Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Non linear estimation in anisotropic multiindex denoising II


Code(s) de Classification MSC:

Résumé: In dimension one, it has long been observed that the minimax rates of convergences in the scale of Besov spaces present essentially two regimes (and a boundary): dense and the sparse zones. In this paper, we consider the problem of denoising a function depending of a multidimensional variable (for instance an image), with anisotropic constraints of regularity (especially providing a possible disparity of the inhomogeneous aspect in the different directions). The case of the dense zone has been investigated in a former paper. Here, our aim is to investigate the case of the sparse region. This case is more delicate in some aspects. For instance, it was an open question to decide whether this sparse case, in the $d$ dimensional context has to be split into different regions corresponding to different minimax rates. We will see here that the answer in negative: we still observe a sparse region but with a unique minimax behavior, except, as usual, on the boundary. It is worthwhile to notice that our estimation procedure admits the choice of its parameters under which it is adaptive up to logarithmic factor in the "dense case" (see the former paper) and minimax adaptive in the "sparse case". It is also interesting to observe that in the "sparse case", the embedding properties of the spaces are fondamental.

Mots Clés: non parametric estimation ; denoising ; anisotropic smoothness ; minimax rate of convergence ; curse of dimensionality ; anisotropic Besov spaces

Date: 2003-03-21

Prépublication numéro: PMA-810

Front pages.