Fast Track Confirmation for Multi-Target Tracking with Doppler Measurements Xuezhi Wang∗ , Darko Muˇsicki† and Richard Ellem‡ ∗ Melbourne
Systems Laboratory Dept of Electrical and Electronic Engineering, University of Melbourne, Australia Email:
[email protected] †
Independent Consultant Email:
[email protected] ‡ Maritime
Operations Division Defence Science and Technology Organisation, PO Box 1500, Edinburgh SA 5111, Australia Email:
[email protected]
Abstract
2
10
Two−Point Method One−Point Method 1
10
NIFT
In this paper, a one-point track initiation method is derived for conventional target tracking systems where noisy sensor measurements of both target position and Doppler are available. Tentative tracks are initiated using position measurements from every single scan and the associated Doppler measurements are incorporated to reduce the velocity uncertainties of these initiated track states. The proposed track initiation method is capable of reducing the computational requirements of the system quadratically compared to the classical two-point method when false measurements are present. We further demonstrate that a tracking system which incorporates Doppler measurements both for track initiation and measurement-to-track association can greatly reduce the number of false tracks that are generated and provides greater flexibility for confirming true tracks without sacrificing the system’s performance in terms of the number of confirmed false tracks. As a consequence, a tracking system that incorporates the proposed method will exhibit a much shorter true track confirmation delay than a similar system based on the standard approach.
all possible pairs of measurements from two consecutive scans. New tentative tracks are subsequently formed with velocity components extracted from these measurement pairs1 . The method is robust in the sense that it guarantees that all target originated tracks will be established with reasonably small velocity uncertainty. It has been popularly applied to many practical systems [3], [4]. However, as seen in Figure 1, the number of false tracks initiated with the two-point method grows quadratically with the number of false measurements whereas it is only expected to grow linearly for the one-point method. With a low clutter density, ρ, the two-point method yields a lower number of false tracks. For higher clutter densities, a larger number of false tracks will be generated, resulting in considerable increases to the computational load.
1. I NTRODUCTION In a standard tracker, tentative tracks may be formed using either “two-point” or “one-point” track initiation methods [1] which differ in the number of scans required to initiate tracks and in the approximation used to determine the initial target velocity state. The classical two-point track initiation method described in [2] is traditionally used to form new tracks using position only measurements. At each scan, a window spanned by the maximum expected target velocity over a scan period selects
0
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10 ρ
0
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Fig. 1: Number of initiated false tracks (NIFT) vs. clutter density (ρ) for a surveillance region of volume 10.
The one-point method initiates tentative tracks from each
0 This
work is supported by Maritime Operation Division, Defence Science and Technology Organisation, Australia and the Melbourne Systems Laboratory, University of Melbourne.
1-4244-1502-0/07/$25.00 © 2007 IEEE
1 It assumes a constant target velocity during the time interval between two consecutive scans.
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measurement in a single scan [5], [6], [7]. It is not as popular as the two-point method because the initial track velocity approximation is poor due to the lack of track velocity information, however, the number of tracks that are initiated is linearly related to the number of measurements, as demonstrated in Figure 1. Usually, a zero-mean Gaussian density is assumed to approximate the underlying target velocity with standard deviation spanned to admit an anticipated maximum target speed [2]. When both target position and Doppler measurements are available, the initial track velocity state can be better approximated by incorporating the associated Doppler information as in [8], [9]. In this paper, a one-point track initiation method is considered. Following our previous work on incorporating Doppler measurements for multi-target tracking in [8], we derive a closed form initial track error covariance, where the Doppler measurement is used to reduce the uncertainty of the initial track velocity. The advantage of the new one-point track initiation method is highlighted by comparing the performance of one-point track initiation both with and without using Doppler measurements in an active sonar tracking scenario. Furthermore, we experimentally show using the Linear Multitarget Integrated PDA (LMIPDA) algorithm [10], that the incorporation of target Doppler information in both track initiation and data association results in a much larger probability margin for confirming true tracks. This allows the track confirmation threshold to be lowered, providing a shorter true track confirmation delay, without sacrificing the system performance in terms of the number of confirmed false tracks (NCFT). The remainder of this paper is arranged as follows. In Section 2, we describe the problem of true track confirmation delay for an IPDA type tracking system and then discuss several practical considerations for reducing the true track confirmation delay. The proposed one-point track initiation method that incorporates target Doppler information is presented in Section 3. In Section 4, the benefit of using Doppler information via the proposed method is demonstrated with an underwater active sonar tracking application. Our conclusions is provided in Section 5.
