Radon needlet thresholding

Gerard Kerkyacharian (1) , Erwan Le Pennec (1) , Dominique Picard (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $\bL_p$ loss. It is 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 as well as to inhomogeneous smoothness. We also perform a numerical study which is of importance since we especially have to discuss the cubature problems and propose an averaging procedure which is mostly in the spirit of the cycle spinning performed for periodic signals.

Mots clés: statistical inverse problems; minimax estimation; second-generation wavelets

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Gerard Kerkyacharian, Erwan Le Pennec, Dominique Picard. Radon needlet thresholding. 2009. ⟨hal-00409903v1⟩