Statistics and Computing

, Volume 10, Issue 2, pp 95–103 | Cite as

Image processing through multiscale analysis and measurement noise modeling

  • F. Murtagh
  • J.-L. Starck


We describe a range of powerful multiscale analysis methods. We also focus on the pivotal issue of measurement noise in the physical sciences. From multiscale analysis and noise modeling, we develop a comprehensive methodology for data analysis of 2D images, 1D signals (or spectra), and point pattern data. Noise modeling is based on the following: (i) multiscale transforms, including wavelet transforms; (ii) a data structure termed the multiresolution support; and (iii) multiple scale significance testing. The latter two aspects serve to characterize signal with respect to noise. The data analysis objectives we deal with include noise filtering and scale decomposition for visualization or feature detection.

wavelet transform multiresolution analysis filtering deconvolution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anscombe F.J. 1948. The transformation of Poisson, binomial and negative-binomial data. Biometrika 15: 246–254.Google Scholar
  2. Bijaoui A., Starck J.-L., and Murtagh F. 1994. Restauration des images multi-échelles par l'algorithme à trous. Traitement du Signal 11: 229–243.Google Scholar
  3. Breen E.J., Jones R., and Talbot H. 2000. Mathematical morphology: A useful set of tools for image analysis. Statistics and Computing, this issue.Google Scholar
  4. Bruce A. and Gao H.-Y. 1994. S+Wavelets User's Manual, Version 1.0, StatSci Division, MathSoft Inc., Seattle.Google Scholar
  5. Bury P. 1995. De la distribution de matière àa grande échelle à partir des amas d'Abell (On the large-scale distribution of matter using Abell clusters). PhD Thesis, University of Nice, Sophia Antipolis.Google Scholar
  6. Combes J.M., Grossmann A., and Tchamitchian Ph. (Eds.). 1989. Wavelets: Time-Frequency Methods and Phase Space. Springer-Verlag, Berlin.Google Scholar
  7. Dasgupta A. and Raftery A.E. 1995. Detecting features in spatial point processes with clutter via model-based clustering. Technical Report 295, Department of Statistics, University of Washington. Available at Journal of the American Statistical Association, in press.Google Scholar
  8. Daubechies I. 1992. Ten Lectures on Wavelets. SIAM (Society for Industrial and Applied Mathematics), Philadelphia.Google Scholar
  9. Donoho D.L., Johnstone I.M., Kerkyacharian G., and Picard D. 1995. Wavelet shrinkage: asymptopia? (with discussion). Journal of the Royal Statistical Society, Series B 57: 301–370.Google Scholar
  10. Holschneider M., Kronland-Martinet R., Morlet J., and Tchamitchian Ph. 1989. A real-time algorithm for signal analysis with the help of the wavelet transform. In: Combes J.M., Grossmann A., and Tchamitchian Ph. (Eds.), Wavelets: Time-Frequency Methods and Phase Space. Springer-Verlag, Berlin, pp. 286–297.Google Scholar
  11. Kolaczyk E.D. 1997. Estimation of intensities of burst-like processes using Haar wavelets. Submitted to Journal of the Royal Statistical Society Series B, Department of Statistics, University of Chicago, 27 pp.Google Scholar
  12. Mallat. 1989. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11: 674–693.Google Scholar
  13. Mccoy E.J. and Walden A.T. 1996. Wavelet analysis and synthesis of stationary long-memory processes. Journal of Computational and Graphical Statistics 5: 26–56.Google Scholar
  14. Meyer Y. 1993.Wavelets: Algorithms and Applications. SIAM (Society for Industrial and Applied Mathematics), Philadelphia. Original French version 1990. Ondelettes, Hermann, Paris.Google Scholar
  15. Meyer Y. 1995. Wavelets: Algorithms and Applications. SIAM, Philadelphia.Google Scholar
