Error estimation for Bregman iterations and inverse scale space methods in image restoration
 M. Burger,
 E. Resmerita,
 L. He
 … show all 3 hide
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In this paper, we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. Burger in Inverse Prob 20: 1411–1421, 2004; Resmerita in Inverse Prob 21: 1303–1314, 2005; Inverse Prob 22: 801–814, 2006), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (nondifferentiable, not strictly convex) are used to achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and timecontinuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.
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 Title
 Error estimation for Bregman iterations and inverse scale space methods in image restoration
 Journal

Computing
Volume 81, Issue 23 , pp 109135
 Cover Date
 20071101
 DOI
 10.1007/s006070070245z
 Print ISSN
 0010485X
 Online ISSN
 14365057
 Publisher
 Springer Vienna
 Additional Links
 Topics
 Keywords

 Primary 47A52
 Secondary 49M30
 94A08
 image restoration
 error estimation
 iterative regularization
 Bregman distance
 total variation
 wavelets
 Industry Sectors
 Authors

 M. Burger ^{(1)}
 E. Resmerita ^{(2)}
 L. He ^{(2)}
 Author Affiliations

 1. Institut für Numerische und Angewandte Mathematik, Westfälische WilhelmsUniversität Münster, Einsteinstr. 62, 48149, Münster, Germany
 2. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria