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Februrary 2003 Thresholding estimators for linear inverse problems and deconvolutions
Jérôme Kalifa, Stéphane Mallat
Ann. Statist. 31(1): 58-109 (Februrary 2003). DOI: 10.1214/aos/1046294458


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 kernelshaving 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.


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Jérôme Kalifa. Stéphane Mallat. "Thresholding estimators for linear inverse problems and deconvolutions." Ann. Statist. 31 (1) 58 - 109, Februrary 2003.


Published: Februrary 2003
First available in Project Euclid: 26 February 2003

zbMATH: 1102.62318
MathSciNet: MR1962500
Digital Object Identifier: 10.1214/aos/1046294458

Primary: 34A55 , 49K35
Secondary: ‎42C40

Keywords: hyperbolic deconvolution , Ill-posed inversed problems , Minimax optimality , wavelet packets

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 1 • Februrary 2003
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