Set JPDA Algorithm for Tracking Unordered Sets of Targets

12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009

Set JPDA algorithm for tracking unordered sets of targets Lennart Svensson Dept. of Signals and Systems Chalmers Univ. of Technology G¨oteborg, Sweden [email protected]

Daniel Svensson Dept. of Signals and Systems Chalmers Univ. of Technology G¨oteborg, Sweden [email protected]

Abstract – In this article we show that traditional tracking algorithms should be adjusted when the objective is to recursively estimate an unordered (unlabeled) set of target state vectors, i.e., when it is not of importance to try to preserve target identities over time. We study scenarios where the number of targets is known, and propose a new version of the Joint Probabilistic Data Association (JPDA) filter that we call Set JPDA (SJPDA). Simulations show that the new filter outperforms the JPDA in a two-target scenario when evaluated according to the Mean Optimal Subpattern Assignment (MOSPA) measure. Keywords: Tracking, JPDA, Random finite sets, OSPA, target identity, SJPDA

1 Introduction Traditional tracking algorithms are tailored to the problem of tracking a—possibly unknown—number of targets with preserved target identity. That is, the interest is not only in estimating the states of each target, but also to keep track of the identities of the targets over time. Examples of such algorithms are the Probabilistic Data Association (PDA) and Joint PDA (JPDA) filters [1, 2], the Multiple Hypothesis Tracking (MHT) algorithm [3, 4, 5, 6], and particle filters [7, 8]. A recently published performance measure for multiobject tracking is the Optimal Subpattern Assignment (OSPA) measure [9, 10]. The measure provides information both regarding how well an algorithm estimates the number of targets, as well as their respective states. In OSPA, the target identities have no impact, unless an identity label is part of the state vector. This implies that the measure describes how well an algorithm answers the question of where there are targets, rather than where target one, two, and so on are. In many cases the target identity must be preserved, for example if information is to be aggregated over time in order to classify. However, in some situations the provenance is unnecessary, and to eschew it makes a combinatorial task much simpler. An appealing example is that of tracking for sensor cueing. In light of Fig. 3, consider a radar situation of a pair

978-0-9824438-0-4 ©2009 ISIF

Peter Willett ECE Dept. Univ. of Connecticut Storrs, USA [email protected]

of targets within a single radar beam. It is not immediately necessary to identify one from the other; but it is important to probe them via monopulse dwells with boresight at their midpoint while they are together and with separate beams as they become resolved in angle. In this article, we consider the problem of tracking multiple targets with the objective of minimizing the Mean OSPA (MOSPA), when the number of targets is known and constant, and when target identity is not part of the state vector. Two algorithms that are designed to perform well under MOSPA-like measures are the Probability Hypothesis Density (PHD) and Cardinalized PHD (CPHD) filters [11, 12, 13, 14]. The filters are derived using Finite Set Statistics (FISST) [15]. Within FISST, the target states are described by a random finite set (RFS), which is a set with a random number of elements, and where each element is a random entity. Due to the RFS description, there is no labeling of targets—instead, the PHD/CPHD filters describe the intensity of targets over the state space. Traditional tracking algorithms, originally designed to perform well according to the Mean Square Error (MSE) measure, have also been used on the MOSPA problem. The purpose has been to compare the performance of PHD/CPHD and traditional tracking algorithms [16, 17, 18], such as MHT and a conventional single-hypothesis tracker. Although target identity is not considered in the MOSPA measure, the algorithms have remained unaltered. There may be several reasons for this, but one reason might be that the MSE and MOSPA problems have been considered to be more similar than they actually are. As we show in the current article, there is substantial room for improvements of the traditional algorithms when evaluated according to MOSPA. The contributions of the current paper are twofold. The first part of the article concerns how the problem of minimizing MOSPA differs from the classic tracking problem, and we discuss how and why traditional algorithms should be adjusted to this problem. We note that there is a relation between the density of the RFS of targets and the ordinary density of ordered targets, and we make three observations:

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1) When we use MOSPA as the performance measure, there is a number of ordered densities that correspond to the same RFS density, and therefore ideally should render the same estimates.

