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research article

Wavelets on the 2-Sphere: A Group-Theoretical Approach

Antoine, J.
•
Vandergheynst, P.  
1999
Applied and Computational Harmonic Analysis

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and +* for dilations on S2, which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X SO0(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO0(3, 1) acting in the space L2(S2, d ) of finite energy signals on S2. This representation is square integrable over X; thus it yields immediately the wavelets on S2 and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S2 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R . Then the parameter space goes into the similitude group of 2 and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.

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Type
research article
DOI
10.1006/acha.1999.0272
Author(s)
Antoine, J.
•
Vandergheynst, P.  
Date Issued

1999

Published in
Applied and Computational Harmonic Analysis
Volume

7

Issue

3

Start page

262

End page

291

Subjects

LTS2

Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
LTS2  
Available on Infoscience
June 14, 2006
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/231029
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