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Probabilistic Data Association Techniques for Target Tracking in Clutter THIAGALINGAM KIRUBARAJAN, SENIOR MEMBER, IEEE, YAAKOV BAR-SHALOM, FELLOW, IEEE

AND

Invited Paper

In tracking targets with less-than-unity probability of detection in the presence of false alarms (FAs), data association—deciding which of the received multiple measurements to use to update each track—is crucial. Most algorithms that make a hard decision on the origin of the true measurement begin to fail as the FA rate increases or with low observable (low probability of target detection) maneuvering targets. Instead of using only one measurement among the received ones and discarding the others, an alternative approach is to use all of the validated measurements with different weights (probabilities), known as probabilistic data association (PDA). This paper presents an overview of the PDA technique and its application for different target tracking scenarios. First, it describes the use of the PDA technique for tracking low observable targets with passive sonar measurements. This target motion analysis is an application of the PDA technique, in conjunction with the maximum-likelihood approach, for target motion parameter estimation via a batch procedure. Then, the PDA technique for tracking highly maneuvering targets and for radar resource management is illustrated with recursive state estimation using the interacting multiple model estimator combined with PDA. Finally, a sliding window (which can also expand and contract) parameter estimator using the PDA approach for tracking the state of a maneuvering target using measurements from an electrooptical sensor is presented. Keywords—Target tracking, estimation, Kalman filter (KF), data association, probabilistic data association filter.

I. INTRODUCTION In tracking targets with less-than-unity probability of detection in the presence of false alarms (FAs) (clutter), data association—deciding which of the received multiple measurements to use to update each track—is crucial. A number of algorithms have been developed to solve this Manuscript received March 15, 2003; revised October 28, 2003. T. Kirubarajan is with the Electrical and Computer Engineering Department, McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: [email protected],). Y. Bar-Shalom is with the Information, Communication and Decision Systems Group, Electrical and Computer Engineering Department, University of Connecticut, Storrs, CT 06269-2157 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JPROC.2003.823149

problem [3], [4], [7], [12]. Two simple solutions are the strongest-neighbor filter (SNF) and the nearest-neighbor filter (NNF). In the SNF, the signal with the highest intensity among the validated measurements (in a gate) is used for track update and the others are discarded. In the NNF, the measurement closest to the predicted measurement is used. While these simple techniques work reasonably well with benign targets in sparse scenarios, they begin to fail as the FA rate increases or with low observable (low probability of target detection) maneuvering targets [28], [35]. Instead of using only one measurement among the received ones and discarding the others, an alternative approach is to use all of the validated measurements with different weights (probabilities), known as probabilistic data association (PDA) [7]. The standard PDA and its numerous improved versions have been shown to be very effective in tracking a single target in clutter [35], [40]. Data association becomes more difficult with multiple targets where the tracks compete for measurements. Here, in addition to a track validating multiple measurements as in the single-target case, a measurement itself can be validated by multiple tracks (i.e., contention occurs among tracks for measurements). Many algorithms exist to handle this contention. The joint PDA (JPDA) algorithm is used to track multiple targets by evaluating the measurement-to-track association probabilities and combining them to find the state estimate [7]. The multiple hypothesis tracking (MHT) is a more powerful (but much more complex) algorithm that handles the multitarget tracking problem by evaluating the likelihood that there is a target given a sequence of measurements [12]. In the tracking benchmark problem [16] designed to compare the performance of different algorithms for tracking highly maneuvering targets in the presence of electronic countermeasures, the PDA-based estimator, in conjunction with the interacting multiple model (IMM) estimator, yielded one of the best solutions. Its performance was comparable to that of the MHT algorithm [11], [35].

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While the original PDA filter was presented in the 1970s [8] in the context of target tracking, it has continued to evolve in its formulation as well as applications. It has been extended handle multiple targets [6], track coalescence [17], multiple model estimation [18], variable dimensions [1], stability issues [33], iterated versions [60], multipattern algorithms [29], and multiframe association [51]. In addition, its effective application has been demonstrated in the areas of speech [23], imaging [57], computer vision [39], [44], [46], [49], [59], robotics [24], [50], [54], soft computing [9], [19], [20], [32], communications [42], [56], navigation [27], transportation [2], biomedical engineering [10], acoustics [43], and radar signal processing processing [31]. This paper presents an overview of the PDA technique and its application for different target tracking scenarios. Section II summarizes the PDA technique. Section III describes the use of the PDA technique for tracking low observable targets with passive sonar measurements. This target motion analysis (TMA) is an application of the PDA technique, in conjunction with the maximum-likelihood (ML) approach, for target motion parameter estimation via a batch procedure. Section IV presents the use of the PDA technique for tracking highly maneuvering targets and for radar resource management. It illustrates the application of the PDA technique for recursive state estimation using the IMM estimator with PDA filter (PDAF) modules (IMMPDAF). Section V presents a state-of-the-art sliding-window (which can also expand and contract) parameter estimator using the PDA approach for tracking the state of a maneuvering target using measurements from an electrooptical (EO) sensor. This, while still a batch procedure, offers the flexibility of varying the batches depending on the estimation results.

• The past information about the target is summarized approximately by (3) where denotes the normal probability density function (pdf) with argu, mean and covariance matrix ment . This assumption of the PDAF is similar to the generalized pseudo-Bayesian (GPB1) approach [5], where a single “lumped” state estimate is a quasi-sufficient statistic. • At each time, a validation region as in [7] is set up (see (6)). • Among the possibly several validated measurements, at most one of them can be target originated—if the target was detected and the corresponding measurement fell into the validation region. • The remaining measurements are assumed to be FAs or clutter and are modeled as independent identically distributed (i.i.d.) measurements with uniform spatial distribution. • The target detections occur independently over time . with known probability These assumptions enable a state estimation scheme to be obtained, which is almost as simple as the Kalman filter (KF), but much more effective in clutter. B. PDAF Approach The PDAF uses a decomposition of the estimation with respect to the origin of each element of the latest set of validated measurements, denoted as (4)

II. PROBABILISTIC DATA ASSOCIATION The PDA algorithm calculates in real-time the probability that each validated measurement is attributable to the target of interest. This probabilistic (Bayesian) information is used in a tracking filter, the PDAF, which accounts for the measurement origin uncertainty.

where is the th validated measurement and is the number of measurements in the validation region at time . The cumulative set (sequence) of measurements1 is

A. Assumptions

C. Measurement Validation

The following assumptions are made to obtain the recursive PDAF state estimator (tracker). • There is only one target of interest whose state evolves according to a dynamic equation driven by process noise. The state of the target of interest, of dimension , is assumed to evolve in time according to

From the Gaussian assumption (3), the validation region is the elliptical region

(5)

(6) where

is the gate threshold and

(1) (7) with the true measurement of dimension

given by (2)

where and are zero-mean mutually independent white Gaussian noise sequences with known coand , respectively. variances matrices • The track has been initialized.

is the covariance of the innovation corresponding to the true measurement. The volume of the validation region (6) is (8) 1When the running index is a time argument, one has a sequence; otherwise, it is a set where the order is not relevant. The context should make it clear which is the case.

