Wavelets, ridgelets, curvelets on the sphere

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Wavelets, ridgelets, curvelets on the sphere and
applications
Y. Moudden, J.-L. Starck P. Abrial Service
dAstrophysique CEA Saclay, France
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Wavelets, ridgelets, curvelets on the sphere and
applications
Y. Moudden, J.-L. Starck P. Abrial Service
dAstrophysique CEA Saclay, France

• Outline
• - Motivations
• - Isotropic undecimated wavelet transform on the
  sphere
• - Ridgelets and Curvelets on the sphere
• - Applications to astrophysical data
  denoising, source separation

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Introduction - Motivations

• Numerous applications in astrophysics,
  geophysics, medical imaging, computer graphics,
  etc. where data are given on the sphere e.g.
• - imaging the Earths surface with POLDER

http//polder.cnes.fr
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Introduction - Motivations

• Numerous applications in astrophysics,
  geophysics, medical imaging, computer graphics,
  etc. where data are given on the sphere e.g.
• - imaging the Earths surface with POLDER
• - mapping CMB fluctuations with WMAP

http//map.gsfc.nasa.gov
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Introduction - Motivations

• Numerous applications in astrophysics,
  geophysics, medical imaging, computer graphics,
  etc. where data are given on the sphere e.g.
• - imaging the Earths surface with POLDER
• - mapping CMB fluctuations with WMAP

Need for specific data processing tools, inspired
from successful methods in flat-land wavelets,
ridgelets and curvelets.
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Wavelet transform on the sphere

• Related work
• P. Schroder and W. Sweldens (Orthogonal Haar
  WT), 1995.
• M. Holschneider, Continuous WT, 1996.
• W. Freeden and T. Maier, OWT, 1998.
• J.P. Antoine and P. Vandergheynst, Continuous
  WT, 1999.
• L. Tenerio, A.H. Jaffe, Haar Spherical CWT,
  (CMB), 1999.
• L. Cayon, J.L Sanz, E. Martinez-Gonzales,
  Mexican Hat CWT, 2001.
• J.P. Antoine and L. Demanet, Directional CWT,
  2002.
• M. Hobson, Directional CWT, 2005.

• Present implementation
• isotropic wavelet transform
• similar to the a trous algorithm, undecimated,
  simple inversion
• algorithm based on the spherical harmonics
  transform

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The isotropic undecimatedwavelet transform on
the sphere

• Spherical harmonics expansion
• We consider an axisymetric bandlimited scaling
  (low pass) function
• Spherical correlation theorem

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The isotropic undecimatedwavelet transform on
the sphere

• Multiresolution decomposition
• Can be obtained recursively
• where
• Possible scaling function

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The isotropic undecimatedwavelet transform on
the sphere

• Wavelet coefficients can be computed as
• Hence the wavelet function
• Recursively

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The isotropic undecimatedwavelet transform on
the sphere

• Reconstruction is a simple sum
• Recursively, using conjugate filters
• where

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The isotropic undecimatedwavelet transform on
the sphere
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The isotropic undecimatedwavelet transform on
the sphere
j1
j2
j3
j4
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Healpix

• Curvilinear hierarchical partition of the
  sphere.
• 12 base resolution quadrilateral faces, each
  has nside2 pixels.
• Equal area quadrilateral pixels of varying
  shape.
• Pixel centers are regularly spaced on
  isolatitude rings.
• Software package includes forward and inverse
  spherical harmonic transform.

K.M. Gorski et al., 1999, astro-ph/9812350http//
www.eso.org/science/healpix
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The isotropic pyramidalwavelet transform on the
sphere
j2
j3
j4
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Warping
Healpix provides a natural invertible mapping of
the quadrilateral base resolution pixels onto
flat square images.
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Ridgelets on the sphere
Obtained by applying the euclidean digital
ridgelet transform to the 12 base resolution
faces.

• Continuous ridgelet transform (Candes, 1998)

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Ridgelets on the sphere
Obtained by applying the euclidean digital
ridgelet transform to the 12 Healpix base
resolution faces.

• Continuous ridgelet transform (Candes, 1998)
• Connection with the Radon transform

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Ridgelets on the sphere
Obtained by applying the euclidean digital
ridgelet transform to the 12 base resolution
faces.
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Ridgelets on the sphere
Back-projection of ridgelet coefficients at
different scales and orientations.
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Digital Curvelet transform

• local ridgelets
• with proper scaling

Width Length2
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Curvelets on the sphere
Obtained by applying the euclidean digital
curvelet transform to the 12 Healpix base
resolution faces.
Algorithm
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Curvelets on the sphere
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Denoising full-sky astrophysical maps

• hard thresholding of spherical wavelet
  coefficients
• hard thresholding of spherical curvelet
  coefficients
• combined filtering

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Results
Top Details of the original and noisy
synchrotrn maps. Bottom Detail of the map
obtained using the combined filtering technique,
and the residual.
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Full-sky CMB data analysis

• CMB is a relic radiation from the early Universe.
• Full-sky observations from WMAP and
  Planck-Surveyor.
• The spectrum of its spatial fluctuations is of
  major importance in cosmology.
• Foregrounds
• Detector noise
• Galactic dust
• Synchrotron
• Free - Free
• Thermal SZ

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A static linear mixture model
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Foreground removal using ICA

• Different classes of ICA methods
• Algorithms based on non-gaussianity i.e.
  higher order statistics.
• Most mainstream ICA techniques fastICA, Jade,
  Infomax, etc.
• Techniques based on the diversity (non
  proportionality)
• of variance (energy) profiles in a given
  representation such as
• in time, space, Fourier, wavelet joint
  diagonalization of
• covariance matrices, SMICA, etc.

• CMB is well modeled by a stationary Gaussian
  random field.
• Use Spectral matching ICA
• But, non stationary noise process and Galactic
  emissions. Strongly emitting regions are masked.
  
• in a wavelet representation, to preserve scale
  space information.

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Spectral Matching ICA in wavelet space

• Apply the undecimated isotropic spherical wavelet
  transform to the multichannel data.
• For each scale j, compute empirical estimates of
  the covariance matrices of the multichannel
  wavelet coefficients (avoiding for instance
  masked regions)

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Spectral Matching ICA in wavelet space

• Apply the undecimated isotropic spherical wavelet
  transform to the multichannel data.
• For each scale j, compute empirical estimates of
  the covariance matrices of the multichannel
  wavelet coefficients (avoiding for instance
  masked regions)
• Fit the model covariance matrices to the
  estimated covariance matrices by minimizing the
  covariance mismatch measure

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Spectral Matching ICA in wavelet space

• The components may be estimated via Wiener
  filtering in each scale before inverting the
  wavelet transform

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Experiment

• Three independent components
• Galactic region masked
• Simulated observations in the six channels of
  the Planck HFI
• Nominal noise standard deviation and 6dB, 3dB
• Separation using wSMICA and SMICA in six scales
  and corresponding spectral bands.

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Results
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Conclusion

• We have introduced new multiscale decompositions
  on the sphere.
• Shown their usefulness in denoising and source
  separation.
• More can be found on http //jstarck.free.fr
• Software package should be released soon !?