Sequential Monte Carlo Methods for Tracking Multiple Targets With

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 3, MARCH 2008

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Sequential Monte Carlo Methods for Tracking Multiple Targets With Deterministic and Stochastic Constraints Ioannis Kyriakides, Student Member, IEEE, Darryl Morrell, Senior Member, IEEE, and Antonia Papandreou-Suppappola, Senior Member, IEEE

Abstract—In multitarget scenarios, kinematic constraints from the interaction of targets with their environment or other targets can restrict target motion. Such motion constraint information could improve tracking performance if effectively used by the tracker. In this paper, we propose three particle filtering methods that incorporate constraint information in their proposal and weighting process; the number of targets is fixed and known in all methods. The reproposed constrained motion proposal (RCOMP) utilizes an accept/reject method to propose particles that meet the constraints. The truncated constraint motion proposal (TCOMP) uses proposal densities truncated to satisfy the constraints. The constraint likelihood independent partitions (CLIP) method simply rejects proposed partitions that do not meet the constraints. We use simulation to evaluate the performance of these three methods for two constrained motion scenarios: a vehicle convoy and soldiers executing a leapfrog motion. Moreover, we demonstrate the utility of constraint information by comparing the proposed algorithms with the independent partition (IP) proposal method that does not use constraint information. The simulation results demonstrate that the root mean square error (RMSE) tracking performance of the RCOMP and the TCOMP methods is much better than the CLIP and IP methods; this is due to their more efficient proposal process. Index Terms—Constrained target motion, efficient proposal processes, Monte Carlo methods, multiple target tracking, particle filtering.

I. INTRODUCTION NFORMATION about constraints on target motion can improve the performance of a multiple target tracker. Some examples of such kinematic constraints include restrictions on the position and velocity of objects; for example, ships must avoid land [1], [2] and vehicles must follow roads [3]. Other constraints may be imposed on a group of targets. For example, in convoy movement [4], the vehicles are constrained to be neither too close nor too far away from each other. Another type of constrained motion is when people are moving in a prescribed manner, such as the leapfrog motion in military applications [5].

I

Manuscript received July 26, 2006; revised July 26, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Jaume Riba. This work was supported by the DARPA Integrated Sensing and Processing program through a contract with the Office of Naval Research N00014-04-C-0437. Approved for Public Release, Distribution Unlimited. I. Kyriakides and A. Papandreou-Suppappola are with the Department of Electrical Engineering, Arizona State University Tempe, AZ 85287 USA (e-mail: [email protected]; [email protected]). D. Morrell is with the Department of Engineering, Arizona State University, Mesa, AZ 85212 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2007.908931

A Bayesian approach to filtering allows constraints on target motion to be incorporated into the filter by computing the conditional density of the target states given observations and constraints. In order to estimate the target state, this density must be computed; however, it is typically not possible to compute it in closed form due to nonlinearity in the resulting motion model. Instead, particle filtering can provide practical implementations of Bayesian filters by approximating the required densities with weighted samples called particles [6]. For tracking multiple targets, particle filtering can estimate the joint multitarget probability density (JMPD) of the targets [1], [7]–[10]. In the JMPD particle filter, each particle includes all the information of interest such as the number of targets present and their states. The independent partition (IP) [7] and the adaptive partition (AP) algorithms [8] incorporate observation information into the proposal, and as such, propose particles that represent the JMPD more accurately. Recently, the rigor of the JMPD approach has been questioned when the number of targets is unknown (and possibly time varying). In [11], the problem of tracking an unknown and varying number of targets was rigorously formulated using finite-set statistics (FISST) [12]. The probability hypothesis density (PHD) filter, a principled approximation strategy to the FISST formulation, is studied in [11] and [13]. Furthermore, particle filtering formulations of the PHD can be found in [14]–[17]. In [18], closed-form solutions to the PHD recursion are demonstrated, under the assumptions of linearity and Gaussianity of the target state and birth process, with extensions to accommodate mild nonlinearities. Higher order PHD approximations are presented in [19] and [20] for which closed-form solutions are developed in [21]. The problem of group motion of targets was considered in [22] and [23] using an approach similar to extended target tracking [24]–[26]. However, there is a need to deal with a more general problem where each target follows an independent trajectory but still affects the motion of other targets. In [27], hard constraints are imposed on the speed and acceleration of a single target using sampling and rejection. In [2], the problem of tracking targets that cannot cross a boundary (e.g., land/sea) is considered. Constraints are imposed in the kinematics by using a reflecting boundary that forces the targets’ position to be inside the allowed region and also reverses the component of the proposed target velocity normal to the boundary. In [3], constraints are incorporated into the measurement likelihood using pseudomeasurements that cause the measurement likelihood to have a high value when constraints are satisfied. A Gaussian