KIRUBARAJAN AND BAR-SHALOM: PROBABILISTIC DATA ASSOCIATION TECHNIQUES FOR TARGET TRACKING IN CLUTTER

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where the coefficient depends on the dimension of the measurement (it is the volume of the -dimensional unit , etc.). hypersphere:

where the combined innovation is (17)

D. State Estimation In view of the assumptions listed, the association events (shown in (9) at the bottom of the page)

The covariance associated with the updated state is

(18)

is the target-originated measurement none of the measurements is target originated (9) . are mutually exclusive and exhaustive for Using the total probability theorem [5] with respect to the above events, the conditional mean of the state at time can be written as

where the covariance of the state updated with the correct measurement is [7] (19) and the spread of the innovations term (similar to the spread of the means term in a mixture [5]) is

(20) (10) is the updated state conditioned on the event where that the th validated measurement is correct, and

F. Prediction Equations The prediction of the state and measurement to done as in the standard filter, i.e.,

(21) (22)

(11) is the conditional probability of this event—the association probability, obtained from the PDA procedure, presented in the next section. The estimate conditioned on measurement being correct is

The covariance of the predicted state is similarly (23) is given by (18). where The innovation covariance (for the correct measurement) is, again, as in the standard filter (24)

(12) where the corresponding innovation is

G. PDA (13)

The gain

is

is the same as in the standard filter (14)

since, conditioned on , there is no measurement origin uncertainty. , i.e., if none of the measurements is correct or if For there is no validated measurement (i.e., ), one has (15)

To evaluate the association probabilities, the conditioning is broken down into the past data and the latest data . A probabilistic inference can be made on both the number of measurements in the validation region (from the clutter density, if known) and on their location, expressed as

(25) Using Bayes’ formula, the above is rewritten as

E. State and Covariance Update Combining (12) and (15) into (10) yields the state update equation of the PDAF (16) 538

(26) The joint density of the validated measurements condi, is the product of the following. tioned on PROCEEDINGS OF THE IEEE, VOL. 92, NO. 3, MARCH 2004

• the (assumed) Gaussian pdf of the correct (target originated) measurements and • the pdf of the incorrect measurements, which are assumed to be uniform in the validation region whose is given in (8). volume factor The pdf of the correct measurement (with the that accounts for restricting the normal density to the validation gate) is

The nonparametric model (33) can be obtained from (32) by setting (34) i.e., replacing the Poisson parameter with the sample spatial of density of the validated measurements. The volume the elliptical (i.e., Gaussian based) validation region is given in (8). H. Parametric PDA

(27)

Using (32) and (28) with the explicit expression of the Gaussian pdf in (26) yields, after some cancellations, the final equations of the parametric PDA with the Poisson clutter model

(28)

(35)

The pdf from (26) is then

The probabilities of the association events conditioned only on the number of validated measurements are

where (36) (37) The last expression above can be rewritten as (38)

(29) I. Nonparametric PDA where is the probability mass function (pmf) of the number of false measurements (FAs or clutter) in the validation region. in a volume Two models can be used for the pmf of interest . i) A Poisson model with a certain spatial density (30) ii) A diffuse prior model (discussed in [5]) (31) where the constant is irrelevant, since it cancels out. Using the (parametric) Poisson model in (29) yields

The nonparametric PDA is the same as above except for in (38) by —this obviates the need to replacing know . III. LOW OBSERVABLE TMA USING THE ML-PDA APPROACH WITH FEATURES This section considers the problem of TMA—estimation of the trajectory parameters of a constant velocity target—with a passive sonar, which does not provide full target position measurements. The methodology presented here applies equally to any target motion characterized by a deterministic equation, in which case the initial conditions (a finite dimensional parameter vector) characterize in full the entire motion. In this case the (batch) ML parameter estimation cab be used; this method is more powerful than state estimation when the target motion is deterministic (it does not have to be linear). Furthermore, the ML-PDA approach makes no approximation, unlike the PDAF in (3). A. Amplitude Information Feature

(32) The (nonparametric) diffuse prior (31) yields (33)

The standard TMA consists of estimating the target’s position and its constant velocity from bearings-only (wideband sonar) measurements corrupted by noise [5]. Narrowband passive sonar tracking, where frequency measurements are also available, has been studied [30]. The advantages of narrowband sonar are that it does not require a maneuver of the


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platform for observability and it greatly enhances the accuracy of the estimates. However, not all passive sonars have frequency information available. In both cases, the intensity of the signal at the output of the signal processor, which is referred to as measurement amplitude or amplitude information (AI), is used implicitly to determine whether there is a valid measurement. This is usually done by comparing it with the detection threshold, which is a design parameter. This section shows that the measurement amplitude carries valuable information and that its use in the estimation process increases the observability even though the AI cannot be correlated to the target state directly. Also, superior global convergence properties are obtained. The pdf of the envelope detector output (i.e., the AI) when the signal is due to noise only is denoted as and the corresponding pdf when the signal originated from the . If the signal-to-noise ratio2 (SNR) is , the target is density functions of noise only and target-originated measurements can be written as (39)

the candidate measurements. The amplitude likelihood ratio is defined as (43) B. Target Models Assume that sets of measurements, made at times , are available. For bearings-only estimation, the target motion is defined by the four-dimensional (4-D) parameter vector (44) and are the distances of the target in where the east and north directions, respectively, from the origin at the reference time . The corresponding velocities, assumed constant, are and , respectively. This assumes deterministic target motion (i.e., no process noise [5]). Any other deterministic motion, e.g., constant acceleration, can be handled within the same framework. is defined by The state of the platform at (45)

(40) respectively. A suitable threshold, denoted by , is used to declare a detection. The probability of detection and the probability of and , respectively. Both and FA are denoted by can be evaluated from the pdf’s of the measurements. , the threshold must be lowClearly, in order to increase ered. However, this also increases . Therefore, depending on the SNR, must be selected to satisfy two conflicting requirements.4 The density functions given above correspond to the signal at the envelope detector output. Those corresponding to the output of the threshold detector are 3

The relative position components in the east and north directions of the target with respect to the platform at are defined by and , respectively. Similarly, and define the relative velocity components. The true bearing of the target from the platform at is given by (46) The range of possible bearing measurements is (47) The set of measurements at

is denoted by (48)

(41)

is the number of measurements at and the pair where is defined by of bearing and amplitude measurements (49)

(42)

The cumulative set of measurements during the entire period is (50)

where is the pdf of the validated measurements that are caused by noise only and is the pdf of those that originated from the target. In the following, is the amplitude of

The following additional assumptions about the statistical characteristics of the measurements are also made [30]. 1) The measurements at two different sampling instants are conditionally independent, i.e.,

2This

is the SNR in a resolution cell, to be denoted later as SNR . is a Rayleigh fading amplitude (Swerling I) model believed to be the most appropriate for shallow water passive sonar. 4For other probabilistic models of the detection process, different SNR values correspond to the same P ; P pair. Compared to the Rician model receiver operating characteristic (ROC) curve, the Rayleigh model ROC curve requires a higher SNR for the same pair (P ; P ), i.e., the Rayleigh model considered here is pessimistic. 3This

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(51) is the pdf. where 2) A measurement that originated from the target at a particular sampling instant is received by the sensor only once during the corresponding scan with probability PROCEEDINGS OF THE IEEE, VOL. 92, NO. 3, MARCH 2004

and is corrupted by zero-mean Gaussian noise of known variance; that is, (52) where is the bearing measurement noise. Due to the presence of false measurements, the index of the true measurement is not known. 3) The false bearing measurements are distributed uniformly in the surveillance region, i.e., (53) 4) The number of false measurements at a sampling instant is generated according to a Poisson law with a known expected number of false measurements in the surveillance region. This is determined by the detection threshold at the sensor (exact equations are given in Section III-E). For narrowband sonar (with frequency measurements) the target motion model is defined by the five-dimensional vector

