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Statistical Challenges in Astronomy pp 343–364Cite as

Threshold Selection in Transform Shrinkage

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Abstract

The transform shrinkage paradigm is reviewed, of which wavelet denoising is a key example, with a focus on the blockwise approach to processing of transform coeffients. Thresholding approaches are surveyed, with special emphasis is placed on an empirical Bayes approach, which promises to adapt well to the demands of both ’dense’ and ’sparse’ signals. Since the author has no significant experience with problems in astronomy, discussion and examples (denoising of signals, images and decon volution) are alas generic.

This paper is followed by a commentary by astronomer Jean-Luc Starck.

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References

  • [ABDJ99] Felix Abramovich, Yoav Benjamini, David Donoho, and Iain Johnstone. Adapting to unknown sparsity by controlling the false discovery rate. Technical Report, Submitted to Annals of Statistics, 1999.

    Google Scholar 

  • [BH95] Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society, Series B., 57:289–300, 1995.

    MathSciNet  MATH  Google Scholar 

  • [Bre95] Leo Breiman. Better subset selection using the non-negative garotte. Technometrics, 37:373–384, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • [CD99a] E. Candès and D. L. Donoho. Curvelets: A surprisingly effective nonadaptive representation of objects with edges. Technical report, Stanford University, 1999.

    Google Scholar 

  • [CD99b] E. Candès and D. L. Donoho. Ridgelets: the key to high dimensional intermittency? Philosophical Transactions of the Royal Society of London, Series A, 357:2495–2509, 1999.

    Article  ADS  MATH  Google Scholar 

  • [CNB98] M. S. Crouse, R. D. Nowak, and R. G. Baraniuk. Wavelet-based statistical signal processing using hidden markov models. IEEE Transactions on Signal Processing, 46(4):886–902, 1998.

    Article  MathSciNet  ADS  Google Scholar 

  • [CS99] T. Cai and B. W. Silverman. Incorporating information on neighboring coefficients into wavelet estimation. Technical report, University of Bristol, 1999.

    Google Scholar 

  • [CT01] L. Cavalier and A. B. Tsybakov. Penalized blockwise stein’s method, monotone oracles and sharp adaptive estimation. Technical report, Université de Paris VI, 2001.

    Google Scholar 

  • [Dau92] I. Daubechies. Ten Lectures on Wavelets. Number 61 in CBMSNSF Series in Applied Mathematics. SIAM, Philadelphia, 1992.

    MATH  Google Scholar 

  • [DJ94] D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation via wavelet shrinkage. Biometrika, 81:425–455, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • [DJ95] D. L. Donohoand I. M. Johnstone. Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc., 90:1200–1224, 1995.

    Article  MathSciNet  Google Scholar 

  • [DJKP95] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard. Wavelet shrinkage: Asymptopia? Journal of the Royal Statistical Society, Series B, 57:301–369, 1995.

    MathSciNet  MATH  Google Scholar 

  • [Don95] D.L. Donoho. Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied Computational and Harmonic Analysis, 2:101–126, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • [EP84] S.Yu. Efroimovich and M.S. Pinsker. A learning algorithm for nonparametric filtering. Automat. i Telemeh., 11:58–65, 1984. (in Russian), translated in Automation and Remote Control, 1985, p 1434–1440.

    MathSciNet  Google Scholar 

  • [HKP99] P. G. Hall, G. Kerkyacharian, and D. Picard. On block thresholding rules for curve estimation using kernel and wavelet methods. Annals of Statistics, 26:922–942, 1999.

    MathSciNet  Google Scholar 

  • [JS] I. M. Johnstone and B. W. Silverman. Empirical bayes estimates of sparse sequences, with applications to transform shrinkage. manuscript in preparation.

    Google Scholar 

  • [JS97] I. M. Johnstone and B. W. Silverman. Wavelet threshold estimators for data with correlated noise. Journal of the Royal Statistical Society, Series B., 59:319–351, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • [KM99] J. Kalifa and S. Mallat. Minimax deconvolution in mirror wavelet bases. Technical report, Ecole Polytechnique, Palaiseau, 1999.

    Google Scholar 

  • [Mal99] Stéphane Mallat. A Wavelet Tour of Signal Processing. Academic Press, 1999. 2nd, expanded, edition.

    Google Scholar 

  • [MC97] Francois G. Meyer and Ronald R. Coifman. Brushlets: A tool for directional image analysis and image compression. Applied Computational and Harmonic Analysis, 4:147–187, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • [PS00] Javier Portilla and Eero P. Simoncelli. A parametric texture model based on joint statistics of complex wavelet coe cients. International Journal of Computer Vision, 40(1):49–71, 2000.

    Article  MATH  Google Scholar 

  • [Vid99] Brani Vidakovic. Statistical Modelling by Wavelets. John Wiley and Sons, 1999.

    Google Scholar 

References

  1. J.L. Starck, F. Murtagh, and A. Bijaoui. Image Processing and Data Analysis: TheMultiscale Approach. Cambridge University Press, Cambridge (GB), 1998.

    Book  Google Scholar 

  2. M. Antonini, M. Barlaud, P. Mathieu and I. Daubechies, Image coding using wavelet transform. IEEE Transactions on Image Processing, 1(2), 1992.

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Stanford University, Stanford, USA

    Iain Johnstone

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  1. Iain Johnstone
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© 2003 Springer-Verlag New York, Inc.

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Johnstone, I. (2003). Threshold Selection in Transform Shrinkage. In: Statistical Challenges in Astronomy. Springer, New York, NY. https://doi.org/10.1007/0-387-21529-8_23

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  • DOI: https://doi.org/10.1007/0-387-21529-8_23

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-95546-9

  • Online ISBN: 978-0-387-21529-7

  • eBook Packages: Springer Book Archive

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