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Denoising by frame thresholding is one of the most basic and efficientmethods for recovering a discrete signal or image from data that are corruptedby additive Gaussian white noise. The basic idea is to select a frame ofanalyzing elements that separates the data in few large coefficients due to thesignal and many small coefficients mainly due to the noise \epsilon_n. Removingall data coefficients being in magnitude below a certain threshold yields areconstruction of the original signal. In order to properly balance the amountof noise to be removed and the relevant signal features to be kept, a preciseunderstanding of the statistical properties of thresholding is important. Forthat purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n}|<\phi_\omega^n,\epsilon_n>| for a wide class of redundant frames(\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we givea rationale for universal extreme value thresholding techniques yieldingasymptotically sharp confidence regions and smoothness estimates correspondingto prescribed significance levels. The results cover many frames used inimaging and signal recovery applications, such as redundant wavelet systems,curvelet frames, or unions of bases. We show that `generically' a standardGumbel law results as it is known from the case of orthonormal wavelet bases.However, for specific highly redundant frames other limiting laws may occur. Weindeed verify that the translation invariant wavelet transform shows adifferent asymptotic behaviour.

Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding