Interacting and Annealing Particle Filters: Mathematics and a Recipe for Applications

  • Jürgen GallEmail author
  • Jürgen Potthoff
  • Christoph Schnörr
  • Bodo Rosenhahn
  • Hans-Peter Seidel


Interacting and annealing are two powerful strategies that are applied in different areas of stochastic modelling and data analysis. Interacting particle systems approximate a distribution of interest by a finite number of particles where the particles interact between the time steps. In computer vision, they are commonly known as particle filters. Simulated annealing, on the other hand, is a global optimization method derived from statistical mechanics. A recent heuristic approach to fuse these two techniques for motion capturing has become known as annealed particle filter. In order to analyze these techniques, we rigorously derive in this paper two algorithms with annealing properties based on the mathematical theory of interacting particle systems. Convergence results and sufficient parameter restrictions enable us to point out limitations of the annealed particle filter. Moreover, we evaluate the impact of the parameters on the performance in various experiments, including the tracking of articulated bodies from noisy measurements. Our results provide a general guidance on suitable parameter choices for different applications.


Interacting particle systems Particle filtering Annealing Motion capture 


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  1. 1.
    Alspach, D., Sorenson, H.: Nonlinear Bayesian estimation using Gaussian sum approximations. IEEE Trans. Autom. Control 17(4), 439–448 (1972) zbMATHCrossRefGoogle Scholar
  2. 2.
    Chigansky, P., Liptser, R.: Stability of nonlinear filters in nonmixing case. Ann. Appl. Probab. 14(4), 2038–2056 (2004) zbMATHCrossRefGoogle Scholar
  3. 3.
    Crisan, D., Doucet, A.: A survey of convergence results on particle filtering methods for practitioners. IEEE Trans. Signal Process. 50(3), 736–746 (2002) CrossRefGoogle Scholar
  4. 4.
    Crisan, D., Grunwald, M.: Large deviation comparison of branching algorithms versus resampling algorithms: application to discrete time stochastic filtering. Technical report, Statistical Laboratory, Cambridge University, UK, 1999 Google Scholar
  5. 5.
    Crisan, D., Del Moral, P., Lyons, T.: Discrete filtering using branching and interacting particle systems. Markov Process. Relat. Fields 5(3), 293–319 (1999) zbMATHGoogle Scholar
  6. 6.
    Deutscher, J., Blake, A., Reid, I.: Articulated body motion capture by annealed particle filtering. In: Proc. Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 1144–1149 (2000) Google Scholar
  7. 7.
    Deutscher, J., Reid, I.: Articulated body motion capture by stochastic search. Int. J. Comput. Vis. 61(2), 185–205 (2005) CrossRefGoogle Scholar
  8. 8.
    Doucet, A., de Freitas, N., Gordon, N. (eds.): Sequential Monte Carlo Methods in Practice. Springer, New York (2001) zbMATHGoogle Scholar
  9. 9.
    Gall, J., Rosenhahn, B., Brox, T., Seidel, H.-P.: Learning for multi-view 3d tracking in the context of particle filters. In: Int. Symposium on Visual Computing (ISVC). Lecture Notes in Computer Science, vol. 4292, pp. 59–69. Springer, Berlin (2006) Google Scholar
  10. 10.
    Gidas, B.: Metropolis-type Monte Carlo simulation algorithms and simulated annealing. In: Topics in Contemporary Probability and Its Applications, pp. 159–232. CRC Press, Boca Raton (1995) Google Scholar
  11. 11.
    Le Gland, F., Oudjane, N.: Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. Ann. Appl. Probab. 14(1), 144–187 (2004) zbMATHCrossRefGoogle Scholar
  12. 12.
    Gordon, N., Salmond, D., Smith, A.: Novel approach to non-linear/non-Gaussian bayesian state estimation. IEE Proc. F 140(2), 107–113 (1993) Google Scholar
  13. 13.
    Hammersley, J., Handscomb, D.: Monte Carlo Methods. Methuen, London (1967) Google Scholar
  14. 14.
    Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970) zbMATHCrossRefGoogle Scholar
  15. 15.
    Isard, M., Blake, A.: Contour tracking by stochastic propagation of conditional density. In: Proc. European Conf. on Computer Vision, vol. 1, pp. 343–356 (1996) Google Scholar
  16. 16.
    Jazwinski, A.: Stochastic Processes and Filtering Theory. Academic Press, London (1970) zbMATHGoogle Scholar
  17. 17.
    Julier, S., Uhlmann, J.: A new extension of the Kalman filter to nonlinear systems. In: Int. Symposium on Aerospace/Defence Sensing, Simulation and Controls (1997) Google Scholar
  18. 18.
    Kalman, R.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82, 35–45 (1960) Google Scholar
  19. 19.
    Kanazawa, K., Koller, D., Russell, S.: Stochastic simulation algorithms for dynamic probabilistic networks. In: Proc. of the Eleventh Annual Conf. on Uncertainty in AI (UAI), pp. 346–351 (1995) Google Scholar
  20. 20.
    Kirkpatrick, S., Gelatt, C. Jr., Vecchi, M.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983) CrossRefGoogle Scholar
  21. 21.
    Kitagawa, G., Gersch, W.: Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics, vol. 116. Springer, New York (1996) zbMATHGoogle Scholar
  22. 22.
    MacCormick, J.: Probabilistic models and stochastic algorithms for visual tracking. PhD thesis, University of Oxford (2000) Google Scholar
  23. 23.
    Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953) CrossRefGoogle Scholar
  24. 24.
    Del Moral, P.: Nonlinear filtering: Interacting particle solution. Markov Process. Relat. Fields 2(4), 555–580 (1996) zbMATHGoogle Scholar
  25. 25.
    Del Moral, P.: Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems. Ann. Appl. Probab. 8(2), 438–495 (1998) zbMATHCrossRefGoogle Scholar
  26. 26.
    Del Moral, P.: Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications. Springer, New York (2004) zbMATHGoogle Scholar
  27. 27.
    Del Moral, P., Doucet, A.: On a class of genealogical and interacting metropolis models. In: Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics, vol. 1832. Springer, New York (2003) Google Scholar
  28. 28.
    Del Moral, P., Guionnet, A.: On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré, B Probab. Stat. 37(2), 155–194 (2001) zbMATHCrossRefGoogle Scholar
  29. 29.
    Del Moral, P., Miclo, L.: Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to nonlinear filtering. In: Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics, vol. 1729, pp. 1–145. Springer, New York (2000) CrossRefGoogle Scholar
  30. 30.
    Del Moral, P., Miclo, L.: Annealed Feynman–Kac models. Commun. Math. Phys. 235, 191–214 (2003) zbMATHCrossRefGoogle Scholar
  31. 31.
    Neal, R.: Annealed importance sampling. Stat. Comput. 11, 125–139 (2001) CrossRefGoogle Scholar
  32. 32.
    Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2002) Google Scholar
  33. 33.
    Rogers, L., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 1, 2nd edn. Cambridge University Press, Cambridge (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Jürgen Gall
    • 1
    Email author
  • Jürgen Potthoff
    • 2
  • Christoph Schnörr
    • 3
  • Bodo Rosenhahn
    • 1
  • Hans-Peter Seidel
    • 1
  1. 1.Max-Planck-Institute for Computer ScienceSaarbrückenGermany
  2. 2.Dept. M & CS, Stochastics GroupUniversity of MannheimMannheimGermany
  3. 3.Dept. M & CS, CVGPR GroupUniversity of HeidelbergHeidelbergGermany

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