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Wavelets, ridgelets, curvelets on the sphere and applications
Y. Moudden, J.-L. Starck & P. AbrialService d’Astrophysique
CEA Saclay, France
Wavelets, ridgelets, curvelets on the sphere and applications
Y. Moudden, J.-L. Starck & P. AbrialService d’Astrophysique
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
Introduction - Motivations
• Numerous applications in astrophysics, geophysics, medical imaging, computer graphics, etc. where data are given on the sphere e.g. :
- imaging the Earth’s surface with POLDER
http://polder.cnes.fr
Introduction - Motivations
• Numerous applications in astrophysics, geophysics, medical imaging, computer graphics, etc. where data are given on the sphere e.g. :
- imaging the Earth’s surface with POLDER
- mapping CMB fluctuations with WMAP
http://map.gsfc.nasa.gov
Introduction - Motivations
• Numerous applications in astrophysics, geophysics, medical imaging, computer graphics, etc. where data are given on the sphere e.g. :
- imaging the Earth’s 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.
Wavelet transform on the sphereRelated 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
The isotropic undecimatedwavelet transform on the sphere
• Spherical harmonics expansion :
• We consider an axisymetric bandlimited scaling (low pass) function :
• Spherical correlation theorem :
The isotropic undecimatedwavelet transform on the sphere
• Multiresolution decomposition :
• Can be obtained recursively :
where
• Possible scaling function :
The isotropic undecimatedwavelet transform on the sphere
• Wavelet coefficients can be computed as :
• Hence the wavelet function :
• Recursively :
The isotropic undecimatedwavelet transform on the sphere
• Reconstruction is a simple sum :
• Recursively, using conjugate filters :
where
The isotropic undecimatedwavelet transform on the sphere
The isotropic undecimatedwavelet transform on the sphere
j=1
j=2
j=3
j=4
Healpix
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.
K.M. Gorski et al., 1999, astro-ph/9812350http://www.eso.org/science/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.
The isotropic pyramidalwavelet transform on the sphere
j=2
j=3
j=4
j=1
Warping
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.
Healpix provides a natural invertible mapping of the quadrilateral base resolution pixels onto flat square images.
Ridgelets on the sphereObtained by applying the euclidean digital ridgelet transform to the 12 base resolution faces.
• Continuous ridgelet transform (Candes, 1998) :
€
R f a,b,θ( ) = ψ a,b,θ∫ x( ) f x( )dx
€
ψa,b,θ x( ) = a1
2ψx1 cos(θ) + x2 sin(θ) − b
a
⎛
⎝ ⎜
⎞
⎠ ⎟
Ridgelets on the sphereObtained by applying the euclidean digital ridgelet transform to the 12 Healpix base resolution faces.
• Continuous ridgelet transform (Candes, 1998) :
• Connection with the Radon transform ;€
R f a,b,θ( ) = ψ a,b,θ∫ x( ) f x( )dx
€
ψa,b,θ x( ) = a1
2ψx1 cos(θ) + x2 sin(θ) − b
a
⎛
⎝ ⎜
⎞
⎠ ⎟
€
R f (a,b,θ) = Rf (θ, t)ψ (t − b
a∫ )dt
Ridgelets on the sphereObtained by applying the euclidean digital ridgelet transform to the 12 base resolution faces.
Ridgelets on the sphere
Back-projection of ridgelet coefficients at different scales and orientations.
Digital Curvelet transform
• local ridgelets
• with proper scaling
Width = Length^2
Curvelets on the sphere
Obtained by applying the euclidean digital curvelet transform to the 12 Healpix base resolution faces.
Algorithm:
Curvelets on the sphere
Denoising full-sky astrophysical maps
• hard thresholding of spherical wavelet coefficients
• hard thresholding of spherical curvelet coefficients
• combined filtering :
Results
Top : Details of the original and noisy synchrotrn maps.
Bottom : Detail of the map obtained using the combined filtering technique, and the residual.
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– …
A static linear mixture model
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
23 GHz
33 GHz
41 GHz
94 GHz
Free-f
ree
CM
B
Syn
chro
tron
Foreground removal using ICADifferent 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 asin 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.
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):
• 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 :
Spectral Matching ICA in wavelet space
• The components may be estimated via Wiener filtering in each scale before inverting the wavelet transform :
Spectral Matching ICA in wavelet space
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.
Results
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 !?