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mixture cardinalized PHD particle filter implementation to track targets with road map constraints is presented in [28]. In this paper, we present techniques for the incorporation of constraints into a JMPD particle filter with a fixed and known number of targets. We first develop the constraint likelihood IP (CLIP) algorithm [29], [30] that accounts for the motion constraint information through a constraint likelihood function similar to the land avoidance function in [1]. This function is designed to null the weights of particles that do not conform with the constraints. Therefore, only particles that have been proposed within the constraints contribute to the estimate of the multitarget state and propagate within the particle filtering process. We also propose two constrained motion proposal (COMP) algorithms that incorporate motion constraints into the process of proposing particles. In the reproposed COMP (RCOMP), individual partitions are repeatedly proposed until they meet the constraints. The RCOMP method also makes use of constraint likelihood functions to remove particles that cannot conform to the constraints even after a number of reproposals. In the truncated Gaussian COMP (TCOMP) method, each of the components of each partition is proposed using a truncated Gaussian proposal density in which the truncation points are set according to the constraints. We use simulation to evaluate the performance of these methods for two constrained motion scenarios: a vehicle convoy and soldiers executing a leapfrog motion. Moreover, we demonstrate the utility of constraint information by comparing the proposed algorithms with the IP proposal method that does not use constraint information. Our simulation results show that both COMP algorithms have better root mean square error (RMSE) performance on problems with motion constraints, than either the IP, which approximates joint target motion as being independent, or the CLIP algorithm. This is due to their more efficient proposal process. The paper is organized as follows. We describe the state space formulation of the multiple target tracking approach in Section II, and we review the IP method in Section III. In Sections IV and V, we propose the CLIP and COMP algorithms, respectively. We provide simulation results of the proposed methods, and we compare them with existing multiple target tracking algorithms in Section VI. II. STATE-SPACE FORMULATION A. Motion Model We consider a fixed and known number of targets, denoted by , moving in a two-dimensional plane. The dynamics of each target is modeled by a nearly constant velocity motion model [31] in Cartesian coordinates. Specifically, the state vector for the th target, , at time step , is given by

where , are the positions in the and directions, and , are the corresponding velocities. The motion is formulated as (1)

where

and is the time difference between observations. is a white, zero-mean vector Gaussian process with covariance matrix that models target deviations from constant velocity; to be diagonal. The model in (1) in this work, we restrict can be used to determine the kinematic prior distribution for target . Our approach can be used as well for other dynamics models (e.g., nearly constant acceleration). The multitarget state vector is expressed in terms of the state vectors of each target as

where denotes the transpose of . Following [7], we refer to of the state vector as a partition. each component

B. Constraint Modeling: Deterministic and Stochastic In this paper, we address target motion constrained both by the interaction between each target and the environment as well as by the interaction between targets. Self-constraints provide information about the relationship of a target and its environment (e.g., information about vehicle motion on a road or a specific elevation of an airplane, as described in [3]). Group constraints provide information imposed on a target from other targets with which it interacts (e.g., convoy motion). Conceptually, we represent constraints as regions in which allowable values of state vectors lie. The constraint region asis denoted by .A sociated with the entire state vector deterministic constraint is one for which the constraint region is completely known to the tracker (i.e., the tracker can defalls within ). A stochastic constraint is termine whether one for which has a functional dependence on random variables whose values are not known to the tracker; we denote these variables as auxiliary variables. If the auxiliary variables were known to the tracker, then the tracker could determine whether falls within . We apply constraints in the context of particle filtering algorithms in which partitions are proposed sequentially and then assembled into particles as described in Section III. Before each partition is proposed, the constraints imposed on the partition by previously proposed partitions are computed. The constraint region associated with the th partition is denoted by . depends on the self constraint region , which represents constraints imposed on the partition by the environment and the group constraint regions , , ; represents the constraint imposed on target by target . is a function of , the actual value proposed Note that for partition . Thus, depends on the partitions that have

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been proposed prior to partition ; it is the intersection of and the group constraints imposed by previous partitions: (2) To implement stochastic constraints, we introduce the set . For a of auxiliary variables stochastic constraint, group constraints for partition are a function of both and and could be denoted as , although we use the simpler notation . We assume that the auxiliary variables evolve in time, and that their evolution can be modeled by a Markov process . We assume that the characterized by: auxiliary variables evolve independently of the target states. As an example of the use of auxiliary variables, consider a scenario in which vehicles travel in convoys whose composition changes over time; vehicles impose constraints only on other vehicles in their convoy. A simple model for this is obtained by defining the auxiliary variable if in

imposes constraints on otherwise.

