## The Annals of Statistics

### Thresholding estimators for linear inverse problems and deconvolutions

#### Abstract

Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set $\Theta$, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new "mirror wavelet" basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images.

#### Article information

Source
Ann. Statist., Volume 31, Number 1 (2003), 58-109.

Dates
First available in Project Euclid: 26 February 2003

https://projecteuclid.org/euclid.aos/1046294458

Digital Object Identifier
doi:10.1214/aos/1046294458

Mathematical Reviews number (MathSciNet)
MR1962500

Zentralblatt MATH identifier
1102.62318

Subjects
Primary: 49K35: Minimax problems 34A55: Inverse problems
Secondary: 42C40: Wavelets and other special systems

#### Citation

Kalifa, Jérôme; Mallat, Stéphane. Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 (2003), no. 1, 58--109. doi:10.1214/aos/1046294458. https://projecteuclid.org/euclid.aos/1046294458

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