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Starlet Transform in Astronomical Data Processing

  • Jean-Luc Starck
  • Fionn Murtagh
  • Mario Bertero

Abstract

We begin with traditional source detection algorithms in astronomy. We then introduce the sparsity data model. The starlet wavelet transform serves as our main focus in this chapter. Sparse modeling, and noise modeling, are described. Applications to object detection and characterization, and to image filtering and deconvolution, are discussed. The multiscale vision model is a further development of this work, which can allow for image reconstruction when the point spread function is not known, or not known well. Bayesian and other algorithms are described for image restoration. A range of examples is used to illustrate the algorithms.

Keywords

Point Spread Function Wavelet Coefficient Sparse Representation Poisson Noise Unknown Object 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References and Further Reading

  1. 1.
    Anscombe FJ (1948) The transformation of Poisson, binomial and negative-binomial data. Biometrika 15:246–254MathSciNetGoogle Scholar
  2. 2.
    Benvenuto F, La Camera A, Theys C, Ferrari A, Lantéri H, Bertero M (2008) The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise. Inverse Probl 24(035016)Google Scholar
  3. 3.
    Bertero M, Boccacci P (1998) Introduction to inverse problems in imaging. Institute of Physics, BristolMATHCrossRefGoogle Scholar
  4. 4.
    Bertero M, Boccacci P, Desiderá G, Vicidomini G (2009) Image deblurring with Poisson data: from cells to galaxies. Inverse Probl 25:123006CrossRefGoogle Scholar
  5. 5.
    Bertin E, Arnouts S (June 1996) Extractor: software for source extraction. Astron Astrophys Suppl Ser 117:393–404CrossRefGoogle Scholar
  6. 6.
    Bijaoui A (Apr 1980) Sky background estimation and application. Astron Astrophys 84:81–84Google Scholar
  7. 7.
    Bijaoui A, Rué F (1995) A multiscale vision model adapted to astronomical images. Signal Process 46:229–243CrossRefGoogle Scholar
  8. 8.
    Buonanno R, Buscema G, Corsi CE, Ferraro I, Iannicola G (Oct 1983) Automated photographic photometry of stars in globular clusters. Astron Astrophys 126:278–282Google Scholar
  9. 9.
    Chen SS, Donoho DL, Saunders MA (1999) Atomic decomposition by basis pursuit. SIAM J Sci Comput 20(1):33–61MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Combettes PL, Wajs VR (2005) Signal recovery by proximal forward-backward splitting. Multiscale Model Simulat 4(4):1168–1200MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Da GS (1992) Costa basic photometry techniques. In: Howel SB (ed) ASP conference series 23, Astronomical CCD Observing and Reduction Techniques, vol 23. Astronical Society of the Pacific, p 90Google Scholar
  12. 12.
    Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math 57:1413–1541MATHCrossRefGoogle Scholar
  13. 13.
    Davoust E, Pence WD (1982) Detailed bibliography on the surface photometry of galaxies. Astron Astrophys Suppl Seri 49:631–661Google Scholar
  14. 14.
    Debray B, Llebaria A, Dubout-Crillon R, Petit M (Jan 1994) CAPELLA: software for stellar photometry in dense fields with an irregular background. Astron Astrophys 281:613–635Google Scholar
  15. 15.
    Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc B 39(1):1–38MathSciNetMATHGoogle Scholar
  16. 16.
    Djorgovski S (Dec 1983) Modelling of seeing effects in extragalactic astronomy and cosmology. J Astrophys Astron 4:271–288CrossRefGoogle Scholar