The measurements resulting from noise only are assumed to be uniformly distributed in the entire surveillance region. C. ML Estimator Combined With PDA—The ML-PDA In this section we present the derivation and the implementation of the ML estimator combined with the PDA technique for both bearings-only tracking and narrowband sonar tracking. detections at one has the following muIf there are tually exclusive and exhaustive events [7]: measurement

is from the target

all measurements are false (60) The pdf of the measurements corresponding to the above events can be written as

(54) where is the unknown emitted frequency assumed constant. Due to the relative motion between the target and platform at this frequency will be Doppler shifted at the platform. The (noise free) shifted frequency, denoted by , is given by

(61) is the area of the surveillance region. where Using the total probability theorem, the likelihood function of the set of measurements at can be expressed as

(55) where is the velocity of sound in the medium. If the band, the width of the signal processor in the sonar is measurements can lie anywhere within this range. As in the case of bearing measurements, we assume that an operator is for scanning. In able to select a frequency subregion addition to the bearing surveillance region given in (47), the region for frequency is defined as (56) The noisy frequency measurements are denoted by the measurement vector is

and

(62)

(57)

is the Poisson probability mass function of the where number of false measurements at . Dividing the above by yields the dimenat . Then sionless likelihood ratio

As for the statistical assumptions, those related to the conditional independence of measurements (assumption 1) and the number of false measurements (assumption 4) are still valid. The equations relating the number of FAs in the surveillance region to detection threshold are given in Section III-E. The noisy bearing measurements satisfy (52) and the noisy satisfy frequency measurements (58)

(63)

is the frequency measurement noise. where It is also assumed that these two measurement noise components are conditionally independent; that is,

where is the expected number of FAs per unit area. Alternately, the log-likelihood ratio at can be defined as

(59)

(64)


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Using conditional independence of measurements, the likelihood function of the entire set of measurements can be written in terms of the individual likelihood functions as (65) Then the dimensionless likelihood ratio for the entire data is given by (66)

The same ML-PDA approach is applicable to the estimation of the trajectory of an exoatmospheric ballistic missile [36], [55]. The modification of this fixed-batch ML-PDA estimator to a flexible (sliding/expanding/contracting) procedure is discussed in Section V and demonstrated with an actual EO data example. D. Cramer–Rao Lower Bound for the Estimate For an unbiased estimate, the Cramer–Rao lower bound (CRLB) is given by (70)

From the above, the total log-likelihood ratio be expressed as

can

where

is the Fisher information matrix (FIM) given by (71)

(67) The ML estimate (MLE) is obtained by finding the state that maximizes the total log-likelihood function. In deriving the likelihood function, the gate probability mass, which is the probability that a target-originated measurement falls within the surveillance region, is assumed to be one. The operator selects the appropriate region. Arguments similar to those given earlier can be used to derive the MLE when frequency measurements are also availas in (60), the pdf of the measurements able. Defining is

Only in simulations will the true value of the state parameter be available. In practice, CRLB is evaluated at the estimate. As expounded in [34], the FIM is given in the present ML-PDA approach for the bearings-only case—wideband sonar—by (72) where is the information reduction factor that accounts for the loss of information resulting from the presence of false measurements and less-than-unity probability of detection [7] and the expected number of FAs per unit volume is denoted by . In deriving (72) only the bearing measurements which fall within the validation region5 (73)

(68) where is the volume of the surveillance region. After some lengthy manipulations, the total log-likelihood function is obtained as

at were considered. The validation region volume ( -sigma region), , is given by (74) The information reduction factor for the present two-dimensional (2-D) measurement situation (bearing and amplitude) is given by

(75)

(69) For narrowband sonar, the MLE is found by maximizing (69). This section demonstrated the essence of the use of the PDA—all the measurements are accounted for and the likelihood function is evaluated using the total probability theorem, similar to (10). However, since (69) is exact (for the parameter estimation formulation), there is no need for the approximation in (3), which is necessary in the PDAF for state estimation. 542

is a -fold integral given in [34] where where numerical values of for different combinaand are also presented. The derivation of tions of the integral is based on [7]. In our implementation, was selected. Knowing and can be determined by using (76) 5The objective of measurement validation is to avoid the consideration of outliers that are far away from the predicted location during data association. In addition, validation reduces the computational load by removing the unvalidated measurements from the measurement list.


where is the resolution cell volume of the signal processor (this is discussed in more detail in Section III-E). Finally, , and . the SNR, can be calculated from The rationale for the term information reduction factor follows from the fact that the FIM for zero FA probability and unity target detection probability, , is given by [5]

with the SNR, determines the probability of detection and the probability of FA. The signal processor was assumed to consist of the frequency band [500 Hz, 1000 Hz] with a 2048-point FFT. This whose size is given by results in a frequency cell

(77)

Regarding azimuth measurements, the sonar is assumed with to have 60 equal beams, resulting in an azimuth cell size

, which is From (72) and (77) it is clear that always less than or equal to unity, represents the loss of information due to clutter. For narrowband sonar (bearing and frequency measurements), the FIM is given by

Hz

(82)

(83) Assuming uniform distribution in a cell, the frequency and azimuth measurement standard deviations are given by6 Hz

(84) (85)

(78) for this three-dimensional (3-D) meawhere surement (bearing, frequency, and amplitude) case is evaluated [34] using

(79) and the numerical values The expression for are also given in [34]. for For narrowband sonar, the validation region is defined by

The SNR in a cell7 was taken as 6.1 dB and The corresponding SNR in a 1-Hz bandwidth SNR dB. These values give

.8 is 0.1 (86) (87)

, the expected number of FAs per unit volume, From denoted by , can be calculated using (88) and

Substituting the values for

gives deg Hz

(89)

The surveillance regions for azimuth and frequency, deand , respectively, are taken as noted by (80) and the volume of the validation region

Hz

, is (81)

E. Results Both the bearings-only and narrowband sonar problems with AI were implemented to track a target moving at constant velocity. The results for the narrowband case are given below, accompanied by a discussion of the advantages of using AI by comparing the performances of the estimators with and without AI. In narrowband signal processing, different bands in the frequency domain are defined by an appropriate cell resolution and a center frequency about which these bands are located. The received signal is sampled and filtered in these bands before applying fast Fourier transform (FFT) and beamforming. Then the angle of arrival is estimated using a suitable algorithm [45]. As explained earlier, the received signal is registered as a valid measurement only if it exceeds the threshold . The threshold value, together

Hz

(90) (91)

The expected number of FAs in the entire surveillance recan be calculated. gion and that in the validation gate These values are 9.8 and 0.2, respectively, where the vali. These values mean that for dation gate is restricted to every true measurement that originated from the target there are about ten FAs which exceed the threshold. The estimated tracks were validated using the hypothesis testing procedure described in [34]. The track acceptance test was carried out with a miss probability of 5%. To check the performance of the estimator, simulations were carried out with clutter only (i.e., without a target) and

p

6The “uniform” factor 12 corresponds to the worst case. In practice, and are functions of the 3-dB bandwidth and of the SNR. 7The commonly used SNR, designated here as SNR , is signal strength divided by the noise power in a 1-Hz bandwidth. SNR is signal strength divided by the noise power in a resolution cell. The relationship between them for C = 0:25 Hz is SNR = SNR 6 dB. SNR is believed to be the more meaningful SNR because this determines the ROC curve. 8The estimator is not very sensitive to an incorrect P . This is verified by running the estimator with an incorrect P on the data generated with a different P . Differences up to 0.15 are tolerated by the estimator.


