Bernoulli

  • Bernoulli
  • Volume 18, Number 2 (2012), 391-433.

Radon needlet thresholding

Gérard Kerkyacharian, Erwan Le Pennec, and Dominique Picard

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Abstract

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 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 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.

Article information

Source
Bernoulli Volume 18, Number 2 (2012), 391-433.

Dates
First available in Project Euclid: 16 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1334580718

Digital Object Identifier
doi:10.3150/10-BEJ340

Mathematical Reviews number (MathSciNet)
MR2922455

Zentralblatt MATH identifier
1243.65152

Keywords
minimax estimation second-generation wavelets statistical inverse problems

Citation

Kerkyacharian, Gérard; Le Pennec, Erwan; Picard, Dominique. Radon needlet thresholding. Bernoulli 18 (2012), no. 2, 391--433. doi:10.3150/10-BEJ340. https://projecteuclid.org/euclid.bj/1334580718


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