Uniform Versus Nonuniform Sampling when Tracking in Clutter

I. INTRODUCTION

Uniform Versus Nonuniform Sampling when Tracking in Clutter X. ZHANG, Student Member, IEEE P. WILLETT, Fellow, IEEE Y. BAR-SHALOM, Fellow, IEEE University of Connecticut

Many target tracking subsystems have the ability to schedule their own data rates; essentially they can “order” new information whenever they need it, and the cost is in terms of the sensor resource. But among the unmanaged schemes, uniform sampling, in which a new measurement is requested periodically and regularly, is the most commonly-used sampling scheme; deliberately nonuniform schemes are seldom given serious consideration. In this paper, however, we show that such schemes may have been discarded prematurely: a nonuniform sampling can have its benefits. Specifically, the nonuniform and uniform sampling schemes are compared for two kind of trackers: the probabilistic data association filter (PDAF), which updates its track based on a single scan of information at a time; and N-D assignment (an optimization-based implementation of the multi-hypothesis tracker (MHT)), in which the sliding window involves many scans of observations. We find that given the ground rule of maintenance of the same overall scan rate (i.e., the same sensor effort) uniform sampling is always optimal for the single-scan tracker in the sense of track life. However, nonuniform sampling can outperform uniform sampling if a more sophisticated multi-scan tracker is used, particularly when 1) the target has a high process noise, and/or 2) the false alarm density is high, and/or 3) the probability of detection is high.

Manuscript received June 1, 2002; revised June 6 and September 9, 2005; released for publication October 15, 2005. IEEE Log No. T-AES/42/2/876417. Refereeing of this contribution was handled by W. D. Blair. This research was supported by AFOSR under Contract F49620-97-1-0198 and by ONR under Contract N00014-97-1-0502. This paper is a modified version of [14]. Authors’ address: Dept. of Electrical and Computer Engineering, U-2157, University of Connecticut, 371 Fairfield Rd., Storrs, CT 06269, E-mail: ([email protected]).

c 2006 IEEE 0018-9251/06/$17.00 ° 388

The basic tracking problem is that of state estimation for a linear system from measurements corrupted by Gaussian observation noise. The optimal (in essentially every sense) solution in this case is the Kalman filter. However, in target tracking the obfuscation in the measurement system, previously modeled only as the introduction of an additive Gaussian observation disturbance, now additionally incorporates measurement-origin uncertainty. That is, there can be false alarms, and also the “true” (i.e., target-generated) measurement may be missed. Consequently the measurement “scan” delivered from the sensor to the tracking subsystem can be null (detection missed and no false alarms), single (perhaps true, but also may be a false alarm with the true measurement missed), or multiple (the true measurement, if present at all, is not labeled as such among those delivered). The key tracking issues are hence how to determine which measurement, if any, is true, and how to reflect the resulting uncertainty in the tracker’s self-assessment. Many sensor systems operate autonomously, and deliver their measurement scans on a fixed-interval basis–most sonar systems work this way, as do conventional “rotator” radars. Some electronically steered radar systems, however, allow for the tracking user to request scans of data on an adaptive, or at least an agile, basis. Interestingly, Daum in [4] has suggested that sampling times, and possibly jittered versions of these, ought to be considered as parameters in the optimization of tracking systems. Notwithstanding, it appears that most tracking designers remain more comfortable with the original regular delivery of scans; irregular scans are an unwanted extra dimension of freedom, and it is easiest to ignore them. Here we explore the more general case of nonuniform sampling. Suppose we have a uniform sampling interval T. The kind of nonuniform sampling considered here is that the system samples at interval T1 and T2 alternatively; but to make the overall sampling rate equal, and to have a fair comparison, we must have T1 + T2 = 2T. To be concrete, we may consider that a periodic-scan system can request measurements each second (T = 1), and that an alternative (but comparable in terms of radar resource cost) scheme requests a pair of scans separated by 0.1 s, with each pair separated by 1.9 s (T1 = 0:1 and T2 = 1:9). See Fig. 1 for an illustration. The timing just discussed, and that to be explored in this paper, is the result of a deliberate staggering of measurement sampling times–basically, this is an option for a radar resource management system. But there is another sort of nonuniform sampling that is not deliberate, that arising from data fusion from asynchronous sensors. Consider Fig. 2, in which centralized processing is accomplished based on

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Fig. 1. Illustration of uniform and nonuniform sampling timing schemes. Fig. 3. Overlay of true and false detections from 2 consecutive scans. Notice that false measurements are unlikely to coincide. However, given that inter-scan time is not too great, true measurements are close.