0

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Fig. 1. Estimated tracks from 100 runs for narrowband sonar with AI.

also with a target present; measurements were generated accordingly. Simulations were done in batches of 100 runs. When there was no target, irrespective of the initial guess, the estimated track was always rejected. This corroborates the accuracy of the validation algorithm given in [34]. For the set of simulations with a target, the following scenario was selected: the target moves at a speed of 10 m/s heading west and 5 m/s heading north starting from (5000 m, 35 000 m). The signal frequency is 750 Hz. The target pam m m/s m/s Hz . rameter is The motion of the platform consisted of two velocity legs in the northwest direction during the first half, and in the northeast direction during the second half of the simulation period with a constant speed of 7.1 m/s. Measurements were taken at regular intervals of 30 s. The observation period was 900 s. Fig. 1 shows the scenario including the target true trajectory (solid line), platform trajectory (dashed line), and the 95% probability regions of the position estimates at the initial and final sampling instants based on the CRLB (78). The initial and the final positions of the trajectories are marked by “I” and “F” respectively. The purpose of the probability region is to verify the validity of the CRLB as the actual parameter estimate covariance matrix from a number of Monte Carlo runs [5]. Fig. 1 shows the 100 tracks formed from the estimates. Note that in all but six runs (i.e., 94 runs), the estimated trajectory endpoints fall in the corresponding 95% uncertainty ellipses. Table 1 gives the numerical results from 100 runs. Here is the average of the estimates, the variance of the estimates evaluated from 100 runs, and the theoretical CRLB derived in Section III-D. The range of initial guesses found by rough grid search to start off the estimator are given by . The efficiency of the estimator was verified using the normalized estimation error squared (NEES) [5] defined by (92) 544

Table 1 Results of 100 Monte Carlo Runs for Narrowband Sonar With AI (SNR = 6:1 dB)

where is the estimate and J is the FIM (78). Assuming approximately Gaussian estimation error, the NEES is chisquare distributed with degrees of freedom where is the number of estimated parameters. For the 94 accepted tracks, the NEES was obtained as 5.46, which lies within the 95% . Also note that each compoconfidence region of the corresponding component of is within . nent of IV. THE IMMPDAF FOR TRACKING MANEUVERING TARGETS Target tracking is a problem that has been well studied and documented. Some specific problems of interest in the single-target, single-sensor case are tracking maneuvering targets [5], tracking in the presence of clutter [7], and electronic countermeasures (ECM). In addition to these tracking issues, to be complete, a tracking system for a sophisticated electronically steered antenna radar has to consider radar scheduling, waveform selection, and detection threshold selection. Although many researchers have worked on these issues and many algorithms are available, there had been no standard problem comparing the performances of the various algorithms. Rectifying this, the first benchmark problem [14] was developed, focusing only on tracking a maneuvering target and pointing/scheduling a phased array radar. Of all the algorithms considered for this problem, PROCEEDINGS OF THE IEEE, VOL. 92, NO. 3, MARCH 2004

the IMM estimator yielded the best performance [13]. The second benchmark problem [15] included FAs and ECM—specifically, a standoff jammer (SOJ) and range gate pull-off (RGPO)—as well as several possible radar waveforms (from which the resource allocator has to select one at every revisit time). Preliminary results for this problem showed that the IMM and MHT algorithms were the best solutions [11], [35]. The MHT algorithm, while one to two orders of magnitude costlier computationally than the IMMPDAF [7] for the problem considered (as many as 40 hypotheses are needed),9 yielded comparable results with the IMMPDAF. The benchmark problem of [15] was upgraded in [16] to require the radar resource allocator/manager to select the operating constant FA rate (CFAR) and include the effects of the SOJ on the direction of arrival (DOA) measurements; also the SOJ power was increased to present a more challenging benchmark problem. While, in [15], the primary performance criterion for the tracking algorithm was minimization of radar energy, the primary performance was changed in [16] to minimization of a weighted combination of radar time and energy. This section presents the IMMPDAF technique for automatic track formation, maintenance, and termination. The coordinate selection for tracking, radar scheduling/pointing and the models used for mode-matched filtering (the modules inside the IMM estimator) are also discussed. These cover the target tracking aspects of the solution to the benchmark problem. These are based on the benchmark problem tracking and sensor resource management [16], [35].

the problem linear [7]. The converted measurement report corresponding to is given by [12] (94) where , and are the three position measurements in the Cartesian frame and their covariance matrix, respectively. The converted values are (95) (96) (97) (98) where is the standard deviation of range measurements at is the spherical-to-Cartesian transformation scan and matrix given by11

(99)

where (100) (101) (102) (103)

A. Coordinate Selection For target tracking in track dwell mode of the radar, the number of detections at scan (time ) is denoted by . conThe th detection report , bearing , elevation sists of a time stamp , range , AI given by the SNR, and the standard deviations and , respecof bearing and elevation measurements tively. Thus (93) where the overbar indicates that this is in the radar’s spherical coordinate system. The AI is used only to declare detections and select the radar waveform for the next scan. Since the use of AI, for example, as in [40], can be counterproductive in discounting RGPO measurements, which generally have higher SNR than target-originated measurements, AI is not utilized in the estimation process itself.10 For target tracking, the measurements are converted from spherical coordinates to Cartesian coordinates and then the IMMPDAF is used on these converted measurements. This conversion avoids the use of extended KFs and makes 9The more recent IMM-MHT (as opposed to KF-based MHT) required six to eight hypotheses. 10Using the AI would require a separate model for the RGPO intensity, which cannot be estimated in real time due to its short duration and variability [40].

B. Track Formation In the presence of FAs, track formation is crucial. Incorrect track initiation will result in target loss. In [7] an automatic track formation/deletion algorithm in the presence of clutter was presented based on the IMM algorithm. In the present benchmark problem, a noisy measurement corresponding to the target of interest is given in the first scan.12 Forming new tracks for each validated measurement (based on a velocity gate), as suggested in [7] and as implemented in [35], at subsequent scans is expensive in terms of both radar energy and computational load. In this implementation, track formation is simplified and handled as follows. s) As defined by the benchmark problem, • Scan 1 ( there is only one (target-originated, noisy) measurement. The position component of this measurement is used as the starting point for the estimated track. s) The beam is pointed at the lo• Scan 2 ( cation of the first measurement. This yields, possibly, more than one measurement and these measurements are gated using the maximum possible velocity of the targets to avoid the formation of impossible tracks. This 11For the scenarios considered here, this transformation is practically unbiased and there is no need for the debiasing procedure of [7]. 12Assuming that this is a search pulse without (monopulse) split-beam processing, the angular errors are uniformly distributed in the beam.


