Complex data processing: fast wavelet analysis on the sphere

In the general context of complex data processing, this paper reviews a recent practical approach to the continuous wavelet formalism on the sphere. This formalism notably yields a correspondence principle which relates wavelets on the plane and on the sphere. Two fast algorithms are also presented for the analysis of signals on the sphere with steerable wavelets.Comment: 20 pages, 5 figures, JFAA style, paper invited to J. Fourier Anal. and Appli

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

Fast directional continuous spherical wavelet transform algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE Trans. Sig. Pro

Complex Data Processing: Fast Wavelet Analysis on the Sphere

In the general context of complex data processing, this article reviews a recent practical approach to the continuous wavelet formalism on the sphere. This formalism notably yields a correspondence principle which relates wavelets on the plane and on the sphere. Two fast algorithms are also presented for the analysis of signals on the sphere with steerable wavelet

Sparse Representation on Graphs by Tight Wavelet Frames and Applications

In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold \cM that is formed by generalized dilations and shifts of a finite collection of wavelet functions $\Psi:=\{\psi_j: 1\le j\le r\}\subset L_2(\R)$. We further require that $\psi_j$ is generated by some refinable function $\phi$ with mask $a_j$. We present the condition needed for the masks $\{a_j: 0\le j\le r\}$ so that the associated quasi-affine system generated by $\Psi$ is a tight frame for L_2(\cM). Then, we discuss how the transition from the continuum (manifolds) to the discrete setting (graphs) can be naturally done. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately by proposing the fast tight wavelet frame transforms for graph data (WFTG). Finally, we consider two specific applications of the proposed WFTG: graph data denoising and semi-supervised clustering. Utilizing the sparse representation provided by the WFTG, we propose $\ell_1$-norm based optimization models on graphs for denoising and semi-supervised clustering. On one hand, our numerical results show significant advantage of the WFTG over the spectral graph wavelet transform (SGWT) by [1] for both applications. On the other hand, numerical experiments on two real data sets show that the proposed semi-supervised clustering model using the WFTG is overall competitive with the state-of-the-art methods developed in the literature of high-dimensional data classification, and is superior to some of these methods

On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields

We consider the correlation structure of the random coefficients for a wide class of wavelet systems on the sphere (Mexican needlets) which were recently introduced in the literature by Geller and Mayeli (2007). We provide necessary and sufficient conditions for these coefficients to be asymptotic uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high resolution sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data.Comment: Revised version for Stochastic Processes and their Application

Spin Needlets for Cosmic Microwave Background Polarization Data Analysis

Scalar wavelets have been used extensively in the analysis of Cosmic Microwave Background (CMB) temperature maps. Spin needlets are a new form of (spin) wavelets which were introduced in the mathematical literature by Geller and Marinucci (2008) as a tool for the analysis of spin random fields. Here we adopt the spin needlet approach for the analysis of CMB polarization measurements. The outcome of experiments measuring the polarization of the CMB are maps of the Stokes Q and U parameters which are spin 2 quantities. Here we discuss how to transform these spin 2 maps into spin 2 needlet coefficients and outline briefly how these coefficients can be used in the analysis of CMB polarization data. We review the most important properties of spin needlets, such as localization in pixel and harmonic space and asymptotic uncorrelation. We discuss several statistical applications, including the relation of angular power spectra to the needlet coefficients, testing for non-Gaussianity on polarization data, and reconstruction of the E and B scalar maps.Comment: Accepted for publication in Phys. Rev.