Fig. 2. Possible timing of measurements from 2 asynchronous but equal-period sensors. At fusion center, effect is same as in Fig. 1.

measurements from a pair of autonomous observers. These observers do not scan in lock-step, and the arrival of a measurement from either at the fusion center is aperiodic, and could even be considered (in the general case) a point-arrival random process. We do not consider explicitly the data fusion situation here, there are simply too many variables to adjust for a clear picture to emerge. Similarly, even in the single-sensor case with sensor management, we could consider more sophisticated staggered-sampling schemes. This paper is exploratory: whatever behaviors are to be found in these more involved cases, the simpler situation of Fig. 1 should hint at them. That is, the situation we study here is we hope a proxy for other cases, and if some advantages of a nonuniform sampling should arise here, then perhaps it is worthwhile to examine other cases. So, is there some benefit to nonuniform sampling? The question is given detailed examination in [6], [7], and [8] for multi-sensor situations without measurement-origin uncertainty. The clean-data assumption allows quite elegant and exact Riccati analyses: it turns out that for identical sensors a “uniform stagger” is best, although for sensors having measurements of dissimilar quality there can be benefit to a richer pattern. We are most interested here in systems for which data association is necessary: in the presence of missed detections and false alarms, can a nonuniform sampling strategy be the best choice even for identical sensors?

We assume false alarms to be uniformly distributed1 in the surveillance region and to be independent of the true observation [1, 2]. Under this assumption, the sampling interval scheme is irrelevant to the false alarm process.2 However, since it is generated by the target, the true observation is affected by the sampling interval. As can be easily shown, when the sampling interval is short, the true observations from two time-adjacent samples are close to each other. Thus the true observation can be made more distinguishable from those that are false by use of a short sampling interval: two measurements in adjacent scans that are very close are likely both true, while any measurement in a first scan that is not repeated by one in the second scan that corroborates it is probably (although not definitely) false. This is illustrated in Fig. 3. Thus, notionally, the nonuniform sampling scheme ought to be an excellent means to combat measurement-origin uncertainty. However, there is a price: the longer sampling interval between the two scan pairs (1.9 s in the above) increases the uncertainty (the “gate”), and the resulting increase in false alarms may actually exacerbate measurement/target association. This is illustrated in Fig. 4. Note that the increase in gate size can mean an increase in the number of false alarms that could be confused with the true measurement, and this may result in a lost track. In this paper, these two sampling schemes are compared. It will turn out that the superiority of one versus another is very much a function not only of 1 In reality, the false alarms are not uniformly distributed in the surveillance region. One obvious reason is the size of resolution cells grows larger as the range increases. There are also studies that show false alarms inside the beam concentrate more toward the boresight. But in the scope of the current paper, we assume the false alarms to be uniform, as is widely accepted. 2 We assume the false alarms are generated by white noises at different sensors, so false alarms in different scans have no correlation.

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course really exists in continuous time, it matters a lot. DWNA is inappropriate, and we use DCWNA. The state equation of DCWNA with sampling period Tk , which is of second order, is x(k + 1) = F(k)x(k) + v(k)

(1)

·

(2)

where F(k) =

Fig. 4. Notional uncertainties with uniform and nonuniform sampling. In the former case, uncertainties (illustrated by covariance ellipses) are relatively constant in size. In the latter (nonuniform) case, the ellipses increase significantly in size between pairs of samples, and allow more false alarms to enter.

