Wavelets, ridgelets and curvelets on the sphere

¹ DAPNIA/SEDI-SAP, Service d'Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France e-mail: jstarck@cea.fr
² Department of Statistics, Sequoia Hall, 390 Serra Mall, Stanford University, Stanford, CA 94305, USA
³ Laboratoire Traitement de l'Image et du Signal, CNRS UMR 8051/ENSEA/Université de Cergy-Pontoise, ENSEA, 6 avenue du ponceau, 95014 Cergy, France

Received: 15 April 2005
Accepted: 8 September 2005

Abstract

We present in this paper new multiscale transforms on the sphere, namely the isotropic undecimated wavelet transform, the pyramidal wavelet transform, the ridgelet transform and the curvelet transform. All of these transforms can be inverted i.e. we can exactly reconstruct the original data from its coefficients in either representation. Several applications are described. We show how these transforms can be used in denoising and especially in a Combined Filtering Method, which uses both the wavelet and the curvelet transforms, thus benefiting from the advantages of both transforms. An application to component separation from multichannel data mapped to the sphere is also described in which we take advantage of moving to a wavelet representation.