Starlet Transform in Astronomical Data Processing

Living reference work entry

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 article. 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 Nonnegative Matrix Factorization Blind Deconvolution Poisson Noise 
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

  1. 1.
    Anscombe, F.J.: The transformation of Poisson, binomial and negative-binomial data. Biometrika 15, 246–254 (1948)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Benvenuto, F., La Camera, A., Theys, C., Ferrari, A., Lantéri, H., Bertero, M.: The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise. Inverse Probl.24(035016), 20pp (2008)Google Scholar
  3. 3.
    Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. Institute of Physics, Bristol/Philadelphia (1998)CrossRefMATHGoogle Scholar
  4. 4.
    Bertero, M., Boccacci, P., Desiderá, G., Vicidomini, G.: Image deblurring with Poisson data: from cells to galaxies. Inverse Probl.25(123006), 26pp (2009)Google Scholar
  5. 5.
    Bertero, M., Boccacci, P., Talenti, G., Zanella, R., Zanni, L.: A discrepancy principle for Poisson data. Inverse Probl.26, 10500 (2010)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bertin, E., Arnouts, S.: SExtractor: software for source extraction. Astron. Astrophys. Suppl. Ser.117, 393–404 (1996)CrossRefGoogle Scholar
  7. 7.
    Bijaoui, A.: Sky background estimation and application. Astron. Astrophys.84, 81–84 (1980)Google Scholar
  8. 8.
    Bijaoui, A., Rué, F.: A multiscale vision model adapted to astronomical images. Signal Process.46, 229–243 (1995)CrossRefGoogle Scholar
  9. 9.
    Bonettini, S.: Inexact block coordinate descent methods with application to non-negative matrix factorization. IMA J. Numer. Anal.31(4), 1431–1452 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bonettini, S., Zanella, R., Zanni, L.: A scaled gradient projection method for constrained image deblurring. Inverse Probl.25(1), 015002 (2009)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Buonanno, R., Buscema, G., Corsi, C.E., Ferraro, I., Iannicola, G.: Automated photographic photometry of stars in globular clusters. Astron. Astrophys.126, 278–282 (1983)Google Scholar
  12. 12.
    Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process.6, 298–311 (1997)CrossRefGoogle Scholar
  13. 13.
    Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput.20(1), 33–61 (1999)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul.4(4), 1168–1200 (2005)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Da Costa, G.S.: Basic photometry techniques. In: Howel, S.B. (ed.) Astronomical CCD Observing and Reduction Techniques. ASP Conference Series 23, vol. 23, p. 90. Astronomical Society of the Pacific, San Francisco (1992)Google Scholar
  16. 16.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math.57, 1413–1541 (2004)CrossRefMATHGoogle Scholar
  17. 17.
    Davoust, E., Pence, W.D.: Detailed bibliography on the surface photometry of galaxies. Astron. Astrophys. Suppl. Ser.49, 631–661 (1982)Google Scholar
  18. 18.
    Debray, B., Llebaria, A., Dubout-Crillon, R., Petit, M.: CAPELLA: software for stellar photometry in dense fields with an irregular background. Astron. Astrophys.281, 613–635 (1994)Google Scholar
  19. 19.
    Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39(1), 1–38 (1977)MATHMathSciNetGoogle Scholar
  20. 20.
    Desiderà G., Carbillet, M.: Strehl-constrained iterative blind deconvolution for post-adaptive-optics data. Astron. Astrophys.507(3), 1759–1762 (2009)CrossRefGoogle Scholar
  21. 21.
    Djorgovski, S.: Modelling of seeing effects in extragalactic astronomy and cosmology. J. Astrophys. Astron.4, 271–288 (1983)CrossRefGoogle Scholar
  22. 22.
    Dupé, F.-X., Fadili, M.J., Starck, J.-L.: A proximal iteration for deconvolving Poisson noisy images using sparse representations. IEEE Trans. Image Process.18(2), 310–321 (2009)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Engl, H. W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Volume 375 of Mathematics and Its Applications. Kuwer Academic, Dordrecht/Boston (1996)Google Scholar
  24. 24.
    Figueiredo, M.A., Nowak, R.: An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process.12(8), 906–916 (2003)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell.6, 721–741 (1984)CrossRefMATHGoogle Scholar
  26. 26.
