Searching For, Initiating and Tracking Multiple Targets Using

12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009

Searching for, initiating and tracking multiple targets using existence probabilities Paul Horridge and Simon Maskell QinetiQ Ltd Malvern Technology Centre St. Andrews Road Malvern, Worcs, UK. [email protected], [email protected] Abstract – We present a framework for searching for and tracking multiple targets, using existence probabilities to perform track management. Unlike other approaches, we also use existence probabilities to initiate new tracks. This is done by having a “search” track which represents the probability of an unconfirmed track being present and the state distribution of such a track. The sensor can have a statedependent probability of detecting a target, and we are particularly motivated by scenarios where a sensor periodically scans a search space, being able to see only part of the space at a time. Results show that this approach is able to confirm tracks in an environment with significant clutter levels and a low probability of detection, with a minimal number of false tracks. Keywords: Tracking, search, initiation, existence probability, particle filter

1 Introduction In this paper, we consider the problem of searching for new targets and initiating tracks on them at the same time as tracking existing targets. We suppose that the target space is bounded but that our sensors have a limited range so can only see part of the search space at once. Track management is performed by maintaining a probability of existence on each track (i.e. a probability that the track corresponds to a real target). A confirmed track is deleted when its probability of existence drops to below a specified threshold. In addition to the confirmed tracks, an additional “search track” is maintained. The existence probability associated with the search track represents the probability of a target being in the search space which has yet to be confirmed, and the probability distribution of the search track represents the state of such a target if it exists. Initiation is achieved by promoting the search track to be confirmed if its existence probability is sufficiently high and creating a new search track to search for additional targets. This is in contrast to most existing initiation schemes which use ad hoc methods such as initiating based on measurements which fall outside the gates of all the current tracks. Vermaak et al. [1] and Mu˘sicki and Evans [2] present similar frameworks for track management with existence prob-

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abilities. The novelty of our approach is in applying existence probabilities to initiation as well as generalizing the approach to the case where the probability of detecting a target is dependent on the target and sensor states. The PHD filter [3] can also perform search and track, but it only maintains a measure of the mean number of targets over the space. Since it does not explicitly maintain the identities of the tracks, it does not allow us to keep a continuous track on a particular target. We derive the necessary equations to manipulate the probability distributions independently of how the distributions are represented. However, the state-dependent probability of detection and the search track make representing the distributions as Gaussian mixtures and using a Kalman filterbased approach problematic so instead a particle filter [4] is recommended. Multiple target particle filters where the prior proposal distribution is used are covered by Schulz et al. [5] and a generalization which allows the use of an arbitrary proposal distribution is given by Vermaak et al. [6]. The rest of the paper is organized as follows. In Section 2 we formulate the tracking problem of interest. In Sections 3 and 4 we show how to perform the prediction and update steps for this tracking problem. In Section 5 we discuss how to promote the search track. In Section 6 we give an illustration of our approach. Finally, in Sections 7 and 8 we give some results and conclusions.

2 Problem formulation Suppose that at time step k ∈ N we have Nk tracks. Each track i = 1, . . . , Nk has a binary existence variable eik representing whether or not it corresponds to a real target. Conditional on the existence event Eki , {eik = 1}, the track has a state xik . At each time step k we receive a scan of Mk measurements Y k = {yk1 , . . . , ykMk }

(1)

of the targets. These measurements may include spurious measurements (clutter) and targets may not necessarily be detected. Also, which measurements originated from which targets is not known.

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Given the measurements Y 1:k = {Y 1 , . . . , Y k } received so far, we wish to maintain for each track i at time step k a probability of existence p(Eki |Y 1:k )

(2)

and a representation of the target state distribution conditional on its existence p(xik |Y 1:k , Eki ).

(3)

We maintain a number of confirmed tracks as well as an extra track which is used to search the space. This search track is initialized with a specified initial distribution and the existence probability of the search track represents the probability of there being a target in the space which has not yet been confirmed. When the existence probability of the search track reaches a specified threshold, the search track is promoted to a confirmed track and a new search track is initialized to search for additional targets. When the existence probability of a confirmed track falls below a deletion threshold, the track is deleted.

clutter measurements is Poisson-distributed with mean λV , where V is the volume of the search space and λ is a known parameter representing the clutter density. Each clutter measurement is uniformly distributed over the search space. The Mk measurements are assumed to be shuffled in some random order, with each of the Mk ! permutations equally likely. Hence which measurements correspond to which targets is not directly known and association probabilities must be inferred.