With Zk being the set of measurements received at time k, and Zk being the sequence of measurement sets

2) In practise, some of these densities are especially easy to work with, for example since they are approximately Gaussian.

up to and including time k, the problem of this article can be formulated as the (approximative) calculation of    p Xk Zk . (3)

3) If we as estimate want to use the expected value of one of the ordered posterior densities, albeit ideally equivalent, the densities will provide different estimates, with different quality with respect to MOSPA. In the second part of the article we present a novel adjustment of the JPDA filter, which we call Set JPDA (SJPDA). Inspired by observations 1) and 2) above, the SJPDA filter switches between ordered densities that correspond to the same RFS density. The switch is designed such that the ordered density is better described by a Gaussian density, to improve upon the approximations of JPDA. The output of the SJPDA filter is the posterior expected value of the (possibly changed) ordered density. We will argue that also the estimates are improved after the switch of densities, which is related to observation 3) above. The SJPDA filter is evaluated on a dense two-target scenario. The simulations indicate that the filter not only improves the approximations of the JPDA filter, but that it also provides better estimates when evaluated according to MOSPA. We notice, in particular, that the proposed filter avoids the track coalescence problem of JPDA, and that it therefore is much faster in detecting the separation of targets.

Zk = {Z1 , Z2 , . . . , Zk }

The propagation  of target  states is described by the process model p xk xk−1 , and the likelihood of a target state xk , given a measurement    vector z k , is given by the measurement model p zk xk .

2.1 Performance measure

In order to assess the performance of a multi-target tracking algorithm, a multi-object measure of performance is required. In this article we use the average OSPA measure— called MOSPA—which for a certain time instant and outcome uses an optimal assignment technique to find the best ˆ ik and true target states xjk , combination of state estimates x where i, j ∈ {1, . . . , n} (and where i is not necessarily equal to j). The measure is constructed to capture the quality of an algorithm both regarding its estimate of the number of targets and the estimates of their respective state vectors. Following [10], we describe the OSPA for a known and constant number of targets. Let X k be the set of true target ˆ k be the set of target estimates at time k, both states and X (c) with cardinality n. The OSPA d¯p is then defined as ˆ d¯(c) p (Xk , Xk ) =

2 Problem formulation In this article we study the problem of tracking a known number n of targets. Further, we are not interested in the identity of the targets, and, as we shall see, this has a large impact on the problem solution. In all other aspects, we consider a standard tracking problem with the possibility to have clutter, a probability of detection P d less than one, and nonlinear process and measurement models. A key aspect of the problem formulation is how to describe the tracking problem without target identity. We use an RFS description of the unordered set of targets, in which the number of targets is a random variable which is associated with a probability mass function called the cardinality distribution. Moreover, each element in the set is a random vector with a probability density function. Let Xk be the RFS of targets at time k, with known cardinality n. The RFS is thus described as   Xk = x1k , x2k , . . . , xnk , (1)

where xik is the state vector of target number i at time k, which we assume does not include a target identity label. Since an RFS is without ordering, the target numbers can be reordered without affecting the RFS, which means that target identity is not preserved over time with this description.

(2)



1 n



min

π∈Πn

n  i=1

(c)

d

π(i) (ˆ xik , xk )p

1/p

.

(4)

Here, d(c) (ˆ xk , xk )  min(c, d(ˆ xk , xk )) is the distance d ˆ k and xk , cut-off at c. Further, Π n is the set of all between x possible permutations of the set of true target states. In this article, we let d be the squared Euclidean distance. Further, we use p equal to 1 and c equal to 1m 2 . For a fixed number of targets, the OSPA measure is similar to the performance measure proposed by Drummond [19].

3 Random finite sets and labeled densities In this section, we discuss the relations between an ordered density and the density of an RFS. We also discuss how and why traditional tracking algorithms could be adjusted to perform better under the MOSPA performance measure. Based on the general findings in this section, we propose an improved version of the JPDA filter in Section 4. The information needed to produce optimal state estimates, in the MOSPA sense, is summarized in the posterior density of the RFS,    (5) p x1k , x2k , . . . , xnk Zk .

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The relation between an RFS density p({x 1k , . . . , xnk }) and a density of ordered state vectors, p(x 1k , . . . , xnk ), is p({x1k , . . . , xnk } = {α1 , . . . , αn }) n!

