Abstract
Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.
Highlights
Wavelet methodologies on the sphere are of considerable theoretical interest in their own right and have important practical application
Wavelets analyses on the sphere have led to many insightful scientific studies in the fields of planetary science (e.g. [4,5]), geophysics (e.g. [13,34,68,69]) and cosmology, in particular for the analysis of the cosmic microwave background (CMB) (e.g. [7,9,15,30,40,41,42,43, 49,53,59,60,61,62,63,66,75,76,77,82,83]), among others
In this article we show that these properties hold for directional scale-discretised wavelets
Summary
Wavelet methodologies on the sphere are of considerable theoretical interest in their own right and have important practical application. The construction of Freeden & Windheuser [19] is based on singular integrals on the sphere, while Antoine and Vandergheynst [2,3] follow a group theoretic approach. In the latter construction dilation is defined via the stereographic projection of the sphere to the plane, leading to a consistent framework that reduces locally to the usual continuous wavelet transform in the Euclidean limit. An implementation and technique to approximate functions on the sphere has been developed for this approach [1] This construction is revisited in [78], independently of the original group theoretic formalism, and fast algorithms are developed in [79,80]. Each approach has been extended to analyse spin functions on the sphere [21,22,23,24,37,46,70] and functions defined on the three-dimensional ball formed by augmenting the sphere with the radial line [17,31,32,45]
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