Spatially Varying Spectral Filtering of Signals on the Unit Sphere

This paper presents a general framework for spatially-varying spectral filtering of signals defined on the unit sphere, as an analogy to joint time-frequency filtering. For this purpose, we first map spherical signals from spatial domain into joint spatio-spectral domain, where a spatio-spectral signal transformation or modification is introduced. For mapping spatial signals into joint spatio-spectral domain, we use the spatially localized spherical harmonic transform (SLSHT) from the literature. We then propose a suitable scheme to transform the modified signal from the spatio-spectral domain back to an admissible signal in the spatial domain using the least squares approach. We also show that the overall action of the SLSHT and spatio-spectral signal modification can be described through a single transformation matrix, which is useful in practice. Finally, we discuss two specific and useful instances of spatially-varying spectral filtering, defined through multiplicative and convolutive modification of the SLSHT distribution, and show through numerical examples their effectiveness in selective spectral filtering of different spatial regions of the signal.

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

[2]  Frederik J. Simons,et al.  Minimum-Variance Multitaper Spectral Estimation on the Sphere , 2007, 1306.3254.

[3]  F. Sansò,et al.  Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere , 1999 .

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

[5]  F. Simons,et al.  Localized spectral analysis on the sphere , 2005 .

[6]  Ronald L. Allen,et al.  Signal Analysis: Time, Frequency, Scale and Structure , 2003 .

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

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

[9]  Sean C. Solomon,et al.  Localization of gravity and topography: constraints on the tectonics and mantle dynamics of Venus , 1997 .

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

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

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

[13]  Salman Durrani,et al.  On the construction of low-pass filters on the unit sphere , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[14]  Michael P. Hobson,et al.  Optimal Filters on the Sphere , 2006, IEEE Transactions on Signal Processing.

[15]  R. Ramamoorthi,et al.  Frequency domain normal map filtering , 2007, SIGGRAPH 2007.

[16]  W. Kozek,et al.  A comparative study of linear and nonlinear time-frequency filters , 1992, [1992] Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis.

[17]  Rodney A. Kennedy,et al.  Commutative Anisotropic Convolution on the 2-Sphere , 2012, IEEE Transactions on Signal Processing.

[18]  Thomas Bülow,et al.  Spherical Diffusion for 3D Surface Smoothing , 2004, 3DPVT.

[19]  Michael Riley Time-frequency filtering , 1989 .

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

[21]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

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

[23]  W. Kozek,et al.  Time-frequency subspaces and their application to time-varying filtering , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[24]  R. C. Williamson,et al.  Theory and design of broadband sensor arrays with frequency invariant far‐field beam patterns , 1995 .

[25]  Rodney A. Kennedy,et al.  Introducing Space into MIMO Capacity Calculations , 2003, Telecommun. Syst..

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

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

[28]  Boualem Boashash,et al.  Time-Frequency Signal Analysis and Processing: A Comprehensive Reference , 2015 .

[29]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.

[30]  Thomas W. Parks,et al.  Time-varying filtering and signal estimation using Wigner distribution synthesis techniques , 1986, IEEE Trans. Acoust. Speech Signal Process..

[31]  Leon Cohen,et al.  Time Frequency Analysis: Theory and Applications , 1994 .

[32]  J. J. Sakurai,et al.  Modern Quantum Mechanics , 1986 .

[33]  Willi Freeden,et al.  Constructive approximation and numerical methods in geodetic research today – an attempt at a categorization based on an uncertainty principle , 1999 .

[34]  F. Simons,et al.  Spectral estimation on a sphere in geophysics and cosmology , 2007, 0705.3083.

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

[36]  Salman Durrani,et al.  Spatio-Spectral Analysis on the Sphere Using Spatially Localized Spherical Harmonics Transform , 2012, IEEE Transactions on Signal Processing.

[37]  Rodney A. Kennedy,et al.  On azimuthally symmetric 2-sphere convolution , 2011, Digit. Signal Process..

[38]  Werner Krattenthaler,et al.  Time-Frequency Design and Processing of Signals Via Smoothed Wigner Distributions , 1993, IEEE Trans. Signal Process..

[39]  Bahaa E. A. Saleh,et al.  Time-variant filtering of signals in the mixed time frequency domain , 1985, IEEE Trans. Acoust. Speech Signal Process..

[40]  Edward J. Wollack,et al.  Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology , 2006, astro-ph/0603449.

[41]  Douglas L. Jones,et al.  Improved time-frequency filtering using an STFT analysis-modification-synthesis method , 1994, Proceedings of IEEE-SP International Symposium on Time- Frequency and Time-Scale Analysis.

[42]  Frederik J. Simons,et al.  Efficient analysis and representation of geophysical processes using localized spherical basis functions , 2009, Optical Engineering + Applications.

[43]  James D. Louck,et al.  Angular Momentum in Quantum Physics: Theory and Application , 1984 .

[44]  J O Hirschfelder,et al.  THE INTEGRAL OF THE ASSOCIATED LEGENDRE FUNCTION. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[45]  B. T. Thomas Yeo,et al.  On the Construction of Invertible Filter Banks on the 2-Sphere , 2008, IEEE Transactions on Image Processing.