parameters, but also of the tracking scheme used. Specifically, we shall investigate the probabilistic data association filter (PDAF) and N-D assignment algorithms [2, 5, 9, 10]. The PDAF associates measurements to tracks one scan at a time, and hence that it will be shown that uniform sampling is always the best approach for the PDAF is perhaps unsurprising. However, when a tracker that makes use of more than one scan of past data is used (i.e., multidimensional assignment), the system with nonuniform sampling can outperform the “conventional” uniform one. The advantage becomes most obvious when the target maneuvers more, when the clutter (average number of false alarms) is high, and particularly when the probability of detection is close to unity. II. MODELS AND TRACKERS A. System Model of a Single Target Here we focus on the 1-dimensional discretized continuous white noise acceleration model (DCWNA) [1]. The alternative is to assume the discrete white noise acceleration model (DWNA), and although which is chosen is often a matter of taste (both are based on reasonable statistical assumptions, but since these assumptions are not necessarily satisfied by a real target one tends to choose one’s favorite) one more commonly sees DWNA since its parameters are slightly easier to relate to real-world effects. Under DWNA, the model is directly in the discrete-time domain, and is tantamount to a continuous-time kinematic object undergoing a constant acceleration during each interval between observations; under DCWNA the model is of discrete-time samples from a continuous-time system driven by continuous-time white noise. Normally there is little difference; but here, since we wish to compare different sampling schemes of the same target that of 390

1 Tk 0

1

¸

and the covariance of the discrete time process noise v(k) is · 1 3 1 2¸ 3 Tk 2 Tk Q(k) = 1 q˜ : (3) 2 Tk 2 Tk It will be assumed that only position measurements are available, that is, z(k) = Hx(k) + w(k)

(4)

H = [1 0]

(5)

where and the measurement noise variance is R = E[w(k)2 ] = ¾w2 :

(6)

In the above equations, Tk is the kth sampling interval. For uniform sampling, the sampling interval is fixed to be Tk = T. For nonuniform sampling, we consider sampling intervals T1 and T2 alternatively, with T1 + T2 = 2T. As discussed in the previous section, other nonuniform sampling schemes are possible. B. PDAF At the outset let us note the main feature of the PDAF: it is entirely optimal for target tracking with a linear Gaussian model except that after each scan its posterior track state probability density function (pdf) (ideally a mixture of Gaussian pdfs) is converted to a single Gaussian having the same mean and variance (i.e., via moment matching). Thus, at each scan, estimation is built upon a Gaussian prior, converted to a Gaussian mixture posterior, which is then forced back to Gaussianity for the succeeding scan. Reference to Fig. 5 may be helpful. The reader is encouraged to examine the derivation in [2]. It should be noted that in practice the predicted measurement is often enclosed pby a “gate” whose volume is proportional to jS(k)j, where S(k) is the innovation covariance matrix at time k, and whose function is to reduce computation by ignoring any measurement whose association probability is negligible. C.

N-D Assignment Algorithm

Recall that nonuniform sampling is one short sampling interval followed by a long one, and although the short sampling interval has its benefits,

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Fig. 5. Representation of one-step tracking as performed by PDAF.

Fig. 6. Procedure of track split filter.

the price is paid in the long one. The notion behind nonuniform sampling is in the added evidence behind the presence of spatially-close detections in two temporally-close scans; and conversely in the reduced weight to be given a measurement in one such scan that is not corroborated in the next. To take greater advantage of nonuniform sampling, one should improve one’s utilization of the benefits of the short sampling interval to overcome the penalty brought by the long sampling interval, and the corresponding association scheme must incorporate multiple scans of data. An N-dimensional assignment algorithm [2, 5, 9, 10] does this. The procedure consists of the following steps. 1) After initialization (at k = 0), for every measurement at k = 1 that falls in the validation region around the location zˆ (1 j 0) where the measurement is expected, the track is split. 2) For each measurement an updated state is computed via the (standard) Kalman filter equations [1] and propagated forward to yield another validation region at k = 2. 3) For each new validation region at k = 2 the procedure is repeated. 4) When the procedure reaches k = N ¡ 1, the procedure is stopped. By calculating the likelihood

function, the measurement of the most likely branch at k = 1 is chosen as the true measurement and the N-dimensional search is restarted at k = 1 and ends at k = N. This (sliding window) block search is repeated. The procedure is illustrated in Fig. 6. From that figure it can be seen that during the short sampling interval T1 , the validation region volume V(k + 1, T1 ) is small, hence the probability of a false alarm falling into this validation region is greatly reduced. Equivalently, during the short sampling interval, the true observation, which is close to the predicted value of the measurement, has fewer “competitors” (false alarms) in its likelihood function evaluation. This can explain, in part, why nonuniform sampling may be a good idea with assignment. III. NUMERICAL RESULTS In the simulations of both the PDAF and the N-D assignment algorithm, we use the one-dimensional linear Gaussian DCWNA target model described in Section IIA. In the presence of uncertain-origin measurements, define the observations at time k to be k Z(k) = fzi (k)gm i=1

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Fig. 7. Average track life in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 0:9, q˜ = 10, ¾w2 = 10.