(3)

can be modeled to transition from time step to according to a first order Markov chain; with a probability , does not change, and with a probability , it toggles. This model is used in Section VI-C.

Fig. 1. Sensor field of view is divided into cells; each cell covers a certain angle, and the sensor reports a detection or no detection for each cell.

clares either a detection or no detection in each cell. The measurement produced by sensor at time is given by the vector , where if a detection is de, otherwise. clared in cell , and represents the resulting measurement vector from all sensors. Next, we derive the likelihood of the measurements conditioned on the true target state . Following [8], we model the energy measurements as Rayleigh-distributed with a signal-to-noise ratio (SNR) denoted by and equal for all targets. With an energy threshold chosen to give a desired probability of false alarm , the probability of detection in a given cell is [8]

(6) C. Constrained Motion Model The standard assumption in multiple target tracking is that all targets move independently without constraints; under this assumption, the joint kinematic prior distribution is the product of the kinematic priors for each target

(4)

where is the number of targets in cell . The likelihood of is if has no targets in cell if has targets in cell The sensor is assumed to obtain measurements in each cell independently. Therefore, the individual sensor likelihoods are assumed to be conditionally independent and the likelihood of is the product of the likelihoods for each cell

We introduce the constraint likelihood function if otherwise. In this work, we assume that the joint kinematic prior distribution with constraints is

Finally, the total likelihood is given by the product of the likelihoods of all the sensors

(5)

III. MEASUREMENT MODEL We assume that the targets are simultaneously observed by passive sensors. The area monitored by each sensor is divided radially into nonoverlapping cells as shown in Fig. 1. The sensor measures the energy received in each cell from targets and noise, compares this energy to a threshold, and de-

Note that our approach is not dependent on this particular sensor model; it can be applied for any sensor for which can be specified. IV. MULTITARGET TRACKING ALGORITHMS The multitarget particle filter [8] proposes particles that consist of partitions, with each partition representing one of the targets of interest. In a straightforward implementation of the par-

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ticle filter, the partitions in a particle are proposed simultaneously using the target dynamics model without any additional assessment of their validity. Many of these particles receive low weights at the particle weighting step because one or more partitions are bad estimates of the target states. In this paper, we consider a class of particle proposal algorithms that we call sequential partition algorithms; this class includes the IP algorithm as well as the RCOMP and TCOMP algorithms. In a sequential partition algorithm, partitions are proposed sequentially and then assembled into particles. A given partition may be proposed independently of previously proposed partitions (as is the case of the IP and CLIP algorithms), or the proposed values for a given partition may depend on previously proposed partitions (as is the case of the RCOMP and TCOMP algorithms). The steps of a sequential partitio algorithm include partition sampling, partition weighting, partition resampling, particle weighting, and particle resampling. In partition sampling, we sample each partition of the th particle from a single partition proposal density that may depend on previously sampled partition values. In partition weighting, we weight the partition with a partition that indicates the parweighting function tition likelihood given measurement information; we then . In parnormalize the partition weight distribution from the tition resampling, we sample a partition index with replacement. The resulting selected distribution of and selection probability partition has value . After partition resampling, we assemble parti. In cles from the sampled partitions as particle weighting, we weight these particles with weights that incorporate prior and measurement information using , and then we normalize the distribution of . In particle resampling, we resample particles according to this distribution. The IP algorithm [7], [8], summarized in Table I, is a sequential partition algorithm that assumes the joint dynamics model in (4). The IP algorithm is an approximation to the JMPD particle filter for this dynamics model; the approximation is accurate when the targets are well separated in the observation space. When targets are close in sensor space, their partitions cannot be independently proposed as described above. In this case, the coupled partition (CP) and the AP methods [8] may be used. V. CONSTRAINT LIKELIHOOD FUNCTION INDEPENDENT PARTITION ALGORITHM The CLIP algorithm[29], [30] that we propose is a natural extension of the IP. It considers both the measurements and the constrained kinematics in proposing particles. The CLIP incorporates information on constrained motion using a constraint likelihood function. The constraint likelihood function is a generalization of the land avoidance function in [1]. The CLIP adds two types of constraint likelihood functions to the IP method: i) partition constraint likelihood functions used in partition weighting, and ii) particle constraint likelihood functions used in particle weighting. For the CLIP, we sample from . the kinematic prior for each partition as

