Abstract
A generalized iterative regularization procedure based on the total variation penalization is introduced for image denoising models with non-quadratic convex fidelity terms. By using a suitable sequence of penalty parameters we solve the issue of solvability of minimization problems arising in each step of the iterative procedure, which has been encountered in a recently developed iterative total variation procedure Furthermore, we obtain rigorous convergence results for exact and noisy data.
We test the behaviour of the algorithm on real images in several numerical experiments using L 1 and L 2 fitting terms. Moreover, we compare the results with other state-of-the art multiscale techniques for total variation based image restoration.
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Lin He received the B.S. degree in pure mathematics from Peking University, China in 1997 and the M.A. degree in mathematics from UCLA in 2003. She will complete her Ph.D. degree in mathematics in June 2006 from UCLA. Lin He has been working on PDE and/or wavelet based inverse problems and image processing as well as level set methods and its applications on material sciences.
Martin Burger received his master (1998) and PhD (2000) from Johannes Kepler University, Linz, Austria. After working at the Universities of Milano, Linz, and UCLA, he is currently assistant professor at Johannes Kepler University and scientific advisor at the Johann Radon Institute for Computational and Applied Mathematics (RICAM, Austrian Academy of Sciences). Dr. Burger has been working on the numerical solution and applications of inverse problems, mathematical imaging, as well as on the modelling and simulation of problems in materials and life sciences.
Stanley Osher received his MS and PhD (1966) from the Courant Institute, NYU. After working at Brookhaven National Laboratory, UC Berkeley and SUNY Stony Brook, he has been at UCLA since 1976. He is Director of Special Projects at the Institute for Pure and Applied Mathematics at UCLA. Dr. Osher is the coinventor of (i) level set methods for computing moving fronts (160,000 references on GOOGLE), (ii) ENO, WENO and other numerical methods for computing solutions to hyperbolic conservation laws and Hamilton-Jacobi equations, (iii) total variation and other PDE based image processing techniques. He has been a Fulbright and Alfred P. Sloan Fellow, received the NASA Public Service Group Achievment Award, Japan Society of Mechanical Engineers Computational Mechanics Award, was an invited speaker at the International Congress of Mathematicians, received the SIAM Pioneer Prize at the last ICIAM conference, the SIAM Kleinman Prize at the last SIAM national meeting and was just (May, 2005) elected to the US National Academy of Sciences.
He has cofounded 3 successful companies, based, in part, on his own research.
His work has been written up numerous times in the scientific and international media, e.g., Science News, Die Zeit. He is a highly cited researcher, according to web-of-science and is the Associate Editor of a number of major journals.
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He, L., Burger, M. & Osher, S.J. Iterative Total Variation Regularization with Non-Quadratic Fidelity. J Math Imaging Vis 26, 167–184 (2006). https://doi.org/10.1007/s10851-006-8302-3
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DOI: https://doi.org/10.1007/s10851-006-8302-3