3 Prediction step The tracking proceeds in two stages: first we predict the target state and existence distributions based on the target models, then we update the distributions based on the received measurements.

3.1 Confirmed tracks From (4) and (5), we have for each confirmed track i, p(Eki |Y 1:k−1 )

p(xik |Eki , Y 1:k−1 ) = (8) Z f (xik |xik−1 )p(xik−1 |Eki , Y 1:k−1 ) dxik−1 .

2.1 Target births, deaths and motion models We assume that an existing target at time step k − 1 will cease to exist by time step k with some known probability k Pdeath , independently of the other targets, i.e. i P (Eki |Ek−1 )

k = 1 − Pdeath .

(4)

Furthermore, if there are no unconfirmed targets in the search space at time step k − 1, one will appear at time k k . We also have a probability distribuwith probability Pbirth tion pbirth (·) for the state of a newly arrived target. Targets which continue to exist are assumed to move independently of each other with a given Markov transition i f (xik |xik−1 ) = p(xik |xik−1 , Ek−1 , Eki ).

(5)

3.2 The search track Since new targets can appear, the prediction calculations for the search track differ slightly from those for the confirmed tracks: p(Eki |Y 1:k−1 ) = +

pT (y|x)

(1 −

(9)

k i Pdeath )p(Ek−1 |Y 1:k−1 ).

i

p(xik |Eki , Y 1:k−1 ) =

(6)

i

k Pbirth p(E k−1 |Y 1:k−1 )

i Here, E k−1 is the complement of Ek−1 . To predict the position of the unconfirmed target, conditional on it existing, we need to condition on whether it existed previously or whether it has newly appeared:

2.2 Measurement model At time step k, we receive a scan of measurements from a sensor. Note that we can have different sensors reporting at different time steps so we are not restricted to having a single sensor. However, since we receive measurements from a single known sensor at each time step, we can omit the sensor index from the notation. We assume that each target is detected independently with some probability Pd (x) depending on its state x. Given that the target is detected, let

k i = (1 − Pdeath )p(Ek−1 |Y 1:k−1 )(7)

1 X

eik−1 =0

(10)

p(xik |eik−1 , Eki , Y 1:k−1 )p(eik−1 |Eki , Y 1:k−1 ).

Here, i

p(xik |E k−1 , Eki , Y 1:k−1 ) i , Eki , Y 1:k−1 ) p(xik |Ek−1

Z

= pbirth (xik )

(11)

=

(12)

i , Y k−1 ) dxik−1 f (xik |xik−1 )p(xik−1 |Ek−1

and p(eik−1 |Eki , Y 1:k−1 )

be the probability distribution of the resulting measurement y. In addition to measurements generated by targets, a number of clutter measurements are generated. The number of

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∝

=

(13) p(Eki |eik−1 , Y 1:k−1 )p(eik−1 |Y 1:k−1 ) ( i k Pbirth p(E k−1 |Y 1:k−1 ) if eik−1 = 0 (14) k i 1:k−1 (1 − Pdeath )p(Ek−1 |Y ) if eik−1 = 1.

4 Update step The update calculations are the same for the confirmed tracks and the search track, except that if the existence probability of a search track exceeds the confirmation threshold, the state of the newly promoted track is conditioned on the target being detected. This avoids the possibility of the initial confirmed track distribution being too widely spread over the search space. This section follows [7] and [8], with the main difference being dealing with the state dependent probability of detection. We make the standard approximation that the joint prior distribution of the target states can be written as a product of marginals: k p(x1:N |Y 1:k−1 ) k

=

Nk Y

i=1

p(xik |Y 1:k−1 )

(15)

The target state distribution conditional on knowing whether or not the target was detected is p(xik |dik , Eki , Y 1:k−1 )

∝ p(dik |xik , Eki )p(xik |Eki , Y 1:k−1 )  (1 − Pd (xik ))p(xik |Eki , Y 1:k−1 ) = Pd (xik )p(xik |Eki , Y 1:k−1 )

=

Nk Y

i=1

p(eik |Y 1:k−1 ).

p(aik , eik |Y 1:k )

(18)

be the set of such valid joint associations. We write the required posterior probabilities as mixtures over the unknown association variables: X ai p(xik |aik , ykk , Eki , Y 1:k−1 ) × p(xik |Eki , Y 1:k ) =

∝

(25)

4.1 Likelihood conditional on association Let ∆ be the set of detected targets, i.e.  ∆ = i ∈ {1, . . . , Nk } : dik = 1 .