TABLE I INDEPENDENT PARTITION ALGORITHM [7], [8]

The samples are weighted by a partition weight function which is the product of the partition weighting function, , and the partition constraint likelihood function, . As described in is the constraint region within which values Section II-B, can exist given previously of the partition state vector proposed partitions. , , Resampling the partition indexes , we obtain from the partition weight distribution and the associated parthe selection probability tition , , for particle . Note that we used the tilde notation for the partitions sampled to differentiate them from the initially from resulting after resampling from the partition partitions weight distribution. After assembling particles from partitions , we weight the th as particle as [6]

where is the proposal density used to propose each of the particles. Each of the partitions of the particle is proposed by the kinematic prior and resampled according to the partition weights. Therefore

(7) where (8) Combining (7) and (8), the weight of each particle becomes


TABLE II CONSTRAINT LIKELIHOOD INDEPENDENT PARTITION ALGORITHM

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TABLE III TRUNCATED AND REPROPOSED COMP ALGORITHMS

which, using (4), simplifies to

The steps of the CLIP algorithm are summarized in Table II. VI. COMP ALGORITHM As the CLIP partition resampling step does not resample partitions that do not meet the constraints, it may reject a significant percentage of the proposed partitions. This, in turn, may increase the number of particles necessary for satisfactory performance or may result in poor estimates. Our next proposed algorithm, the COMP [29], [30] proposes partitions that meet constraints, thus eliminating the rejection of particles due to constraint violations. The COMP accounts for target motion constraints by proposal densities that incorporate the constraints and the constraint likelihood function that reflects the constraints as part of the JMPD. In the COMP, the measurements are incorporated into the partition proposal as with the CLIP; however, the kinematics include the constraints on the motion of each target. We present two different methods to achieve this, as summarized in Table III. The reproposed Gaussian COMP (RCOMP) proposes partitions repeatedly until the constraints are satisfied. The truncated Gaussian COMP (TCOMP) proposes partitions using truncated Gaussian distributions, where the truncation points are chosen to satisfy the constraints. In both approaches, each partition of a given particle is proposed to satisfy the constraints imposed by the partitions proposed earlier in the particle. When group constraints exist, the order in which partitions are proposed is varied to ensure that the partitions values do not lead to constraints that cannot be satisfied by subsequent partitions, as explained next. This is not necessary if there are only self-constraints. A. Partition Ordering and Formulation of Constraints We next describe how the order of proposed partitions is partition indexes can be sequentially proposed in varied. different orders. Therefore, the set of particles can be subsets, and each subset can separated into a maximum of have its partitions proposed in a different order; we denote such subsets of particles as permutation subsets. A practical

consideration that limits the number of subsets is that the number of particles in each subset should be large since several particles are sampled from each subset in the sampling step of the proposal. The proposal orders can be chosen randomly. We proceed with the mathematical formulation of the constraints applied to each partition and the notation and organization of the different proposal orders, for a specific particle and time step . The th permutation, , at time is denoted by the vector of partition indexes . As described in Section II-B, we use the following method to obtain the constraints applied to a partition. , is proposed according to its The first partition, indexed by , we use self-constraints. For the th partition, indexed by its self-constraints together with the group constraints from the partitions. The drawback of this sequential approach first is that proposed partition values might lead to constraints that cannot be satisfied by subsequent partitions. Variability in partition proposal ordering alleviates this problem. With this proposal scheme, (2) becomes

(9) B. Importance Density Both COMP methods require that the kinematic prior density of partitions is Gaussian. The RCOMP method repeatedly proposes partitions with this density until or a the proposed value falls within the constraint set set number of proposals have been made. The TCOMP method samples from a truncated Gaussian density where upper and . Next, we first lower truncation limits approximate describe the steps common to both methods before providing more details for each.