=

1  p(x1k = αmi1 , . . . , xnk = αmin ), n! i=1

(6)

where mij , for j = 1, . . . , n, is index j in permutation number i. For instance, when n = 2 a natural choice is to set m11 = 1, m12 = 2 and m21 = 2 and m22 = 1. Using (6), we can calculate the RFS density from an ordered density. One important consequence of this relation is described in the following proposition.

the same RFS family1 . When these densities are updated using the same set of measurements, Z k , the updated densities still belong to the same RFS family. Proof of proposition 2 The key to this result is the fact that given the state vector, the likelihood is the same for all terms in (6), i.e., for all permutations of the target positions. Intuitively, this means that once the information about the target identities is lost, it can not be recovered from future data. Consequently, after the Bayesian update we get p1 ({x1k , . . . , xnk } = {α1 , . . . , αn }Zk )

 1 n! n 1  p(Zk xk = αmi1 , . . . , xk = αmin ) = n! i=1 p(Zk ) 

Proposition 1 For n > 1, the mapping from densities of ordered state vectors, pi (x1k , . . . , xnk ), to RFS densities, p({x1k , . . . , xnk }), is many-to-one.

× p1 (x1k = αmi1 , . . . , xnk = αmin )

 p(Zk x1k = α1 , . . . , xnk = αn ) = p(Zk )

The relevance of this proposition is related to the fact that all such densities should result in the same RFS state estimates.

n!

1  p1 (x1k = αmi1 , . . . , xnk = αmin ) n! i=1  = p2 ({x1k , . . . , xnk } = {α1 , . . . , αn }Zk ),

Definition 1 When two labeled densities, p 1 (x1k , . . . , xnk ) and p2 (x1k , . . . , xnk ), correspond to the same RFS density we say that they belong to the same RFS family. That is, using (6), we obtain the same RFS density regardless if p1 (x1k , . . . , xnk ) or p2 (x1k , . . . , xnk ) is used. Due to properties of RFS families, traditional algorithms (like JPDA and MHT) can be improved in two different ways. Both improvements rely on changing from one density in an RFS family to another one. The first improvement is to make algorithmic approximations more accurate. In many algorithms, a Gaussian approximation is performed on a multimodal density. When a density change can make the density less multimodal and more similar to a Gaussian density, the approximation will be improved. Second, traditional algorithms calculate Minimum MSE (MMSE) estimates, i.e., posterior means, from an ordered density. By switching between densities, within an RFS family, we can obtain MMSE estimates which are closer to the optimal MOSPA estimates. In the following section, we further discuss the properties of RFS families. The two subsequent sections then give examples that illustrate the advantages of switching between densities of an RFS family.

3.1 Choice of labeled density—a design issue All densities that belong to the same RFS family should result in the same state estimates. Some of these densities may be difficult to handle and some convenient, for instance because they can be accurately approximated by a Gaussian density. In situations where we have a labeled density at hand which is troublesome to process, it may seem tempting to replace it with a different one within the same RFS family. But can we really do this? Due to the following proposition, the answer is yes! and Proposition 2 Suppose p 1 (x1k , x2k , . . . , xnk ) 1 2 n p2 (xk , xk , . . . , xk ) are two labeled densities within

(7)

×

(8) (9)

where the last equality is due to p 1 (x1k , x2k , . . . , xnk ) and p2 (x1k , x2k , . . . , xnk ) being part of the same RFS family. As we can see, all labeled densities that correspond to the n! 1 1 n same RFS density, n! i=1 p(xk = αmi1 , . . . , xk = αmin ) before measurement update, still belong to the same (only  updated) RFS density, p({x 1k = α1 , . . . , xnk = αn }Zk ), after the measurement update. To phrase this differently, all densities that belong to the same RFS family should result in the same estimates, both now and for all future times, i.e., also when new data is available. We can therefore replace our density at hand with any other density within the RFS family, without influencing the estimates. This assumes, of course, that the estimates are computed using optimal algorithms. In practise, however, the algorithms are not optimal and the choice of density therefore matters. As already mentioned, there are at least two reasons why a clever switch of densities can improve performance. In Sections 3.2 and 3.3, we discuss these possibilities in more detail and illustrate them using two examples.

3.2 Improved density approximations In this section, we discuss how traditional algorithms that use merging approximations can be improved when evaluated using MOSPA. We focus the discussion around a simple example that illustrates how the selection of densities of ordered sets of targets can affect the accuracy of a Gaussian approximation. 1 We

also assume that all targets have identical properties conditioned on the state vector, i.e., that the measurement equations are the same, independently of the target number.

