. Using this map, any plane wavelet may be lifted to a wavelet on the sphere. In this work we quickly review some existing constructions of spherical wavelets, then we apply the new procedure to orthogonal compactly supported wavelet bases in the plane and we get continuous, locally supported orthogonal wavelet bases on the sphere. As an example, we perform a singularity detection, where the other constructions of spherical wavelet bases fail