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.

[1]  P. Baldi,et al.  Spherical Needlets for CMB Data Analysis , 2007, 0707.0844.

[2]  P. Vielva,et al.  Detection of Non-Gaussianity in the Wilkinson Microwave Anisotropy Probe First-Year Data Using Spherical Wavelets , 2004 .

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

[4]  Pascal Audet,et al.  Directional wavelet analysis on the sphere: Application to gravity and topography of the terrestrial planets , 2011 .

[5]  P. Vielva,et al.  Detection of non-Gaussianity in the WMAP 1-year data using spherical wavelets , 2003 .

[6]  Daryl Geller,et al.  Nearly tight frames and space-frequency analysis on compact manifolds , 2007, 0706.3642.

[7]  Y Wiaux,et al.  Global universe anisotropy probed by the alignment of structures in the cosmic microwave background. , 2006, Physical review letters.

[8]  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.

[9]  Yves Wiaux,et al.  A Novel Sampling Theorem on the Rotation Group , 2015, IEEE Signal Processing Letters.

[10]  Ignace Loris,et al.  Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion , 2011, Optical Engineering + Applications.

[11]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[12]  Michael P. Hobson,et al.  Fast Directional Continuous Spherical Wavelet Transform Algorithms , 2005, IEEE Transactions on Signal Processing.

[13]  Jason D. McEwen,et al.  Ieee Transactions on Signal Processing 1 Exact Wavelets on the Ball , 2022 .

[14]  P. Baldi,et al.  Asymptotics for spherical needlets , 2006, math/0606599.

[15]  Roger Penrose,et al.  Note on the Bondi-Metzner-Sachs Group , 1966 .

[16]  Laurent Jacques,et al.  Fast spin +-2 spherical harmonics transforms , 2005 .

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

[18]  Pierre Vandergheynst,et al.  On the computation of directional scale-discretized wavelet transforms on the sphere , 2013, Optics & Photonics - Optical Engineering + Applications.

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

[20]  J. D. McEwen,et al.  Probing dark energy with steerable wavelets through correlation of WMAP and NVSS local morphological measures , 2007, 0704.0626.

[21]  J.-L. Starck,et al.  Spherical 3D isotropic wavelets , 2012 .

[22]  Hrushikesh Narhar Mhaskar,et al.  Polynomial frames on the sphere , 2000, Adv. Comput. Math..

[23]  Domenico Marinucci,et al.  Integrated Sachs-Wolfe effect from the cross-correlation of WMAP 3 year and NVSS: new results and constraints on dark energy , 2006 .

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

[25]  A. M. M. Scaife,et al.  Simulating full‐sky interferometric observations , 2008, 0803.2165.

[26]  Gerard Kerkyacharian,et al.  Spin Needlets for Cosmic Microwave Background Polarization Data Analysis , 2008, 0811.2881.

[27]  J. D. McEwen,et al.  Data compression on the sphere , 2011, 1108.3900.

[28]  Domenico Marinucci,et al.  Spin Needlets Spectral Estimation , 2009, 0907.3369.

[29]  Michael P. Hobson,et al.  A directional continuous wavelet transform on the sphere , 2006, ArXiv.

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

[31]  Domenico Marinucci,et al.  The needlets bispectrum , 2008, 0802.4020.

[32]  Stefano Trapani,et al.  Calculation of spherical harmonics and Wigner d functions by FFT. Applications to fast rotational matching in molecular replacement and implementation into AMoRe. , 2006, Acta crystallographica. Section A, Foundations of crystallography.

[33]  Daniel Potts,et al.  Interpolatory Wavelets on the Sphere , 1995 .

[34]  M. P. Hobson,et al.  Cosmological Applications of a Wavelet Analysis on the Sphere , 2007, 0704.3158.

[35]  Krzysztof M. Gorski,et al.  NeedATool: A NEEDLET ANALYSIS TOOL FOR COSMOLOGICAL DATA PROCESSING , 2010, 1010.1371.

[36]  Pierre Vandergheynst,et al.  On spin scale-discretised wavelets on the sphere for the analysis of CMB polarisation , 2014, Proceedings of the International Astronomical Union.

[37]  Jean-Pierre Antoine,et al.  Discrete Wavelet Frames on the sphere , 2004, 2004 12th European Signal Processing Conference.

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

[39]  Paolo Baldi,et al.  Spherical needlets for cosmic microwave background data analysis , 2008 .

[40]  Domenico Marinucci,et al.  Integrated Sachs-Wolfe effect from the cross correlation of WMAP 3 year and the NRAO VLA sky survey data: New results and constraints on dark energy , 2006 .

[41]  Pierre Vandergheynst,et al.  S2LET: A code to perform fast wavelet analysis on the sphere , 2012, ArXiv.

[42]  R. Wilson.  Modern Cosmology , 2004 .

