Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 124904, 13 pages
Research Article

Locally Supported Orthogonal Wavelet Bases on the Sphere via Stereographic Projection

1Department of Mathematics, Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania
2Institute of Theoretical Physics, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

Received 22 April 2009; Revised 25 July 2009; Accepted 27 July 2009

Academic Editor: Victoria Vampa

Copyright © 2009 Daniela Roşca and Jean-Pierre Antoine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.