where ωkτ ∼ N (ωkτ , 0, Rkc,τ ). The Doppler (range rate) measurement from target τ can be expressed as ykd
τ
h(xk ) =
=
+
νkτ ,
νkτ
∼
N (νkτ , 0, Qτk )
s s s s (xk − xsk )(x˙ k − x˙ sk ) + (yk − yk )(y˙ k − y˙ k ) + (zk − zk )(z˙ k − z˙ k ) s )2 + (z − z s )2 (xk − xsk )2 + (yk − yk k k
(4)
where xsk = [xsk , x˙ sk , yks , y˙ ks , zks , z˙ks ] is the known sensor state vector at k. We assume that the system process noise νjτ and measurement noises ωkτ and nτi are independent of each other for all j, k, i and τ . In addition to target measurements, the sensor also receives clutter measurements, which are assumed to be uniformly distributed both in position and Doppler measurement spaces. The multi-target tracking problem is to find the conditional PDF p(xτk |Z k ), τ ∈ 1, 2, · · · , where Z k = {Z1 , Z2 , · · · , Zk } is the sensor measurement sequence up to current scan k and Zk = {zk,1 , zk,2 , · · · } is the set of measurements at scan k. Obviously, false tracks can be initiated. In the presence of false tracks, track confirmation is a necessary procedure to differentiate true target tracks from false tracks. For the LMIPDA algorithm, this can be simply handled by comparing the probability of target existence (PTE) associated with a track to a probability threshold. If the PTE of a track exceeds this threshold the track is promoted to a confirmed track. Only confirmed tracks are believed to be true tracks and are displayed as tracking system output. In order to remove uncertainties in the sensor observation domain, the confirmation threshold must be set to a value such that the tracker confirms all true tracks whilst maintaining the NCFT below an acceptable amount. This inevitably results in a true track confirmation delay which must be kept as short as possible. The true track confirmation delay (TTCD) may be defined as the total number of scans required for an initiated true track to be confirmed. In the example shown in Figure 2, the track confirmation delay is 4 − 1 = 3 scans for the given track PTE and track confirmation threshold. The average TTCD PTE 1
Let xτk denote the state of target τ at time k, which is assumed to be a vector of 6 components consisting of position and velocity for each axis. Its trajectory is modeled by Fk xτk
track confirmation delay track confirmation time
track confirmation threshold
(1)
We assume that an active sonar sensor transmits continuous waveform (CW) pulses at interval of T and obtains both position and Doppler measurements from returns. In other words, a measurement consists of bothposition and Doppler c d , where the position , yk,j components, i.e., zk,j = yk,j measurement from target τ at time k is linearly related to the target state, ykc = Hkc xτk + ωkτ (2)
(3)
where the measurement error is assumed to be a Gaussian nτk ∼ N (nτk ; , 0, Rkd,τ ) and h(xτk ) is given by [5]
2. T RACK C ONFIRMATION D ELAY
xτk+1
= h(xτk ) + nτk
initial probability for new tracks
Scan
0 k=1
k=2
k=3
k=4
k=5
... ...
Fig. 2: Probability of target existence vs. scan. We assume that target measurements are available from k = 1.
can also be used as a measure when Monte Carlo runs are involved. Figure 2 indicates that the use of the one-point track
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initiation method in a tracking system can also potentially reduce the TTCD compared to the two-point method. As we will see later, this depends on the uncertainty of the initial track state and clutter density. When Doppler measurements are available, it is possible to provide better estimates of the uncertainties in track initial velocity and measurement-to-track data association, which reduces the system TTCD. In general, a longer TTCD will allow the tracking system to assess more information from measurements before making a decision on whether or not a track should be confirmed. Clearly, a tracker with good data association is able to provide more track information for use in track confirmation decision making than one with poor data association. In many real applications, the TTCD is a critical requirement. One may adjust the track confirmation threshold for a required TTCD, however, a balance between the TTCD and NCFT is typically necessary as shorter TTCD can lead to unacceptable increases in the NCFT of the tracker. This issue will be discussed in Section 4 using simulation examples. 3. I NITIAL T RACK V ELOCITY A PPROXIMATION
vx
Target
vy vz
RR
Hkd = rk =
(6)
Therefore, our best guess for the initial relative velocity state XV (1) after receiving the Doppler measurement y1d at k = 1 can be derived based on the LMMSE criterion as
−1 ˆ V (1) = ΣV (0)(H d ) H d ΣV (0)(H d ) + Rd X y1d 1 1 1 1 ⎡ ⎤ aσx2 1⎣ bσy2 ⎦ y1d = (7) δ cσz2
Vmax
a=
x1 − xs1 y1 − y1s z1 − z1s , b= , c= , r1 r1 r1 δ = a2 σx2 + bσy2 + cσz2 + R1d
B −−− Bearing E −−− Elevation R −−− Range RR −− Doppler
and the associated covariance matrix is given by
X
Fig. 3: For One-Point tracking initialization, the possible target velocity may be expressed as the slant height of a right circular cone with its apex at the current target position and height representing the Doppler measured by the sensor. Clearly, if a zero speed for the sensor is assumed, the amplitude of the possible target velocity is bounded by [RR, Vmax ], where RR is the measured Doppler and Vmax is the maximum target speed.
