Wavelets on the Two-Sphere and Other Conic Sections

Abstract

ABSTRACT. We survey the construction of the continuous wavelet transform (CWT) on the two-sphere. Then we discuss the discretization of the spherical CWT, obtaining various types of discrete frames. Finally we give some indications on the construction of a CWT on other conic sections.

Spin Wavelets on the Sphere

“…(This is analogous to the usual situation in which all the wavelets at each scale are translates of each other.) Again, w t,x is given by (2). On the sphere S 2 , we have…”

“…In recent years a rapidly growing literature has focussed on the construction of wavelet systems on the sphere, see for instance [1][2][3]48] and the references therein. These attempts have been motivated by strong interest from the applied sciences, for instance in the areas of Geophysics, Medical Imaging and especially Cosmology/Astrophysics.…”

The Continuous Wavelet Transform on Conic Sections

“…40 In the sequel of this section, we shall give a rapid overview of the series of works mentioned above, following mostly Ref. 7. As a general reference on 2-D wavelets, we use our recent monograph.…”

“…7 (1) Unlike the usual 2-D CWT, which is fully covariant with respect to translations, rotations and dilations, the spherical CWT is only partially covariant. It is covariant under motions on S 2 : for any o ∈ SO(3), the transform of the rotated signal f (…”

S2LET: A code to perform fast wavelet analysis on the sphere

“…In particular, wavelets on the sphere (Antoine & Vandergheynst 1998, 1999Baldi et al 2009;Marinucci et al 2008;McEwen et al 2006a;Narcowich et al 2006;Starck et al 2006a;Wiaux et al 2006aWiaux et al , 2005Wiaux et al , 2007Wiaux et al , 2008aYeo et al 2008) have been applied very successfully to problems in astrophysics and cosmology, where data-sets are increasingly large and need to be analysed at high resolution in order to confront accurate theoretical predictions (e.g. Barreiro et al 2000;Basak & Delabrouille 2012;Cayón et al 2001;Deriaz et al 2012;Labatie et al 2012;Lan & Marinucci 2008;McEwen et al 2006bMcEwen et al , 2007aMcEwen et al ,b, 2008Pietrobon et al 2008;Starck et al 2006b;Schmitt et al 2010;Vielva et al 2004Vielva et al , 2006aVielva et al ,b, 2007Wiaux et al 2006bWiaux et al , 2008b.…”