Abstract
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 L2 spaces. This map in turn leads to equivalence between the continuous wavelet transform formalisms on the plane and on the sphere. More precisely, any plane wavelet may be lifted, by inverse stereographic projection, to a wavelet on the sphere. In this work we apply this procedure to orthogonal compactly supported wavelet bases in the plane, and we get continuous, locally supported orthogonal wavelet bases on the sphere. As applications, we give three examples. In the first two examples, we perform a singularity detection, including one where other existing constructions of spherical wavelet bases fail. In the third example, we show the importance of the local support, by comparing our construction with the one based on kernels of spherical harmonics.
Highlights
Two-dimensional wavelets are a standard tool in image processing, under the two concurrent approaches, the Discrete Wavelet Transform DWT, based on the concept of multiresolution analysis, and the Continuous Wavelet Transform CWT
Η3 1 cos θ, Let p : S 2 → R2 denote the stereographic projection from the North Pole N 0, 0, 2 and let ν : S 2 → R, : R2 → R be defined as νη 2 − η3
The corresponding spherical wavelets obtained here will have a support that almost vanishes when they are in the neighborhood of the North Pole, which corresponds to regions far away from the origin in the plane
Summary
Two-dimensional wavelets are a standard tool in image processing, under the two concurrent approaches, the Discrete Wavelet Transform DWT , based on the concept of multiresolution analysis, and the Continuous Wavelet Transform CWT. This case brings us back to the topic of this paper It has been shown in 4 that the reconstruction of the orthographic i.e., vertical projection from a parabolic mirror can be computed as the inverse stereographic projection from the image plane onto the unit sphere—which is precisely the tool we are going to use in the sequel for designing wavelets on the sphere. The method we propose consists in lifting wavelets from the tangent plane to the sphere by inverse stereographic projection It yields simultaneously smoothness, orthogonality, local support, vanishing moments. We aim at detecting and quantizing local singularities in data, at small scales This is, from the practical point of view, the main purpose of wavelet analysis.
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