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  Extreme value analysis of frame coefficients and implications for image denoising.

Haltmeier, M., & Munk, A. (2014). Extreme value analysis of frame coefficients and implications for image denoising. Applied and Computational Harmonic Analysis, 36(3), 434-460. doi:10.1016/j.acha.2013.07.004.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-000F-9D6D-D Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0027-CA91-0
Genre: Journal Article

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 Creators:
Haltmeier, M.1, Author              
Munk, A.1, Author              
Affiliations:
1Research Group of Statistical Inverse-Problems in Biophysics, MPI for biophysical chemistry, Max Planck Society, ou_1113580              

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Free keywords: Denoising; Thresholding estimation; Extreme value analysis; Gumbel distribution; Berman's inequality; Wavelet thresholding; Curvelet thresholding; Translation invariant wavelets; Frames; Redundant dictionaries
 Abstract: Denoising by frame thresholding is one of the most basic and efficient methods for recovering a discrete signal or image from data that are corrupted by additive Gaussian white noise. The basic idea is to select a frame of analyzing elements that separates the data in few large coefficients due to the signal and many small coefficients mainly due to the noise ϵnϵn. Removing all data coefficients being in magnitude below a certain threshold yields a reconstruction of the original signal. In order to properly balance the amount of noise to be removed and the relevant signal features to be kept, a precise understanding of the statistical properties of thresholding is important. For that purpose we derive the asymptotic distribution of View the MathML sourcemaxω∈Ωn|〈ϕωn,ϵn〉| for a wide class of redundant frames View the MathML source(ϕωn:ω∈Ωn). Based on our theoretical results we give a rationale for universal extreme value thresholding techniques yielding asymptotically sharp confidence regions and smoothness estimates corresponding to prescribed significance levels. The results cover many frames used in imaging and signal recovery applications, such as redundant wavelet systems, curvelet frames, or unions of bases. We show that ‘generically’ a standard Gumbel law results as it is known from the case of orthonormal wavelet bases. However, for specific highly redundant frames other limiting laws may occur. We indeed verify that the translation invariant wavelet transform shows a different asymptotic behaviour.

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Language(s): eng - English
 Dates: 20122014-05
 Publication Status: Published in print
 Pages: 27
 Publishing info: -
 Table of Contents: -
 Rev. Method: Peer
 Identifiers: DOI: 10.1016/j.acha.2013.07.004
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Title: Applied and Computational Harmonic Analysis
Source Genre: Journal
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Pages: - Volume / Issue: 36 (3) Sequence Number: - Start / End Page: 434 - 460 Identifier: -