The difficulty in initiating tracks using position measurements from one scan is the lack of knowledge of the underlying target velocity. An initial track state with large errors can potentially lead to filter divergence [11]. In addition, the number of confirmed false tracks is potentially higher as a larger track gate will be formed for use in measurement selection. If Doppler measurements are available, then they can be used to reduce the initial target velocity uncertainty. In the proposed method, the initial position state and covariance of a track is approximated using the associated position measurement and covariance. The velocity of a track is approximated as follows. In view of Equation (4), we can re-write (3) as ykd
s zk −zk rk
Clearly, if the underlying position state XP (1) is known, the Doppler measurement y1d may be seen as the measurement of the velocity state XV (1) at k = 1 observed via (5). Furthermore, we assume that the prior relative velocity of a target has a zero-mean Gaussian distribution2 with variance ⎤ ⎡ 2 σx 0 0 ΣV (0) = ⎣ 0 σy2 0 ⎦ . 0 0 σz2
E
Sensor
s yk −yk rk ,
xk − xsk , yk − yks , zk − zks , XV (k) = x˙ k − x˙ sk , y˙ k − y˙ ks , z˙k − z˙ks .
possible target velocity
Z
xk −xsk rk ,
(xk − xsk )2 + (yk − yks )2 + (zk − zks )2
XP (k) =
B Y
where
V
R
where
1 X (k)XV (k) + nk rk P = Hkd XV (k) + nk
ΣV (1) = ΣV (0) − ΣV (0)(H1d ) −1
× H1d ΣV (0)(H1d ) + R1d H1d ΣV (0) (8) ⎤ ⎡ 2 1 2 2 1 1 2 2 2 2 σx (1 − δ a σx ) − δ abσx σy − δ acσx σz ⎣ − 1 abσx2 σy2 σy2 (1 − 1δ b2 σy2 ) − 1δ bcσx2 σz2 ⎦ . δ 1 1 − δ acσx2 σz2 − δ bcσx2 σz2 σz2 (1 − 1δ c2 σz2 )
=
In real applications, target motion may be constrained by a maximum speed Vmax and sensor speed Vs is known. Let σ = Vmax + Vs and σx = σy = σz = σ, (7) and (9) can be written as ⎡ ⎤ a 2 σ ˆ V (1) = ⎣ b ⎦ y1d X (9) σ 2 + R1d c ΣV (1) =
=
⎡ σ2
⎢ ⎢ ⎣
a2 σ 2 ) σ 2 +R1d abσ 2 − σ2 +Rd 1 2 − σacσ 2 +Rd 1
(1 −
2
− σabσ 2 +Rd 1
b2 σ 2 ) σ 2 +R1d bcσ 2 − σ2 +Rd 1
(1 −
⎤
2
− σacσ 2 +Rd 2
1
− σbcσ 2 +Rd (1 −
1
c2 σ 2 ) σ 2 +R1d
⎥ ⎥ ⎦
(10) (5)
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2 This
might be the best guess without target Doppler information.
In fact, the vector [a, b, c] is the unit vector which preserves the direction from the sensor location to target location and ˆ V (1) is the mean velocity based on therefore, the value of X the measured Doppler information [8]. As shown in Figure 3, the unit vector can be written as a function of the observed target bearing B and elevation E, i.e., ⎡ ⎤ ⎡ ⎤ a sin B cos E ⎣ b ⎦ = ⎣ cos B cos E ⎦ (11) c sin E
A. One-Point versus Two-Point track initiation At each scan, all received measurements are used to initiate new tracks. For the one-point track initiation method, the Doppler measurements are used for initial track velocity approximation. The comparison results are presented in Table 1 and Figure 4. As seen from Table 1, in the presence of low clutter, little performance difference can be identified between the use of the one-point and two-point track initiation methods. However, the two-point track initiation method has attracted a significant amount of computational load in heavy clutter as the number of initiated tracks grows exponentially.
