The Annals of Statistics

Thresholding estimators for linear inverse problems and deconvolutions

Jérôme Kalifa and Stéphane Mallat

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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

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

First available in Project Euclid: 26 February 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Ill-posed inversed problems minimax optimality wavelet packets hyperbolic deconvolution


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.

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