Fig. 8. Average track life in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 1, q˜ = 10, ¾w2 = 10.

where mk is the number of the observations, in general a random quantity itself. In target tracking, Z(k) is comprised of all observations collected at time k, and these mk observations can all be false alarms, or there is one true detection and (mk ¡ 1) false alarms, or there might even be no observations (mk = 0). The true target is detected with probability PD . False alarms are modeled as independent and uniformly distributed over the observation volume (or gated volume) V with pdf 1 ¢ p0 (z) = p(z j z is a false measurement) = (8) V and the distribution of the number of false alarms is Poisson (¸V)n PFA (n) = e¡¸V (9) n! where ¸ is the spatial density of the false alarms (number of false alarms per unit volume) [2]. The performance of the tracking algorithm is evaluated in terms of track life, i.e., how long the track can be maintained before it is lost. Throughout this paper, we decide a track is lost if, for five 392

consecutive scans, the estimated target position falls out of the “five sigma” region centered around the true target position, that is, when [H(xˆ (k) ¡ x(k))]T (25R)¡1 [H(xˆ (k) ¡ x(k))] > 1: (10) Figs. 7—12 show the average track life (in seconds) evaluated as a function of false alarm density; 500 Monte Carlo runs are used to obtain the results in these figures. Nonuniform sampling is compared to uniform sampling in these figures, in which for the nonuniform case we have T1 = 0:1 s and T2 = 1:9 s, as compared with the uniform-sampling case’s T = 1 s. In Figs. 7, 9, and 11, probability of detection is set to 0.9, while in Figs. 8, 10, and 12, we have PD = 1. We fix the state noise power spectral density q˜ = 10 and the measurement noise variance takes the values of ¾w2 = 0:1, ¾w2 = 1, and ¾w2 = 10. For the DCWNA model of Section IIA, the maneuvering index, defined in [1] as T3 q˜ ´= 2 ¾w

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Fig. 9. Average track life in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 0:9, q˜ = 10, ¾w2 = 1.

Fig. 10. Average track life in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 1, q˜ = 10, ¾w2 = 1.

Fig. 11. Average track life in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 0:9, q˜ = 10, ¾w2 = 0:1.

(in which T is the uniform inter-sampling time), provides a convenient scalar measure of a target’s elusiveness. For example, Figs. 8, 10, and 12 have maneuvering indices ´ = 1, 10, and 100, respectively. From Figs. 7—12, the clearest conclusion is that for the PDAF, and presumably for any frame-by-frame

tracker, a uniform sampling rate is better. These figures depict only a few sets of parameters, but we have been unable to find a case in which its conclusion is contradicted. Another observation is that nonuniform sampling can be preferable for N-D assignment or other

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Fig. 12. Average track life in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 1, q˜ = 10, ¾w2 = 0:1.

Fig. 13. Number of lost tracks (out of 500 runs) in 100 s in PDAF and 3-D assignment, as function of false alarm density, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 1, q˜ = 10, ¾w2 = 0:1.

multi-frame algorithms. This is confirmed by Fig. 12, where we have high probability of detection (PD = 1), high maneuvering index (¾w2 = 0:1 and ´ = 100). Comparing Figs. 7—12, we can clearly see that high probability of detection helps the nonuniform sampling, and we also reveal that any benefits of nonuniform sampling are magnified by a more maneuvering target. The effect of false alarm rate on the selection of sampling schemes is not obvious with the above figures, but with the help of Fig. 13, where the number of lost tracks (out of 500 runs) in 100 s is plotted, we can see that the nonuniform sampling performance improves, relatively speaking, as the false alarm density increases. An interesting observation from Figs. 7—12 is that it appears that the performance of both PDAF and 3-D assignment trackers degrade as the measurement noise is reduced. This is because when measurement noise is reduced and the process noise remains the same, we have a higher maneuvering index and the motion of the target is harder to be predicted by the trackers [1]. The benefit of nonuniform sampling is amplified in Fig. 14, which shows the average track life as a function of the ratio T1 =T2 , with of course the 394

