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Multi-Label Markov Random Fields as an Efficient and Effective Tool for Image Segmentation, Total Variations and Regularization

Published online by Cambridge University Press:  28 May 2015

Dorit S. Hochbaum
Dorit S. Hochbaum*
Affiliation:
Department of Industrial Engineering and Operations Research, University of California, Berkeley, Ca 94720, USA
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Abstract

One of the classical optimization models for image segmentation is the well known Markov Random Fields (MRF) model. This model is a discrete optimization problem, which is shown here to formulate many continuous models used in image segmentation. In spite of the presence of MRF in the literature, the dominant perception has been that the model is not effective for image segmentation. We show here that the reason for the non-effectiveness is due to the lack of access to the optimal solution. Instead of solving optimally, heuristics have been engaged. Those heuristic methods cannot guarantee the quality of the solution nor the running time of the algorithm. Worse still, heuristics do not link directly the input functions and parameters to the output thus obscuring what would be ideal choices of parameters and functions which are to be selected by users in each particular application context.

We describe here how MRF can model and solve efficiently several known continuous models for image segmentation and describe briefly a very efficient polynomial time algorithm, which is provably fastest possible, to solve optimally the MRF problem. The MRF algorithm is enhanced here compared to the algorithm in Hochbaum (2001) by allowing the set of assigned labels to be any discrete set. Other enhancements include dynamic features that permit adjustments to the input parameters and solves optimally for these changes with minimal computation time. Several new theoretical results on the properties of the algorithm are proved here and are demonstrated for images in the context of medical and biological imaging. An interactive implementation tool for MRF is described, and its performance and flexibility in practice are demonstrated via computational experiments.

We conclude that many continuous models common in image segmentation have discrete analogs to various special cases of MRF and as such are solved optimally and efficiently, rather than with the use of continuous techniques, such as PDE methods, that restrict the type of functions used and furthermore, can only guarantee convergence to a local minimum.

Keywords

05C2105C8568W4068U10Total variationMarkov random fieldsimage segmentationparametric cuts
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 169 - 198
Copyright
Copyright © Global Science Press Limited 2013

