Locally supported orthogonal wavelet bases on the sphere via stereographic projection
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. This map in turn leads to equivalence between the continuous wavelet transform fo...
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ftunivlouvain:oai:dial.uclouvain.be:boreal:35163 2024-05-12T08:08:36+00:00 Locally supported orthogonal wavelet bases on the sphere via stereographic projection Rosca, Daniela Antoine, Jean-Pierre UCL - SC/PHYS - Département de physique 2009 http://hdl.handle.net/2078.1/35163 https://doi.org/10.1155/2009/124904 eng eng Hindawi Publishing Corporation boreal:35163 http://hdl.handle.net/2078.1/35163 doi:10.1155/2009/124904 urn:ISSN:1024-123X info:eu-repo/semantics/restrictedAccess Mathematical Problems in Engineering : theory, methods and applications, Vol. 2009, p. article n°124904 (2009) info:eu-repo/semantics/article 2009 ftunivlouvain https://doi.org/10.1155/2009/124904 2024-04-17T17:33:55Z 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. 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. Copyright (C) 2009 D. Rosca and J.-P. Antoine. Article in Journal/Newspaper North Pole South pole DIAL@UCLouvain (Université catholique de Louvain) South Pole North Pole Mathematical Problems in Engineering 2009 1 13 |
institution |
Open Polar |
collection |
DIAL@UCLouvain (Université catholique de Louvain) |
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ftunivlouvain |
language |
English |
description |
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. 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. Copyright (C) 2009 D. Rosca and J.-P. Antoine. |
author2 |
UCL - SC/PHYS - Département de physique |
format |
Article in Journal/Newspaper |
author |
Rosca, Daniela Antoine, Jean-Pierre |
spellingShingle |
Rosca, Daniela Antoine, Jean-Pierre Locally supported orthogonal wavelet bases on the sphere via stereographic projection |
author_facet |
Rosca, Daniela Antoine, Jean-Pierre |
author_sort |
Rosca, Daniela |
title |
Locally supported orthogonal wavelet bases on the sphere via stereographic projection |
title_short |
Locally supported orthogonal wavelet bases on the sphere via stereographic projection |
title_full |
Locally supported orthogonal wavelet bases on the sphere via stereographic projection |
title_fullStr |
Locally supported orthogonal wavelet bases on the sphere via stereographic projection |
title_full_unstemmed |
Locally supported orthogonal wavelet bases on the sphere via stereographic projection |
title_sort |
locally supported orthogonal wavelet bases on the sphere via stereographic projection |
publisher |
Hindawi Publishing Corporation |
publishDate |
2009 |
url |
http://hdl.handle.net/2078.1/35163 https://doi.org/10.1155/2009/124904 |
geographic |
South Pole North Pole |
geographic_facet |
South Pole North Pole |
genre |
North Pole South pole |
genre_facet |
North Pole South pole |
op_source |
Mathematical Problems in Engineering : theory, methods and applications, Vol. 2009, p. article n°124904 (2009) |
op_relation |
boreal:35163 http://hdl.handle.net/2078.1/35163 doi:10.1155/2009/124904 urn:ISSN:1024-123X |
op_rights |
info:eu-repo/semantics/restrictedAccess |
op_doi |
https://doi.org/10.1155/2009/124904 |
container_title |
Mathematical Problems in Engineering |
container_volume |
2009 |
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1 |
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13 |
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1798851645806739456 |