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Wavelet deconvolution in a periodic setting

Author

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  • Iain M. Johnstone
  • Gérard Kerkyacharian
  • Dominique Picard
  • Marc Raimondo

Abstract

Summary. Deconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with O{n log (n)2} steps. In the periodic setting, the method applies to most deconvolution problems, including certain ‘boxcar’ kernels, which are important as a model of motion blur, but having poor Fourier characteristics. Asymptotic theory informs the choice of tuning parameters and yields adaptivity properties for the method over a wide class of measures of error and classes of function. The method is tested on simulated light detection and ranging data suggested by underwater remote sensing. Both visual and numerical results show an improvement over competing approaches. Finally, the theory behind our estimation paradigm gives a complete characterization of the ‘maxiset’ of the method: the set of functions where the method attains a near optimal rate of convergence for a variety of Lp loss functions.

Suggested Citation

  • Iain M. Johnstone & Gérard Kerkyacharian & Dominique Picard & Marc Raimondo, 2004. "Wavelet deconvolution in a periodic setting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 547-573, August.
    • Handle: RePEc:bla:jorssb:v:66:y:2004:i:3:p:547-573
    DOI: 10.1111/j.1467-9868.2004.02056.x
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    Citations

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    Cited by:

    1. Christophe Chesneau, 2011. "On adaptive wavelet estimation of a quadratic functional from a deconvolution problem," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(2), pages 405-429, April.
    2. Nicolai Bissantz & Hajo Holzmann, 2013. "Asymptotics for spectral regularization estimators in statistical inverse problems," Computational Statistics, Springer, vol. 28(2), pages 435-453, April.
    3. Benhaddou, Rida, 2017. "On minimax convergence rates under Lp-risk for the anisotropic functional deconvolution model," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 120-125.
    4. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    5. J. Andrés Christen & Bruno Sansó & Mario Santana-Cibrian & Jorge X. Velasco-Hernández, 2016. "Bayesian deconvolution of oil well test data using Gaussian processes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(4), pages 721-737, March.
    6. repec:jss:jstsof:21:i02 is not listed on IDEAS
    7. Bissantz, Nicolai & Holzmann, Hajo, 2007. "Statistical inference for inverse problems," Technical Reports 2007,40, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Jérémie Bigot & Sébastien Van Bellegem, 2009. "Log‐density Deconvolution by Wavelet Thresholding," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 749-763, December.
    9. Abbaszadeh, Mohammad & Chesneau, Christophe & Doosti, Hassan, 2012. "Nonparametric estimation of density under bias and multiplicative censoring via wavelet methods," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 932-941.
    10. Nicolai Bissantz & Gerda Claeskens & Hajo Holzmann & Axel Munk, 2009. "Testing for lack of fit in inverse regression—with applications to biophotonic imaging," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 25-48, January.
    11. Birke, Melanie & Bissantz, Nicolai, 2007. "Shape constrained estimators in inverse regression models with convolution-type operator," Technical Reports 2007,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    12. Petsa, Athanasia & Sapatinas, Theofanis, 2009. "Minimax convergence rates under the Lp-risk in the functional deconvolution model," Statistics & Probability Letters, Elsevier, vol. 79(13), pages 1568-1576, July.
    13. Fabienne Comte & Charles-A. Cuenod & Marianna Pensky & Yves Rozenholc, 2017. "Laplace deconvolution on the basis of time domain data and its application to dynamic contrast-enhanced imaging," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 69-94, January.
    14. Chesneau, Christophe, 2007. "Regression with random design: A minimax study," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 40-53, January.
    15. Raimondo, Marc & Stewart, Michael, 2007. "The WaveD Transform in R: Performs Fast Translation-Invariant Wavelet Deconvolution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i02).
    16. Benhaddou, Rida, 2016. "Deconvolution model with fractional Gaussian noise: A minimax study," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 201-208.
    17. Maarten Jansen & Guy P. Nason & B. W. Silverman, 2009. "Multiscale methods for data on graphs and irregular multidimensional situations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 97-125, January.

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