Multi-target Tracking with PHD Filters
Contact Info
Michele Pace

INRIA Bordeaux - Sud-Ouest
Université Bordeaux I
IMB - Bat A33
351, Cours de la Libération
33405 Talence Cedex, France

EMail:
Michele.Pace AT inria.fr
pace AT math.u-bordeaux1.fr
Michele Pace

PhD  Student, equipe ALEA
Centre INRIA Bordeaux Sud-Ouest
Institut de Mathématiques de Bordeaux
Université de Bordeaux I
Objective:
Estimate an unknown, time varying number of targets and their states from noisy observations avaiable at discrete intervals of time.

General problems:
Tracking multiple maneuvering targets in a surveilled region involves at each time step the joint estimation of the number of targets as well as their state vectors. Clutter, uncertainties in the dynamic of the maneuvers, in data association and in target detection make this process very difficult.The intrinsic problem of multiple-target tracking is the impossibility to known with certitude the correct association between the measurements and the targets that have generated them.

Traditional approaches
An alternative solution to traditional approaches like Multiple Hypotheses tracking (MHT), Joint Probabilistic Data Association Filter (JPDAF) or (PMHT) Probabilistic MHT , is based on the Random Finite Set (RFS) formulation in which the collection of individual targets is treated as a set-valued state and the collection of observations as a set-valued observation.
This approach, joined with Mahler's finite set statistics (FISST), has lead to the development of an effective class of multiple-target filters such as the Probability Hypothesis Density (PHD) filters.
You can find here some example of the application of two approximations of the PHD recursion: a Sequential Monte Carlo PHD filter (SMC-PHD) and a Gaussian Mixture PHD filter (GM-PHD) to a realistic naval and aerial scenario.

PHD Filters

The Probability Hypothesis Density (PHD) filter is a multiple-target filter for recursively estimating the number and the state of a set of targets given a set of observations. It is able to operate in environments with false alarms and missdetections and it works by propagating in time the intensity of the targets RFS instead of the full multi-target posterior density.
In literature, various implementations have been proposed and their performance compared using different levels of clutter and model uncertainty. The generic sequential Monte Carlo implementation (SMC-PHD) filter, proposed by Vo et al. in [4], generally suffers of an high computational cost as it requires a large number of particles and relies on clustering techniques to provide state estimates. The unreliability of estimates due to inaccuracy introduced by the clustering step and the computational complexity constitute its main
drawbacks. To alleviate these problem a closed form solution to the PHD filter recursion, called Gaussian Mixture PHD (GM-PHD), has been proposed by Vo and Ma in [7]. This approach does not require clustering procedures but, since
it makes use of the Kalman filter equations, it is restricted to linear-Gaussian target dynamic. When targets show a mildly non-linear dynamic it is generally possible to rely on extensions for the GM-PHD filter using the Extended Kalman filter (EK-PHD) or the Unscented Kalman filter (UK-PHD) or to use the Gaussian Particle Implementations of the PHD filter.


Scenario:
GMPHD Filter:
SMCPHD Filter with K-Means Clustering:
Useful references

[1] Donald B. Reid ; An algorithm for tracking multiple targets IEEE
Conference on Decision and Control, Jan. 1978, Volume: 17, On
page(s): 1202-1211

[2] Clark, D.; Ba-Tuong Vo; Ba-Ngu Vo; Gaussian Particle Implementations
of Probability Hypothesis Density Filters Aerospace Conference,
2007 IEEE, 3-10 March 2007 Page(s):1 - 11

[3] R. Mahler Multitarget Bayes filtering via first-order multitarget moments
IEEE Transactions on Aerospace and Electronic Systems, 39,
No.4:1152-1178, 2003.

[4] Vo, B. and Singh, S.S. and Doucet, A. Sequential Monte Carlo
Implementation of the PHD Filter for Multi-target Tracking The
International Conference on Information Fusion, 2003.

[5] Vo, B. and Singh, S.S. and Doucet, A. Sequential Monte Carlo
Methods for Multi-target Filtering with Random Finite Sets IEEE
transactions on aerospace and electronic systems 2005, vol. 41, no4,
pp. 1224-1245

[6] Panta, Kusha. Multi-target tracking using 1st moment of random finite
sets Thesis (Ph.D.) University of Melbourne, Dept. of Electrical and
Electronic Engineering, 2008.

[7] Ba-Ngu Vo , Wing-Kin Ma The Gaussian Mixture Probability Hypothesis
Density Filter Signal Processing, IEEE Transactions on
Publication Date: Nov. 2006 Volume: 54, Issue: 11 On page(s): 4091-
4104

[8] Ba-Ngu Vo; Pasha, A.; Hoang Duong Tuan A Gaussian Mixture PHD
Filter for Nonlinear Jump Markov Models 45th IEEE Conference
on Decision and Control, 2006 Volume , Issue , 13-15 Dec. 2006
Page(s):3162 - 3167

[9] Julier, Simon J.; Uhlmann, Jeffrey K. New extension of the Kalman
filter to nonlinear systems Proc. SPIE Vol. 3068, p. 182-193, Signal
Processing, Sensor Fusion, and Target Recognition VI, Ivan Kadar;
Ed.

[10] Julier, Simon J.; Uhlmann, Jeffrey K. Unscented Filtering and Nonlinear
Estimation Proceedings of the IEEE, 92(3):401-422, March
2004.

[11] Clark, Daniel; Vo, Ba-Ngu; Bell, Judith GM-PHD filter multitarget
tracking in sonar images Signal Processing, Sensor Fusion, and
Target Recognition XV. Edited by Kadar, Ivan. Proceedings of the
SPIE, Volume 6235, pp. 62350R (2006).

