Thresholding estimators for linear inverse problems and deconvolutions
Jérôme Kalifa and Stéphane Mallat
Source: Ann. Statist. Volume 31, Number 1 (2003), 58-109.
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.
Primary Subjects: 49K35, 34A55
Secondary Subjects: 42C40
Keywords: Ill-posed inversed problems; minimax optimality; wavelet packets; hyperbolic deconvolution
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1046294458
Digital Object Identifier: doi:10.1214/aos/1046294458
Mathematical Reviews number (MathSciNet):
MR1962500
Zentralblatt MATH identifier:
01934465
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