  16. Murtagh F. and Starck J.-L. 1998. Pattern clustering based on noise modeling in wavelet space. Pattern Recognition 31: 847–855.Google Scholar
  17. Murtagh F., Starck J.-L., and Bijaoui A. 1995. Image restoration with noise suppression using a multiresolution support. Astronomy and Astrophysics Supplement Series 112: 179–189.Google Scholar
  18. Olsen S.I. 1993. Estimation of noise in images: an evaluation. CVGIP: Graphical Models and Image Processing 55: 319–323.Google Scholar
  19. Powell K.J., Sapatinas T., Bailey T.C., and Krzanowski W.J. 1995. Application of wavelets to pre-processing of underwater sounds. Statistics and Computing 5: 265–273.Google Scholar
  20. Seales W.B., Yuan C.J., Hu W., and Cutts M.D. 1996. Content analysis of compressed video. University of Kentucky Computer Science Department Technical Report No. 2, 28 pp.Google Scholar
  21. Shensa M.J. 1992. The discrete wavelet transform: wedding the à trous and Mallat algorithms. IEEE Transactions on Signal Processing 40: 2464–2482.Google Scholar
  22. Slezak E., de Lapparent V., and Bijaoui A. 1993. Objective detection of voids and high density structures in the first CfA redshift survey slice. Astrophysical Journal 409: 517–529.Google Scholar
  23. Snyder D.L., Hammoud A.M., and White R.L. 1993. Image recovery from data acquired with a charge-coupled camera. Journal of the Optical Society of America 10: 1014–1023.Google Scholar
  24. Starck J.-L. and Bijaoui A. 1994. Filtering and deconvolution by the wavelet transform. Signal Processing 35: 195–211.Google Scholar
  25. Starck J.-L. and Murtagh F. 1994. Image restoration with noise suppression using the wavelet transform. Astronomy and Astrophysics 288: 342–348.Google Scholar
  26. Starck J.-L. and Murtagh F. 1998a. Automatic noise estimation from the multiresolution support. Publications of the Astronomical Society of the Pacific, 110: 193–199.Google Scholar
  27. Starck J.-L. and Murtagh F. 1998b. Image filtering from the combination of multi-vision models, submitted.Google Scholar
  28. Starck J.-L. and Murtagh F. 1999. Multiscale entropy filtering. Signal Processing, 76: 147–165.Google Scholar
  29. Starck J.-L., Murtagh F., and Bijaoui A. 1995. Multiresolution support applied to image filtering and deconvolution. Graphical Models and Image Processing 57: 420–431.Google Scholar
  30. Starck J.-L., Murtagh F., and Bijaoui A. 1998. Image Processing and Data Analysis: The Multiscale Approach. Cambridge University Press, Cambridge (GB).Google Scholar
  31. Starck J.-L., Murtagh F., and Gastaud R. 1998. A new entropy measure based on the wavelet transform and noise modeling. IEEE Transactions on Circuits and Systems II on Multirate Systems, Filter Banks, Wavelets, and Applications 45: 1118–1124.Google Scholar
  32. Starck J.-L., Murtagh F., Pirenne B., and Albrecht M. 1996. Astronomical image compression based on noise suppression. Publications of the Astronomical Society of the Pacific 108: 446–455.Google Scholar
  33. Starck J.-L. and Pierre M., 1998. Structure detection in low intensity X-ray images. Astronomy and Astrophysics, Suppl. Ser. 128: 397–407.Google Scholar
  34. Strang G. and Nguyen T. 1996. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley.Google Scholar
  35. Tekalp A.M. and Pavlović G. 1991. Restoration of scanned photographic images. In: Katsaggelos A.K. (Ed.), Digital Image Restoration. Springer-Verlag, New York, pp. 209–239.Google Scholar
  36. Wickerhauser M.V. 1994. Adapted Wavelet Analysis from Theory to Practice. A.K. Peters, Wellesley.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • F. Murtagh
    • 1
  • J.-L. Starck
    • 2
  1. 1.School of Computer ScienceThe Queen's University of BelfastBelfastNorthern Ireland
  2. 2.CEA-SaclayDAPNIA/SEI-SAPGif-sur-Yvette CedexFrance

Personalised recommendations