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validation region volume, which is centered on the initial measurement, is given by

(104)

coordinates, respectively. The measurement vector consists and is denoted of the Cartesian position components at . by Assuming that the target motion is linear in the Cartesian coordinate system, the true state of the target can be written as (105)

where

s is the sampling interval and , and are the maximum speeds in the , and directions, respectively; , and are the variances of position measurements in these directions, obtained from the diagonal components of (98). The maximum speed in each direction is assumed to be 500 m/s. The measurement in the first scan and the measurement with the highest SNR in the second scan are used to form a track with the two-point initialization technique [5]. The track splitting used in [7] and [35] was found unnecessary—the strongest validated measurement was adequate. This technique yields the position and velocity estimates and the associated covariance matrices in all three coordinates. s) The pointing direction for the radar • Scan 3 ( s using is given by the predicted position at the estimates at scan 2. An IMMPDA filter with three models discussed in the sequel is initialized with the estimates and covariance matrices obtained at the second scan. The acceleration component for the third-order , where model is assumed zero with variance m/s is the maximum expected acceleration of the target. From scan 3 on, the track is maintained using the IMMPDAF as described in Section IV-C. In order to maintain a high SNR for the target-originated measurement during track formation, a high energy waveform is used. Also, at the second scan, three dwells are used to ensure target detection. This simplified approach cannot be used if the target-originated measurement is not given at the first scan. In that case, the track formation technique in [7] can be used. Immediate revisit with sampling interval 0.1 s is carried out during track formation because the initial velocity of the target is not known—in the first scan only the position is measured and there is no a priori velocity. This means that in the second scan, the radar must be pointed at the first scan position, assuming zero velocity. Waiting longer to obtain the second measurement could result in the loss of the targetoriginated measurement due to incorrect pointing. Also, in order to make the IMM mode probabilities converge to the correct values as quickly as possible, the target is revisited at a high rate. C. Track Maintenance The true state of the target at

is

where , and are the positions, , are the velocities, and and and are the accelerations of the target in the corresponding 546

and the target-originated measurement is related to the state according to (106) . The white Gaussian noise sequences and are independent and their covariances are and , respectively. at time With the above matrices, the predicted state is

where

(107) and the predicted measurement is (108) with associated innovation covariance (109) is the predicted state covariance to be defined where is the (expected) measurement noise coin (123) and variance. 1) PDA: During track maintenance, each measurement at scan is validated against the established track. This is achieved by setting up a validation region centered on the predicted measurement at . The validation region is (110) is the expected covariance of the innovation corwhere (0.9989 responding to the correct measurement and probability mass [7]) is the gate size. The set of measurements validated for the track at is (111) is the number of measurements validated and aswhere sociated with the track. Also, the cumulative set of validated measurements up to and including scan is denoted by . All unvalidated measurements are discarded. validated measurements at one has the With these following mutually exclusive and exhaustive events: measurement

is from the target

all measurements are false (112) Using the nonparametric version of the PDAF [7] the validated measurements are associated probabilistically to the track. The combined target state estimate is obtained as (113)


where is the probability that the th validated meais the updated state condisurement is correct and tioned on that event. The conditionally updated states are given by

is the predicted state covariance and the term

(114) is the filter gain and is the innovation associated with the th validated measurement. The gain, which depends on the measurement noise covariance, is

where

(115) where depends on the observed SNR for measurement [16]. are given by The association event probabilities

(124) is analogous to the spread of the innovations in the standard PDA [7]. Monopulse processing results in different accuracies (standard deviations) for different measurements within the same dwell. This accounts for the difference in the above equations from the standard PDA, where the measurement accuracies are assumed to be the same for all of the validated measurements. , the following estimates are To initialize the filter at used [5]:

(116)

(117) (125) where (118) (119) is the probability of detection of a target-originated and measurement. The probability that a target-originated measurement, if detected, falls within the validation gate is asdenotes the normal pdf sumed to be unity. Also, with argument , mean zero, and covariance matrix . The is the union of the valicommon validation volume used to validate the individual meadation volumes is given by surements associated with the target and (120) is the volume of the unit hypersphere of dimenwhere sion , the dimension of the measurement . For the 3-D [7]. position measurements The state estimate is updated as (121) and the associated covariance matrix is updated as

(122)

where is the index corresponding to the validated measurement with the highest SNR in the second scan and the , and denote the components in the corresuperscripts sponding directions, respectively. The associated covariance matrix can be derived [5] using the measurement covariance and the maximum target acceleration . If the two point differencing results in a velocity component that exceeds the corresponding maximum speed, it is replaced by that speed. Similarly, the covariance terms corresponding to the velocity components are upper bounded by the corresponding maximum values. 2) IMM Estimator Combined With the PDA Technique: In the IMM estimator it is assumed that at any time the target trajectory evolves according to one of a finite number of models, which differ in their noise levels and/or structures [5]. By probabilistically combining the estimates of the filters, typically Kalman, matched to these modes, an overall estimate is found. In the IMMPDAF, the KF is replaced with the PDA filter (given in Section IV-C1 for mode-conditioned filtering of the states), which handles the data association. be the number of mode-matched filters used, Let the index of the mode in effect in the semiopen , and be the probability that mode interval is in effect in the above interval. Thus (126)

where The mode transition probability is defined as (123) KIRUBARAJAN AND BAR-SHALOM: PROBABILISTIC DATA ASSOCIATION TECHNIQUES FOR TARGET TRACKING IN CLUTTER

(127) 547

The state estimates and their covariance matrix at conand , ditioned on the th mode are denoted by respectively. The steps of the IMMPDAF are as follows [7]. • Step 1—Mode interaction or mixing. The mode-conditioned state estimate and the associated covariances from the previous iteration are mixed to obtain the initial condition for the mode-matched filters. for the PDAF The initial condition in cycle matched to the th mode is computed using

• Step 4—State combination. The mode-conditioned estimates and covariances are combined to find the and its covariance matrix , overall estimate as follows:

(128)

3) Models in the IMM Estimator: The selection of the model structures and their parameters is one of the critical aspects of the implementation of IMMPDAF. Designing a good set of filters requires a priori knowledge about the target motion, usually in the form of maximum accelerations and sojourn times in various motion modes [5]. The tracks considered in the benchmark problem span a wide variety of motion modes—from benign constant velocity motions to maneuvers up to 7 g. To handle all possible motion modes and to handle automatic track formation and termination, the following models are used. . This second-order model • Benign motion model with low noise level (to be given later) has a probability given by the target’s expected of target detection SNR and corresponds to the nonmaneuvering intervals of the target trajectory. For this model, the process noise is typically assumed to model air turbulence. . This second-order model • Maneuver model with high noise level corresponds to ongoing maneuvers. For this white noise acceleration model, the is obtained using process noise standard deviation

where

(129) are the mixing probabilities. The covariance matrix associated with (128) is given by

(130) • Step 2—Mode-conditioned filtering. A PDAF is used for each mode to calculate the mode-conditioned state estimates and covariances. In addition, we evaluate the of each mode at using the likelihood function Gaussian-uniform mixture

(131)

(132) where is defined in (118) and in (119). Note that the likelihood function, as a pdf, has a physical . Since ratios of these dimension that depends on likelihood functions are to be calculated, they all must . Thus, have the same dimension, i.e., the same a common validation region (110) is vital for all the models in the IMMPDAF. Typically the “largest” innovation covariance matrix corresponding to “noisiest” model covers the others; therefore, this can be used in (110) and (120). • Step 3—Mode update. The mode probabilities are updated based on the likelihood of each mode using (133)

548

(134)

(135)

(136) where is the maximum acceleration in the corre[5]. sponding modes and . This is a • Maneuver detection model third-order (Wiener process acceleration) model with high-level noise. For highly maneuvering targets, like military attack aircraft, this model is useful for detecting the onset and termination of maneuvers. For civilian air traffic surveillance [58], this model is not necessary. For a Wiener process acceleration model, the stanis chosen using dard deviation (137) where is the maximum acceleration increment per unit time (jerk), is the sampling interval and [5]. For the targets under consideration, m/s and m/s . Using these values, the process noise standard deviations were taken as m/s m/s

for nonmaneuvering intervals for maneuvering intervals for maneuver start/termination


