Wavelets, ridgelets, curvelets on the sphere and applications

1. Wavelets, ridgelets, curvelets on the sphere and applications Y. Moudden, J.-L. Starck & P. Abrial Service d’Astrophysique CEA Saclay, France

2. Wavelets, ridgelets, curvelets on the sphere and applications Y. Moudden, J.-L. Starck & P. Abrial Service 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

3. 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

4. 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

5. 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.

6. 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

7. The isotropic undecimatedwavelet transform on the sphere • Spherical harmonics expansion : • We consider an axisymetric bandlimited scaling (low pass) function : • Spherical correlation theorem :

8. The isotropic undecimatedwavelet transform on the sphere • Multiresolution decomposition : • Can be obtained recursively : where • Possible scaling function :

9. The isotropic undecimatedwavelet transform on the sphere • Wavelet coefficients can be computed as : • Hence the wavelet function : • Recursively :

10. The isotropic undecimatedwavelet transform on the sphere • Reconstruction is a simple sum : • Recursively, using conjugate filters : where

11. The isotropic undecimatedwavelet transform on the sphere

12. j=1 The isotropic undecimatedwavelet transform on the sphere j=2 j=3 j=4

13. 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

14. j=1 The isotropic pyramidalwavelet transform on the sphere j=2 j=3 j=4

15. Warping Healpix provides a natural invertible mapping of the quadrilateral base resolution pixels onto flat square images.

16. Ridgelets on the sphere Obtained by applying the euclidean digital ridgelet transform to the 12 base resolution faces. • Continuous ridgelet transform (Candes, 1998) :

17. 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 ;

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

19. Ridgelets on the sphere Back-projection of ridgelet coefficients at different scales and orientations.

20. Digital Curvelet transform • local ridgelets • with proper scaling Width = Length^2

21. Curvelets on the sphere Obtained by applying the euclidean digital curvelet transform to the 12 Healpix base resolution faces. Algorithm:

22. Curvelets on the sphere

23. Denoising full-sky astrophysical maps • hard thresholding of spherical wavelet coefficients • hard thresholding of spherical curvelet coefficients • combined filtering :

30. Results Top : Details of the original and noisy synchrotrn maps. Bottom : Detail of the map obtained using the combined filtering technique, and the residual.

31. 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 • …

32. 23 GHz Synchrotron Free-free CMB 33 GHz 41 GHz 94 GHz A static linear mixture model

33. 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.

34. 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):

35. 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 :

36. Spectral Matching ICA in wavelet space • The components may be estimated via Wiener filtering in each scale before inverting the wavelet transform :

37. 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.

38. Results

39. 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 !?