HAL : hal-00409903, version 2

arXiv : 0908.2514

DOI : 10.3150/10-BEJ340

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Bernoulli 18, 2 (2012) 391-433

Versions disponibles :

v1 (18-08-2009)

v2 (09-05-2012)

Radon needlet thresholding

Gerard Kerkyacharian 1, Erwan Le Pennec 1, 2, 3, Dominique Picard 1

(2012)

We provide a new algorithm for the treatment of the noisy inversion of the Radon transform using an appropriate thresholding technique adapted to a well-chosen new localized basis. We establish minimax results and prove their optimality. In particular, we prove that the procedures provided here are able to attain minimax bounds for any $\mathbb {L}_p$ loss. It s important to notice that most of the minimax bounds obtained here are new to our knowledge. It is also important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover and to inhomogeneous smoothness. We perform a numerical study that is of importance since we especially have to discuss the cubature problems and propose an averaging procedure that is mostly in the spirit of the cycle spinning performed for periodic signals.

1 :	Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
	CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
2 :	SELECT (INRIA Saclay - Ile de France)
	INRIA – Université Paris XI - Paris Sud – CNRS : UMR
3 :	Laboratoire de Mathématiques d'Orsay (LM-Orsay)
	CNRS : UMR8628 – Université Paris XI - Paris Sud

Domaine

:

Mathématiques/Statistiques Statistiques/Théorie

Mots Clés : statistical inverse problems – minimax estimation – second-generation wavelets

hal-00409903, version 2

http://hal.archives-ouvertes.fr/hal-00409903

oai:hal.archives-ouvertes.fr:hal-00409903

Contributeur : Erwan Le Pennec <>

Soumis le : Mardi 8 Mai 2012, 10:27:13

Dernière modification le : Mercredi 9 Mai 2012, 09:56:44