Abstract
Active-set methods based on augmented Lagrangian smoothing of nondifferentiable optimization problems arising in image restoration are studied through numerical experiments. Implemented algorithms for solving both one-dimensional and two-dimensional image restoration problems are described. A new, direct way for solving the constrained optimization problem appearing in the inner iteration of the monotone algorithms is presented. Several numerical examples are included.
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KÄrkkÄinen, T., and Majava, K., Nonmonotone and Monotone Active Set Methods for Image Restoration, Part 1: Convergence Analysis, Journal of Optimization Theory and Applications, Vol. 106, pp. 61-80, 2000.
Rudin, L. I., Osher, S., and Fatemi, E., Nonlinear Total-Variation Based Noise Removal Algorithms, Physica D, Vol. 60, pp. 259-268, 1992.
Vogel, C. R., and Oman, M. E., Iterative Methods for Total-Variation Denoising, SIAM Journal on Scientific Computing, Vol. 17, pp. 227-238, 1996.
Vogel, C. R., and Oman, M. E., Fast, Robust Total-Variation Based Reconstruction of Noisy, Blurred Images, IEEE Transactions on Image Processing, Vol. 7, pp. 813-824, 1998.
Chan, T., Golub, G., and Mulet, P., A Nonlinear Primal—Dual Method for Total-Variation Based Image Restoration, SIAM Journal on Scientific Computing, Vol. 20, pp. 1964-1977, 1999.
Chambolle, A., and Lions, P. L., Image Recovery via Total-Variation Minimization and Related Problems, Numerische Mathematik, Vol. 76, pp. 167-188, 1997.
Ito, K., and Kunisch, K., An Active Set Strategy Based on the Augmented Lagrangian Formulation for Image Restoration, Mathematical Modelling and Numerical Analysis, Vol. 33, pp. 1-21, 1999.
Majava, K., Kohinanpoisto Lagrange-Menetelmillä, MS Thesis, University of Jyväskylä, 1998 (in Finnish).
KÄrkkÄinen, T., and Majava, K., Nonmonotone and Monotone Active Set Methods for Image Restoration, Technical Report B9y/999, Department of Mathematical Information Technology, University of Jyväskylä, 1999.
Casas, E., Kunisch, K., and Pola, C., Regularization by Functions of Bounded Variation and Applications to Image Enhancement, Applied Mathematics and Optimization, Vol. 40, pp. 229-257, 1999.
Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.
Hackbusch, W., Multigrid Methods and Applications, Springer Series in Computational Mathematics, Springer Verlag, Berlin, Germany, 1985.
Ito, K., and Kunisch, K., Augmented Lagrangian Formulation of Nonsmooth, Convex Optimization in Hilbert Spaces, Control of Partial Differential Equations and Applications, Edited by E. Casas, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, Vol. 174, pp. 107-117, 1995.
Ekeland, I., and Temam, R., Convex Analysis and Variational Problems, Studies in Mathematics and Its Applications, North-Holland, Amsterdam, Holland, Vol. 1, 1976.
Glowinski, R., Lions, J. L., and Tremolieres, R., Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981.
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KÄRKKÄINEN, T., Majava, K. Nonmonotone and Monotone Active-Set Methods for Image Restoration, Part 2: Numerical Results. Journal of Optimization Theory and Applications 106, 81–105 (2000). https://doi.org/10.1023/A:1004607123926
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DOI: https://doi.org/10.1023/A:1004607123926