Suboptimal Target Tracking in Clutter Using a Generalized Probabilistic Data Association Algorithm

Simple tracking algorithms based upon nearest neighbor filtering do not correctly consider measurement origin uncertainty and, therefore, fail to perform well in situations of high target density and clutter. The optimal tracking algorithm for commonly used targetclutter models computes the posterior density of the target state conditioned on the past history of' observations. This posterior density is a Gaussian mixture with the number of terms equal to the number of possible ways to associate observations and targets. Though a recursive algorithm may be developed for the optimal estimator, it requires exponentially growing rlemory and computation and is, therefore, unimplementable. In this paper a new suboptimal algorithm is proposed where approximation is done by naturally partitioning and grouping the target state estimates into a set of approximate sufficient statistics. A new critel-ion fu~lc t io~l is iiltroduced in this approximation process. The well-known Probabilistic Data Association filter (PDAF) turns out to be a special case of the new algorithm. Comparisons are made for the proposed estimator versus the PDAF. Target tracking is an old problem with origins going back as far as eighteenth century astronom,ers who first attempted to determine the orbits of the visible planets. More modern work can trace its ancestry to the early 1960s where the problem was driven by applicaticms in ballistic missile defense, orbital vehicle tracking, and air traffic control. Certain applications of target tracking have become relatively more important in the last few years. These include air traffic control [I] fueled by large growth in civilian aviation and the resulting traffic congestion near major airports, and highway veh.icle surveillance [2], motivated by current interest in intelligent transportation systems. A major issue in the design of target tracking systems is the uncertainty associated with the origin of measurements. Such uncertainty arises due to the presence of clutter, receiver related false alarms, and other nearby targets. In many situations the measurement origin uncertainty is a far more important impairment to tracking performance than is the noise associated with the measurements themselves. The use of standard trajectory estimation algorithms where the measurement nearest (in some metric) .to the predicted measurement is chosen to update a track can lead to very poor perfori-nance when the density o-E spurious measurements is high. Such an algorithm (the nearest. neighbor filter) does not properly account for the fact that the measurement chosen for track update may be unrelated to the target. The recent introduction of advanced sensors which capture new types of target information (target signatures, images, etc.) in addition to position and velocity have made advances in tracking algorithms possible. At the same time these new data collection possibilities complicate implementation by adding to the huge amount of data that must be processed. Processing power will therefore continue to be a bottleneck in the implementation of sophisticated tracking algorithms. Some of the earliest tracking research done in the modern spirit can be traced to the 1964 paper by Sittler [3]. In order to account for measurement origin uncertainty, he proposed splitting the track whenever more than one observation was mad,e in the vicinity of a predicted measurement. A likelihood function for each trajectory was computed and those falling below a threshold were dropped in order to avoid an exponential growth in complexity. Sittler's work was done before the I 0 is the number of validated measurements) is produced by the sensor at time k. If the target measurement is detected, then it is one of the measurements yk,i and any others are due to clutter. Note that the ordering is assumed to be random. Let T/i, be the volume of G k and let nk be the number of validated clutter measureineilts in scan 1: (0 5 7zk 5 mk). Using the spatial Poisson model for clutter generation, it follows that the conditional distribution of nk given Ifk is Poisson with parameter XcVk. Furthermore, if (assuming nk > 0) denotes the vector of clutter measurements validated in scan k, then the probability density of c k given nk and G k is specified by noting that the ck.i are independently and uniformly distributed over the gate Gk. At a :particular scan time k, the target related measurement z k will be present in the observation set only if the target is actually detected. To model the detection process let { d k : k I? 0) be an i.i.d. sequence taking values 0 and 1 with the interpretation that the target is detected at time k if and only if dk = 1. Define the detection probability to be Po = pd(l). We include the possibility that z k will fall outside of the gate in the detection probability. Of course, the validation gates are designed to contain the target related rr~easurement with high probability. The clbservation process is of varying dimension depending upon the detection of the target and the number of clutter measurements. The observation at scan time k has the form ( m k , y k ) where mk is a non-negative integer indicating the number of measurements and y k is a real vector of dimension mkn,. This notation is used to emphasize the fact the observation contains the number of measurements information. To co-mpletely specify the measurements in an observation scan we need to know if the scan contains a target related measurement, and, if so, where that measurement is located in the observa.tion set. To keep track of the actual measurement situation. we hypothesize a data as.sociation random process { rk : k 2 0). Define rn. as a random function of the random va.riables dk and nk in the sense that The first component jk indicates the position of the target related measurement in the vector y k ; it may take values between 0 and mk = nk + d k . If dk = 0, then j k = 0 and (for If dk = 1: then jk takes a value between 1 and mk = nk + 1 and in otherwords, zn. appea.rs a.s the element y k , j k in the vector y k of dimension nzkn., = (nk + dk)n, . If nk = 0, then