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References

[1] Ahuja, R. K., Hochbaum, D. S. and Orlin, J. B., A cut-based algorithm for the convex dual of the minimum cost network flow problem, Algorithmica, 39(3) (2004) pp. 189–208. Also, UC Berkeley manuscript, (1999).CrossRefGoogle Scholar
[2] Ahuja, R. K., Hochbaum, D. S. and Orlin, J. B., Solving the convex cost integer dual network flow problem, Management Science, 49(7) (2003), pp. 950–964. Extended abstract in, Proceedings of IPCO’99, Cornuejols, G., Burkard, R. E. and Woeginger, G. J. (Eds.), Lecture Notes in Computer Science, 1610 (1999), pp. 31–34.Google Scholar
[3] Bae, E. and Tai, X.-C., Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation, Scale Space and Variational Methods in Computer Vision (SSVM 2009), Tai, X.-C. and Mórken, K. and Lysaker, M. and Lie, K.-A. eds. LNCS, 5567 (2009), pp. 1–13.CrossRefGoogle Scholar
[4] Blake, A. and Zisserman, A., Visual Reconstruction, MIT Press, 1987.CrossRefGoogle Scholar
[5] Boykov, Y. and Jolly, M. P., Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images, International Conference on Computer Vision, (ICCV), I (2001), pp. 105–112.Google Scholar
[6] Boykov, Y. and Kolmogorov, V, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, IEEE Trans. Pattern Anal. Machine Intelligence, (PAMI), 26(9) (2004), pp. 1124–1113.CrossRefGoogle ScholarPubMed
[7] Boykov, Y., Veksler, O. and Zabih, R., Markov random fields with efficient approximations, Proc IEEE Conference CVPR, Santa Barbara CA, (1998), pp. 648–655.Google Scholar
[8] Boykov, Y., Veksler, O. and Zabih, R., Fast approximate energy minimization via graph cuts, Proc 7th IEEE International Conference on Computer Vision, (1999), pp. 377–384.Google Scholar
[9] Chan, T. F. and Vese, L. A., Active contours without edges, IEEE Trans. Image Processing, 10(2) (2001), pp. 266–277.CrossRefGoogle ScholarPubMed
[10] Chan, T. F. and Esedoglu, S., Aspects of total variation regularized l1 function approximation, SIAM J. Appl. Math., 65(5) (2005), pp. 1817–1837.CrossRefGoogle Scholar
[11] Chandran, B. G. and Hochbaum, D. S., A computational study of the pseudoflow and push-relabel algorithms for the maximum flow Problem, Operations Research, 57(2) (2009), pp. 358–376.CrossRefGoogle Scholar
[12] Chandran, B. G. and Hochbaum, D. S., Solver, Pseudoflow, accessed January 2012, http://riot.ieor.berkeley.edu/Applications/Pseudoflow/maxflow.html. Google Scholar
[13] Collins, D. L., Zijdenbos, A. P., Kollokian, V, Sled, J. G., Kabani, N. J., Holmes, C. J. and Evans, A. C., Design and construction of a realistic digital brain phantom, IEEE Trans. Med. Imaging, 17(3) (1998), pp. 463–468.CrossRefGoogle ScholarPubMed
[14] Cox, I. J., Rao, S. B. and Zhong, Y., Ratio regions: a technique for image segmentation, Proc. Int. Conf. Pattern Recognition, B (1996), pp. 557–564.CrossRefGoogle Scholar
[15] Darbon, J. and Sigelle, M., Image Restoration with discrete constrained total variation Part I: Fast and exact optimization, J. Math. Imaging Vis., 26(3) (2006), pp. 261–276.CrossRefGoogle Scholar
[16] Fishbain, B. Hochbaum, D. S. and Mueller, S., The Pseudoflow algorithm for minimum-Cut in vision problems, http://arxiv.org/abs/1007.4531 and UC Berkeley manuscript, 2011.Google Scholar
[17] Ford, L. R. and Fulkerson, D. R., Maximal flow through a network, Canadian J. Math., 8(3) (1956), pp. 339–404.CrossRefGoogle Scholar
[18] Gallo, G., Grigoriadis, M. D. and Tarjan, R. E., A fast parametric maximum flow algorithm and applications, SIAM J. Comput., 18 (1989), pp. 30–55.CrossRefGoogle Scholar
[19] Geiger, D. and Girosi, F., Parallel and deterministic algorithms for MRFs: surface reconstruction, IEEE Trans. Pattern Anal. Machine Interlligence, PAMI, 13 (1991), pp. 401–412.CrossRefGoogle Scholar
[20] Geman, S. and Geman, D., Stochastic relaxation, Gibbs distributions and the bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intelligence, PAMI, 6 (1984), pp. 721–741.CrossRefGoogle ScholarPubMed
[21] Goldberg, A., The partial augmentÜelabel algorithm for the maximum flow problem, ESA, (2008), pp. 466–477.Google Scholar
[22] Greig, D. M. and Porteous, B. T. and Seheult, A. H., Exact maximum a posteriori estimation for binary images, J. Royal Statis. Society Ser. B, 51(2) (1989), pp. 271–279.Google Scholar
[23] Hochbaum, D. S. and Orlin, J. B., Simplifications and speedups of the pseudoflow algorithm, Networks, to appear 2012. Online in http://onlinelibrary.wiley.com/doi/10.100 2/net.21467/abstract.Google Scholar
[24] Hochbaum, D. S., Qranfal, J. and Tanoh, G., A fast computational algorithm for segmentation of noisy medical images, Algorithmic Operations Res., 6 (2011), pp. 79–90.Google Scholar
[25] Hochbaum, D. S., The Pseudoflow algorithm: a new algorithm for the maximum flow problem, Operations Res., 58(4) (2008), pp. 992–1009.CrossRefGoogle Scholar
[26] Hochbaum, D. S., An efficient algorithm for image segmentation, Markov Random Fields and related problems, Acm, J., 48(4) (2001), pp. 686–701.Google Scholar
[27] Hochbaum, D. S., Complexity and algorithms for nonlinear optimization problems, Annal. Operations Res., 153 (2007), pp. 257–296.CrossRefGoogle Scholar
[28] Ishikawa, H. and Geiger, D., Segmentation by grouping junctions, IEEE Conference on Computer Vision and Pattern Recognition CVPR98, (1998), pp. 125–131.Google Scholar
[29] Ishikawa, H., Exact optimization for Markov Random fields with convex priors, IEEE Trans. Pattern Anal. Machine Intelligence, 25(10) (2003), pp. 1333–1336.CrossRefGoogle Scholar
[30] Kolmogorov, V. and Zabih, R., What energy functions can be minimized via graph cuts, IEEE Trans. Pattern Anal. Machine Intelligence, 26(2) (2004), pp. 147–159.CrossRefGoogle ScholarPubMed
[31] Li, S. Z., Chan, K. L. and Wang, H., Bayesian image restoration and segmentation by constrained optimization, IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’96), (1996).Google Scholar
[32] Malik, J., Belongie, S., Leung, T. and Shi, J., Contour and texture analysis for image segmentation, Int. J. Comput. Vision, 43 (2001), pp. 7–27.CrossRefGoogle Scholar
[33] Osher, S. J. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer New York, 2003.CrossRefGoogle Scholar
[34] Pock, T., Chambolle, A., Cremers, D. and Bischof, H., A convex relaxation approach for computing minimal partitions, IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’09), (2009) pp. 810–817.Google Scholar
[35] Pretorius, P. H., King, M. A., Tsui, B. M. W., Lacroix, K. J. and Xia, W., A mathematical model of motion of the heart for use in generating source and attenuation maps for simulating emission imaging, Med. Phys., 26 (1999), pp. 2323–2332.CrossRefGoogle ScholarPubMed
[36] Rudin, L. I., Osher, S. J. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.CrossRefGoogle Scholar
[37] Shi, J. and Malik, J., Normalized cuts and image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 22(8) (2000), pp. 888–905.Google Scholar