Evaluation of a Probabilistic Multihypothesis Tracking Algorit Cluttered Environments R. G. Hutchins and D. T. Dunham Electrical and Computer Engineering Department Naval Postgraduate School Monterey, CA 93943
Abstract This research examines the probabilistic multi-hypothesis tracker (PMHT), a batch-mode, empirical, Bayesian data association and tracking algorithm. Like a traditional multihypothesis tracker (MHT), track estimation is deferred until more conclusive data is gathered. However, unlike a traditional algorithm, PMHT does not attempt to enumerate all possible combinations of feasible data association links, but uses a probabilistic structure derived using expectation-maximization. This study is focusing on two issues: the behavior of the PMHT algorithm in clutter and algorithm initialization in clutter. We also compare performance between this algorithm and other algorithms, including a nearest neighbor tracker, a probabilistic data association filter (PDAF), and a traditional measurement-oriented MHT algorithm.
1. Introduction Traditional data association and tracking algorithms make use of relatively simple techniques to reject clutter, such as gating and nearest neighbor association [1,2]. As clutter density increases, these techniques simply do not work well. During the 1970s several multiple hypothesis data association algorithms were developed to improve system performance in the presence of clutter [1,2,3]. These algorithms are cumbersome and require large amounts of memory to store the combinatorial number of possible data associations required by these techniques. They have only been practical in situations where data rates are relatively low and computing power has been large, and they have primarily been in use for sonar applications aboard submarines and surface ships. In response to these shortcomings, researchers have developed more sophisticated "single hypothesis" data association strategies, such as the probabilistic data association filter (PDAF) [l]. These strategies do not make hard data association assignment decisions, but include all measurements in the data association and track update algorithm, ranking them by their "likelihood" of association. These algorithms have enjoyed a degree of success, but they do not look ahead beyond the current collection (or "scan") of measurements before track update decisions are made.
1058-6393/97$10.00 0 1997 IEEE
In 1995, Streit and Luginbuhl at the Naval Undersea Warfare Center proposed a new algorithm called the probabilistic multi-hypothesis tracking (PMHT) algorithm [4,5]. This algorithm processed the data in batch mode, looking over multiple scans of data. Hence, like the traditional multihypothesis tracking algorithm, decisions and track updates are deferred until more conclusive data is gathered. However, unlike a traditional MHT, PMHT does not attempt to enumerate all possible combinations of feasible data association links. Rather, it begins with an initial guess of measurement-to-track assignment probabilities and iterates over the data, refining these assignment probabilities at each iteration, until convergence is reached. Such a technique, wlule requiring more computations than "single hypothesis" methods, can be much faster than traditional multihypothesis algorithms. However, actual performance of the algorithm in clutter has not been adequately studied. Nor has algorithm initialization been adequately examined, especially as this process relates to cluttered environments. This study is focusing on two issues: the behavior of the PMHT algorithm in clutter and algorithm initialization in clutter. We also compare performance between the PMHT and other algorithms, including a nearest neighbor tracker, a probabilistic data association filter (PDAF), and a traditional measurement-oriented MHT.
