Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring

M. Fornasier, Y. Kim, A. Langer, C.-B. Schönlieb

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Abstract

In this paper, we show additional properties of the limit of a sequence produced by the subspace correction algorithm proposed by Fornasier and Schönlieb [SIAM J. Numer. Anal., 47 (2009), pp. 3397-3428 for L 2/TV-minimization problems. An important but missing property of such a limiting sequence in that paper is the convergence to a minimizer of the original minimization problem, which was obtained in [M. Fornasier, A. Langer, and C.-B. Schönlieb, Numer. Math., 116 (2010), pp. 645-685 with an additional condition of overlapping subdomains. We can now determine when the limit is indeed a minimizer of the original problem. Inspired by the work of Vonesch and Unser [IEEE Trans. Image Process., 18 (2009), pp. 509-523], we adapt and specify this algorithm to the case of an orthogonal wavelet space decomposition for deblurring problems and provide an equivalence condition to the convergence of such a limiting sequence to a minimizer. We also provide a counterexample of a limiting sequence by the algorithm that does not converge to a minimizer, which shows the necessity of our analysis of the minimizing algorithm. © 2012 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)857-885
Number of pages29
JournalSIAM Journal on Imaging Sciences
Volume5
Issue number3
DOIs
StatePublished - Jul 19 2012
Externally publishedYes

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