Discrete Wavelet Frames on the Sphere

In this paper the Continuous Wavelet Transform on the sphere is exploited to build the asociated Discrete Wavelet Frames. First, the half-continuous frames, i.e. frames where the position remains a continuous variable, is presented and then the fully discrete theory is presented. The notion of controlled frames is introduced, which reflects the particular nature of the underlying theory, particularly the apperant conflict between dilation and the compacity of the sphere. The paper is concluded with some numerical illustrations and future work

Stereographic Frames of Wavelets on the Sphere

In this technical report the Discrete Wavelet Frames are build on the already existing Spherical Continuous Wavelet Transform. The spherical half-continuous frames are explored, i.e when the position on the sphere is kept continuous variable. Then, the controlled frames are introduced, which comes from the particular nature of the underlying theory, namely the conflict between dilation and the compacity of the spherical manifold. The perspectives for the future work are given

Continuous Wavelets on Compact Manifolds

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say 0 \neq f \in \mathcal{S}(\RR^+), and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on \RR^n. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on \RR^n in \mathcal{S}(\RR^n). In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus \TT^2 or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small

Localisation of directional scale-discretised wavelets on the sphere

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.Comment: 28 pages, 8 figures, minor changes to match version accepted for publication by ACH