2. Description of the PMHT The PMHT is a batch-mode, empirical, Bayesian data association and tracking algorithm. A derivation of the algorithm, based on the probabilistic notion of marginalizing out the measurement to track assignments (i.e., treating them as nuisance variables), appears in [4]. The derivation of the updated target state estimate relies on the expectation-maximization (EM) method. The linear Gaussian case is assumed for the target probabilistic description as implemented here (Section 5 of [4]). This research focuses on tracking slow moving targets in the plane using range-bearing measurements that are converted to Cartesian coordinates prior to use in the Kalman smoother that is an integral part of the PMHT algorithm. Straight line target motion with nominal white noise acceleration is assumed for actual target tracks, along with a linear measurement model based on preprocessing range-bearing measurements as discussed in
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[ 6 ] . Hence, this study is modeled on the process of tracking slow moving submarines using scans of rangebearing acoustic measurements. The first step in the PMHT algorithm is to obtain an initial sequence of target state estimates and target measurement probabilities for each target at each measurement scan time:
These quantities i$,? and R$ are now used in the standard Kalman smoothing algorithm to generate updated estimates for {k$> and {P$]. The algorithm now iterates until convergence is achieved (up to a maximum of 100 iterations in this research). Convergence is achieved when
Here, t specifies the target model (t = 1,...,M),k specifies the time (k = 0,...,T), m specifies the measurement at time k (m = 1,...,nk), and the superscript (i = 0 above) indicates the algorithm iteration number. Also, the matrix Plfkis the covariance associated with the smoothed state estimate 2Ifkfor any iteration i of the PMHT. The values n$ specify the estimated probability that a measurement at scan time k is assigned to target model t after i iterations of the algorithm. At each iteration, the set of assignment weights is computed using: m;$+i) = N ( z m , k l z t , k , P(j), t , k i i t , k ) for a target track, W t : g L ) = p (an adjustable parameter) for clutter, and
2
2") t , k = HPItbHT + k t , k
Here Zm,k is measurement m at scan time k, and the
Rifi
is derived by computing a covariance matrix covariance matrix from a hypothetical range-bearing measurement that arises at the location of the current estimate of target position. The superscript i in these equations is the PMHT iteration index. These weights are used to define the updated target measurement probabilities: where - a(i)n(i-l), t,k t,k
~ ( -~ 1 t,k
where nkis the number of measurements in the scan at time k. The centroid measurement is now computed using:
This algorithm differs from the baseline described in Streit and Luginbuhl ([4], Section 5) in that each measurement has a distinct covariance matrix, which is why the matrix Ri:: has been defined above. Other possibilities have been tried for a more general covariance structure, but this has worked the best in our testing to date.
3. Implementation The above algorithm has been implemented in a simulation scenario to test performance in clutter. We have assumed target speeds are in the range of 2 to 10 knots, and that range and bearing measurements are made with 1-sigma errors of 100 meters and 3 degrees, respectively. First, a realistic initialization strategy was needed. We have tested initializations of the form N-of-N in this research. For small values of N, the initial heading estimate can be very wrong, even when computed using the correct target measurements. Also, the algorithm is least likely to converge when initial bearing estimates are normal to the true motion of the target. To minimize these cases, an N 2 5 measurements was required. There still remained problems with clutter measurements creating unrealistically fast tracks. Hence, a speed limit has been imposed on the initial track estimate. Limits we have tested are 2 knots and 10 knots. Another significant adjustment to the algorithm has been the truncation of the weights @t:t!k when they fall below a threshold, preventing outlying measurements from affecting convergence, especially during initialization. Also, as implemented here, the algorithm is run in batches of 5 data points, so that convergence is accomplished incrementally as data is received. This is also a more realistic processing strategy for a deployed system than is pure batch processing. Figure 1 demonstrates the scenarios employed in our simulation tests. Here, the straight line is actual target motion, the * symbols represent true target measurements, and the circles are clutter. This is a moderate clutter density, representing 5 clutter points per ping cycle (a density of 1.67 x lou2clutter points per square kilometer).
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The results reported here include clutter loads between 1 to 10 clutter points per ping (densities between 3 . 3 ~ 1 0 to -~ 3 . 3 ~ 1 0clutter -~ returns per square kilometer).
4.
