An iterative method with general convex fidelity term for image restoration
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part I
Primal and Dual Bregman Methods with Application to Optical Nanoscopy
International Journal of Computer Vision
Dual Norm Based Iterative Methods for Image Restoration
Journal of Mathematical Imaging and Vision
A Variational Approach for Sharpening High Dimensional Images
SIAM Journal on Imaging Sciences
A Combined First and Second Order Variational Approach for Image Reconstruction
Journal of Mathematical Imaging and Vision
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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 (non-differentiable, 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 time-continuous 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.