Table 2 Performance of IMMPDAF in the Presence of FAs, Range Gate Pull-Off, and the Standoff Jammer

In addition to the process noise levels, the elements of the Markov chain transition matrix between the modes, defined in (127), are also design parameters. Their selection depends on the sojourn time in each motion mode. The transition probability depends on the expected sojourn time via (138) where is the expected sojourn time of the th mode, is the probability of transition from th mode to the same mode, and is the sampling interval [5]. For the above models, are calculated using (139) and are the lower and upper limits, where respectively, for the th model transition probability. The expected sojourn times of 15, 4, and 2 s are assumed for modes , and , respectively. The selection of the off-diagonal elements of the Markov transition matrix depends on the switching characteristics among the various modes and is done as follows:

The components of target dynamics are uncoupled and the same process noise is used in each coordinate. D. Track Termination According to the benchmark problem, a track is declared lost if the estimation error is greater than the two-way beamwidth in angles or 1.5 range gates in range. In addition to this problem-specific criterion, the IMMPDAF declares (on its own) track loss if the track is not updated for 100 s. Alternatively, one can include a “no target” model [7], which is useful for automatic track termination, in the IMM mode set. In a more general tracking problem, where the true target state is not known, the “no target” mode probability or the track update interval would serve as the criterion for track termination and the IMMPDAF would provide a unified framework for track formation, maintenance and termination.

E. Simulation Results This section presents the simulation results obtained using the algorithms described earlier. The computational requirements and root-mean-square errors (RMSE) are given. The tracking algorithm using the IMMPDAF is tested on the following six benchmark tracks.13 • Target 1 A large military cargo aircraft with maneuvers up to 3 . • Target 2 A Learjet or commercial aircraft which is smaller and more maneuverable than target 1 with maneuvers up to 4 . • Target 3 A high speed medium bomber with maneuvers up to 4 . • Target 4 Another medium bomber with good maneuverability up to 6 . • Targets 5 and 6 Fighter or attack aircraft with very high maneuverability up to 7 . In Table 2, the performance measures and their averages of the IMMPDAF (in the presence of FA, RGPO, and SOJ [16], [35]) are given. The averages are obtained by adding the corresponding performance metrics of the six targets (with those of target 1 added twice) and dividing the sum by seven. In the table, the maneuver density is the percentage of the total time that the target acceleration exceeds 0.5 . The average floating point operation (FLOP) count per second was obtained by dividing the total number of floating point operations by the target track length. This is the computational requirement for target and jammer tracking, neutralizing techniques for ECM, and adaptive parameter selection for the estimator, i.e., it excludes the computational load for radar emulation. The average FLOP14 requirement is 25 kFLOPS, which can be compared with the FLOP rate of 78 MFLOPS of a Pentium processor running at 133 MHz. Thus, the real-time implementation of the complete tracking system is possible. With the average revisit interval of 2.5 s, the FLOP requirement of the IMMPDAF is 62.5 kFLOP/radar cycle. With the revisit time calculations taking about the same amount of 13The tracking algorithm does not know the type of the target under track—the parameters are selected to handle any target. 14The FLOP count is obtained using the built-in MATLAB function flops. Note that these counts, which are given in terms of thousands of floating point operations per second (kFLOPS) or millions of floating point operations per second (MFLOPS), are rather pessimistic—the actual FLOP requirement would be considerably lower.


549

computation as a cycle of the IMMPDAF, but running at half the rate of the KF (which runs at constant rate), the IMMPDAF with adaptive revisit time is about ten times costlier computationally than a KF. Due to its ability to save radar resources, which are much more expensive than computational resources, the IMMPDAF is a viable alternative to the KF, which is the standard “workhorse” in many current tracking systems.15 V. A FLEXIBLE-WINDOW ML-PDA ESTIMATOR FOR TRACKING LO TARGETS One difficulty with the ML-PDA approach of Section III, which uses a set of scans of measurements as a batch, is the incorporation of noninformative scans when the target is not present in the surveillance region for some consecutive scans. For example, if the target appears within the surveillance region of the sensor after the first few scans, the estimator can be misled by the pure clutter in those scans—the earlier scans contain no relevant information and the incorporation of these into the estimator not only increases the amount of processing (without adding any more information) but also results in less accurate estimates or even track rejection. Also, a target could disappear from the surveillance region for a while during tracking and reappear sometime later. Again, these intervening scans contain little or no information about the target and can potentially mislead the tracker. In addition, the standard ML-PDA estimator assumes that the target SNR, the target velocity, and the density of FAs over the entire tracking period remain constant. In practice, this may not be the case and then the standard ML-PDA estimator will not yield the desired results. For example, the average target SNR may vary significantly as the target gets closer to or moves away from the sensor. Also, the target might change its course and/or speed intermittently over time. For EO sensors, depending on the time of the day and weather, the number of FAs may vary as well. Because of these concerns, an estimator capable of handling time-varying SNR (with online adaptation), FA density, and slowly evolving course and speed is needed. While a recursive estimator like the IMM-PDA is a candidate, in order to operate under low SNR conditions in heavy clutter, a batch estimator is still preferred. In this section, the above problems are addressed by introducing an estimator that uses the ML-PDA with AI adaptively in a sliding-window fashion [22], rather than using all the measurements in a single batch as the standard ML-PDA estimator does [34]. The initial time and the length of this sliding window are adjusted adaptively based on the information content in the measurements in the window. Thus, scans with little or no information content are eliminated and the window is moved over to scans with “informative” measurements. This algorithm is also effective when the target is temporarily lost and reappears later. In contrast, recursive algorithms will diverge in this situation and may require an expensive track reinitiation. The standard batch estimator will be oblivious to the disappearance and may lose the whole track. 15Some

550

0 filter as their “workmule.”

systems still use the

This section demonstrates the performance of the adaptive sliding-window ML-PDA estimator on a real scenario with heavy clutter for tracking a fast-moving aircraft using an EO sensor. This problem can be solved using hidden Markov model or expectation–maximization-based algorithms as well [21], [53]. A. Scenario The adaptive ML-PDA algorithm was tested on an actual scenario consisting of 78 frames of long wave infrared (LWIR) IR data collected during the Laptex data collection, which occurred in July 1996 at Crete, Greece. The sequence contains a single target—a fast-moving Mirage F1 fighter jet. The 920 480 pixel frames, taken at a rate of 1 Hz, were registered to compensate for frame-to-frame line-of-sight (LOS) jitter. Fig. 2 shows the last frame in the F1 Mirage sequence. A sample detection list for the Mirage F1 sequence obtained at the end of preprocessing is shown in Fig. 3. Each “ ” in the figure represents a detection above the threshold. B. Formulation of the ML-PDA Estimator This section describes the target models used by the estimator in the tracking algorithm and the statistical assumptions made by the algorithm. The ML-PDA estimator for these models is introduced, and, finally, the CRLB for the estimator and the hypothesis test used to validate the track are presented. 1) Target Models: The ML-PDA tracking algorithm is used on the detection lists after the data preprocessing phase. It is assumed that there are detection lists obtained at times . The th detection list, where , detections at pixel positions along consists of and directions. In addition to locations, the signal the strength or amplitude of the th detection in the th list, , is also known. Thus, assuming conwhere stant velocity over a number of scans, the problem can be formulated as a 2-D scenario in space with the target motion defined by the 4-D vector (140) where and are the horizontal and vertical pixel positions of the target, respectively, from the origin at the reference time . The corresponding velocities along these directions are assumed constant at pixel/s and pixel/s, respectively. The set of measurements in list at time is denoted by (141) where is the number of measurements at . The measureis denoted by ment vector (142) where tively.

and

are observed

and

positions, respec-


Fig. 2.