[12] Maggio, E. Piccardo, E. Regazzoni, C. Cavallaro, A. Particle PHD
Filtering for Multi-Target Visual Tracking ICASSP 2007. IEEE
International Conference on Acoustics, Speech and Signal Processing,
2007 Volume: 1, On page(s): I-1101-I-1104

[13] S.S. Blackman Multiple target tracking with radar applications
Artech House, 1986



Video example of GMPHD-Filtering

In this example the GMPHD Filter is used to filter the position of 99 naval targets moving in the sourveillance zone.
During each time step 50 false alarms are registered (clutter)

Example II

This section considers the application of the PHD filter to an aereal and naval multiple-target tracking scenario. The purpose is to track an unknown, time-varying number of aircrafts and ships in a region of sourveillance determined by the characteristics and location of the radar mounted on a naval platform. The target evolution is modeled using a Constant Velocity model perturbed by random accelerations.The radar is mounted on a ship located at the origin of the reference system and considered not moving. It
completes a 360 degrees scan in a fixed amount of time T and collects measurements of targets that are within a certain range and whose elevation angle is between a maximal and a minimal value. The radar is unable to localize targets that are too close (within a blind distance of 300 mt.); the measurements collected are affected by a Gaussian, zero centred random noises with known variances.

The clutter model takes into account the geometry of the sourveilled region and, to a lesser degree, its physical properties. It is modelled as the superposition of two Poisson point processes with different intensities. The first is used to model the false alarms generated by the reflection of the electromagnetic beam over the surface of the sea and the second is used to model the spourious measurements generated by termic and athmospheric noises in the sourveilled
region of the sky. To better define the clutter process the rest of this section details the clutter probability distribution over
the sourveillance region.



In this scenario, ten airplanes (of the same type) travel across the sourveillance zone during a period of 10 minutes.
The measurements are collected by a radar mounted on a ship located at P = [0; 0; 0] and are available every 2 seconds.
The SMC-PHD filter is configured to utilize 300 particles for each target active and an average of 100 particles for
each measurement to approximate the birth intensity and pick up new targets. The GM-PHD filter is configured with
a truncation thresholds for the Gaussian terms of 0.4 and a merging threshold of 500mt.




Example I


An unknown, varying number of targets move along the line segment [-100,100]. The state of the targets consist of position and velocity; only the position is observed. Targets may appear or disappear at any time during and are subjected to random accelerations.






The system evolution is partially observed by the following observation model:
The PHD intensity is rebuilt summing the weigths of the particles at each time step:
Clutter intensity: 0, Probability of detection: 0.9
Clutter intensity: Poisson process with average of 5, Probability of detection: 0.9
The importance of having a good approximation of the birth intensity (yellow particles):
[14] S.S. Blackman Multiple hypothesis tracking for multiple target tracking
Blackman, S.S. Aerospace and Electronic Systems Magazine,
IEEE Volume 19, Issue 1, Jan. 2004 Page(s):5 - 18

[15] Bar-Shalom, Yaakov; Fortmann, Thomas Tracking and Data Association
San Diego, CA Academic, 1988

[16] Fortmann, T. Bar-Shalom, Y. Scheffe, M. Sonar tracking of multiple
targets using joint probabilistic data association Oceanic Engineering,
IEEE Journal of Publication Date: Jul 1983 Volume: 8, Issue: 3
Page(s): 173- 184

[17] Streit, Roy L.; Luginbuhl, Tod E. Maximum likelihood method for
probabilistic multihypothesis tracking Proc. SPIE Vol. 2235, p. 394-
405, Signal and Data Processing of Small Targets 1994, Oliver E.
Drummond; Ed.

[18] I. R. Goodman , Ronald P. Mahler Mathematics of Data Fusion
Kluwer Academic Publishers 1997

[19] Mahler R. Random set theory for target tracking and identification
Data Fusion Hand Book, D. Hall and J. Llinas (eds.)

[20] D. Daley and D. Vere-Jones. An Introduction to the Theory of Point
Processes. Springer-Verlag, 1988.

[21] Schuhmacher, D.; Vo, B.-T.; Vo, B.-N. A Consistent Metric for
Performance Evaluation of Multi-Object Filters IEEE Transactions
on Signal Processing, Volume 56, Issue 8, Aug. 2008 Page(s):3447 -
3457
[22] Schuhmacher, D.; Vo, B.-T.; Vo, B.-N. Random Finite Sets in Multi
Object Filtering PhD Thesis, The University of Western Australia.
October 2008

[23] J. B. MacQueen Some Methods for classification and Analysis of
Multivariate Observations Proceedings of 5-th Berkeley Symposium
on Mathematical Statistics and Probability, Berkeley, University of
California Press, 1:281-297

[24] XR Li, VP Jilkov Survey of maneuvering target tracking. Part I:
Dynamic models IEEE Transactions on Aerospace and Electronic
Systems, 39(4): 1333-1364, October 2003

[25] Sidenbladh, H. Multi-target particle filtering for the probability hypothesis
density Proceedings of the Sixth International Conference
of Information Fusion, 2003. Volume: 2, On page(s): 800- 806

[26] Tobias, M. Lanterman, A.D. Techniques for birth-particle placement
in the probability hypothesis density particle filter applied to passive
radar Radar, Sonar and Navigation, IET Publication Date: October
2008 Volume: 2, Issue: 5 Page(s): 351-365

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