Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Sauer, K., Bouman, C.: Bayesian estimation of transmission tomograms using segmentation based optimization. IEEE Nuclear Science 39, 1144–1152 (1992)
Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Chambolle, A.: An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision 20, 89–97 (2004)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE PAMI 23, 1222–1239 (2001)
Pollak, I., Willsky, A., Huang, Y.: Nonlinear evolution equations as fast and exact solvers of estimation problems. To appear in IEEE Signal Processing (2004)
Amini, A., Weymouth, T., Jain, R.: Using dynamic programming for solving variational problems in vision. IEEE PAMI 12, 855–867 (1990)
Ishikawa, H.: Exact optimization for Markov random fields with priors. IEEE PAMI 25, 1333–1336 (2003)
Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Signal Processing 45, 913–917 (1997)
Chan, T., Esedog̃lu, S.: Aspect of total variation regularized l 1 function approximation. Technical Report 7, UCLA (2004)
Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. SIAM J. Num. Anal. 40, 965–994 (2002)
Nguyen, H., Worring, M., van den Boomgaard, R.: Watersnakes: Energy-driven watershed segmentation. IEEE PAMI 23, 330–342 (2003)
Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Applications of mathematics. Springer, Heidelberg (2003)
Guichard, F., Morel, J.: Mathematical morphology ”almost everywhere”. In: Proceedings of ISMM, pp. 293–303. Csiro Publishing (2002)
Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and statistical mechanics. Random Structures and Algorithms 9, 223–252 (1996)
Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE PAMI 6, 721–741 (1984)
Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistics Society 51, 271–279 (1989)
Roy, S.: Stereo without epipolar lines: A maximum-flow formulation. International Journal of Computer Vision 34, 147–162 (1999)
Kolmogorov, V., Zabih, R.: What energy can be minimized via graph cuts? IEEE PAMI 26, 147–159 (2004)
Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE PAMI 26, 1124–1137 (2004)
Strong, D., Chan, T.: Edge preserving and scale-dependent properties of total variation regularization. Inverse Problem 19, 165–187 (2003)
Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Darbon, J., Sigelle, M. (2004). Exact Optimization of Discrete Constrained Total Variation Minimization Problems. In: Klette, R., Žunić, J. (eds) Combinatorial Image Analysis. IWCIA 2004. Lecture Notes in Computer Science, vol 3322. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30503-3_40
Download citation
DOI: https://doi.org/10.1007/978-3-540-30503-3_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23942-0
Online ISBN: 978-3-540-30503-3
eBook Packages: Computer ScienceComputer Science (R0)