The last frame in the F1 Mirage sequence.

Fig. 3. Detection list corresponding to the frame in Fig. 2.

The cumulative set of measurements made in scans through is given by

rupted by zero-mean additive Gaussian noise of known variance, i.e., (144) (145)

(143) A measurement can either originate from a true target or from a spurious source. In the former case, each measurement is assumed to have been received only once in each and to have been corscan with a detection probability

where and are the zero-mean Gaussian noise components with variances and along the and directions, respectively.


551

Thus, the joint pdf of the position components of given by

is

(146) The FAs are assumed to be distributed uniformly in the surveillance region and their number at any sampling instant obeys the Poisson probability mass function (147) is the area of surveillance and is the expected where number of FAs per unit of this area. It has been shown in [34] that the performance of the ML-PDA estimator can be improved by using AI of the received signal in the estimation process itself, in addition to thresholding. After signal has been passed through the matched filter, an envelope detector can be used to obtain the amplitude of the signal. The noise at the matched filter is assumed to be narrowband Gaussian. When this is fed through the envelope detector, the output is Rayleigh distributed. Given the detection threshold , the probability and the probability of FA are of detection Target-originated measurement exceeds threshold

(148)

and Measurement due to noise only exceeds threshold (149) is the probability of an event. where The pdf’s at the output of the threshold detector, which corresponds to signals from the target and FAs, are denoted and , respectively. Then the amplitude likeliby hood ratio can then be written as [7] (150) where is the detection threshold. 2) ML-PDA Estimator: As shown earlier, the total log, expressed in terms of the individual likelihood ratio at sampling time instants , log-likelihood ratios becomes

(151) 552

that The MLE is obtained by finding the vector maximizes the total log-likelihood ratio given in (151). This maximization is performed using a quasi-Newton (variable metric) method. This can equivalently be done by minimizing the negative log-likelihood function. In our implementation of the MLE, the Davidon–Fletcher–Powell variant of the variable metric method is used. This method is a conjugate gradient technique which finds the minimum value of the function iteratively [47]. However, the negative log-likelihood function may have several local minima, i.e., it has multiple modes. Due to this property, if the search is initiated too far away from the global minimum, the line search algorithm may converge to a local minimum. To remedy this, a multipass approach is used as in [34]. C. Adaptive ML-PDA Often, the measurement process begins before the target becomes visible; that is, the target enters the surveillance region of the sensor some time after the sensor started to record measurements. Also, the target may disappear from the surveillance region for a certain period before reappearing. During these periods of blackout, the received measurements are purely noise only, and the scans of data contain no information about the target under track. Incorporating these scans into a tracker reduces its accuracy and efficiency. Thus, detecting and rejecting these scans is important to ensure the fidelity of the estimator. This subsection presents a method which uses the ML-PDA algorithm in a sliding-window fashion. In this case, the algorithm uses only a subset of the data at a time rather than all of the frames at once, to eliminate the use of scans that have no target. The initial time and the length of the sliding window are adjusted adaptively based on the information content of the data—the smallest window, and thus the fewest number of scans, required to identify the target is determined online and adapted over time. The key steps in the adaptive ML-PDA estimator are as follows. 1) Start with a window of minimum size. 2) Run the ML-PDA estimator within this window and carry out the validation test on the estimates. 3) If the estimate is accepted (i.e., if the test is passed), and if the window is of minimum size, accept the window. The next window is the present window advanced by one scan. Go to step 2. 4) If the estimate is accepted, and if the window is greater than minimum size, try a shorter window by removing the initial scan. Go to step 2 and accept the window only if estimates are better than those from the previous window. 5) If the test fails and if the window is of minimum size, increase the window length to include one more scan of measurements and, thus, increase the information content in the window. Go to step 2. 6) If the test fails and if the window is greater than minimum size, eliminate the first scan, which could contain pure noise only. Go to step 2. 7) Stop when all scans are used. PROCEEDINGS OF THE IEEE, VOL. 92, NO. 3, MARCH 2004

Fig. 4. Sliding-window adaptive ML-PDA algorithm.

Fig. 5. Scenario with a target being present for only a partial time during observation.

The algorithm is described below. In order to specify the exact steps in the estimator, the following variables are defined. Current window length Minimum window length Scan (set) of measurements at time With these definitions, the algorithm is shown in Fig. 4. To illustrate the adaptive algorithm, consider a scenario where a sensor records ten scans of measurements over a surveillance region. The target, however, appears in this region (i.e., its intensity exceeds the threshold) only after the second scan (i.e., from the third scan onwards). This case is illustrated in Fig. 5. Consider the smallest window size required for a detection to be five. Then the algorithm will evolve as shown in Fig. 6. First, for the sake of illustration, assume that a single “noisy” scan present in the data set is sufficient to cause the MLE to fail the hypothesis test for track acceptance. The algorithm tries to expand the window to include an additional scan if a track detection is not made. This is done because an addi-

Fig. 6. Adaptive ML-PDA algorithm applied to the scenario illustrated in Fig. 5.

tional scan of data may bring enough additional information to detect the target track. The algorithm next tries to cut down the window size by removing the initial scans. This is done to check whether a better estimate can be obtained without this scan. If this initial scan is noise only, then it degrades the accuracy of the estimate. If a better estimate is found (i.e., a more accurate estimate) without this scan, the latter is eliminated. Thus, as in the example given above, the algorithm expands at the front (most recent scan used) and contracts at the rear end of the window to find the best window that produces the strongest detection, based on the validation test. D. Results 1) Estimation Results: The first two scans are useless, because they contain only noise. The Mirage F1 data set consists of 78 scans or frames of LWIR IR data. The target ap-


553

Fig. 7. Progress of the algorithm showing windows with detections. Table 3 Parameters Used in the ML-PDA Algorithm for the F1 Mirage Jet

pears late in this scenario and moves toward the sensor. There are about 600 detections per frame. In this implementation the parameters shown in Table 3 were chosen. The choice of these parameters is explained as follows. and are, as in (146), the standard deviations along • the horizontal and vertical axes respectively. The value of 1.25 for both variables models the results of the preprocessing. , was chosen to be • The minimum window size, ten. The algorithm will expand this window if a target is not detected in ten frames. Initially a shorter window was used, but the estimates appeared to be unstable. Therefore, less than ten scans is assumed to be ineffective at producing an accurate estimate. • The initial target SNR was chosen as 9.5 dB because the average SNR of all the detections over the frames is approximately 9.0 dB. However, in most frames, 554

random spikes were noted. In the first frame, where a target is unlikely to be present, a single spike of 15.0 dB is noted. These spikes, however, cannot and should not be modeled as the target SNR. of 0.7 was • A constant probability of detection chosen. A value that is too high would bring down the . detection threshold and increase • is the parameter used to update the estimated target SNR with an filter. A high value is chosen for the purpose of detecting a distant target that approaches the sensor over time and to account for the presence of occasional spikes of noise. Thus, the estimated SNR is less dependent on a detection that could originate from a noisy source and, thus, set the bar too high for future detections. is the miss probability. • are used in the multipass approach of the op• and timization algorithm [30], [34]. • The number of passes in the multipass approach of the optimization algorithm was chosen as four. To get a better idea of the detection process, Fig. 7 depicts the windows where the target has been detected. From the above results, note the following. • The first detection uses 22 scans and occurs at scan 28. This occurs because the initial scans have low information content, as the target appears late in the frame of surveillance. The IMM-MHT algorithm [52] required 38 scans for a detection, while the IMMPDA [41] required 39 scans. Some spurious detections were noticed at earlier scans, but these were rejected. PROCEEDINGS OF THE IEEE, VOL. 92, NO. 3, MARCH 2004

