Statistics and Computing

, Volume 25, Issue 3, pp 595–608 | Cite as

Calibrating the Gaussian multi-target tracking model

  • Sinan Yıldırım
  • Lan Jiang
  • Sumeetpal S. Singh
  • Thomas A. Dean
Article

Abstract

We present novel batch and online (sequential) versions of the expectation–maximisation (EM) algorithm for inferring the static parameters of a multiple target tracking (MTT) model. Online EM is of particular interest as it is a more practical method for long data sets since in batch EM, or a full Bayesian approach, a complete browse of the data is required between successive parameter updates. Online EM is also suited to MTT applications that demand real-time processing of the data. Performance is assessed in numerical examples using simulated data for various scenarios. For batch estimation our method significantly outperforms an existing gradient based maximum likelihood technique, which we show to be significantly biased.

Keywords

Multiple target tracking Parameter estimation Online Expectation–maximization Particle filters 

References

  1. Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B-stat. Methodol. 72, 269–342 (2010). doi:10.1111/j.1467-9868.2009.00736.x CrossRefMathSciNetGoogle Scholar
  2. Bar-Shalom, Y., Li, X.: Multitarget-Multisensor Tracking: Principles and Techniques. YBS, Bradford (1995)Google Scholar
  3. Cappé, O.: Online sequential Monte Carlo EM algorithm. In: Proc. IEEE Workshop Stat. Signal Process. (2009).Google Scholar
  4. Cappé, O., Moulines, E., Rydén, T.: Inference in Hidden Markov Models. Springer, New York (2005)MATHGoogle Scholar
  5. Chopin, N., Jacob, P., Papaspiliopoulous, O.: SMC\(^2\): an efficient algorithm for sequential analysis of state-space models. JRSSB 75, 397–426 (2012)CrossRefGoogle Scholar
  6. Cox, I.J., Miller, M.L.: On finding ranked assignments with application to multi-target tracking and motion correspondence. IEEE Trans. Aerosp. Electron. Syst. 32, 48–49 (1995)Google Scholar
  7. Del Moral, P., Doucet, A., Singh, S.S.: Forward smoothing using sequential Monte Carlo. Tech. Rep. 638, Univ. Cambridge, Eng. Dep. (2009).Google Scholar
  8. Delyon, B., Lavielle, M., Moulines, E.: Convergence of a stochastic approximation version of the EM algorithm. Ann. Stat. 27(1), 94–128 (1999)CrossRefMATHMathSciNetGoogle Scholar
  9. Ehrlich, E., Jasra, A., Kantas, N.: Gradient free parameter estimation for hidden Markov models with intractable likelihoods. Methodol. Comput. Appl. Probab. 2, 1–35 (2013)Google Scholar
  10. Elliott, R.J., Krishnamurthy, V.: New finite-dimensional filters for parameter estimation of discrete-time linear Gaussian models. IEEE Trans. Autom. Control 44(5), 938–951 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. Fearnhead, P.: MCMC, sufficient statistics and particle filters. J. Comput. Graph. Stat. 11, 848–862 (2002)CrossRefMathSciNetGoogle Scholar
  12. Hue, C., Le Cadre, J.P., Perez, P.: Sequential Monte Carlo methods for multiple target tracking and data fusion. IEEE Trans. Signal Process. 50(2), 309–325 (2002)CrossRefGoogle Scholar
  13. Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME—Ser. D 82, 35–45 (1960)CrossRefGoogle Scholar
  14. Lee, A., Yau, C., Giles, M., Doucet, A., Holmes, C.: On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. JCGS 19(4), 769–789 (2010). 14Google Scholar
  15. Mahler, R.: Multitarget Bayes filtering via first-order multitarget moments. IEEE Trans. Aerosp. Electron. Syst. 39(4), 1152–1178 (2003) Google Scholar
  16. Mahler, R.P.S., Vo, B.T., Vo, B.N.: CPHD filtering with unknown clutter rate and detection profile. IEEE Trans. Signal Process. 59(8), 3497–3513 (2011)CrossRefMathSciNetGoogle Scholar