  17. 17.
    Dupé F-X, Fadili MJ, Starck J-L (2009) A proximal iteration for deconvolving Poisson noisy images using sparse representations. IEEE Trans Image Process 18(2):310–321MathSciNetCrossRefGoogle Scholar
  18. 18.
    Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, vol 375 of Mathematics and its applications. Kuwer AcademicGoogle Scholar
  19. 19.
    Figueiredo MA, Nowak R (2003) An EM algorithm for wavelet-based image restoration. IEEE Trans Image Process 12(8):906–916MathSciNetCrossRefGoogle Scholar
  20. 20.
    Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741MATHCrossRefGoogle Scholar
  21. 21.
    Irwin MJ (June 1985) Automatic analysis of crowded fields. Monthly Notices Roy Astron Soc 214:575–604Google Scholar
  22. 22.
    Kron RG (June 1980) Photometry of a complete sample of faint galaxies. Astrophys J Suppl Ser 43:305–325CrossRefGoogle Scholar
  23. 23.
    Kurtz MJ (1983) Classification methods: an introductory survey. In: Statistical methods in astronomy. European Space Agency Special Publication 201, pp 47–58Google Scholar
  24. 24.
    Lefèvre O, Bijaoui A, Mathez G, Picat JP, Lelièvre G (1986) Electronographic BV photometry of three distant clusters of galaxies. Astron Astrophys 154:92–99Google Scholar
  25. 25.
    Lucy LB (1974) An iteration technique for the rectification of observed distributions. Astron J 79:745–754CrossRefGoogle Scholar
  26. 26.
    Maddox SJ, Efstathiou G, Sutherland WJ (Oct 1990) The APM galaxy survey – Part Two – Photometric corrections. Monthly Notices Roy Astron Soc 246:433Google Scholar
  27. 27.
    Mallat S (2008) A wavelet tour of signal processing, the sparse way, 3rd edn. Academic, New YorkGoogle Scholar
  28. 28.
    Mallat S, Zhang Z (1993) Matching pursuits with time-frequency dictionaries. IEEE Trans Signal Process 41(12):3397–3415MATHCrossRefGoogle Scholar
  29. 29.
    Moffat AFJ (Dec 1969) A theoretical investigation of focal stellar images in the photographic emulsion and application to photographic photometry. Astron Astrophys 3:455Google Scholar
  30. 30.
    Molina R, Ripley BD, Molina A, Moreno F, Ortiz JL (Oct 1992) Bayesian deconvolution with prior knowledge of object location – applications to ground-based planetary images. Astrophys J 104:1662–1668Google Scholar
  31. 31.
    Murtagh F, Starck J-L, Bijaoui A (1995) Image restoration with noise suppression using a multiresolution support. Astron Astrophys Suppl Ser 112:179–189Google Scholar
  32. 32.
    Natterer F, Wûbbeling F (2001) Mathematical methods in image reconstruction. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
  33. 33.
    Naylor T (May 1998) An optimal extraction algorithm for imaging photometry. Monthly Notices Roy Astron Soc 296:339–346CrossRefGoogle Scholar
  34. 34.
    Okamura S (1985) Global structure of Virgo cluster galaxies. In: ESO workshop on the virgo cluster of galaxies, pp 201–215Google Scholar
  35. 35.
    Pence WD, Davoust E (1985) Supplement to the detailed bibliography on the surface photometry of galaxies. Astron Astrophys Suppl Ser 60:517–526Google Scholar
  36. 36.
    Pierre M, Valtchanov I, Altieri B, Andreon S, Bolzonella M, Bremer M, Disseau L, Dos Santos S, Gandhi P, Jean C, Pacaud F, Read A, Refregier A, Willis J, Adami C, Alloin D, Birkinshaw M, Chiappetti L, Cohen A, Detal A, Duc P, Gosset E, Hjorth J, Jones L, LeFevre O, Lonsdale C, Maccagni D, Mazure A, McBreen B, McCracken H, Mellier Y, Ponman T, Quintana H, Rottgering H, Smette A, Surdej J, Starck J, Vigroux L, White S (Sept 2004) The XMM-LSS survey. Survey design and first results. J Cosmol Astropart Phys 9:11CrossRefGoogle Scholar