stipulation T1 + T2 = 2 s in all cases; 1000 Monte Carlo runs were used to obtain the results. It is clear from this that when pairs of samples are close (i.e., a small ratio) there can be a significant improvement from a nonuniform approach. The difference can be substantial (fifty percent!) when the ratio approaches zero; interestingly, this is the (synchronous) “data fusion” case discussed earlier. Fig. 14 is particularly kind to nonuniformity because it depicts the case of perfect detection: there are false alarms, but no measurements are missed. Fig. 15 (again from 1000 Monte Carlo runs) explores this further. Apparently, the improvement for the nonuniform sampling case is greatest when the target is faithfully present in the data. We discuss this in the next section. Generally, though, we have found that nonuniform sampling is preferable only when PD > 95%. IV. DISCUSSION It has been observed that uniform sampling is preferable to nonuniform when the tracking algorithm

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Fig. 14. Average track life in PDAF and 3-D assignment, as function of T1 =T2 , with uniform and nonuniform sampling schemes, false alarm density ¸ = 0:01, T = 1 s, PD = 1, q˜ = 10, ¾w2 = 0:1.

Fig. 15. Average track life in PDAF and 3-D assignment, as function of probability of detection, with uniform and nonuniform sampling schemes, T1 = 0:1 s, T2 = 1:9 s, T = 1 s, ¸ = 0:01, q˜ = 10, ¾w2 = 0:1.

associates measurements to targets on a scan-by-scan basis, as with a PDAF. However, and on the contrary, if association is based on several scans worth of measurements, then nonuniform sampling actually has something to offer. This is especially true when the probability of detection is high, when the clutter is high, and when the target’s maneuvering index is high. Now, this study of nonuniform sampling was motivated by the idea that data association might be helped by a cross-check between returns from two contiguous scans of observations. Presumably this is indeed a beneficial effect. However, association is hampered by false alarms, and the number of these can be thought of as proportional to the size of the association “gate” surrounding each measurement’s predicted location, and inside which candidate associations are sought. This gate has volume itself proportional to the innovations’ covariance S (actually the square root of its determinant), and is hence a quadratic polynomial of the time between scans. Presumably, then, this favors a uniform sampling. The point is that these two effects are in conflict, and it is difficult to tell at the outset which will

dominate. In fact, we cannot even envision proofs of the earlier observations, given the complexity of the tracking task. However, the tendencies observed (e.g., high PD is best for nonuniform) are reasonable, and in the following sections we try to explain why. We begin with a study of the Kalman filter, and in particular its average gate size. A. Kalman Filter Case Underlying most tracking systems is the Kalman filter, the tracker that would be optimal were there no measurement-origin uncertainty. Above it is argued that the performance of a tracking system is degraded by a larger gate size, since the greater the gate, the more false alarms can enter. Prediction of the gate size is difficult, but in the simplistic Kalman filter situation it can be done. For the usual Kalman filter, the steady-state gate size is straightforward to estimate from the discrete time Riccati equation [1]. Here, however, the gate size varies due to the irregular sampling times. During the longer inter-sampling interval the gate will increase more than it would have increased during a uniform

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Fig. 16. Average gate size for Kalman filter. Figure is relative, as compared with maximum; curves are parameterized by q˜ =r with unit-time between each pair of samples.

sampling time, but of course it begins from a smaller uncertainty due to the prior short sampling interval. The question is: which effect wins? Let us consider the model of Section IIA. To incorporate nonuniform sampling, we stack the state for consecutive pairs of scans. Specifically, let us write, with reference to (1) and its sequel, · ¸ x((k + 1)(T1 + T2 )) x((k + 1)(T1 + T2 ) + T2 ) · ¸ ¸· 0 F1 x(k(T1 + T2 )) = + v(k) 0 F2 F1 x(k(T1 + T2 ) + T2 )

sampling is best in terms of average uncertainty. However, the “AEV” in [8] is time-integrated and our analysis here is only the averaged uncertainty at the sampling times. Presumably the AEV is a better global performance predictor; but here we are interested in the average number of gated false alarms (i.e., the effect on systems with data association), so the average uncertainty at the sampling times is more meaningful. We also observe that the benefit from nonuniform sampling increases as the target exhibits greater maneuverability, a corroboration of the observations in Figs. 8, 10, and 12.