yn. consists of the single measurement z k when dk = 1, or is empty when dk = 0. The ~narginal distributions of the jk will be specified in such a way that they are identically distributed, and, in addition, we will assume that they are independent. The distribution of jk is specified by conditioning on dk and nk. If no target detection is made p ( j k = O ( l & = 0 , n k ) = 1 regardless of the value of nk while if the target is detected p( jk Idk = 1 , nk) = l / ( n k + 1 ) for 1 5 j k 5 nk + 1 otherwise As specified above the data association process 7rk is iid. Each must take values of the form 7rk == ( j k , mk) where mk > 0 and 0 5 jk 5 mk. The unconditional distribution of 7rk is found from the work given above as 2.4 The Optimal Single Target Tracking Algorithm For tracking a single target in clutter it is natural to choose the mean-squa.red error ( M S E ) as the criterion for optimality. In this section we summarize the M M S E estimator and illustrate a recursive algorithm for its computation. The goal is the conlputation of the estimate of the state a t time k given the sequence rk = { Y ~ , . . . , ~ k } of observations up to time k:. The algorithm which results is not practical because it requires exponentially growing memory and computation. By conditioning on the past data association sequence we can write E { x k I r k ) = E { x k l r k , n k } p ( n k I r k ) n k where the sum is over all lIk compatible with rk . Recall that rk contains the information on past numbers of validacted measureillents Mk = { m o , . . . , m k } . Given this we may just as well index the data association nk using Jk = { j o , . . . , j k ) the sequence of locations of the target related measurements. The optimal algorithm [ l l , 1 4 1 computes the estimate in Equation ( 4 ) by setting up recursive equations based upon the Kalman filter for the computation of the terms E { x k l r k , n k } and p(nklrk). At the end of the processing of the measurements in observation scan k, there is a term in the sum (4) for every possible data a~soci~ation hypothesis Ilk given Mk. In all, there are flf",o(nzl + 1 ) terms. 3 Hypothesis Clustering for Tracking 3.1 Background on Hypothesis Clustering A variety. of techniques have been proposed in the literature for dealing with the central problem in target tracking in clutter: the exponentially growing memory and computational requirements of the optimal algorithm [ 7 ] . All such methods involve a combination of validation gates, pruning of extremely unlikely hypotheses, and the cornbination of hypotheses with similar trajectory estimates. These operations may be viewed as a type of clustering algorithm [15] applied to the individual trajectory estimates. Though such notions are central to a tracking algorithm and its performance, the deta~ils of individual algorithms are often not precisely laid out in the literature. Since hypotheses may be viewed as branches on a tree, hypothesis reduction techniques are often viewed a.s either branch pruning or branch combination. Many i~lgorithms use a combination of both techniques. A typical approach [14] uses a threshold on the probability of each data association hypothesis, only those with sufficiently large probability are retained. The same pruning technique has been proposed for the N-scan filters. Pruning with a fixed threshold is not enough by itself to eliminate all complexity problems. Neither is pruning enough to ensure good performance. Hypotheses must also be combined either using the method of Singer [ll] to coillbiile data association hypotheses having the last N scans in common or by combining hypotheses which have "similar effects" as in Reid's paper [14]. The former is really a method ba.sed on clustering. This is the approach taken in this pa.per. 3.2 A Tracking Based Clustering Criterion The structure of computation for the proposed tracking algorithm (one ta.rget case) is shown in Figure 1. In the same spirit as was done for the PDA filter [12], we derive the new algorithm based upon the assumption that the conditional density of the state xk given the observations l?k-I up to scan time k 1 is an L component Gaussian mixture. In this paper the number of terms L in the mixture is considered to be fixed from scan to scan. Varying i; allows a tradeoff between performance and complexity; the case L = 1 actually corresponds to the PDA filter. With the notation n/(xl/r? P) to denote the multivariate Gaussian density with inean p and covariance P, the assumption on the posterior density of the state xk can be written Let the statistics parameterizing the terms in the mixture above be denoted by for 1 I i I L. Then the algorithm shown in Figure 1 processes the input statistics {Uklk-l(i))f=l and the scan k observations {yk,,);"=: to produce: 1. The filtered state estimate iklk and the error covariance Cklk . 2. The statistics needed to propagate the solution to the next scan {UkS.l lk(i))~=l , where u k + , l l k ( i ) = (?k+llk(i), xk+l,k(i), ~ k + l ( i ) ) The details concerning each block in Figure 1 are presented in the fol1owi:ng sections. 3.3 H[ypothesis Tree Construction Given the validity of the assumption in Equation (5), the one step prediction density f(xklrk-.[) is updated by the processing of the scan k observations {yk,,);"=*l to give the posterior density f ( x k ) r k ) as an L(mk + 1) component Gaussian mixture. The details, which art: by now quite standard [ l l , 14, 71, are illustrated by the diagram of Figure 2. Each component i in the mixture (5) is the root of a tree formed from the scan k observati'm ~k and the possible data association hypotheses. These are indexed by 0 < j < mk where j = 0 corresponds to the hypothesis that the target was not detected in scan k and j > 0 corresponds to the hypothesis that the target related measurernent is yk,j. The Kalman lilter is applied yielding L(mk + 1) trajectory estimates f i k ~ k ( ~ , j ) = ?klk-l(i) + G(i, j)(yk,j H?klk-l ( i ) ) Sklk = xklk-l(i) G(i, j )Hxk(kl ( i ) where 1 :i i 5 L, 0 < j 5 mk, and the gain term is given by A probability a k ( i , j) is associated with each branch in the tree. These are calculated from the measurement model and the data association hypothesis by defining