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p(x1k , x2k )

Example 1: Consider a scenario where we have two Gaussian distributed targets; the probability of detection is one for both targets (P d = 1), and we have received two detections. We represent the two possible data associations by the hypotheses H1 and H2 . Now, suppose that calculations yield the numbers 2,

x1k |H1 x2k |H1

2

(10)

∼ N (3, 1)

(12) (13)

x1k |H2 ∼ N (−0.2, 1)

(14)

x2k |H2

(15)

∼ N (2.7, 1). β1 , x2k

1

(11)

∼ N (−0.5, 1) Pr{H2 } = 0.7

{x1k

3

xk2

Pr{H1 } = 0.3

4

0

-1

-2 -2

x1k

2

3

4

2

3

4

3

x2k

2

1

x1k |H1 ∼ N (3, 1)

(17)

0

x2k |H1

∼ N (−0.5, 1) Pr{H2 } = 0.7

(18) (19)

−1

x1k |H2 ∼ N (2.7, 1)

(20)

−2 −2

x2k |H2

(21)

In other words, both the density p 1 (x1 , x2 ), described by eq. (10)–(15); and p 2 (x1 , x2 ), described by eq. (16)–(21); belong to the same RFS family and should, ideally, render the same estimates. As illustrated in Fig. 1, the suggested switch leads to a simpler problem, since the marginalized densities of x 1k and x2k can be approximated by a Gaussian density more accurately. The dash-dotted line in Fig. 1 is a symmetry line. When probability mass is moved from one labeled point to another, the movement is through this line to the mirror point on the other side. Of course, the example is selected to highlight the advantages with a switch, and one can easily construct situations when it is better not to switch indices. Still, the example illustrates a general technique that can be employed by most tracking algorithms that use merging. To further improve the understanding of the concepts in Example 1, we stress the relation to the RFS densities. Let p˜1 (x1 , x2 ) and p˜2 (x1 , x2 ) denote the Gaussian approximations of p1 (x1 , x2 ) and p2 (x1 , x2 ), respectively. Both p1 (x1 , x2 ) and p2 (x1 , x2 ) correspond to the same RFS density, i.e., p1 ({x1 , x2 }) = p2 ({x1 , x2 }). As the approximation p˜2 (x1 , x2 ) ≈ p2 (x1 , x2 ) is fairly accurate, it follows that p˜2 ({x1 , x2 }) ≈ p2 ({x1 , x2 }) = p1 ({x1 , x2 }). It is likely that the approximation p˜1 ({x1 , x2 }) ≈ p1 ({x1 , x2 }) is less accurate. Hence, by switching densities we will make

1

4

(16)

∼ N (−0.2, 1).

0

p(x1k , x2k )

{x1k

We realize that the two sets = = β2 } and = β2 , x2k = β1 } represent the same set of targets. We may therefore move density from one such labeled point to the other, without changing the RFS density. One way to do this is by switching the indexes under H 2 , Pr{H1 } = 0.3

-1

−1

0

1

x1k

Figure 1: Marginalized posterior densities before (above) and after (below) switching the indexes under H 2 . In our opinion, a Gaussian approximation is much more appropriate in the bottom figure. The symmetry line is dash-dotted. approximations that better preserve the information about the desired RFS density, p 1 ({x1 , x2 }).

3.3 Enhanced RFS state estimation It is not obvious how to compute MOSPA estimates from a given RFS density. Instead, we propose to use the MMSE estimates, i.e., the posterior means, of a density of ordered targets. We intend, however, to select a density within the RFS family such that the MMSE estimates are close to the optimal MOSPA estimates. In the following examples, we illustrate the importance that the choice of density has on the posterior means and the MOSPA performance. Example 2: Consider two scalar state variables, distributed according to x1k = 1, x2k = −1

with probability θ

(22)

x1k

with probability 1 − θ.

(23)

= −1,

x2k

=1

Each value of θ (0 ≤ θ ≤ 1) represents a probability mass function. All these mass functions correspond to the RFS

2 All probabilities and densities are conditioned on data, but this is omitted for notational convenience.

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X = {1, −1},

(24)

and all such distributions therefore belong to the same RFS family. One can argue that the situations θ = 0 and θ = 1 are easier to treat, and if our aim is to estimate x 1k and x2k well, according to the MOSPA measure, we can select θ ourselves (independently of the actual value). If we select θ = 1 (θ = 0) the posterior means correspond to x 1k = 1 and x2k = −1 (x1k = −1 and x2k = 1), which is the optimal RFS estimate. For general values of θ we get the posterior ˆ2k = 1 − 2θ, which may be far means, x ˆ1k = 2θ − 1 and x from correct, especially for θ ≈ 1/2. Example 1, revisited: We return to Example 1, and study the MMSE state vector estimates before and after the index switch. In the original indexation, the posterior means are x ˆ1k = 0.76,

x ˆ2k = 1.74

(25)

whereas the posterior means after the switch are x ˆ1k = 2.79,

x ˆ2k = −0.29.