(26)

(Note that a detected target must exist, i.e. if dik = 1 then eik = 1.) Then k k p(yk1:Mk |a1:N , e1:N , Mk , Y 1:k−1 ) k Z k 1:Nk 1:Nk = p(yk1:Mk |x∆ , ek , Mk , Y 1:k−1 ) × k , ak

=

V

1:Nk 1:Nk p(x∆ , ek , M, Y 1:k−1 ) dx∆ k |ak k  Z 1 Y aik i i i i 1:k−1 i )dxk . pT (yk |xk )p(xk |dk , Ek , Y Mc i∈∆

(27)

(20)

Here, Mc = Mk − |∆| is the number of clutter measurements and p(xik |dik , Eki , Y 1:k−1 )

(28)

has been calculated already in (22).

4.2 Prior association probabilities By generalizing equation (12) in [7], it can be shown that

ai

p(xik |aik , ykk , Eki , Y 1:k−1 ) i

pT (yak |xik , Eki )p(xik |dik , Eki , Y 1:k−1 ).

1:k−1 p(ak1:Nk , e1:T ) k , Mk |Y

(19)

Calculating the first term in the product in right side of (19) is relatively straightforward since it is conditional on the measurement association, although the state dependent probability of detection means that whether or not the track is detected influences the state distribution:

∝

k k p(yk1:Mk |a1:N , e1:N , Y 1:k−1 , Mk ) × k k

k where the proportionality constant does not depend on a1:N k 1:Nk and ek . The two following subsections each deal with one of the terms in the product of the right hand side of (25).

aik

p(Eki |Y 1:k ) =

(24)

are calculated by marginalizing the joint association probabilities:

(16)

be the joint association over all the targets, and analogously k for d1:N and ek1:Nk . k Note that each measurement can be generated by at most k one track, so the only joint associations a1:N which can k occur are those where the nonzero values are distinct. Let

p(aik |Eki , Y 1:k ) X p(aik , Eki |Y 1:k ). i ak

=0 (23) = 1.

p(ak1:Nk , ek1:Nk |Y 1:k )

Let aik be a random variable representing the measurement associated with track i (or 0 if the track is not detected). We also let dik be the detection variable dik = [aik 6= 0] and let eik be the existence variable [Eki ]. 1 Let   Nk 1 k = a , . . . , a (17) a1:N k k k

Ω ⊂ {0, . . . , Mk }Nk

if if

The proportionality constants in (21) and (22) do not depend on xik . Hence we can compute the posterior target state distributions conditional on the measurement association. The marginal association probabilities

and similarly for the existence variables: k p(e1:N |Y 1:k−1 ) k

(22) dik dik

(21)

1 We

use the Iverson notation: if P is a statement, [P ] equals 1 if the statement is true and 0 otherwise.

613

k k p(a1:N , e1:N , Mk |Y 1:k−1 ) k k

=

e−λV (λV )Mc Mk !

Nk Y

i=1

(29)

p(dik |eik , Y 1:k−1 )p(eik |Y 1:k−1 )

k for a1:N ∈ Ω, where k

serious problem, but otherwise we may have to deal with this by, for example, checking whether the distribution looks p(dik |eik , Y k−1 ) (30) multimodal and if so, initiating more than one confirmed  R track by clustering the particles. This should be addressed p(dik |xik , Eki )p(xik |Eki , Y k−1 ) dxik if eik = 1, = in future work. dik = eik otherwise. To replace the newly promoted track with a new search Note that p(dik |Eki , Y k−1 ) appears as the proportionality track, we initialize its target state distribution with the birth constant in (22). distribution pbirth (·). Other choices are possible: for example we could use the old search track distribution condi4.3 Combined likelihood tioned on the null hypothesis. This has low density in the Combining (27) and (30), we see that for a1:Nk ∈ Ω, areas recently (unsuccessfully) searched, but it was found that, even conditioning on the null hypothesis, this distriNk Y 1:Nk 1:Nk 1:k (31) bution still has considerable mass in the same area as the βi,aik ,eik p(ak , ek |Y ) ∝ confirmed track, especially if the probability of detection i=1 is low. Alternative methods of replacing the search track where should also be considered as a possible avenue of future research. i i 1:k−1 i 1:k−1 βi,0,eik = λp(D k |ek , Y )p(ek |Y ), (32) = 0 for aik 6= 0

βi,aik ,0

(33)

and, for aik = 1, . . . , M βi,ai ,1

6 Illustration of the approach

= p(Dki |Eki , Y 1:k−1 )p(Eki |Y 1:k−1 ) × Z pT (ykai |xik )p(xik |Dki , Y 1:k−1 ) dxik .