Therefore, for each measurement zk = {ykc , ykd } at time k, a tentative track is initiated with the state x0 = ykc (1), vt (1), ykc (2), vt (2), ykc (3), vt (3) ˆ V (1) + vt = X
xs (2),
The associated state covariance ⎡ c c R11 0 R12 ⎢ 0 0 ⎢ c Σ11 c ⎢ R21 0 R22 c ⎢ P =⎢ 0 ⎢ 0c Σ21 c ⎣ R31 0 R32 0 0 Σ31
xs (4), xs (6)
is defined as 0 Σ12 0 Σ22 0 Σ32
c R13 0 c R23 0 c R33 0
0 Σ13 0 Σ23 0 Σ33
.
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Prob. False Alarm Pf = 0.00005 Pf = 0.0001 Method Two-Point One-Point Two-Point One-Point Average CPU (sec) 0.2855 0.2446 3.9855 0.9907 Peak CPU (sec) 0.3519 0.3228 4.9162 1.3680 Average NT 17.2183 23.0400 82.7001 62.2861 Peak NT 19.3300 25.1400 91.3600 66.2700 Average NCFT 0.0097 0.0128 0.0285 0.0175 Peak NCFT 0.2500 0.2500 0.4600 0.4200 TABLE 1: C OMPARISON OF THE TIME AVERAGE PERFORMANCE .
(12)
where Rc is the covariance matrix associated with each position measurement and Σ is given by (9). Note that the form of equation (9) is virtually identical to that found in [9] where a 2D case was considered. In this paper, this has been extended to the 3D case and the associated velocity covariance (10) has been explicitly derived. As previously mentioned, if no target Doppler measurement is available, a zero-mean Gaussian distribution is used to characterize the underlying track initial velocity. All possible target velocities are included within a 3D sphere of zero origin and with radius Vmax . In the next section, the advantage of incorporating Doppler measurements into one-point track initiation is highlighted by comparing the track initiation performance with and without using Doppler measurements.
Figure 4 clearly demonstrates that the average TTCD for the one-point method is at least one scan shorter than that of the two-point method. In this example, the true track is confirmed at scan 9 with confirmation threshold 0.997 for both cases. 1.05 1.025 1 Probability of Target Existence
where
a) With or without using Doppler measurements in One-Point track initiation, that is, the initial track velocity is approximated by either the mean velocity via (7) and (9), or, a zero-mean Gaussian distribution. b) With or without using the Doppler data association technique for target tracking. An LMIPDA tracker with complete track management is implemented for the simulation.
4. S IMULATION R ESULT A NALYSIS In order to demonstrate the reduction of TTCD by incorporating Doppler information, we compare the following two cases using computer Monte Carlo simulations of an active sonar underwater target tracking scenario. 1) Two-Point versus One-Point track initiation method for an LMIPDA tracker in low and high clutter densities respectively. The tracker performance is measured in terms of computational load (CPU time), the number of tracks held by tracker (NT), the number of confirmed false tracks (NCFT) and the root mean squared (RMS) track error. 2) Tracking performance comparison for trackers with or without using Doppler measurements. This includes:
0.975
Two−Point Method One−Point Method Conf Threshold
0.95 0.925 0.9 0.875 0.85 0.825 0.8
1
2
3
4
5
6
7
8 9 Scan
10
11
12
13
14
15
Fig. 4: Probability of target existence comparison during first 15 scans.
B. One-point track initiation with or without Doppler measurements The track initial velocity state generated with Doppler measurements is determined using (7) and (9) in Section 3 and
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is labeled as Mean-Velocity, while the initial track velocity generated without Doppler measurements has a zero-mean Gaussian distribution with standard deviation equal to the assumed target maximum speed and is labeled as ZeroVelocity. The comparison results averaged over 100 runs are shown in Figures 5 and 6.
