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

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are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

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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 !?