    Green, P.J.-F.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imaging 9, 84–93 (1990)CrossRefGoogle Scholar
  27. 27.
    Holmes, T.J.: Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach. J. Opt. Soc. Am.A-9, 1052–1061 (1992)CrossRefGoogle Scholar
  28. 28.
    Irwin, M.J.: Automatic analysis of crowded fields. Mon. Not. R. Astron. Soc.214, 575–604 (1985)CrossRefGoogle Scholar
  29. 29.
    Kron, R.G.: Photometry of a complete sample of faint galaxies. Astrophys. J. Suppl. Ser.43, 305–325 (1980)CrossRefGoogle Scholar
  30. 30.
    Kurtz, M.J.: Classification methods: an introductory survey. In: Statistical Methods in Astronomy. European Space Agency Special Publication 201, pp. 47–58. ESA Scientific & Technical Publications Branch, Noordwijk (1983)Google Scholar
  31. 31.
    Lantéri, H., Roche, M., Aime, C.: Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms. Inverse Probl.18, 1397–1419 (2002)CrossRefMATHGoogle Scholar
  32. 32.
    Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. Adv. Neural Inf. Process.13, 556–562 (2001)Google Scholar
  33. 33.
    Lefèvre, O., Bijaoui, A., Mathez, G., Picat, J.P., Lelièvre, G.: Electronographic BV photometry of three distant clusters of galaxies. Astron. Astrophys.154, 92–99 (1986)Google Scholar
  34. 34.
    Lucy, L.B.: An iteration technique for the rectification of observed distributions. Astron. J.79, 745–754 (1974)CrossRefGoogle Scholar
  35. 35.
    Maddox, S.J., Efstathiou, G., Sutherland, W.J.: The APM galaxy survey – part two – photometric corrections. Mon. Not. R. Astron. Soc.246, 433 (1990)Google Scholar
  36. 36.
    Mallat, S.: A Wavelet Tour of Signal Processing, The Sparse Way, 3rd edn. Academic, Boston (2008)Google Scholar
  37. 37.
    Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process.41(12), 3397–3415 (1993)CrossRefMATHGoogle Scholar
  38. 38.
    Moffat, A.F.J.: A theoretical investigation of focal stellar images in the photographic emulsion and application to photographic photometry. Astron. Astrophys.3, 455–461 (1969)Google Scholar
  39. 39.
    Molina, R., Ripley, B.D., Molina, A., Moreno, F., Ortiz, J.L.: Bayesian deconvolution with prior knowledge of object location – applications to ground-based planetary images. Astrophys. J.104, 1662–1668 (1992)Google Scholar
  40. 40.
    Murtagh, F., Starck, J.-L., Bijaoui, A.: Image restoration with noise suppression using a multiresolution support. Astron. Astrophys. Suppl. Ser.112, 179–189 (1995)Google Scholar
  41. 41.
    Natterer, F., Wûbbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)CrossRefMATHGoogle Scholar
  42. 42.
    Naylor, T.: An optimal extraction algorithm for imaging photometry. Mon. Not. R. Astron. Soc.296, 339–346 (1998)CrossRefGoogle Scholar
  43. 43.
    Okamura, S.: Global structure of Virgo cluster galaxies. In: ESO Workshop On The Virgo Cluster of Galaxies, Garching, pp. 201–215 (1985)Google Scholar
  44. 44.
    Pence, W.D., Davoust, E.: Supplement to the detailed bibliography on the surface photometry of galaxies. Astron. Astrophys. Suppl. Ser.60, 517–526 (1985)Google Scholar
  45. 45.
    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.: The XMM-LSS survey. Survey design and first results. J. Cosmol. Astro-Part. Phys.9, JCAP09(2004)011 (2004)Google Scholar
  46. 46.
    Prato, M., Cavicchioli, R., Zanni, L., Boccacci, P., Bertero, M.: Efficient deconvolution methods for astronomical imaging: algorithms and IDL-GPU codes. Astron. Astrophys.539, A133 (2012)CrossRefGoogle Scholar
  47. 47.
    Prato, M., La Camera, A., Bonettini, S., Bertero, M.: A convergent blind deconvolution method for post-adaptive-optics astronomical imaging. Inverse Probl.29(6), 065017 (2013)CrossRefGoogle Scholar
  48. 48.