(11) in which x(¢) is a stacked vector of position and velocity, and ¸ ¸ · · 1 T1 1 T2 F1 = , F2 = 0 1 0 1 · ¸ Q1 Q1 F20 Q= F2 Q1 F2 Q1 F20 + Q2

B. Case with Data Association

(12)

where Q1 and Q2 are from (3). Solution of the discrete algebraic Riccati equation with the above parameters yields prediction uncertainties when estimation is in steady state: the (1,1) element is required for the gate volume for the T1 interval, and the same procedure with T1 and T2 interchanged goes toward the T2 -interval’s gate volume. In Fig. 16 the average gate volume (averaged over long and short intervals) is plotted as a function of the ratio T1 =(T1 + T2 ). We see that this average is actually greatest for the uniform sampling case and least for the “data fusion” scenario. At first glance this appears to contradict the conclusion in [8] that uniform 396

As discussed earlier, analytic results with data association are sparse. However, there is a recent development of some interest: a Cramer-Rao lower bound (CRLB) has been developed for single-target tracking. As given in [13], this CRLB has elements of the Riccati equation, although the parameters are modified somewhat by an “information reduction factor” that reduces measurement fidelity as a function of the uncertainty in provenance. Now, the CRLB shows the accuracy of an optimal maximum a posteriori (MAP) estimator. The truly MAP estimator never loses track, regardless of the harshness of the tracking scenario; as such, we cannot expect any corroboration of the “track-length” behavior of a practical tracker such as the PDAF or hard-assignment. However, the CRLB does provide a measure of the curvature of the log-likelihood function near the true data, and is in that regard a refinement of the Kalman filter results in that data association is accounted for.

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Fig. 17. CRLB for mean square error of uniform and nonuniform sampling schemes, with T1 = 0:1 s, T2 = 1:9 s, T = 1 s, PD = 0:9, q˜ = 5, ¾w2 = 0:25.

Example results are shown in Fig. 17, which plots the CRLB after both long- and short-interval predictions, as a function of clutter density. Perhaps of greatest interest here is that the short-interval uncertainty is significantly less than that when uniform sampling is used; and that even for the longer interval the inflation in this quantity is not huge. That is, the less comprehensive but more exact analysis for the Kalman filter does seem to apply. C. Multi-Scan Tracker Results Our contention, at the outset, was that if one was presented with two measurements relatively close in time, one could cross-check detections in one with those in the other, and thereby eliminate from serious consideration many false alarms. The disadvantage from doing this, however, is the lengthened time between such pairs of samples when one assumes a constant average sensor resource budget: the uncertainty gate grows, and more false alarms enter the picture. However, it appears that a multi-scan tracker can see some benefits from nonuniform sampling in the situations listed below. 1) High Maneuvering Index: This is not immediately intuitive. However, the Kalman filter analysis does indicate that it will be so. A more maneuvering target engenders a more quickly increasing gate, which magnifies effects. A more maneuvering target’s motion is also hard to be predicted by the trackers, and thus makes the benefits from the cross-check effect of the nonuniform sampling scheme more obvious. 2) High False Alarm Density: Similar to target maneuver, a greater clutter density magnifies any effects from a small or a large gate. 3) High Probability of Detection: Any possible benefit from the proposed cross-check is lost if the

true observation is not detected frequently, or is not expected to do so. The first two considerations are tantamount to the target being difficult to track, whereas the last (a high PD ) indicates an easier tracking task (usually heavy clutter and high maneuvering index go with low PD , although both clutter and PD can be adjusted together by the detection threshold). The reason that a high probability of detection is necessary to observe the benefits of nonuniform sampling is that if a measurement is not repeated in succeeding scans at a lower PD , then it may be either a miss or clutter, but if PD is sufficiently high, the overwhelming likelihood is for the latter. D. Single-Scan Tracker Results The PDAF (and presumably other scan-by-scan data association algorithms such as JPDAF or 2-D assignment) cannot, as observed in our simulations, realize any benefit from nonuniform sampling’s cross-validation between scan pairs. As to whether the they are hindered by an increase in average gate size, the Kalman filter and CRLB analysis suggest that this ought not be so. However, both these analyses are quite idealized. In the former there is no data association (the PDAF must associate false alarms and missed detections) and in the latter estimation is optimal based on all received data (the PDAF is certainly not that). It turns out that the imperfections of the PDAF dominate. V.

