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A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration (2007)
| Venue: | IEEE TRANSACTIONS ON IMAGE PROCESSING |
| Citations: | 182 - 26 self |
BibTeX
@MISC{Bioucas-Dias07anew,
author = {José M. Bioucas-Dias and Mário A. T. Figueiredo},
title = { A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration },
year = {2007}
}
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Abstract
Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these IST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step IST (TwIST) algorithms, exhibiting much faster convergence rate than IST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers ( norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.
Keyphrases
image restoration new twist two-step iterative shrinkage thresholding algorithm convergence rate total variation new method linear observation operator ist algorithm linear observation model objective function convergence proof vast class image deconvolution linear inverse problem similar speed gain nonquadratic regularizer problem result experimental evidence nonquadratic convex regularizers wavelet-based regularization monotonic version besov norm convex unconstrained optimization problem two-step ist ill-conditioned problem iterative shrinkage thresholding noninvertible observation operator