    Richardson, W.H.: Bayesian-based iterative method of image restoration. J. Opt. Soc. Am.62, 55–59 (1972)CrossRefGoogle Scholar
  49. 49.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imaging MI-2, 113–122 (1982)CrossRefGoogle Scholar
  50. 50.
    Starck, J.-L., Aussel, H., Elbaz, D., Fadda, D., Cesarsky, C.: Faint source detection in ISOCAM images. Astron. Astrophys. Suppl. Ser.138, 365–379 (1999)CrossRefGoogle Scholar
  51. 51.
    Starck, J.-L., Bijaoui, A., Murtagh, F.: Multiresolution support applied to image filtering and deconvolution. CVGIP: Graph. Models Image Process.57, 420–431 (1995)Google Scholar
  52. 52.
    Starck, J.-L., Elad, M., Donoho, D.L.: Redundant multiscale transforms and their application for morphological component analysis. Adv. Imaging Electron Phys.132, 287–348 (2004)CrossRefGoogle Scholar
  53. 53.
    Starck, J.-L., Fadili, J., Murtagh, F.: The undecimated wavelet decomposition and its reconstruction. IEEE Trans. Image Process.16, 297–309 (2007)CrossRefMathSciNetGoogle Scholar
  54. 54.
    Starck, J.-L., Murtagh, F.: Image restoration with noise suppression using the wavelet transform. Astron. Astrophys.288, 343–348 (1994)Google Scholar
  55. 55.
    Starck, J.-L., Murtagh, F.: Automatic noise estimation from the multiresolution support. Publ. Astron. Soc. Pac.110, 193–199 (1998)CrossRefGoogle Scholar
  56. 56.
    Starck, J.-L., Murtagh, F.: Astronomical Image and Data Analysis. Springer, Berlin (2002).CrossRefGoogle Scholar
  57. 57.
    Starck, J.-L., Murtagh, F.: Astronomical Image and Data Analysis, 2nd edn. Springer, Berlin (2006)CrossRefGoogle Scholar
  58. 58.
    Starck, J.-L., Murtagh, F., Bijaoui, A.: Image Processing and Data Analysis: The Multiscale Approach. Cambridge University Press, Cambridge/New York (1998)CrossRefGoogle Scholar
  59. 59.
    Starck, J.-L., Pierre, M.: Structure detection in low intensity X-ray images. Astron. Astrophys. Suppl. Ser.128, 397–407 (1998).CrossRefGoogle Scholar
  60. 60.
    Starck, J.-L., Siebenmorgen, R., Gredel, R.: Spectral analysis by the wavelet transform. Astrophys. J.482, 1011–1020 (1997)CrossRefGoogle Scholar
  61. 61.
    Takase, B., Kodaira, K., Okamura, S.: An Atlas of Selected Galaxies. University of Tokyo Press, Tokyo (1984)Google Scholar
  62. 62.
    Thonnat, M.: INRIA Rapport de Recherche, Centre Sophia Antipolis, No. 387 (1985). Automatic morphological description of galaxies and classification by an expert systemGoogle Scholar
  63. 63.
    Tikhonov, A.N., Goncharski, A.V., Stepanov, V.V., Kochikov, I.V.: Ill-posed image processing problems. Sov. Phys. – Dokl.32, 456–458 (1987)Google Scholar
  64. 64.
    Watanabe, M., Kodaira, K., Okamura, S.: Digital surface photometry of galaxies toward a quantitative classification. I. 20 galaxies in the Virgo cluster. Astron. Astrophys. Suppl. Ser.50, 1–22 (1982)Google Scholar
  65. 65.
    Zanella, R., Boccacci, P., Zanni, L., Bertero, M.: Efficient gradient projection methods for edge-preserving removal of Poisson noise. Inverse Probl.25, 045010 (2009)CrossRefMathSciNetGoogle Scholar
  66. 66.
    Zhang, B., Fadili, M.J., Starck, J.-L.: Wavelets, ridgelets and curvelets for Poisson noise removal. IEEE Trans. Image Process.17(7), 1093–1108 (2008)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CEA, Laboratoire AIM CEA/DSM-CNRS-Université Paris Diderot, CEA, IRFU, Service d’Astrophysique, Centre de SaclayGif-Sur-Yvette CedexFrance
  2. 2.School of Computer Science and Informatics De Montfort UniversityLeicesterUK
  3. 3.DIBRIS Università di GenovaGenovaItaly

Personalised recommendations