(34)

From this, the EHM2 algorithm [7] can be applied to calculate the marginal probabilities (24) without the need for evaluating each of the joint probabilities (25). This gives us everything we need to calculate (2) and (3).

5 Promoting the search track and initializing a new search track As previously stated, if the existence probability of the search track exceeds a specified threshold, the search track is promoted to be a confirmed track and a new search track is initialized to search for any remaining targets. In order to ensure that the distributions of the newly confirmed tracks are localized around the measurements, when a search track is promoted we condition the target state distribution so that one of the measurements is generated by the target (i.e. we disregard the null hypothesis of a missed detection for a newly promoted target): p(xik |Eki , aik 6= 0, Y 1:k ) ∝

Mk X

aik =1

(35)

ai

p(xik |aik , ykk , Eki , , Y 1:k−1 )p(aik |Eki , Y 1:k ).

This equation replaces (19) in this case. Note that if there is more than one target in the field of view of the sensor, it is possible that the initiated track will be multimodal. If the target density is comparatively low compared to the volume of the field of view, this is not a

To illustrate the approach, Figure 1 shows the particle clouds at points during runs. (When plotting the figures, to account for the fact that the particles have different weights we plot particles sampled according to their weight. Hence high weight particles are more likely to be shown. Also, the number of particles shown for each track is proportional to its existence probability.) The scenario used is explained in more detail in Section 7 below, but briefly it involves a sensor scanning through the search space [0, 1] × [0, 1] with a limited range outside which it cannot detect any targets. Figure 1(a) is from a run where the probability of detecting a target in the range of the sensor is 0.3. Clumps of search (black) particles appear where measurements have indicated the possible existence of a target but where there is not yet sufficient evidence to confirm a track. Note that one of the clumps actually contains a target, but confirming tracks at this stage would result in a large number of false tracks. Figures 1(b) and 1(c) are from a run with a 0.9 probability of in-range detection. 1(b) is at a stage where a target (being tracked by the magenta particle cloud) has just died. Here, the existence probability on the magenta track is 0.76. Figure 1(c) shows the later stage when the existence probability has dropped to 0.08 (below the deletion threshold) and the track is about to be deleted. Since we exploit the knowledge that the sensor has a limited range, the particles in the range of the sensor where there are no detections get low weight whereas the weights of the particles outside the sensor range are largely unchanged. Hence our approach can deduce that the track has a low probability of existence when most of the track’s probability mass has been unsuccessfully searched for measurements.

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(c) Figure 1: Illustrations of the particles during a search. The particles of the confirmed tracks are shown in green, red and magenta. and the search particles are in black. The blue crosses show the true target positions and the brown asterisk shows the sensor position.

To test this approach, we run an example scenario as follows: The search space is the region [0, 1] × [0, 1] in (x, y) space. Targets move according to a random walk with independent Gaussian-distributed jumps of standard deviation 0.005 at each time step, but are constrained to remain within the search space. At each time step, there is a death probability of 0.005 for each target, and a new target will appear in the search space with probability 0.01. New target arrivals are uniformly distributed over the search space. We have a single sensor which fails to detect a target if the distance between the sensor and the target is greater than 0.2. Otherwise, the target is detected with probability 0.9. This means that the sensor can only see at most 0.22 π ≈ 13% of the search space at a time. Targetgenerated measurements are of the target’s position, with Gaussian noise of standard deviation 0.01 in each of the x and y components. The number of clutter measurements is Poisson-distributed with mean 5, and clutter measurements are uniformly distributed over the search space. The sensor sweeps through the space along the path shown in Figure 2, starting from (0.1, 0.1). When it reaches the end of the path, it follows the path in the opposite direction back to the start. It then repeats the circuit 4 more times, for a total of 10 passes through the space. We start by having 3 targets initially in the space, and births and deaths of targets are simulated using the probabilities above. The tracker has no prior knowledge of the number or positions of the targets and is initialized with only a search track. The threshold for promoting a search track to confirmed status is taken to be 0.9 and the threshold for deleting confirmed tracks is 0.1. New search tracks are initiated with existence probability 0.5 and particles distributed uniformly over the search space. 10000 particles are used for each track and the prior proposal distribution is used.

