Exact Optimization of Discrete Constrained Total Variation Minimization Problems
This paper deals with the total variation minimization problem when the fidelity is either the L 2-norm or the L 1-norm. We propose an algorithm which computes the exact solution of these two problems after discretization. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems associated with each level set. Exact solutions of these binary problems are found thanks to minimum-cut techniques. We prove that these binary solutions are increasing and thus allow to reconstruct the solution of the original problems.
KeywordsMonotone Property Markov Random Field Level Line Gradient Descent Algorithm Binary Problem
Unable to display preview. Download preview PDF.
- 5.Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE PAMI 23, 1222–1239 (2001)Google Scholar
- 6.Pollak, I., Willsky, A., Huang, Y.: Nonlinear evolution equations as fast and exact solvers of estimation problems. To appear in IEEE Signal Processing (2004)Google Scholar
- 7.Amini, A., Weymouth, T., Jain, R.: Using dynamic programming for solving variational problems in vision. IEEE PAMI 12, 855–867 (1990)Google Scholar
- 8.Ishikawa, H.: Exact optimization for Markov random fields with priors. IEEE PAMI 25, 1333–1336 (2003)Google Scholar
- 10.Chan, T., Esedog̃lu, S.: Aspect of total variation regularized l 1 function approximation. Technical Report 7, UCLA (2004)Google Scholar
- 12.Nguyen, H., Worring, M., van den Boomgaard, R.: Watersnakes: Energy-driven watershed segmentation. IEEE PAMI 23, 330–342 (2003)Google Scholar
- 13.Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Applications of mathematics. Springer, Heidelberg (2003)Google Scholar
- 14.Guichard, F., Morel, J.: Mathematical morphology ”almost everywhere”. In: Proceedings of ISMM, pp. 293–303. Csiro Publishing (2002)Google Scholar
- 17.Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistics Society 51, 271–279 (1989)Google Scholar
- 19.Kolmogorov, V., Zabih, R.: What energy can be minimized via graph cuts? IEEE PAMI 26, 147–159 (2004)Google Scholar
- 20.Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE PAMI 26, 1124–1137 (2004)Google Scholar
- 22.Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22 (2001)Google Scholar