Abstract
We introduce a class of polygonal Markov fields driven by local activity functions. Whereas the local rather than global nature of the field specification ensures substantial additional flexibility for statistical applications in comparison to classical polygonal fields, we show that a number of simulation algorithms and graphical constructions, as developed in our previous joint work with M.N.M. van Lieshout and R. Kluszczynski, carry over to this more general framework. Moreover, we provide explicit formulae for the partition function of the model, which directly implies the availability of closed form expressions for the corresponding likelihood functions. Within the framework of this theory we develop an image segmentation algorithm based on Markovian optimization dynamics combining the simulated annealing ideas with those of Chen-style stochastic optimization, in which successive segmentation updates are carried out simultaneously with adaptive optimization of the local activity functions.
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References
Aarts, E., Korst, J.: Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing. Wiley, New York (1989)
Arak, T.: On Markovian random fields with finite number of values. In: Abstracts of Communications of the 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics, Tbilisi, p. 1 (1982)
Arak, T., Surgailis, D.: Markov fields with polygonal realizations. Probab. Theory Relat. Fields 80(4), 543–579 (1989)
Arak, T., Surgailis, D.: Consistent polygonal fields. Probab. Theory Relat. Fields 89(3), 319–346 (1991)
Arak, T., Clifford, P., Surgailis, D.: Point-based polygonal models for random graphs. Adv. Appl. Probab. 25(2), 348–372 (1993)
Beucher, S., Lantuéjoul, C.: Use of watersheds in contour detection. In: Proceedings of the International Workshop on Image Processing, Real-Time Edge and Motion Detection (1979)
Chen, K.: Simple learning algorithm for the traveling salesman problem. Phys. Rev. E 55, 7809–7812 (1997)
Clifford, P., Middleton, R.D.: Reconstruction of polygonal images. J. Appl. Stat. 16, 409–422 (1989)
Green, P.: Reversible jump MCMC computation and Bayesian model determination. Biometrika 82(4), 711–732 (1995)
Kluszczyński, R., van Lieshout, M.N.M., Schreiber, T.: An algorithm for binary image segmentation using polygonal Markov fields. In: Roli, F., Vitulano, S. (eds.) Proceedings of the 13th International Conference on Image Analysis and Processing. Lecture Notes in Computer Science, vol. 3617, pp. 383–390 (2005)
Kluszczyński, R., van Lieshout, M.N.M., Schreiber, T.: Image segmentation by polygonal Markov fields. Ann. Inst. Stat. Math. 59(3), 465–486 (2007)
Liggett, T.: Interacting Particle Systems. Springer, New York (1985)
Lloyd, S.P.: Least square quantization in PCM. IEEE Trans. Inf. Theory 28(2), 129–137 (1982)
Møller, J., Skare, Ø.: Bayesian image analysis with coloured Voronoi tesselations and a view to applications in reservoir modelling. Stat. Model. 1, 213–232 (2001)
Nicholls, G.K.: Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models. J. R. Stat. Soc., Ser. B, Stat. Methodol. 60, 643–659 (1998)
Nicholls, G.K.: Spontaneous magnetization in the plane. J. Stat. Phys. 102(5–6), 1229–1251 (2001)
Peretto, P.: An introduction to the modeling of neural networks. Collect. Alia-Saclay, Monogr. Texts Stat.. Phys. 2 (1992)
Schreiber, T.: Random dynamics and thermodynamic limits for polygonal Markov fields in the plane. Adv. Appl. Probab. 37, 884–907 (2005)
Schreiber, T.: Dobrushin-Kotecký-Shlosman theorem for polygonal Markov fields in the plane. J. Stat. Phys. 123, 631–684 (2006)
Schreiber, T.: Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations. J. Stat. Phys. 132(4), 669–705 (2008)
Schreiber, T., van Lieshout, M.N.M.: Disagreement loop and path creation/annihilation algorithms for binary planar Markov fields with applications to image segmentation. Scand. J. Stat. 37(2), 264–285 (2010)
Surgailis, D.: Thermodynamic limit of polygonal models. Acta Appl. Math. 22(1), 77–102 (1991)
van Lieshout, M.N.M., Schreiber, T.: Perfect simulation for length-interacting polygonal Markov fields in the plane. Scand. J. Stat. 34(3), 615–625 (2007)
Winkler, G.: Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction, 2nd edn. Applications of Mathematics, Stochastic Modelling and Applied Probability, vol. 27. Springer, Berlin (2003)
Yang, F., Jiang, T.: Pixon based image segmentation with Markov random fields. IEEE Trans. Image Process. 12(12), 1552–1559 (2003)
Acknowledgements
We gratefully acknowledge the support from the Polish Minister of Science and Higher Education grant N N201 385234 (2008–2010).
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Matuszak, M., Schreiber, T. (2012). Locally Specified Polygonal Markov Fields for Image Segmentation. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_15
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