### 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.

Original language | English |
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Pages (from-to) | 857-885 |

Number of pages | 29 |

Journal | SIAM Journal on Imaging Sciences |

Volume | 5 |

Issue number | 3 |

DOIs | |

Publication status | Published - 17 Jul 2012 |

### Keywords

- image deblurring
- wavelet decomposition method
- convex optimization
- oblique thresholding
- total variation minimization
- alternating minimization

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## Cite this

Fornasier, M., Kim, Y., Langer, A., & Schönlieb, C-B. (2012). Wavelet decomposition method for L2/TV-image deblurring.

*SIAM Journal on Imaging Sciences*,*5*(3), 857-885. https://doi.org/10.1137/100819801