Electronic Journal of Statistics

Statistical consistency of the data association problem in multiple target tracking

Curtis B. Storlie, Jan Hannig, and Thomas C.M. Lee

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Simultaneous tracking of multiple moving objects extracted from an image sequence is an important problem which finds numerous applications in science and engineering. In this article we conduct an investigation of the theoretical properties a statistical model for tracking such moving objects, or targets. This tracking model allows for birth, death, splitting and merging of targets, and uses a Markov model to decide the times at which such events occur. This model also assumes that the track traveled by each target behaves like a Gaussian process. The estimated tracking solution is obtained via maximum likelihood. One of the contributions of this article is to establish the almost sure consistency to the data association problem by using these maximum likelihood tracking estimates. A major technical challenge for proving this consistency result is to identify the correct track (data association) amongst a group of similar (but incorrect) track proposals that are results of various combinations of target birth, death, splitting and/or merging. This consistency property of the tracking estimates is empirically verified by numerical experiments. To the best of our knowledge, this is the first time that a comprehensive study is performed for the large sample properties of a multiple target tracking method. In addition, the issue of how to quantify the confidence of a tracking estimate is also addressed.

Article information

Electron. J. Statist., Volume 5 (2011), 1227-1275.

First available in Project Euclid: 19 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]
Secondary: 60G17: Sample path properties 60G15: Gaussian processes 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Data association in-fill asymptotics multiple hypothesis tracking multiple target tracking merging splitting Brownian motion


Storlie, Curtis B.; Hannig, Jan; Lee, Thomas C.M. Statistical consistency of the data association problem in multiple target tracking. Electron. J. Statist. 5 (2011), 1227--1275. doi:10.1214/11-EJS639. https://projecteuclid.org/euclid.ejs/1319028568

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  • Bar-Shalom, Y., Li, X. & Kirubarajan, T. (2001), Estimation with Applications to Tracking and Navigation, John Wiley & Sons, Inc., New York, NY.
  • Blackman, S. (2004), ‘Multiple hypothesis tracking for multiple target tracking’, IEEE A&E Systems Magazine 19, 5–18.
  • Blackman, S. & Popoli, R. (1999), Design and Analysis of Modern Tracking Systems, Artech House, Boston.
  • Chang, K. & Bar-Shalom, Y. (1984), ‘Joint probabilistic data association for multitarget tracking with possibly unresolved measurements and maneuvers’, IEEE Transactions on Automatic Control 29, 585–594.
  • Chen, H., Li, X. & Bar-Shalom, Y. (2004), ‘On joint track initiation and parameter estimation under measurement origin uncertainty’, IEEE Transactions on Aerospace and Electronic Systems 40, 675–694.
  • Cong, S. & Hong, L. (1999), ‘Computational complexity analysis for multiple hypothesis tracking’, Mathematical and computer modeling 29, 1–16.
  • Daum, F. E. (1994), ‘The importance of resolution in multiple target tracking’, Signal and Data Processing of Small Targets 1994, SPIE Proceedings 2235, 329–338.
  • Doucet, A., de Freitas, N. & Gordon, N. (2001), Sequential Monte Carlo Methods in Practice, Springer Verlag, New York, NY.
  • Doucet, A., Godsill, S. & Andrieu, C. (2000), ‘On sequential Monte Carlo sampling methods for Bayesian methods’, Statistics and Computing 10, 197–208.
  • Fortmann, T., Bar-Shalom, Y. & Scheffe, M. (1983), ‘Sonar tracking of multiple targets using joint probabilistic data association’, IEEE Journal of Oceanic Engineering 8, 173–184.
  • Gelfand, A. (1990), ‘Sampling-based approaches to calculating marginal densities’, Journal of the American Statistical Association 85(410), 398–409.
  • Genovesio, A. & Olivo-Marin, J. (2004), ‘Split and merge data association filter for dense multi-target tracking’, Proceedings of the 17th International Conference on Pattern Recognition pp. 677–680.
  • Gordon, N., Salmond, D. & Smith, A. (1993), ‘Novel approach to nonlinear/non-Gaussian Bayesian state estimation’, IEE Proceedings-F 140, 107–113.
  • Hall, P., Peng, L. & Rau, C. (2001), ‘Local likelihood tracking of fault lines and boundaries’, Journal of the Royal Statistical Society B 63, 569–582.
  • Hall, P., Qiu, P. & Rau, C. (2007), ‘Tracking edges, corners and vertices in an image’. unpublished, manuscript.
  • Hall, P. & Rau, C. (2000), ‘Tracking a smooth fault line in a response surface’, Annals of Statistics 28, 713–733.
  • Kallenberg, O. (2002), Foundations of Modern Probability, 2nd edn, Springer-Verlag, New York, NY.
  • Khoshnevisan, D. (2002), Multiparameter Processes; An Introduction to Random Fields, Springer-Verlag, New York, NY.
  • Kitagawa, G. (1996), ‘Monte Carlo filter and smoother for non-Gaussian nonlinear state models’, Journal of Computational and Graphical Statistics 5, 1–25.
  • Koch, W. & van Keuk, G. (1997), ‘Multiple hypothesis track maintenance with possibly unresolved measurements’, IEEE Transactions on Aerospace and Electronic Systems 33, 883–892.
  • Li, J. & Jing, Z. (2003), ‘Practical system for tracking multiple maneuvering targets’, Optical Engineering 42, 2439–2451.
  • Liu, J. & Chen, R. (1998), ‘Sequential monte carlo methods for dynamic systems’, Journal of the American Statistical Association 93, 1032–1044.
  • Mahler, R. (2003), ‘Multitarget Bayes filtering via first-order multitarget moments’, IEEE Transactions on Aerospace and Electronic Systems 39, 1152–1178.
  • Mori, S., Chong, C., Tse, E. & Wishner, R. (1986), ‘Tracking and classifying multiple targets without a priori identification’, IEEE Transactions on Automatic Control 31, 401–409.
  • Reid, D. (1979), ‘An algorithm for tracking multiple targets’, IEEE Transactions on Automatic Control 24, 843–854.
  • Revuz, D. & Yor, M. (1999), Continuous Martingales and Brownian Motion, Springer-Verlag, New York, NY.
  • Shepp, L. (1966), ‘Radon nikodym derivatives of Gaussian measures’, Annals of Mathematical Statistics 37, 321–354.
  • Storlie, C., Lee, T., Hannig, J. & Nychka, D. (2009), ‘Tracking of multiple merging and splitting targets: A statistical perspective (with discussion)’, Statistica Sinica 19, 1–52.
  • Trunk, G. & Wilson, J. (1981), ‘Track initiation of occasionally unresolved radar targets’, IEEE Transactions on Aerospace and Electronic Systems 17, 122–130.
  • Vo, B., Singh, S. & Doucet, A. (2005), ‘Sequential Monte Carlo methods for Bayesian multitarget filtering with random finite sets’, IEEE Transactions on Aerospace and Electronic Systems 41, 1224–1245.