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

[44]  Wim Sweldens,et al.  The lifting scheme: a construction of second generation wavelets , 1998 .

[45]  Edward J. Wollack,et al.  NINE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: FINAL MAPS AND RESULTS , 2012, 1212.5225.

[46]  P. Vielva,et al.  Cross-correlation of the cosmic microwave background and radio galaxies in real, harmonic and wavelet spaces: detection of the integrated Sachs–Wolfe effect and dark energy constraints , 2004 .

[47]  Jochen Göttelmann,et al.  Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere , 1999 .

[48]  Guust Nolet,et al.  A new approach to global seismic tomography based on regularization by sparsity in a novel 3D spherical wavelet basis , 2010 .

[49]  Guust Nolet,et al.  Inversion with a sparsity constraint: Application to mantle tomography , 2012 .

[50]  D. Mattis.  Quantum Theory of Angular Momentum , 1981 .

[51]  Domenico Marinucci,et al.  Spin Wavelets on the Sphere , 2008, 0811.2935.

[52]  Yves Wiaux,et al.  Directional spin wavelets on the sphere , 2015, ArXiv.

[53]  P. Vielva,et al.  Non-Gaussianity analysis on local morphological measures of WMAP data , 2007, 0706.2346.

[54]  Marcos López-Caniego,et al.  Wavelets on the sphere. Application to the detection problem , 2006, 2006 14th European Signal Processing Conference.

[55]  Pencho Petrushev,et al.  Localized Tight Frames on Spheres , 2006, SIAM J. Math. Anal..

[56]  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.

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

[58]  Laurent Jacques,et al.  Fast spin ±2 spherical harmonics transforms and application in cosmology , 2007, J. Comput. Phys..

[59]  Y. W. Iaux.  FAST DIRECTIONAL CORRELATION ON THE SPHERE WITH STEERABLE FILTERS , 2006 .

[60]  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.

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

[62]  Jean-Luc Starck,et al.  Wavelets, ridgelets and curvelets on the sphere , 2006 .

[63]  G. W. Pratt,et al.  Planck 2013 results Special feature Planck 2013 results . XXV . Searches for cosmic strings and other topological defects , 2014 .

[64]  Y. W IAUX,et al.  Fast Spin ±2 Spherical Harmonics Transforms , 2008 .

[65]  Anthony N. Lasenby,et al.  Testing the Gaussianity of the COBE DMR data with spherical wavelets , 2000 .

[66]  F. Rohrlich,et al.  Spin‐s Spherical Harmonics and ð , 1967 .

[67]  S. Voronin,et al.  Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity , 2010, 1104.3151.

[68]  P. Vielva,et al.  Steerable wavelet analysis of CMB structures alignment , 2006 .

[69]  G. Peccati,et al.  Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications , 2011 .

[70]  Edward H. Adelson,et al.  The Design and Use of Steerable Filters , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[71]  C. A. Oxborrow,et al.  Planck 2015 results: XXIII. The thermal Sunyaev-Zeldovich effect-cosmic infrared background correlation , 2015, 1509.06555.

[72]  Pascal Audet,et al.  Toward mapping the effective elastic thickness of planetary lithospheres from a spherical wavelet analysis of gravity and topography , 2014 .

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

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

[75]  Domenico Marinucci,et al.  Mixed Needlets , 2010, 1006.3835.

[76]  J. Starck,et al.  Polarized wavelets and curvelets on the sphere , 2009 .

[77]  J. R. Bond,et al.  The statistics of cosmic background radiation fluctuations , 1987 .

[78]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[79]  G. W. Pratt,et al.  Planck 2015 results - XVII. Constraints on primordial non-Gaussianity , 2014 .

[80]  J. Fadili,et al.  Poisson denoising on the sphere: application to the Fermi gamma ray space telescope , 2010, 1003.5613.

[81]  Edward J. Wollack,et al.  NINE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: COSMOLOGICAL PARAMETER RESULTS , 2012, 1212.5226.

[82]  J. Cardoso,et al.  A full sky, low foreground, high resolution CMB map from WMAP , 2008, 0807.0773.

[83]  Jason D. McEwen,et al.  Fourier-Laguerre transform, convolution and wavelets on the ball , 2013, ArXiv.

[84]  C. A. Oxborrow,et al.  Planck2013 results. XII. Diffuse component separation , 2013, Astronomy & Astrophysics.

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

[86]  Yves Wiaux,et al.  A Novel Sampling Theorem on the Sphere , 2011, IEEE Transactions on Signal Processing.

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

[88]  O. Blanc,et al.  Exact reconstruction with directional wavelets on the sphere , 2007, 0712.3519.

[89]  Isaac Z. Pesenson,et al.  Simple proposal for radial 3D needlets , 2014, 1408.1095.