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1) Importance Density Formulation: The proposal process for a given partition consists of: sampling from a single parthat accounts for tition proposal constraints and evaluating the validity of the sample with that approximates the a partition weighting function . single target measurement likelihood function, The partition weighting function and the subsequent sampling from the resulting particle weight distribution will bias the particle proposal towards the measurements. The combination of sampling and weighting leads to the single . partition importance density The multitarget importance density is . We now describe the partition proposal and resampling process in detail. The sequential proposal of partitions for permutation is described mathematically as follows. For each particle in the permutation subset, we propose the first partition as , with index where the proposal distribution is an approximation of the kinematic prior that includes constraints, i.e.,

and the multitarget proposal is

(11) As discussed next, this density depends on the specific COMP algorithm. 2) RCOMP Algorithm: The RCOMP is an incremental accept/reject partition proposal. Specifically, for each parusing the ticle , we propose a given partition . This partition is rekinematic prior peatedly sampled until it falls within the constraint region or until a specified number of sampling attempts have been made. We weight each partition sample as Using (11) and with the conthe fact that we propose from (unless the proposal process halted dition that without meeting the constraints), we obtain the multitarget proposal density for the RCOMP as

(12) The normalization factor equals to the integral of over the allowable region of partition values given by . The partition constraint likelihood function is defined as if otherwise.

(10)

3) TCOMP Algorithm: The TCOMP algorithm can be applied when the constraint on a partition can be formed independently in terms of each component ( position, position, etc.) of the partition. In this approach, we propose each element using a truncated Gaussian distribuof each partition tion. For example, for the position coordinate of the partition , we obtain

After sampling the first partition for all particles in the permutation subset, the samples are weighted by the partition weighting function . The weights are normalized and partitions are sampled based on the weight distribution to with a selection probability . We propose obtain as the second partition with index

The proposal of the rest of the partitions follows in a similar , we obtain fashion until for the last partition

where denotes a truncated Gaussian probability disa Gaussian distribution, is the tribution, is the process noise for the posimean of the distribution, tion coordinate, is the lower bound and the upper . is the stanbound of the position obtained from dard normal cumulative distribution function [32] and is an indicator function that is defined as if otherwise. We proceed similarly for the rest of the elements of the state vector. As the process noise is independent for each component of the state ( is diagonal), this process proposes partition from the truncated kinematic prior distribution

Thus, the single partition proposal density is

(13)


where is the product of the kinematic priors is the of each element of the partition state vector and product of the normalization factors of each element. will Partitions with small normalization factors appear at the numerator of the final weight equation, giving near-zero weights to particles that may have other good partitions. Therefore, to remove those partitions at the proposal in favor of better partitions, we choose to use . Using (11) and (13), the multitarget proposal for the TCOMP becomes

(14) where is defined in (10). 4) Comparison Between RCOMP and TCOMP: The benefit of the TCOMP method is that partitions meeting the constraints are proposed in one sampling step. However, the TCOMP requires that the constraint regions can be formed independently for each element of a partition, which may not be possible in some scenarios. The RCOMP method may require a large number of repetitions to generate a partition sample that meets the constraints, and could fail to meet the constraints altogether. This is particularly an issue if the constraint region is very small. The important advantage of the RCOMP is that it places no restrictions on the structure of the constraint region; it simply checks if the proposed partition is within the constraint region and reproposes if it does not conform with the constraints. The choice of the maximum number of proposals to allow depends on how restrictive the constraints are and the available computational capabilities. This choice will determine the computational load of the RCOMP; our experience has shown that it is not prohibitive in many scenarios. Therefore, the RCOMP may be preferable to the TCOMP due to its simplicity and applicability to any kind of constraint scenario. C. Computation of Particle Weights After the partitions are assembled into particles, the particle weights are computed using the standard particle filter weight equation

With (12) for the RCOMP and (14) for the TCOMP for the multitarget proposal density, we have

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TABLE IV DEMONSTRATION OF PARTICLE CONSTRUCTION USING THE COMP

, the particle weight equation for both normalization factor the RCOMP and TCOMP simplifies to