• The next few detection windows produce similar target estimates. This is because a large number of scans repeat themselves in these windows. • After the initial detections, there is a “jump” in the scan number at which a detection is made. In addition, the estimates, particularly the velocity estimates, deteriorate. This could either indicate that the target has suddenly disappeared (became less visible) from the region of surveillance or that the target made a maneuver. • From scan 44 onward, the algorithm stabilizes for several next windows. At scan 52, however, there is another “jump” in detection windows. This is also followed by a drop in the estimated target SNR, as explained above. This, however, indicates that the algorithm can adjust itself and restart after a target has become suddenly invisible. Recursive algorithms will diverge in this case. • From scan 54 onward, the algorithm stabilizes, as indicated by the estimates. Also, a detection is made for every increasing window, because the target has come closer to the sensor and, thus, is more visible. This is noted by the sharp rise in the estimated target SNR after scan 54. • The above results provide an understanding the target’s behavior. The results suggest that the Mirage F1 fighter jet appears late in the area of surveillance and moves toward the sensor. However, initially it remains quite invisible and possibly undergoes maneuvers. As it approaches the sensor, it becomes more and more visible and, thus, easier to detect. 2) Computational Load: The adaptive ML-PDA tracker took 442 s, including the time for data input/output, on a Pentium III Processor running at 550 MHz to process the 78 scans of the Mirage F1 data. This translates into about 5.67 s per frame (or 5.67 s running time for 1-s data), including input/output time. A more efficient implementation on a dedicated processor can easily make the algorithm real-time capable on a similar processor. Also, by paralellizing the initial grid search, which required more than 90% of the time, the adaptive ML-PDA estimator can be made even more efficient. VI. SUMMARY This paper presented the use of the PDA technique for different tracking problems. Specifically, the PDA approach was used for parameter estimation as well as recursive state estimation. As an example of parameter estimation, track formation of a low observable target using a nonlinear ML estimator in conjunction with the PDA technique with passive (sonar) measurements was presented. The use of the PDA technique in conjunction with the IMM estimator, resulting in the IMMPDAF, was presented as an example of recursive estimation on a radar tracking problem in the presence of ECM. Also presented was an adaptive sliding-window PDA-based estimator that retains the advantages of the batch (parameter) estimator while being capable

of tracking the motion of maneuvering targets. This was illustrated on an EO surveillance problem. These applications demonstrate the usefulness of the PDA approach for a wide variety of real tracking problems.

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Thiagalingam Kirubarajan (Senior Member, IEEE) was born in Sri Lanka in 1969. He received the B.A. and M.A. degrees in electrical and information engineering from Cambridge University, Cambridge, U.K., in 1991 and 1993, respectively, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Connecticut, Storrs, in 1995 and 1998, respectively. Currently, Dr. Kirubarajan is an Assistant Professor in the Electrical and Computer Engineering Department at McMaster University, Hamilton, Ontario. He is also serving as an Adjunct Assistant Professor and the Associate Director of the Estimation and Signal Processing Research Laboratory at the University of Connecticut. He has served as a consultant to a number of companies, including Motorola Corporation, Northrop-Grumman Corporation, Pacific-Sierra Research Corporation, Lockheed Martin Corporation, Qualtech Systems, Inc., Orincon Corporation, and BAE Systems. He has worked on the development a number of number of engineering software programs, including BEARDAT for target localization from bearing and frequency measurements in clutter, FUSEDAT for fusion of possibly heterogeneous multisensor data for tracking, and MTI* for large-scale multisensor ground target tracking. He has also worked with Qualtech Systems, Inc., to develop advanced fault diagnosis and fusion techniques for the TEAMS-RT fault diagnosis engine. He has published about 150 articles, in addition to one book on estimation, tracking, and navigation and four edited volumes. His research interests are in estimation, target tracking, multisource information fusion, sensor resource management, signal detection and fault diagnosis. Dr. Kirubarajan received the Ontario Premier’s Research Excellence Award in 2002 and the Barry Carlton Award for the Best Paper Published in the IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS in 2000. He is an Associate Editor for the IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS (Tracking and Fusion). Previously, he was an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS.


Yaakov Bar-Shalom (Fellow, IEEE) was born on May 11, 1941. He received the B.S. and M.S. degrees in electrical engineering from the Technion—Israel Institute of Technology, Haifa, in 1963 and 1967 and the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ, in 1970. From 1970 to 1976 he was with Systems Control, Inc., Palo Alto, CA. He is currently Board of Trustees Distinguished Professor in the Department of Electrical and Computer Engineering and Director of the Estimation and Signal Processing (ESP) Lab at the University of Connecticut, Storrs. He has consulted for numerous companies, and originated the series of multitarget–multisensor tracking short courses offered via University of California, Los Angeles, Extension and at government laboratories, private companies, and overseas. He has also developed the commercially available interactive software packages MULTIDAT for automatic track formation and tracking of maneuvering or splitting targets in clutter, VARDAT for data association from multiple passive sensors, BEARDAT for target localization from bearing and frequency measurements in clutter, IMDAT for image segmentation and target centroid tracking, and FUSEDAT for fusion of possibly heterogeneous multisensor data for tracking. He has published over 250 papers. He coauthored the monograph Tracking and Data Association (New York: Academic, 1988), the graduate text Estimation and Tracking: Principles, Techniques and Software (Norwood, MA: Artech House, 1993), and the text Multitarget–Multisensor Tracking: Principles and Techniques (Storrs, CT: YBS, 1995), and edited Multitarget–Multisensor Tracking: Applications and Advances [Norwood, MA: Artech House, 1990 (vol. 1), 1992 (vol. 2), 2000, vol. 3)]. From 1978 to 1981, he was Associate Editor of Automatica. His research interests are in estimation theory and stochastic adaptive control. 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. His other interests are stochastic control of vertical airfoils and of pairs of inclined foot supports on crystals. Dr. Bar-Shalom received the IEEE Control Systems Society (CSS) Distinguished Member Award in 1987. He was Corecipient of the M. Barry Carlton Award for the best paper in the IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS in 1995 and the 1998 University of Connecticut AAUP Excellence Award for Research. Since 1995 he has been a Distinguished Lecturer of the IEEE Aerospace and Electronic System Society (AESS) and has given several keynote addresses at major national and international conferences. In 1976 and 1977, he was Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. He was Program Chairman of the 1982 American Control Conference (ACC), General Chairman of the 1985 ACC, and Cochairman of the 1989 IEEE International Conference on Control and Applications. From 1983 to 1987, he was Chairman of the Conference Activities Board of the IEEE CSS. From 1987 to 1989, he was a member of the Board of Governors of the IEEE CSS. He served as Y2K President of the International Society of Information Fusion.


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