  37. 37.
    Richardson WH (1972) Bayesian-based iterative method of image restoration. J Opt Soc Am 62:55–59CrossRefGoogle Scholar
  38. 38.
    Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans Med Imaging MI-2:113–122Google Scholar
  39. 39.
    Starck J-L, Aussel H, Elbaz D, Fadda D, Cesarsky C (1999) Faint source detection in ISOCAM images. Astron Astrophys Suppl Ser 138:365–379CrossRefGoogle Scholar
  40. 40.
    Starck J-L, Bijaoui A, Murtagh F (1995) Multiresolution support applied to image filtering and deconvolution. CVGIP: Graph Models Image Process 57:420–431CrossRefGoogle Scholar
  41. 41.
    Starck J-L, Elad M, Donoho DL (2004) Redundant multiscale transforms and their application for morphological component analysis. Adv Imaging Electron Phys 132:287–348CrossRefGoogle Scholar
  42. 42.
    Starck J-L, Fadili J, Murtagh F (2007) The undecimated wavelet decomposition and its reconstruction. IEEE Trans Image Process 16:297–309MathSciNetCrossRefGoogle Scholar
  43. 43.
    Starck J-L, Murtagh F (1994) Image restoration with noise suppression using the wavelet transform. Astron Astrophys 288:343–348Google Scholar
  44. 44.
    Starck J-L, Murtagh F (1998) Automatic noise estimation from the multiresolution support. Publ Astron Soc Pacific 110:193–199CrossRefGoogle Scholar
  45. 45.
    Starck J-L, Murtagh F (2002) Astronomical image and data analysis. Springer, New YorkGoogle Scholar
  46. 46.
    Starck J-L, Murtagh F (2006) Astronomical image and data analysis, 2nd edn. Springer, BerlinGoogle Scholar
  47. 47.
    Starck J-L, Murtagh F, Bijaoui A (1998) Image processing and data analysis: the multiscale approach. Cambridge University Press, New YorkCrossRefGoogle Scholar
  48. 48.
    Starck J-L, Pierre M (1998) Structure detection in low intensity X-ray images. Astron Astrophys Suppl Ser 128:397–407CrossRefGoogle Scholar
  49. 49.
    Starck J-L, Siebenmorgen R, Gredel R (1997) Spectral analysis by the wavelet transform. Astrophys J 482:1011–1020CrossRefGoogle Scholar
  50. 50.
    Starck J-L, Murtagh F, Fadili J (2010) Sparse image & signal processing. Cambridge University Press, Cambridge (UK)MATHCrossRefGoogle Scholar
  51. 51.
    Takase B, Kodaira K, Okamura S (1984) An atlas of selected galaxies. University of Tokyo Press, TokyoGoogle Scholar
  52. 52.
    Thonnat M (1985) INRIA Rapport de Recherche. Centre Sophia Antipolis, No. 387 Automatic morphological description of galaxies and classification by an expert systemGoogle Scholar
  53. 53.
    Tikhonov AN, Goncharski AV, Stepanov VV, Kochikov IV (1987) Ill-posed image processing problems. Soviet Phys Doklady 32:456–458Google Scholar
  54. 54.
    Watanabe M, Kodaira K, Okamura S (1982) Digital surface photometry of galaxies toward a quantitative classification. I. 20 galaxies in the Virgo cluster. Astron Astrophys Suppl Ser 50:1–22Google Scholar
  55. 55.
    Zhang B, Fadili MJ, Starck J-L (2008) Wavelets, ridgelets and curvelets for Poisson noise removal. IEEE Trans Image Process 17(7):1093–1108MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jean-Luc Starck
    • 1
  • Fionn Murtagh
    • 2
  • Mario Bertero
    • 3
  1. 1.CEA/SaclayGif-sur-Yvette CedexFrance
  2. 2.Science Foundation IrelandDublinIreland
  3. 3.Università diGenovaGenovaItaly

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