D. Example of Particle Construction Using the COMP Algorithm In order to clarify how the particles are constructed using the COMP algorithm, we provide a two-target example. The particles are divided into two permutation subsets, and each subset is constructed with a different partition ordering. We propose the first subset using the order given by the first permutation, ( and ) and the second subset using the ( and order given by the second permutation, ). As this occurs at each time step , the subscript has been omitted for simplicity. The example is shown in Table IV for the first permutation, . In Step (i), we propose partition for . We weight and sample from the population to obtain the shown in Step (ii). In Step (iii), we prosampled version based on constraints provided by partitions pose partitions . This results in the augmented particles of Step (iv). In Step (v), we weight and sample from that population of parti. Here, cles, based on the weight distribution of partition the weights of the particles of Step (iv) have been obtained based only, and not on the combion the validity of partitions nation of and . It is also worth mentioning that if has survived the sampling from Steps (i) and a sample of (ii), it may not survive the sampling from Steps (iv) and (v) if a whose proposal was biased by the constraints sample of of , is not accurate. For example, while has survived the first sampling, it fails to appear among the selection of the second sampling; this is because the sample (whose proposal is affected by its constraints) does not obtain a high weight. For the second permutation, the same steps are taken but by first and then . proposing VII. SIMULATION RESULTS

for both the RCOMP and TCOMP. Considering the fact that (5) becomes an equality by dividing the right-hand side by the

We chose two scenarios to assess the performance of the algorithms proposed in this paper: vehicles moving in a convoy

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Fig. 2. Five targets moving one behind the other, and from left to right, throughout the entire motion in the scenario.


Fig. 3. RMSE versus the time that the simulation takes to complete 81 time steps using 95% confidence intervals.

[4], and a leapfrog motion [5] that military vehicles or personnel employ to move safely in certain directions. A. Convoy Motion and Deterministic Constraints In convoy motion, vehicles move as groups; individual vehicles obey certain rules such as a fixed speed and fixed separation distances between vehicles. The knowledge that a group of vehicles moves in a convoy can be used to formulate deterministic constraints in the target motion model. We use the following scenario to demonstrate the convoy constraint performance. Five vehicles move in a convoy in a fixed order as shown in Fig. 2. The average speed of the convoy is 90 km/h, and the actual distance between the vehicles maintained between 40 and 60 m. The angle-only sensors described in Section II-D are positioned as shown in Fig. 2. Their cells radians each. The probability of false cover an angle of and the SNR, denoted by in (6), is 20 alarm is set to dB. The tracker uses the constant velocity model in (1) with a for each target. The noise covariance number of target vehicles and the constraints on their motion are assumed to be known and fixed. The constraints used by the tracker are constraints on location that restrict vehicles to separations between 45 m and 55 m. Therefore, the constraint region for each partition consists of only group constraints that restrict the distance of to its nearest (in position) partition as . To reduce the complexity of the RCOMP algorithm, we used 24 subsets out possible orderings. of the maximum number of Using 300 Monte Carlo simulations, we evaluated the relationship between the computational requirements of the IP, CLIP, and RCOMP proposal methods and the resulting track accuracy. The computational complexity was measured by the execution time necessary to track the targets over 81 time steps using MATLA running on a 3-GHz Pentium IV workstation with 1-GB memory. Track accuracy was measured by the total RMSE in the position estimates of the targets. The computational complexity was adjusted by varying the number of particles used in the tracker.

Fig. 4. RMSE versus the time steps for the entire motion of the targets.

Fig. 3 shows the total RMSE as a function of execution time for the 300 Monte Carlo simulations; the plot shows the mean RMSE and 95% confidence intervals. The IP and CLIP do not perform as well as the RCOMP. The confidence intervals provide a qualitative measure of the consistency of the performance of the respective schemes. The RCOMP clearly reduces the variability in the resulting performance, providing the most reliable results as it effectively makes use of the available constraints. Note that although we include the IP for comparison, the COMP is more fairly compared to the CLIP as the IP was not designed to include constraint motion information. We also plotted the RMSE versus time averaged over 300 Monte Carlo simulations for the IP, CLIP and RCOMP in Fig. 4. For a fair comparison, we adjusted the number of particles for each of the methods so that their run times are similar. The number of particles that result in a run time of 4000 s is 1200 for the RCOMP and 2640 for both the IP and the CLIP. The RMSE increases as the time steps increase since the convoy moves away from the angle-only sensors which decreases the resolution of the sensor cells. In general, the COMP has a lower RMSE


Fig. 5. Position coordinates at time steps k = 1; . . . ; 49. Stage 1: Group A has started to move forward, while group B remains in position. Stage 2: Group A has paused its forward motion and group B has began advancing. Stage 3: Group B has paused its forward motion, while group A is again advancing forward.

