Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T23:13:51.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic annealing for nearest-neighbour point processes with application to object recognition

Part of: Artificial intelligence (68Txx) Geometric probability and stochastic geometry Markov processes Parametric inference

Published online by Cambridge University Press:  01 July 2016

M. N. M. Van Lieshout
M. N. M. Van Lieshout*
Affiliation:
University of Warwick
*
* Present address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study convergence in total variation of non-stationary Markov chains in continuous time and apply the results to the image analysis problem of object recognition. The input is a grey-scale or binary image and the desired output is a graphical pattern in continuous space, such as a list of geometric objects or a line drawing. The natural prior models are Markov point processes found in stochastic geometry. We construct well-defined spatial birth-and-death processes that converge weakly to the posterior distribution. A simulated annealing algorithm involving a sequence of spatial birth-and-death processes is developed and shown to converge in total variation to a uniform distribution on the set of posterior mode solutions. The method is demonstrated on a tame example.

Keywords

CONVERGENCE IN TOTAL VARIATIONHOUGH TRANSFORMLIKELIHOOD RATIOMAXIMUM A POSTERIORI ESTIMATIONNEAREST-NEIGHBOUR MARKOV PROCESSESSPATIAL BIRTH-AND-DEATH PROCESSESSTOCHASTIC ANNEALING

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 62F15: Bayesian inference 68T10: Pattern recognition, speech recognition
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 281 - 300
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research carried out at the Free University, Amsterdam, and CWI, Amsterdam.

References

[1] Baddeley, A. and Møller, J. (1989) Nearest-neighbour Markov point processes and random sets. Internat. Statist. Rev. 57, 89–121.CrossRefGoogle Scholar
[2] Baddeley, A. J. (1992) An error metric for binary images. In Robust Computer Vision, ed. Forstner, W. and Ruwiedel, S. pp. 59–78. Wichmann, Karlsruhe.Google Scholar
[3] Baddeley, A. J. and Van Lieshout, M. N. M. (1991) Recognition of overlapping objects using Markov spatial models. CWI Report BS-R9109, March 1991.Google Scholar
[4] Baddeley, A. J. and Van Lieshout, M. N. M. (1992) ICM for object recognition. In Computational Statistics, ed. Dodge, Y. and Whittaker, J., Vol. 2, pp. 271–286, Physica/Springer, Heidelberg.CrossRefGoogle Scholar
[5] Baddeley, A. J. and Van Lieshout, M. N. M. (1992) Object recognition using Markov spatial processes. In Proc. 11th IAPR Internat. Conf. on Pattern Recognition, Volume B, pp. 136–139, IEEE Computer Society Press, Los Alamitos.Google Scholar
[6] Baddeley, A. J. and Van Lieshout, M. N. M. (1993) Stochastic geometry models in high-level vision. In: Statistics and Images, Vol. 1, ed. Mardia, K. V. and Kanji, G. K.. Special issue J. Appl. Statist. 20, 233–258.Google Scholar
[7] Besag, J. (1986) On the statistical analysis of dirty pictures (with discussion). J. R. Statist. Soc. Ser. B 48, 259–302.Google Scholar
[8] Dobrushin, R. L. (1956) Central limit theorem for non-stationary Markov chains I, II. Theory Prob. Appl. 1, 65–80, 329-383.Google Scholar
[9] Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence 6, 721–741.Google Scholar
[10] Geman, S. and Hwang, C. R. (1986) Diffusions for global optimization. SIAM. J. Control and Optimization 24, 1031–1043.Google Scholar
[11] Geman, D. (1990) Random fields and inverse problems in imaging. In Ecole d'été de Probabilités de Saint-Flour XVIII 1988. Lecture Notes in Mathematics 1427. Springer-Verlag, Berlin.Google Scholar
[12] Geman, D., Geman, S., Graffigne, C. and Dong, P. (1990) Boundary detection by constrained optimization. IEEE Trans. Pattern Analysis and Machine Intelligence 12, 609–628.Google Scholar
[13] Greig, D. M., Porteous, B. T. and Seheult, A. H. (1989) Exact maximum a posteriori estimation for binary images. J. R. Statist. Soc. B 51, 271–279.Google Scholar
[14] Grenander, U. (1976) Lectures on Pattern Theory. Vol. 1; Pattern Synthesis. Applied Mathematical Sciences Vol. 18, Springer-Verlag, Berlin.Google Scholar
[15] Grenander, U. (1978) Lectures on Pattern Theory. Vol. 2: Pattern Analysis Applied Mathematical Sciences Vol. 24, Springer-Verlag, Berlin.Google Scholar
[16] Grenander, U. (1981) Lectures on Pattern Theory. Vol. 3; Regular Structures. Applied Mathematical Sciences, Vol. 33. Springer-Verlag, Berlin.Google Scholar
[17] Grenander, U. and Keenan, D. M. (1989) A computer experiment in pattern theory. Commun. Statist.-Stoch. Models 5, 531–533.Google Scholar
[18] Haario, H. and Saksman, E. (1991) Simulated annealing process in general state space. Adv. Appl. Prob. 23, 866–893.CrossRefGoogle Scholar
[19] Hough, P. V. C. (1962) Method and means for recognizing complex patterns. US Patent 3,069,654.Google Scholar
[20] Illingworth, J. and Kittler, J. (1988) A survey of the Hough transform. Computer Vision, Graphics and Image Processing 44, 87–116.Google Scholar
[21] Ten Kate, T. K., Van Balen, R., Smeulders, A. W. M., Groen, F. C. A. and Den Boer, G. A. (1990) SCILAIM: a multi-level interactive image processing environment. Pattern Recognition Letters 11, 429–441.CrossRefGoogle Scholar
[22] Lotwick, H. W. and Silverman, B. W. (1981) Convergence of spatial birth-and-death processes. Math. Proc. Camb. Phl. Soc. 90, 155–165.Google Scholar
[23] Miller, M. et al. (1991) Automated segmentation of biological shapes in electron microscopic autoradiography. In 25th Annual Conf. Inf. Sci. Systems, ed. Davidson, F. and Goutsias, J..Google Scholar
[24] Møller, J. (1989) On the rate of convergence of spatial birth-and-death processes. Ann. Inst. Statist. Math. 41, 565–581.Google Scholar
[25] Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
[26] Preston, C. J. (1977) Spatial birth-and-death processes. Bull. Internat. Statist. Inst. 46, 371–391.Google Scholar
[27] Ripley, B. D. (1977) Modelling spatial patterns (with discussion). J. R. Statist. Soc. B 39, 172–212.Google Scholar
[28] Ripley, B. D. and Kelly, F. P. (1977) Markov point processes. J. London Math. Soc. 15, 188–192.CrossRefGoogle Scholar
[29] Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar
[30] Winkler, G. (1990) An ergodic L2 theorem for simulated annealing in Bayesian image reconstruction. J. Appl. Prob. 28, 779–791.Google Scholar