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.
"Thresholding estimators for linear inverse problems and deconvolutions." Ann. Statist. 31 (1) 58 - 109, Februrary 2003. https://doi.org/10.1214/aos/1046294458