(26)

The latter posterior means are probably close to the optimal estimates since, under both hypotheses, one target is fairly close to 2.79 whereas the other target is reasonably close to −0.29. Hence, although the initial objective was to improve the Gaussian approximation, we seem to also obtain MMSE estimates which are closer to the MOSPA-optimal estimates. For the general n-target problem, it appears that the posterior means of x ik , i = 1, . . . , n are better estimates, in the MOSPA sense, when their individual covariances are smaller (studying Fig. 1 reveals that the variances of x 1k and x2k are smaller after the switch). We claim, thereby, that if we reorder the indices such that the density is more similar to a Gaussian density, we will also obtain better estimates.

4 Set Joint Probabilistic Data Association (SJPDA) algorithm In Section 3, we discussed the differences between the MMSE and MOSPA problems in general, and noted that there is room for improvements of traditional algorithms, such as JPDA, MHT and particle filters. In this section, we propose an adjusted JPDA algorithm for the MOSPA problem (assuming a known number of targets). We call this algorithm Set JPDA (SJPDA). The algorithm works as follows: 0) Approximate the predicted densities of all targets using Gaussian densities. 1) Formulate global measurement hypotheses, H i , and calculate conditional densities of all targets as well as the probabilities of all hypotheses. 2) Reorder the target indexes under the different hypotheses with the objective to make the marginalized densities resemble Gaussian densities (by minimizing the covariances, see Section 4.1 for details). 3) Approximate the marginalized posterior densities of all targets using Gaussian densities. Then go back to 0.

Like PHD and CPHD, this algorithm is not specifically designed to maintain the track identity, but instead it is likely to perform better (sometimes even much better) than JPDA when evaluated using the MOSPA performance measure. We can also speculate that this improved performance also translates to improved tracking using traditional metrics such as track-life. An important part of the algorithm is the reordering of target indexes under the global hypotheses. In the following, we propose a procedure for label switching, which is used in the evaluation of the SJPDA algorithm. When there is no label switching, the SJPDA algorithm calculates identical estimates as JPDA. The label switching procedure is derived with the objective to produce better Gaussian approximations, just as in Example 1 in Section 3. In agreement with the revised version of that example, the reordering of target labels also leads to better state estimates, i.e., MMSE estimates which are closer to the MOSPA-optimal estimates.

4.1 Finding the optimal indexation The labeled density is a Gaussian mixture that the JPDA approximates with a single Gaussian density. All targets are approximated as independent with mean and covariance, ˆ tk = x

#H 

ˆ t,h βh x k

(27)

h=1

Ptk =

#H 

h=1

 T   t,h t,h t t ˆ ˆ ˆ ˆ x βh Pt,h x − x − x + , (28) k k k k k

where t = 1, . . . , n is the target number, #H is the number of global data association hypotheses, and β h is the probat,h ˆ t,h bility of hypothesis h. Further, x k and Pk are the posterior mean and covariance matrix of target number t, under global hypothesis h. With the above description of the posterior mean and covariance matrix of each target, the JPDA and the SJPDA algorithms perform identical calculations, except for the label switching in SJPDA. However, the JPDA algorithm is often described in an alternative, but equivalent, fashion [5], where the computation of the state estimates includes the calculation of a weighted measurement residual that is used in an ordinary Kalman filter update. Generally speaking, a Gaussian mixture can be accurately approximated by a single Gaussian density as long as the Gaussian mixture is not too distinctly multimodal. We wish to adjust the indexation within each global hypothesis, in order to find a labeled density which is less multimodal. As a measure of the multimodality we use, n 

tr{Ptk }

(29)

t=1

where Ptk is given in (28). Hence, our objective is to find an

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˜ tk0 = ˆ tk1,h0 ) and x ˆ tk1 + βh0 (ˆ ˜ tk1 = x xtk0,h0 − x By utilizing that x ˆ tk0 + βh0 (ˆ ˆ tk0,h0 ) the test simplifies to x xtk1,h0 − x

indexation that minimizes

#H n   T    t,h t,h t t ˆ ˆ ˆ ˆ x βh tr Pt,h + x − x − x k k k k k

 ˜ = 2βh0 (ˆ ˆ tk0,h0 ) 0