Thresholding estimators for linear inverse problems and deconvolutions
Kalifa, Jérôme ; Mallat, Stéphane
Ann. Statist., Tome 31 (2003) no. 1, p. 58-109 / Harvested from Project Euclid
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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.
Publié le : 2003-02-14
Classification:  Ill-posed inversed problems,  minimax optimality,  wavelet packets,  hyperbolic deconvolution,  49K35,  34A55,  42C40 
@article{1046294458,
     author = {Kalifa, J\'er\^ome and Mallat, St\'ephane},
     title = {Thresholding estimators for linear inverse problems and deconvolutions},
     journal = {Ann. Statist.},
     volume = {31},
     number = {1},
     year = {2003},
     pages = { 58-109},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1046294458}
}

Kalifa, Jérôme; Mallat, Stéphane. Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist., Tome 31 (2003) no. 1, pp.  58-109. http://gdmltest.u-ga.fr/item/1046294458/