Locally Supported Orthogonal Wavelet Bases on the Sphere via Stereographic Projection

• Published in: Mathematical Problems in Engineering
• Main Authors: Daniela Roşca, Jean-Pierre Antoine
• Format: Article in Journal/Newspaper
• Language: English
• Published: Mathematical Problems in Engineering 2009
• Subjects: North Pole; South Pole; South pole
• Online Access: https://doi.org/10.1155/2009/124904

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.