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Nonlinear estimation in anisotropic multi-index denoising
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  • Published: October 2001

Nonlinear estimation in anisotropic multi-index denoising

  • Gérard Kerkyacharian1,
  • Oleg Lepski2 &
  • Dominique Picard3 

Probability Theory and Related Fields volume 121, pages 137–170 (2001)Cite this article

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  • 59 Citations

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Abstract.

In the framework of denoising a function depending of a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our method is a generalization of the Lepski's method of adaptation, and roughly consists in choosing the “coarsest” bandwidth such that the estimated bias is negligible. However, this notion becomes more delicate in a multidimensional setting. We will particularly focus on functions with inhomogeneous smoothness properties and especially providing a possible disparity of the inhomogeneous aspect in the different directions. We show, in particular that our method is able to exactly attain the minimax rate or to adapt to unknown degree of anisotropic smoothness up to a logarithmic factor, for a large scale of anisotropic Besov spaces.

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599; et Université Paris X, 200, avenue de la République, F-92001 Nanterre, France, , , , , , FR

    Gérard Kerkyacharian

  2. Laboratoire d'Analyse, Probabilitiés, Topologie, CNRS-UMR 6632, Université de Provence, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France. e-mail: Oleg.Lepski@gyptis.univ-mrs.fr, , , , , , FR

    Oleg Lepski

  3. Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599; et Université Paris VII, 16, rue de Clisson, F-75013 Paris, France, , , , , , FR

    Dominique Picard

Authors
  1. Gérard Kerkyacharian
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  2. Oleg Lepski
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  3. Dominique Picard
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Additional information

Received: 11 November 1999 / Revised version: 14 November 2000 / Published online: 24 July 2001

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Cite this article

Kerkyacharian, G., Lepski, O. & Picard, D. Nonlinear estimation in anisotropic multi-index denoising. Probab Theory Relat Fields 121, 137–170 (2001). https://doi.org/10.1007/PL00008800

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  • Issue Date: October 2001

  • DOI: https://doi.org/10.1007/PL00008800

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Keywords

  • Besov Space
  • Kernel Estimation
  • Nonlinear Estimation
  • Local Selection
  • Logarithmic Factor
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