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To reduce computation in the data association, we only accept as candidate measurements for a track those which are within 5 standard deviations of a particle for the track. We can also immediately reject measurements which are √ further than 0.2 + 5 2 × 0.012 ≈ 0.27 away from the sensor, since they cannot be within 5 standard deviations of a detected target. This significantly cuts down on the number of clutter measurements we need to consider. Figure 3 shows the tracks resulting from a run of the scenario, with the true target positions shown in black and the tracks in different colours to make distinguishing the tracks easier. In Figure 3(a) we show the tracks in (x, y) space and in Figure 3(b) we show the x coordinates of the tracks plotted against time. Figure 4 shows the number of targets in the scenario over time compared with the number of confirmed tracks. We can see from these figures that the tracker is generally successful at tracking the targets and determining how many targets are present at a particular time. There are periods where track position estimated stay almost constant over time, but this is due to periods where the sensor is unable to see a particular target. Similarly, tracks sometimes persist after the lifetime of a target, but are terminated when the sensor is able to see that the target is no longer there.

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(b) Figure 3: Tracking with a detection probability of 0.9. The targets (black) and confirmed tracks (other colours) in (a) x and y coordinates, (b) x coordinate and time. 5

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To see how the approach deals with a target which is much more difficult to spot, we repeat the run with the same target paths and the same parameters except that the probability of detecting a target within range of the sensor is reduced to 0.3. Because of the increased difficultly in confirming the presence of a target, we also reduce the threshold existence probability for confirming tracks from 0.9 to 0.7. The resulting tracks are shown in Figure 5 and the number of confirmed tracks compared with the number of targets in Figure 6. Reducing the probability of detection results in some degradation of performance, but most of the targets are still successfully tracked.

8 Conclusions We have developed a framework for performing joint target search, initiation, tracking and termination using existence variables and with a target state-dependent probability of detection. This is shown to be successful for a simple example scenario, even with a low probability of detecting the targets in the range of the sensor.

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9 Acknowledgements The authors gratefully acknowledge the funding and support received from UK MoD. They would also like to thank the anonymous reviewer who pointed out the possibility of initiating a multimodal track mentioned in Section 5.

Figure 4: The number of targets in the scenario (black) and number of confirmed tracks (red) against time, with a probability of detection of 0.9.

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[1] J. Vermaak, S. Maskell and M. Briers, “A unifying framework for multi-target tracking and existence”, International Conference on Information Fusion, 2005.

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[2] D. Mu˘sicki and R. Evans, “Linear Joint Integrated Probabilistic Data Association - LJIPDA”, Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, 2002.

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[3] R. Mahler, “A theoretical foundation for the SteinWinter Probability Hypothesiws Density (PHD) multitarget tracking approach”, MSS National Symposium on Sensor and Data Fusion, Vol 1, San Antonio, Texas, 2000.

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[4] S. Arulampalam, S. Maskell, N. Gordon and T. Clapp, “A tutorial on particle filters for on-line nonlinear/nonGaussian Bayesian tracking”, IEEE Transactions on Signal Processing, Vol. 50, pp. 174–188, 2002.

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[5] D. Schulz, W. Burgard, D. Fox and A. Cremers, “Tracking multiple moving objects with a mobile robot using particle filters and statistical data association”, IEEE International Conference on Robotics and Automation, pp. 1665–1670, 2001.

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[6] J. Vermaak, S. Godsill and P. P´erez, “Monte Carlo filtering for multi-target tracking and data association”, IEEE Transactions on Aerospace and Electronic Systems, Vol. 41, pp. 309–332, 2005.

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(b) Figure 5: Tracking with a detection probability of 0.3. The targets (black) and confirmed tracks (other colours) in (a) x and y coordinates, (b) x coordinate and time.

[7] P. Horridge and S. Maskell, “Real-Time Tracking of Hundreds of Targets With Efficient Exact JPDAF Implementation”, International Conference on Information Fusion, 2006. [8] P. Horridge and S. Maskell, “Tracking with intervisibility variables”, IET Seminar on Target Tracking and Data Fusion, 2008.

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Figure 6: The number of targets in the scenario (black) and number of confirmed tracks (red) against time, with a probability of detection of 0.3.

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