CONCLUSION

In this paper, nonuniform sampling and uniform sampling schemes are compared in different trackers (PDAF and N-D assignment). In some situations, such as when data comes from a rotating radar, it

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is moot to discuss the benefits from, or drawbacks to, a nonuniform data-taking structure. But in some situations a nonuniform structure is imposed: an example of this is data fusion, in which measurements arrive from asynchronous sensors. One tends to treat this as an unavoidable evil; but is it really that bad? In fact, it can be beneficial. This indicates, perhaps, that intelligent sensor-resource management systems ought to consider deliberately staggering sampling times. We have explored only the simplest case of nonuniformity, that in which measurements arrive in pairs separated by a short interval. We take this as a proxy for more complicated situations. It has been shown that for the PDAF, uniform sampling works better in our experiments. However, when the tracker utilizes more than one scan of data from the past (as with multiple-frame assignment or, presumably, the multi-hypothesis tracker (MHT)) nonuniform can outperform uniform sampling. This is particularly so when:

[4]

[5]

[6]

[7]

[8]

[9]

the target is highly maneuverable; and/or the false alarm density is high; and/or the probability of detection is high. In the above, we measure in terms of percentage of lost tracks and average track lifetime. In all cases the aggregate sensor resource is kept constant, meaning, for example, that one nonuniform scheme with short interval 0.1 s and long interval 1.9 s can be fairly compared with a uniform scheme in which the constant inter-sample interval is 1 s. Further, although management of the “resource” of a single sensor has provided the results here, the implications on data fusion should be clear. That is, given the conditions of the list above, it may be better to purchase low-rate sensors that scan simultaneously and fuse, as opposed to a single high-rate sensor.

[2]

[3]

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[11]

[12]

[13]

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Bar-Shalom, Y., Li, X. R., and Kirubarajan, T. Estimation with Application to Tracking and Navigation. New York: Wiley, 2001. Bar-Shalom, Y., and Li, X. R. Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS Publishing, 1995. Bar-Shalom, Y., and Li, X. R. Stability evaluation and track life of the PDAF for tracking in clutter. IEEE Transactions on Automatic Control, 36 (May 1991), 588—602.

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Daum, F. A system approach to multiple target tracking. In Y. Bar-Shalom (Ed.), Multitarget-Multisensor Tracking: Applications and Advances, Vol. II, Norwood, MA: Artech House, 1996. Morefield, C. L. Application of 0-1 integer programming to multi-target tracking problems. IEEE Transactions on Automatic Control, 22 (June 1977), 302—312. Niu, R., Varshney, P., Mehrotra, K., and Mohan, C. Temporal fusion in multi-sensor target tracking systems. In Proceedings of the International Conference on Information Fusion, Annapolis, MD, July 2002. Niu, R., Varshney, P., Mehrotra, K., and Mohan, C. Sensor staggering in multi-sensor target tracking systems. In Proceedings of the 2003 IEEE Radar Conference, Huntsville, AL, May 2003. Niu, R., Varshney, P., Mehrotra, K., and Mohan, C. On temporally staggered sensors in multi-sensor target tracking systems. IEEE Transactions on Aerospace and Electronic Systems, 41, 2 (July 2005), 794—808. Pattipati, K. R., Deb, S., Bar-Shalom, Y., and Washburn, R. B. Passive multisensor data association using a new relaxation algorithm. In Y. Bar-Shalom (Ed.), Multitarget-Multisensor Tracking: Advanced Applications, Norwood, MA: Artech House, 1990. Pattipati, K. R., Deb, S., Bar-Shalom, Y., and Washburn, R. B. A new relaxation algorithm and passive sensor data association. IEEE Transactions on Automatic Control, 37, 2 (Feb. 1992). Smith, P., and Buechler, G. A branching algorithm for discriminating and tracking multiple objects. IEEE Transactions on Automatic Control, 20 (Feb. 1975), 101—104. Willett, P., Alford, M., and Vannicola, V. The case for like-sensor pre-detection fusion. IEEE Transactions on Aerospace and Electronic Systems, (Oct. 1994), 986—1000. Zhang, X., and Willett, P. Crame´ r-Rao bounds for discrete-time linear filtering with measurement origin uncertainties. In Proceedings of the Workshop on Estimation, Tracking and Fusion: A Tribute to Yaakov Bar-Shalom, Monterey, CA, May 2001. Zhang, X., Willett, P., and Bar-Shalom, Y. Aspects of measurement scheduling for tracking. In Proceedings of the 2002 IEEE Aerospace Conference, Big Sky MT, Mar. 2002.

