Multitarget Tracking with Random Finite Sets Trevor Wood, David Allwright , Philip Bond, Glen Davidson, Stephen Long, Irene Moroz 1. Tracking
3. Random Finite Sets
4. The PHD filter
When tracking using either sonar or radar, a transmitter sends out a pulse. The pulse propagates outwards, bouncing off anything it encounters, and returns to a detector. The echoed signal is then analysed to build a picture of what is present in the area scanned.
Mathematical jargon - Random Finite Set - A set with a random number of elements, which are themselves random.
We use the following model for multitarget state Xk and measurement Zk : [ Xk = Sk|k−1 (x) ∪ Γk
The analysis of the returning signal will yield a set of detections for potential objects. Many may be false alarms, but some may originate from targets that we want to track. The job of the tracker is to determine from this set of noisy data how many targets are present and where they are.
We model the target and measurement states as random finite sets. This is because we do not know how many targets or measurements there will be or where they will be. Instead of being limited to updating a single target by a single measurement, we want to update the whole set of tracks by the new set of measurements in order to unify the process. • The data update will be done using a Bayesian framework (which is also the basis for the Kalman filter). • This approach removes the need for data association and simplifies track maintenance.
5. Simulated Tracking Results
Can you identify the target moving in a straight line amongst the false alarms?
The scenario: Simulated data for initialisation of a target in high clutter (500 per time step). Confirmed tracks are shown in red, tentative tracks in black. It is known that any new target will appear inside the green box.
In the case with a single target and a single set of measurements, the methods used for tracking, such as the Kalman filter, are well known.
References [1] S. Blackman, R. Popoli Design and Analysis of Modern Tracking Systems Artech House, 1999 [2] D.E. Clark Target Tracking with The Probability Hypothesis Density Filter PhD Thesis, Heriot-Watt University ,2005 [3] R.P.S. Mahler Statistical Multisource-Multitarget Information Fusion Artech House, 2007 [4] C. Yates Private communication
Acknowledgements This work was generously funded by the Engineering and Physical Sciences Research Council (EPSRC), Thales Aerospace and Thales Underwater Systems. I would also like to thank Christian Yates for introducing me to the bacteria tracking problem and for his help.
Zk
=
Kk ∪
[
Θk (x)
x∈Xk−1
- Sk|k−1 represents targets surviving from the previous time step - Γk represents new targets - Θk represents target detections - Kk represents false alarms • Using random finite sets, a range of phenomena relevant to the tracking scenario are simply incorporated. • The filter derived using this model and a moment matching approximation is known as the Probability Hypothesis Density (PHD) filter[3]. • Implementations typically have linear computational complexity with respect to numbers of targets and measurements.
6. Bacteria Tracking Results Mathematical biologists interested in modelling the motion of self-propelled bacteria recorded their motion for study[4]. The tracking techniques described here were used as an aid to the experimental process, automatically tracking the bacteria. Information describing the motion is then extracted from these tracks and analysed.
2. Existing Methods
The solution when confronted with the multitarget tracking problem has traditionally been to use a heuristic method for reducing the problem to the single target case. This usually means using a data association algorithm, which can be computationally intensive, as well as a track management algorithm to decide when to initialise or delete a track.
x∈Xk−1
a) Nearest Neighbour
b) PHD filter
The figures show the PHD filter initialising the track (on the right) more quickly in high clutter than nearest neighbour data association using current methods. In 100 such tests, the PHD filter initialised and maintained the track successfully each time. Nearest neighbour did so only 59 times. These results show the PHD filter performing well in clutter levels higher than previously reported for the PHD filter. Improvement over nearest neighbour is comparable to the reported improvement made by the more advanced data association algorithms[1].
The figure shows current tracks and past paths of the bacteria captured in one snapshot. Application of these tracking techniques has allowed real time bacteria tracking in higher concentrations and with fewer broken paths than previously possible.
7. Conclusions • Use of random finite sets is a new approach to tracking which handles the whole process in a unified way. • PHD filter has lower computational complexity than the more advanced data association algorithms and is easier to implement, while being able to handle comparable levels of clutter.
Contact This is what I look like. I’ll be happy to talk to you about this work. Feel free to come and find me if I’m not standing here. Alternatively, you can email me on:
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• PHD filter was designed to be implemented on a range of non-traditional problems but has only being tried on a narrow range so far. • Future work - Investigating higher order approximations to the Bayesian update, multiple models for manoeuvring targets, testing on real SONAR data containing targets.
