Abstract
This paper copes with the optimization of Markov Random Fields with pairwise interactions defined on arbitrary graphs. The set of labels is assumed to be linearly ordered and the priors are supposed to be submodular. Under these assumptions we propose an algorithm which computes an exact minimizer of the Markovian energy. Our approach relies on mapping the original into a combinatorial one which involves only binary variables. The latter is shown to be exactly solvable via computing a maximum flow. The restatement into a binary combinatorial problem is done by considering the level-sets of the labels instead of the label values themselves. The submodularity of the priors is shown to be a necessary and sufficient condition for the applicability of the proposed approach.
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
Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)
Amini, A., Weymouth, T., Jain, R.: Using Dynamic Programming for Solving Variational Problems in Vision. IEEE Transactions on Pattern Analysis and Machine Interaction 12(9), 855–867 (1990)
Boykov, Y., Kolmogorov, V.: An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision. IEEE Transactions on Pattern Analysis and Machine Interaction 26(9), 1124–1137 (2004)
Boykov, Y., Veksler, O., Zabih, R.: Fast Approximate Energy Minimization via Graph Cuts. IEEE Transactions on Pattern Analysis and Machine Interaction 23(11), 1222–1239 (2001)
Darbon, J., Sigelle, M.: Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization. Journal of Mathematical Imaging and Vision 26(3), 261–276 (2006)
Darbon, J., Sigelle, M.: Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases. Journal of Mathematical Imaging and Vision 26(3), 277–291 (2006)
Darbon, J., Sigelle, M., Tupin, F.: The use of levelable regularization functions for MRF restoration of SAR images while preserving reflectivity. In: The proceedings of the IS&T/SPIE 19th Annual Symposium Electronic Imaging (oral presentation). Conference on Computational Imaging (E112), San Jose, CA, USA (2007)
Durand, S., Nikolova, M.: Stability of the Minimizers of Least Squares with a Non-Convex Regularization. Part II: Global Behavior. Journal of Applied Mathematics and Optimization 53(3), 259–277 (2006)
Geman, S., Geman, D.: Stochastic Relaxation, Gibbs Distributions, And The Bayesian Restoration Of Images. IEEE Transactions on Pattern Analysis and Machine Interaction 6(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(2), 271–279 (1989)
Guichard, F., Morel, J.M.: Mathematical Morphology “Almost Everywhere”. In: Proceedings of International Symposium on Mathematical Morphology, pp. 293–303. Csiro Publishing (2002)
Hochbaum, D.S.: An efficient algorithm for image segmentation, Markov random fields and related problems. Journal of the ACM 48(2), 686–701 (2001)
Ishikawa, H.: Exact optimization for Markov random fields with convex priors. IEEE Transactions on Pattern Analysis and Machine Interaction 25(10), 1333–1336 (2003)
Kolmogorov, V., Zabih, R.: What Energy can be Minimized via Graph Cuts? IEEE Transactions on Pattern Analysis and Machine Interaction 26(2), 147–159 (2004)
Maragos, P., Ziff, R.: Threshold superposition in morphological image analysis systems. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(5), 498–504 (1990)
Murota, K.: Discrete Convex Optimization. SIAM Society for Industrial and Applied Mathematics (2003)
Nikolova, M.: Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares. SIAM Journal on Multiscale Modeling and Simulation 4(3), 960–991 (2005)
Nikolova, M.: Model distortions in Bayesian MAP reconstruction. AIMS Journal on Inverse Problems and Imaging 1(2), 399–422 (2007)
Picard, J., Ratlif, H.: Minimum cuts and related problem. Networks 5, 357–370 (1975)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear Total Variation Based Noise Removal Algorithms. Physica D. 60, 259–268 (1992)
Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. A Mathematical Introduction. In: Applications of mathematics, 3rd edn., Springer, Heidelberg (2006)
Zalesky, B.: Efficient Determination of Gibbs Estimator with Submodular Energy Functions. Technical report, United Institution of Information Problem (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Darbon, J. (2008). Global Optimization for First Order Markov Random Fields with Submodular Priors. In: Brimkov, V.E., Barneva, R.P., Hauptman, H.A. (eds) Combinatorial Image Analysis. IWCIA 2008. Lecture Notes in Computer Science, vol 4958. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78275-9_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-78275-9_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78274-2
Online ISBN: 978-3-540-78275-9
eBook Packages: Computer ScienceComputer Science (R0)