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Xin Zhang (S’01) received the B.S. and M.S. degree in electrical engineering from Fudan University, Shanghai, China, in 1997 and 2000, respectively. He received his Ph.D. degree in the Department of Electrical and Computer Engineering at the University of Connecticut in 2005. His research interests include statistical signal processing, detection, and target tracking. Currently he is a postdoctoral fellow at Princeton University.

Peter Willett (S’83–M’86–F’03) received the B.A.Sc. in 1982 from the University of Toronto, Toronto, Canada, and the Ph.D. from Princeton University, Princeton, NJ, in 1986. He is a professor at the University of Connecticut, Storrs, where he has worked since 1986. His interests are generally in the areas of detection theory and signal processing, and, lately, particularly in the area of data fusion. ZHANG ET AL.: UNIFORM VERSUS NONUNIFORM SAMPLING WHEN TRACKING IN CLUTTER

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Yaakov Bar-Shalom (S’63–M’66–SM’80–F’84) was born on May 11, 1941. He received the B.S. and M.S. degrees from the Technion, Israel Institute of Technology, in 1963 and 1967 and the Ph.D. degree from Princeton University, Princeton, NJ, in 1970, all in electrical engineering. From 1970 to 1976 he was with Systems Control, Inc., Palo Alto, CA. Currently he is Board of Trustees Distinguished Professor in the Dept. of Electrical and Computer Engineering and Marianne E. Klewin Professor in Engineering. He is also director of the ESP Lab (Estimation and Signal Processing) at the University of Connecticut. His research interests are in estimation theory and stochastic adaptive control and he has published over 320 papers and book chapters in these areas. In view of the causality principle between the given name of a person (in this case, “(he) will track,” in the modern version of the original language of the Bible) and the profession of this person, his interests have focused on tracking. He coauthored the monograph Tracking and Data Association (Academic Press, 1988), the graduate text Estimation with Applications to Tracking and Navigation (Wiley, 2001), the text Multitarget-Multisensor Tracking: Principles and Techniques (YBS Publishing, 1995), and edited the books Multitarget-Multisensor Tracking: Applications and Advances (Artech House, Vol. I 1990; Vol. II 1992, Vol. III 2000). He has been elected Fellow of IEEE for “contributions to the theory of stochastic systems and of multitarget tracking.” He has been consulting to numerous companies, and originated the series of Multitarget-Multisensor Tracking short courses offered via UCLA Extension, at Government Laboratories, private companies, and overseas. During 1976 and 1977 he served as associate editor of the IEEE Transactions on Automatic Control and from 1978 to 1981 as associate editor of Automatica. He was program chairman of the 1982 American Control Conference, general chairman of the 1985 ACC, and cochairman of the 1989 IEEE International Conference on Control and Applications. During 1983—1987 he served as chairman of the Conference Activities Board of the IEEE Control Systems Society and during 1987—1989 was a member of the Board of Governors of the IEEE CSS. Currently he is a member of the Board of Directors of the International Society of Information Fusion and served as its Y2K and Y2K2 President. In 1987 he received the IEEE CSS distinguished Member Award. Since 1995 he is a distinguished lecturer of the IEEE AESS. He is corecipient of the M. Barry Carlton Awards for the best paper in the IEEE Transactions on Aerospace and Electronic Systems in 1995 and 2000, and received the 1998 University of Connecticut AAUP Excellence Award for Research and the 2002 J. Mignona Data Fusion Award from the DoD JDL Data Fusion Group. 400

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