Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases
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In Part II of this paper we extend the results obtained in Part I for total variation minimization in image restoration towards the following directions: first we investigate the decomposability property of energies on levels, which leads us to introduce the concept of levelable regularization functions (which TV is the paradigm of). We show that convex levelable posterior energies can be minimized exactly using the level-independant cut optimization scheme seen in Part I. Next we extend this graph cut scheme to the case of non-convex levelable energies.We present convincing restoration results for images corrupted with impulsive noise. We also provide a minimum-cost based algorithm which computes a global minimizer for Markov Random Field with convex priors. Last we show that non-levelable models with convex local conditional posterior energies such as the class of generalized Gaussian models can be exactly minimized with a generalized coupled Simulated Annealing.
Keywordstotal variation level sets convexity Markov Random fields graph cuts levelable functions
Kluwer Academic Publishers
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- 1.R. Ahuja, T. Magnanti, and J. Orlin, Network Flows: Theory, Algorithms and Applications. Prentice Hall, 1993.Google Scholar
- 4.A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, 1987.Google Scholar
- 5.C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Transactions on Transactions on Signal Processing, Vol. 2, No. 3, pp. 296–310, 1993.Google Scholar
- 6.S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.Google Scholar
- 11.G. Giorgi, A. Guerraggio, and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier Science, 2004.Google Scholar
- 12.D. Greig, B. Porteous, and A. Seheult, “Exact maximum a posteriori estimation for binary images,” Journal of the Royal Statistics Society, Vol. 51, No. 2, pp. 271–279, 1989.Google Scholar
- 13.F. Guichard and J. Morel, “Mathematical morphology “almost everywhere,”” in Proceedings of ISMM, 2002, pp. 293–303.Google Scholar
- 19.S. Osher, A. Solé, and L. Vese, “Image decomposition and restoration using total variation minimization and the H −1 norm,” J. Mult. Model. and Simul., Vol. 1, No. 3, 2003.Google Scholar
- 23.B. Zalesky, “Efficient determination of Gibbs estimators with submodular energy functions,” http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:math/0304041, 2003.