Results
Figures 2, 3, and 4 show examples of algorithm convergence. All of these results were obtained at a clutter density of 1.67 x Figure 2 demonstrates convergence for a straight line track moving predominantly across the field of view at a speed of 5 knots at ranges above 40 kilometers (the sensor is at (0,O) and circles are truth here). In this particular case convergence at the end of the scenario (lower right) is very good. Figure 3 shows an example of convergence for two crossing tracks, and figure 4 shows the failure of the algorithm to track through a 90 degree turn on one run. Figure 5 is a more complete analysis of algorithm convergence for crossing tracks. All curves represent mean distance errors from the actual target positions at the measurement times. The 4 curves in this figure are as follows, the lowest curve is the result of the Kalman smoother algorithm utilizing the correct target measurements and the correct covariances in the absence of clutter. The curve is an average of 1000 simulation runs under these ideal conditions. Hence, this curve represents a theoretical minimum in terms of algorithm performance. The highest curve represents the mean target measurement errors, again averaged over 1000 runs. Generally, a tracking algorithm should perform better than distance errors associated with the raw measurement data. Of course, clutter and target maneuvers can seriously degrade a tracking algorithm. Finally, the two middle curves represent the mean distance errors for the estimated track positions once algorithm convergence has occurred. These means are taken over 100 simulation runs, as are all other algorithm mean distance error curves. The lower curve in the center represents the mean distance errors for a track heading along the bearing line of the sensor. The upper curve in the center represents the mean distance errors for a target moving across the bearing line of the sensor. Hence, it is clear that cross-bearing tracks create more problems for the algorithm than do tracks moving along a fixed bearing relative to the sensor. Figure 6 compares the performance of the PMHT with other algorithms. Here only the cross-bearing track was used. Note that since the PMHT generates a smoothed estimate for the entire track, while the other algorithms generate only filtered estimates, the only fair position to judge the competing algorithms with the PMHT is at the end point. Included in this figure are the two performance marker curves, the theoretical minimum and the measurement average distance. The remaining curves, from bottom to top on the right half of the figure are as follows: the PMHT, two curves associated with a traditional MHT, the PDAF and a nearest neighbor tracker. Note that the PMHT and the MHT give virtually
equivalent performance. The two curves associated with the MHT are as follows: the lower curve is the track with the most target points that is being carried in one of the active hypotheses. The upper curve is the track closest to the target that is being carried by the top hypothesis, where the track is required to have at least three measurements. Figure 7 is a repetition of figure 6, except the clutter density has been doubled. Also, the nearest neighbor tracker has been eliminated. Note that now the MHT (either track) is outperforming the PMHT at the end point. The PDAF is the worst performer in this group. Figure 8 is an examination of tracking through a 90 degree turn at a clutter density of 3 . 3 ~ 1 0 clutter -~ returns per square lulometer. Here, both the PMHT and the PDAF break and go off at the upper right. The MHT is able to reacquire the target following a break in the tracks during the maneuver.
5.
Conclusions
The above results indicate that the PMHT is a solid performer at these clutter densities and is competitive with other algorithms. The algorithm does have some quirks, especially with respect to initialization, that has made it more fragile than the other algorithms in our tests, but that is to be expected with a new algorithm. We continue to investigate its strengths and weaknesses.
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5.
References
[l] Bar-Shalom, Yaakov, and Xiao-Rong Li, MultitargetMultisensor Tracking: Principles and Techniques, 1995, Published by Yaakov Bar-Shalom and obtainable from him at
(203) 486-4823 or by contacting him at
[email protected]. [2] Blackman, Samuel S., Multiple-Target Tracking with Radar Applications, Artech House, Norwood, MA, 1986 [3] D. B. Reid, "An Algorithm for Tracking Multiple Targets," IEEE Transactions on Automatic Control, AC-24, pp. 843-854, December 1979.
[4] Streit, Roy L., and Luginbuhl, Tod E., Probabilistic Multi-Hypothesis Tracking , NUWC-NPT Technical Report 10,428, 15 February 1995. [5] R. L. Streit and T. E. Luginbuhl, "Maximum Likelihood Method for Probabilistic Multi-Hypothesis Tracking," Proceedings of the SPIE International Symposium on Signal and Data Processing of Small Targets, SPIE Proceedings Vol. 2335-24, pp. 394-405, Orlando, FL, 5-7 April 1994. [6] Don Lerro and Yaakov Bar-Shalom, "Tracking with Debiased Consistent Converted Measurements Versus EKF", IEEE Transactions on Aerospace and Electronic Systems, Vol. 29, No. 3, July 1993.
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