1. 21/22 February 2003Granada iAstro Worshop1 Analysis of Astrophysical Data Cubes using Cross-correlations and Wavelet Denoisings A.Bijaoui 1, D.Mékarnia 1, J.P.Maillard 2, C.Delle Luche 1 1 Observatoire de la Côte d'Azur (Nice) 2 Institut d’Astrophysique de Paris

2. 21/22 February 2003Granada iAstro Worshop2 Outlines The Astrophysical Data Cubes –BEAR and IFTS The Karhunen-Loève expansion (KL/PCA) –The KL basis –The noise of the basis /components Wavelet denoising of the basis/components The residues and their denoising An application on NGC 7027 cube Conclusion

3. 21/22 February 2003Granada iAstro Worshop3 The Integral-Field Spectrographs Different optical devices –Scanning Fabry-Perot –Optical fibers (VIMOS, GIRAFFE) –Cylindrical lenses + Grating (TIGRE, OASIS) –Multislit (SAURON, MUSE) –Imaging Fourier Transform Spectrograph Resulting Data Cubes –Size depending on the device –From Megapixel to Gigapixel Need of specific analysis methods

4. 21/22 February 2003Granada iAstro Worshop4 BEAR : an IFTS device

5. 21/22 February 2003Granada iAstro Worshop5 BEAR at the CFHT focus

6. 21/22 February 2003Granada iAstro Worshop6 The example of NGC 7027 A post AGB planetary nebula –Observations  Cox et al. 2002 –The resampled data cube: 128x128x1024 What information? –Different spectral lines  Abundance –Velocity field  3D view –Continuum Necessity to denoise the data cube –To increase the SNR –To observe fainter objects

7. 21/22 February 2003Granada iAstro Worshop7 The data cube

8. 21/22 February 2003Granada iAstro Worshop8 Spectra sample

9. 21/22 February 2003Granada iAstro Worshop9 Elements of the data reduction We can take into account –The cross correlation between the images  PCA / KL expansion –The significant details image / image –The significant details spectrum / spectrum Different possible ways –Wavelet Transform + KL exp. + Denoising + Reconstruction (Starck et al. 2001) –KL exp. + Denoising + Reconstruction + Residue + Denoising (Mékarnia et al. 2003)

10. 21/22 February 2003Granada iAstro Worshop10 KL and PCA Search of uncorrelated images The Principal Component Analysis –Iterative extraction of the linear combinations having the greatest variance PCA application to images  KL The eigenvalue = the energy / order

11. 21/22 February 2003Granada iAstro Worshop11 The noisy KL basis

12. 21/22 February 2003Granada iAstro Worshop12 Denoising the KL expansion Each KL component is noisy –Depends on the order / eigenvalue Each KL spectrum is noisy The reconstruction from noisy components leads to a noisy restoration Each KL component / spectrum is denoised –Wavelet denoising –Redundant transform –Soft wavelet shrinkage

13. 21/22 February 2003Granada iAstro Worshop13 The denoised KL basis

14. 21/22 February 2003Granada iAstro Worshop14 The residues and their analysis Do not forget to denoise the mean ! The reconstruction with the denoised KL is limited: –Not enough components –Adding components = increase the noise –The denoising can remove local significant feature Use of the residues between the original data and the restored one

15. 21/22 February 2003Granada iAstro Worshop15 After the residue denoising

16. 21/22 February 2003Granada iAstro Worshop16 Spectra Sample

17. 21/22 February 2003Granada iAstro Worshop17 The velocity field

18. 21/22 February 2003Granada iAstro Worshop18 3D visualisation

19. 21/22 February 2003Granada iAstro Worshop19 A spectrum in a cavity

20. 21/22 February 2003Granada iAstro Worshop20 A continuum image

21. 21/22 February 2003Granada iAstro Worshop21 The integrated continuum

22. 21/22 February 2003Granada iAstro Worshop22 CONCLUSION Data cube can be denoised from KL Limitation of the number of components –We could use more components with denoising –Too local information (spectral/spatial) Residue denoising –Could be improved (best basis, softening rule, regularisation,..) Artifact removal –Use of ICA/SOBI blind source separation Help for astrophysical interpretation