  17. Murray, L.: GPU acceleration of the particle filter: the Metropolis resampler. In: Distributed machine learning and sparse representation with massive data sets. (2011). http://arxiv.org/abs/1202.6163
  18. Murty, K.G.: An algorithm for ranking all the assignments in order of increasing cost. Oper. Res. 16(3), 682–687 (1968)CrossRefMATHGoogle Scholar
  19. Nemeth, C., Fearnhead, P., Mihaylova, L.: Particle approximations of the score and observed information matrix for parameter estimation in state space models with linear computational cost, preprint (2013). http://arxiv.org/abs/1306.0735
  20. Ng, W., Li, J., Godsill, S., Vermaak, J.: A hybrid approach for online joint detection and tracking for multiple targets. In: Aerospace Conference, 2005 IEEE, pp. 2126–2141 (2005).Google Scholar
  21. Oh, S., Russell, S., Sastry, S.: Markov chain Monte Carlo data association for multi-target tracking. IEEE Trans. Autom. Control 54(3), 481–497 (2009)CrossRefMathSciNetGoogle Scholar
  22. Poyiadjis, G., Doucet, A., Singh, S.S.: Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika 98(1), 65–80 (2011)CrossRefMATHMathSciNetGoogle Scholar
  23. Racine, V., Hertzog, A., Jouanneau, J., Salamero, J., Kervrann, C., Sibarita, J.B.: Multiple-target tracking of 3d fluorescent objects based on simulated annealing. In: IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006. 3rd, pp. 1020–1023 (2006).Google Scholar
  24. Reid, D.B.: An algorithm for tracking multiple targets. IEEE Trans. Autom. Control 24, 843–854 (1979)CrossRefGoogle Scholar
  25. Särkkä, S., Vehtari, A., Lampinen, J.: Rao-Blackwellized particle filter for multiple target tracking. Inf. Fusion 8, 2–15 (2007)CrossRefGoogle Scholar
  26. Sergé, A., Bertaux, N., Rigneault, H., Marguet, D.: Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes. Nat. Methods 5, 687–694 (2008)CrossRefGoogle Scholar
  27. Singh, S.S., Whiteley, N., Godsill, S.: An approximate likelihood method for estimating the static parameters in multi-target tracking models. In: Barber, D., Cemgil, T., Chiappa, S. (eds.) Bayesian Time Series Models, pp. 225–244. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  28. Storlie, C.B., Lee, T.C., Hannig, J., Nychka, D.W.: Tracking of multiple merging and splitting targets: A statistical perspective. Stat. Sin. 19, 1–52 (2009)MATHMathSciNetGoogle Scholar
  29. Streit, R., Luginbuhi, T.: Probabilistic multi-hypothesis tracking. Tech. Rep. 10,428, Naval Undersea Warfare Center Division, Newport, Rhode Island (1995).Google Scholar
  30. Vihola, M.: Rao-Blackwellised particle filtering in random set multitarget tracking. IEEE Trans. Aerosp. Electron. Syst. 43, 689–705 (2007)CrossRefGoogle Scholar
  31. Vo, B.N., Ma, W.K.: The Gaussian mixture probability hypothesis density filter. IEEE Trans. Signal Process. 54(11), 4091–4104 (2006)CrossRefGoogle Scholar
  32. Vo, B.T., Vo, B.N., Cantoni, A.: Analytic implementations of the cardinalized probability hypothesis density filter. IEEE Trans. Signal Process. 55, 3553–3567 (2007)CrossRefMathSciNetGoogle Scholar
  33. Yoon, J.W., Singh, S.S.: A Bayesian approach to tracking in single molecule fluorescence microscopy. Tech. Rep. CUED/F-INFENG/TR-612, University of Cambridge, Eng. Dep. (2008).Google Scholar
  34. Yoon, J., Bruckbauer, A., Fitzgerald, W.J., Klenerman, D.: Bayesian inference for improved single molecule fluorescence tracking. Biophys. J. 94, 4932–4947 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sinan Yıldırım
    • 1
  • Lan Jiang
    • 2
  • Sumeetpal S. Singh
    • 2
  • Thomas A. Dean
    • 2
  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK

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