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Fig. 7. RMSE versus the time that the simulation takes to complete for 81 time steps using 95% confidence intervals.

Fig. 8. RMSE versus the time steps for the entire motion of the targets. Fig. 6. Complete motion of the four targets in our leapfrog scenario together with the positioning of the three angle only sensors.

when compared to the corresponding values of the RMSE for the CLIP and IP for the same time index. B. Leapfrog Leapfrog is a commonly used technique [5] in which groups of personnel, vehicles, or aircraft move in a desired direction while one or more other groups remain stationary, providing cover to the moving group by suppressing enemy fire. A tracker can utilize information coming from this kinematic dependence of the targets as constraints to improve its performance. In the first leapfrog scenario used in this paper, we track the motion of four soldiers moving in groups of two. Two of the soldiers advance while the other two remain stationary. The advancing soldiers zigzag while not deviating too much from their mean position to their left or right. Both moving soldiers advance forward by the same amount. Various stages of this motion are shown in Fig. 5.

The deterministic constraint region used by the tracker is formed by the rules: moving targets will not cross in front of stationary targets (a constraint in the position coordinate), and moving targets will advance forward by approximately the same amount (a constraint in the position coordinate). To implement these constraints, the position coordinate of partition is not allowed to be closer than m to the position of any other partition . Thus, . For any two targets in the same group, and , the position coordinates should not . Our simuladiffer by more than 2 m. That is, tion runs for 81 time steps. The constant velocity model in (1) . with constraints is applied using The measurement model of Section II-D is used; each of the radians. The positioning sensor cells cover an angle of of the sensors, together with the complete motion of the targets, is shown in Fig. 6. We use the maximum possible orderings for 4 partitions, that orderings. In Fig. 7, the RMSE versus the program is, execution time for the 81 time steps is provided for the IP, CLIP,

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Fig. 9. True and estimated position in the x coordinate versus the time steps for the leapfrog scenario for the RCOMP. The indicated group names and stage numbers correspond to those in Fig. 5.

Fig. 11. True and estimated position in the x coordinate versus the time steps for the leapfrog scenario for the CLIP. The indications of the group and stages correspond to those in Fig. 5.

Fig. 10. True and estimated position in the y coordinate versus the time steps for the leapfrog scenario for the RCOMP. The indications of the group and stages correspond to those in Fig. 5.

Fig. 12. True and estimated position in the y coordinate versus the time steps for the leapfrog scenario for the CLIP. The indications of the group and stages correspond to those in Fig. 5.

RCOMP and TCOMP. As before, the COMP algorithms outperform the IP and CLIP. Note that the RCOMP and TCOMP perform similarly. In Fig. 8, we plot the RMSE as a function of the time steps for the four different algorithms using different number of particles for each scheme for a similar execution time of about 3000 s for each method. This execution time corresponds to 960 particles for both the RCOMP and TCOMP, and 2400 particles for both the IP and the CLIP. We have used 300 Monte Carlo runs. It is shown that, while the RCOMP and TCOMP have an almost constant RMSE throughout the scenario, the IP and CLIP result in an increasing error as the time steps progress. To show more detail in the performance of the methods in estimating the and position coordinate of the targets in different stages of the motion, we have plotted the RMSE for one target of group A and one target of group B in

the and coordinates for the RCOMP and CLIP methods. In Fig. 9, the estimate of the position coordinate of each target by the RCOMP is shown together with the true target locations. In Fig. 10, the estimate of the position coordinate of each target by the RCOMP is shown together with the true target locations. Similar results are obtained for the CLIP in Figs. 11 and 12. The plots for the TCOMP are similar to those of the RCOMP (and for the IP similar to those of the CLIP) and have thus been omitted. C. Stochastic Constraints In the previous examples, we assumed that the tracker had complete knowledge of the constraints and that this knowledge was correct. However, in realistic scenarios, the constraints may not be completely known. To demonstrate the application of our


Fig. 13. RMSE versus the time that the simulation takes to complete for 81 time steps using 95% confidence intervals.

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Fig. 15. RMSE of the RCOMP versus the grouping state change probability.

