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 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 Goacuterski, provides a savings of O(radicNpix) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 106 pixel map on a personal computer

[1]  Mark S. Gordon,et al.  Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion , 1999 .

[2]  T. Risbo.  Fourier transform summation of Legendre series and D-functions , 1996 .

[3]  E. Saff,et al.  Discretizing Manifolds via Minimum Energy Points , 2004 .

[4]  Roger J. Phillips,et al.  Potential anomalies on a sphere: Applications to the thickness of the lunar crust , 1998 .

[5]  Matthias Holschneider,et al.  Continuous wavelet transforms on the sphere , 1996 .

[6]  W. Sweldens.  The Lifting Scheme: A Custom - Design Construction of Biorthogonal Wavelets "Industrial Mathematics , 1996 .

[7]  S. Swenson,et al.  Methods for inferring regional surface‐mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time‐variable gravity , 2002 .

[8]  J. D. McEwen,et al.  Non-Gaussianity detections in the Bianchi VIIh corrected WMAP one-year data made with directional spherical wavelets , 2005, astro-ph/0510349.

[9]  R. J. Willemann,et al.  Role of membrane stresses in the support of planetary topography , 1981 .

[10]  Mark A. Wieczorek,et al.  Spatiospectral Concentration on a Sphere , 2004, SIAM Rev..

[11]  Edward J. Wollack,et al.  First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results , 2003, astro-ph/0302207.

[12]  J. D. McEwen,et al.  A high-significance detection of non-Gaussianity in the WMAP 5-yr data using directional spherical wavelets , 2006, 0803.2157.

[13]  Peter Schröder,et al.  Spherical wavelets: efficiently representing functions on the sphere , 1995, SIGGRAPH.

[14]  J. D. McEwen,et al.  A high-significance detection of non-Gaussianity in the Wilkinson Microwave Anisotropy Probe 1-yr data using directional spherical wavelets , 2004, astro-ph/0406604.

[15]  Robert G. Crittenden,et al.  Exactly azimuthal pixelizations of the sky , 1998 .

[16]  J. D. McEwen,et al.  Detection of the ISW effect and corresponding dark energy constraints , 2006 .

[17]  Laurent Demanet,et al.  Gabor wavelets on the sphere , 2003, SPIE Optics + Photonics.

[18]  Thomas Bülow,et al.  Surface Representations Using Spherical Harmonics and Gabor Wavelets on the Sphere , 2001 .

[19]  Graham J. L. Kemp,et al.  Fast computation, rotation, and comparison of low resolution spherical harmonic molecular surfaces , 1999, J. Comput. Chem..

[20]  Willi Freeden,et al.  Constructive Approximation on the Sphere: With Applications to Geomathematics , 1998 .

[21]  Pascal Frossard,et al.  Progressive Coding of 3-D Objects Based on Overcomplete Decompositions , 2006, IEEE Transactions on Circuits and Systems for Video Technology.

[22]  Martin J. Mohlenkamp.  A fast transform for spherical harmonics , 1997 .

[23]  Willi Freeden,et al.  Combined Spherical Harmonic and Wavelet Expansion—A Future Concept in Earth's Gravitational Determination , 1997 .

[24]  Gabriele Steidl,et al.  Kernels of spherical harmonics and spherical frames , 1996 .

[25]  Laurent Jacques,et al.  Stereographic wavelet frames on the sphere , 2005 .

[26]  K. Ruedenberg,et al.  Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion , 1998 .

[27]  K. Gorski,et al.  HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere , 2004, astro-ph/0409513.

[28]  O. V. Verkhodanov,et al.  Gauss - Legendre sky pixelization (GLESP) for CMB maps , 2005 .

[29]  Kathy Whaler,et al.  Downward continuation of Magsat lithospheric anomalies to the Earth's surface , 1994 .

[30]  Syed Twareque Ali,et al.  Two-Dimensional Wavelets and their Relatives , 2004 .

[31]  Belgium,et al.  Correspondence principle between spherical and euclidean wavelets , 2005, astro-ph/0502486.

[32]  E. Maksimov,et al.  Group theory and its application in physics , 1991 .

[33]  E. Hivon,et al.  Analysis issues for large CMB data sets , 1998, astro-ph/9812350.

[34]  Boris Rubin.  Continuous Wavelet Transforms on a Sphere , 1998 .

[35]  Qiang Du,et al.  Constrained Centroidal Voronoi Tessellations for Surfaces , 2002, SIAM J. Sci. Comput..

[36]  E. Saff,et al.  Distributing many points on a sphere , 1997 .

[37]  N. Turok,et al.  Looking for a cosmological constant with the Rees-Sciama effect. , 1996, Physical review letters.

[38]  Joseph Silk,et al.  The Wiener-Filtered COBE DMR Data and Predictions for the Tenerife Experiment , 1995 .

[39]  Klaus Ruedenberg,et al.  Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion , 1996 .

[40]  A. Stebbins,et al.  Microwave anisotropy due to cosmic strings , 1984, Nature.

[41]  T. Inui,et al.  Group theory and its applications in physics , 1990 .

[42]  Laurent Jacques,et al.  Wavelets on the sphere: implementation and approximations , 2002 .

[43]  F. J. Narcowich,et al.  Nonstationary Wavelets on them-Sphere for Scattered Data , 1996 .

[44]  Pat Hanrahan,et al.  A signal-processing framework for reflection , 2004, ACM Trans. Graph..

[45]  P. Vandergheynst,et al.  Wavelets on the 2-sphere: A group-theoretical approach , 1999 .

[46]  D. Healy,et al.  Computing Fourier Transforms and Convolutions on the 2-Sphere , 1994 .

[47]  Bruno Torrésani,et al.  Position-frequency analyis for signals defined on spheres , 1995, Signal Process..

[48]  M. A. Blanco,et al.  Evaluation of the rotation matrices in the basis of real spherical harmonics , 1997 .

[49]  Pierre Vandergheynst,et al.  Wavelets on the n-sphere and related manifolds , 1998 .

[50]  Benjamin D. Wandelt,et al.  Fast convolution on the sphere , 2001 .

[51]  E. L. Wright,et al.  Four-Year COBE DMR Cosmic Microwave Background Observations: Maps and Basic Results , 1996, astro-ph/9601067.

[52]  Sean S. B. Moore,et al.  FFTs for the 2-Sphere-Improvements and Variations , 1996 .