1.05 1 0.95 0.9
Track Confirmation Threshold = 0.997
0.85 0.8 Mean Velocilty Zero Velocity
0.75
150 140
0.7
130 120
0.65
110 0.6
100 Second
90
1
6
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31
Mean Velocilty Zero Velocity
80
36 41 Scan
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(a)
70 60
Number of In−Gate Measurements
50
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40
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8
10 0
7 1
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6 5
(a)
4 350
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225 200
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36 41 Scan
46
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71 75
(b)
Mean Velocilty Zero Velocity
175
1
Fig. 6: (a) True target PTE versus scan and (b) Average number of measurements in track gate.
150 125 100 75 50
1
6
11
16
21
26
31
36 41 Scan
46
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71 75
(b) Fig. 5: Performance comparison of One-Point track initiation with or without using Doppler measurements at Pf = 0.0003 using (a) CPU time comparison and (b) NT comparison.
Clearly, using Doppler measurements to approximate the initial track velocity reduces the expected initial track velocity noise, which in turn reduces the number of tracks held by the tracker and the total computational load of the tracker. In addition, the mean-velocity method also results in a shorter average TTCD. As indicated in the sub-figure (a) in Figure 6, the TTCD with an LMIPDA tracker using Doppler velocity initiation is at least 5 scans shorter than one using zero velocity initiation. This is most likely due to the track gate having a smaller volume as evidenced in sub-figure (b) in Figure 6, which can help the PTE of true track to rise more quickly. C. Tracker with Doppler Data Association The Doppler data association (DDA) presented in [8], [12] incorporates target Doppler information into an IPDA based tracker framework and results in a combined measurement
likelihood for track update in an LMIPDA algorithm. A significant data association improvement over the case without DDA has been observed in our simulation. The improved data association also generates enormous potential for improving TTCD performance. To appreciate this, a comparison is given in Figure 7, where the PTEs of all tracks held by a LMIPDA tracker for tracking the target without and with DDA at Pf = 0.0003 are plotted in the sub-figures (a) and (b) respectively, where black solid lines represent PTE of the true (target) track, while colored dots represent PTEs of false tracks. Clearly, the LMIPDA tracker with DDA presents a large (probability threshold) margin for confirming a true track, while such a margin for the tracker without DDA is very narrow. In fact, as indicated in Figure 8, we can actually set the track confirmation threshold as low as 0.35 for the tracker using DDA in this example without confirming a false track. This reduces the TTCD by at least 4 scans compared to the case without using DDA. It is worth mentioning that lifting the initial track PTE that is assigned to a new track can also reduce the TTCD. However, this would require a re-adjustment of the track confirmation threshold and possibly some other parameters to balance the required NCFT, especially when heavy clutter is present.
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1 1 0.9
Probability of Track Existence
Probability of Track Existence
0.8 0.7 0.6 0.5 0.4 0.3
0.6 LMIPDA−D track confirmation delay 0.5 0.4
0.1 6
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0.35
0.3 0.2
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LMIPDA track confirmation delay
0.7
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(a)
1
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8 9 Scan
10
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Fig. 8: Average true track confirmation delay comparison at Pf = 0.0003. In this example, when DDA is used, the track confirmation threshold may be dropped from 0.997 to 0.35 without increase of NCFT and therefore, the track confirmation delay can be reduced by 4 scans compared to the case without using DDA.
1 0.9 0.8 Probability of Track Existence
0.8
0.2
0
0.997
0.9
True Trakc Pte 0.7
R EFERENCES
0.6 0.5 0.4 0.3 Initial Pte of Tentative Tracks 0.2 0.1 0
1
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36 41 Scan
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(b) Fig. 7: PTEs of all tracks held by a LMIPDA tracker vs. scan at Pf = 0.0003. (a) Without DDA. (b) With DDA. In both cases, the initial PTE assigned to a new track is 0.05.
5. C ONCLUSION Following our previous work in [13], [8], we derived initial track velocity using Doppler measurement under LMMSE criterion for one-point track initiation. Simulation results have shown that one-point track initiation using Doppler measurements has significantly better performance in terms of computational load and TTCD over initiation without using Doppler measurements. It is also evident that both LMIPDA and LMITS trackers with DDA are able to strike a better performance balance between lower NCFT and shorter TTCD, while keeping a reasonably low computational load. From an information point of view, incorporating Doppler measurements into the process of target state estimation results in a higher information flow from the measurements and the tracker may obtain sufficient information for confirming a true track in a shorter period of time. Our simulation shows that by incorporating Doppler measurements, the track confirmation threshold may be considerably reduced without increasing the NCFT. As a consequence, a reduced TTCD (typically less than 4 scans at Pf = 0.0003) can be achieved.
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