Figs. 13 and 14 show that, similarly to the deterministic constraint case, the RCOMP exhibits an improved performance from 0 to 0.9 and plot over the CLIP. In Fig. 15, we vary the RMSE of the RCOMP. As expected, setting the probability of change of grouping state too high brings a deterioration in performance. Keeping the constraints constant, on the other hand, does not capture the change in target grouping, causing the performance to degrade. The faster the constraints are expected to change in a given application, the larger the grouping state change probability should be set. That, however, must be matched by the use of a higher number of particles. VIII. CONCLUSION

Fig. 14. RMSE versus the time steps for the entire motion of the targets.

technique to a scenario with stochastic constraints, we considered the following modified leapfrog scenario. At a certain instant, one of the targets leaves its group and at a later time, joins another group. The parameters of the constraints, namely the grouping of the targets and the distance kept among the targets that are grouped together in the direction, are now unknown. To implement these constraints, we use an auxiliary variable that represents status (grouped or not grouped) for each partition with respect to each of the other partitions. As described in Section II-B, the grouping status is modeled as a Markov chain for all partitions. The variance associated with with . We the change in the direction constraint was set to also constrain the value of this distance to be between 0.5 and 3 m. Since the increase in the number of unknowns requires more particles, we offset the extra computational expense by reducing the number of particle orderings to eight per time step. These eight orderings are randomly selected out of the possible 24 orderings.

In this paper, we propose the constrained motion proposal (COMP) algorithm that incorporates deterministic or stochastic kinematic constraint information into a particle filter. The COMP uses sampling methods and likelihood functions that take into account motion constraint information. We also proposed the constraint likelihood IP (CLIP) method that incorporates motion constraints into the formulation of the IP method through a constraint likelihood function. We demonstrated using Monte Carlo simulations that effectively incorporating constraint motion information improves the RMSE performance of the tracker. The CLIP does not work as well as the COMP because it does not propose particles using the constraints directly in the proposal. The gains in error performance by the COMP come at the expense of algorithmic and computational complexity. Note that both the CLIP and the COMP multitarget tracking algorithms were designed for a fixed and known number of targets in the surveillance area. The rigorous formulation of the multitarget constraint region of the state space under the uncertainty in the number of targets is a topic of further research. REFERENCES [1] L. D. Stone, C. A. Barlow, and T. L. Corwin, Bayesian Multiple Target Tracking. Norwood, MA: Artech House, 1999.

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Ioannis Kyriakides (S’04) received the B.S. degree in electrical engineering from Texas A&M University, College Station, in 2003. He is currently working towards the Ph.D. degree in electrical engineering at Arizona State University, Tempe, where he is also a Research Associate. His research interests include Bayesian target tracking, sequential Monte Carlo methods, radar waveform design, and time-varying signal processing.

Darryl Morrell (S’82–M’83–SM’04) received the Ph.D. degree in electrical engineering from Brigham Young University, Provost, UT, in 1988. He is currently an Associate Professor in the Department of Engineering at Arizona State University, Tempe, at the Polytechnic campus, where he is participating in the design and implementation of a multidisciplinary undergraduate engineering program using innovative, research-based pedagogical and curricular approaches. His research interests include stochastic decision theory applied to sensor scheduling and information fusion. He has received funding from the Army Research Office, the Air Force Office of Scientific Research, and DARPA to investigate different aspects of Bayesian decision theory, with applications to target tracking, target identification, and sensor configuration and scheduling problems in the context of complex sensor systems and sensor networks. His publications include over 70 refereed journal articles, book chapters, and conference papers.

Antonia Papandreou-Suppappola (S’87–M’91– SM’93) received the Ph.D. degree in electrical engineering from the University of Rhode Island, Kingston, in 1995. From 1995 to 1999, she held a research faculty position with Navy funding. In 1999, she joined Arizona State University, Tempe, where she was promoted to Associate Professor in 2004. She is the Editor of a Applications in Time-Frequency Signal Processing (CRC, 2002). Her research interests are in the areas of integrated sensing and processing, waveform design for agile sensing, time-frequency signal processing, and signal processing for wireless communications. Prof. Papandreou-Suppappola was the recipient of the 2002 NSF CAREER Award and 2003 IEEE Phoenix Section Outstanding Faculty for Research award. She is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and she was an Associate Editor for the IEEE SIGNAL PROCESSING LETTERS from 2003 to 2005. She is a Technical Committee Member of the IEEE Signal Processing Society on Signal Processing Theory and Methods. She will be the Special Sessions Chair of the 2010 IEEE International Conference of Acoustics, Speech and Signal Processing that will be held in Dallas, TX. She held the position of Treasurer of the IEEE Signal Processing Society Conference Board from 2004 to 2006.