Research Article  Open Access
Feng Lian, Chongzhao Han, Jing Liu, Hui Chen, "Convergence Results for the Gaussian Mixture Implementation of the ExtendedTarget PHD Filter and Its Extended Kalman Filtering Approximation", Journal of Applied Mathematics, vol. 2012, Article ID 141727, 20 pages, 2012. https://doi.org/10.1155/2012/141727
Convergence Results for the Gaussian Mixture Implementation of the ExtendedTarget PHD Filter and Its Extended Kalman Filtering Approximation
Abstract
The convergence of the Gaussian mixture extendedtarget probability hypothesis density (GMEPHD) filter and its extended Kalman (EK) filtering approximation in mildly nonlinear condition, namely, the EKGMEPHD filter, is studied here. This paper proves that both the GMEPHD filter and the EKGMEPHD filter converge uniformly to the true EPHD filter. The significance of this paper is in theory to present the convergence results of the GMEPHD and EKGMEPHD filters and the conditions under which the two filters satisfy uniform convergence.
1. Introduction
The problem of extendedtarget tracking (ETT) [1, 2] arises because of the sensor resolution capacities [3], the high density of targets, the sensortotarget geometry, and so forth. For targets in near field of a highresolution sensor, the sensor is able to receive more than one measurement (observation, or detection) at each time from different corner reflectors of a single target. In this case, the target is no longer known as a point object, which at most causes one detection at each time. It is called extended target. ETT is very valuable for many real applications [4, 5], such as ground or littoral surveillance, robotics, and autonomous weapons.
The ETT problem has attracted great interest in recent years. Some approaches [6, 7] have been proposed for tracking a known and fixed number of the extended targets without clutter. Nevertheless, for the problem of tracking an unknown and varying number of the extended targets in clutter, most of the associationbased approaches [8], such as nearest neighbor, joint probabilistic data association, and multiple hypothesis tracking, would no longer be applicable straightforwardly owing to their underlying assumption of point objects.
Recently, the randomfiniteset (RFS) based tracking approaches [9] have attracted extensive attention because of their lots of merits. The probability hypothesis density (PHD) [10] filter, developed by Mahler for tracking multiple point targets in clutter, has been shown to be computationally tractable alternative to full multitarget filter in the RFS framework. The sequential Monte Carlo (SMC) implementation for the PHD filter was devised by Vo et al. [11]. Then, Vo and Ma [12] devised the Gaussian mixture (GM) implementation for the PHD filter under the linear, Gaussian assumption on target dynamics, birth process, and sensor model. Actually the original intention of the PHD filter devised by Clark and Godsill is to address nonconventional tracking problems, that is, tracking in high target density, tracking closely spaced targets, and detecting targets of interest in a dense clutter background [13]. So it is especially suitable for the ETT problem.
Given the Poisson likelihood model for the extended target [14], Mahler developed the theoretically rigorous PHD filter for the ETT problem in 2009 [15]. Under the linear, Gaussian assumption, the GM implementation for the extendedtarget PHD (EPHD) filter was proposed by Granström et al. [16], in 2010. Similar to the pointtarget GMPHD filter, the GMEPHD filter can also be extended to accommodate mildly nonlinear target dynamics using the extended Kalman (EK) filtering [17] approximation. The extension is called EKGMEPHD filter. Experimental results showed the EKGMEPHD filter was capable of tracking multiple humans, each of which gave rise to, on average, 10 measurements at each scan and was therefore treated as an extended target, using a SICK LMS laser range sensor [16].
Although the GMEPHD and EKGMEPHD filters have been successfully used for many realworld problems, there have been no results showing the convergence for the two filters. The convergence results on pointtarget particlePHD and GMPHD filters [18, 19] do not apply directly for the GMEPHD and EKGMEPHD filters because of the significant difference between the measurement update steps of the PHD and EPHD filters. Therefore, to ensure the more successful and extensive applications of the EPHD filter to “reallife” problems, it is necessary to answer the following question: do the GMEPHD and EKGMEPHD filters converge asymptotically toward the optimal filter and in what sense?
The answer can actually be derived from Propositions 3.2 and 3.3 in this paper. Proposition 3.2 demonstrates the uniform convergence [20–22] of the errors for the measurement update step of the GMEPHD filter. In other words, given simple sufficient conditions, the approximation error of the measurementupdated EPHD by a sum of Gaussians is proved to converge to zero as the number of Gaussians in the mixture tends to infinity. In addition, the uniform convergence results for the measurement update step of the EKGMEPHD filter are derived from Proposition 3.3.
2. EPHD and GMEPHD Filters
At time , let be the state vector of a single extended target, and a single measurement vector received by sensor. Multiple extendedtarget states and sensor measurements can, respectively, be represented as finite sets and , where and denote the number of the extended targets and sensor measurements, respectively. A Poisson model is used to describe the likelihood function for the extended target according to Gilholm et al. [14]: where denotes the singlemeasurement singletarget likelihood density; denotes the expected number of measurements arising from an extended target.
The clutter is modeled as a Poisson RFS with the intensity , where is the average clutter number per scan and is the density of clutter spatial distribution.
Given the Poisson likelihood model for the extended targets, Mahler derived the EPHD filter using finiteset statistics [15, 23]. The prediction equations of the EPHD filter are identical to those of the pointtarget PHD filter [10]. The EPHD measurement update equations are where denotes the probability that the set of observations from the extended target will be detected at time ; denotes that partitions set [15], for example, let , then the partitions of are , , , , and ; denotes the cardinality of the set ; if , and otherwise; the notation is the usual inner product. The measure in of (2.4) is continuous, it defines the integral inner product
By making the same six assumptions that are made in [12] and the additional assumption that can be approximated as functions of the mean of the individual Gaussian components, Granström et al. proposed the GMEPHD filter [16]. At time , let denote the GM approximation to the predicted EPHD with Gaussian components, and the GM approximation to the measurementupdated EPHD with Gaussian components. The prediction equations of the GMEPHD filter are identical to those of pointtarget GMPHD filter [12]. The GMEPHD measurement update equations are as follows.
Let the predicted EPHD be a GM of the form where denotes the density of Gaussian distribution with the mean and covariance .
Then, the measurementupdated EPHD is a GM given by where denotes the Gaussian components handling no detections, and denotes the Gaussian components handling detected targets where denotes the identity matrix; has been assumed to be state independent; and denote the observation matrix and the observation noise covariance, respectively; denotes block diagonal matrix, the measure in of (2.11) is discrete, and it defines the summation inner product
3. Convergence of the GMEPHD and EKGMEPHD Filters
The convergence properties and corresponding proof of the initialization step, prediction step, and pruning and merging step for the GMEPHD filter are identical to those for pointtarget GMPHD filter [19]. The main difficulty and greatest challenge is to prove the convergence for the measurement update step of the filter.
In order to derive the convergence results of the measurement update step for the GMEPHD filter, the following lemma is first presented.
Consider the following assumptions.B1: After the prediction step at time , converges uniformly to . In other words, for any given and any bounded measurable function , where is the set of bounded Borel measurable functions on , there is a positive integer such that for , where denotes norm. , denotes the supremum.B2: The clutter intensity is known a priori.B3: , where denotes the set of the continuous bounded functions on .
Lemma 3.1. Given a partition and suppose that assumptions B1–B3 hold, then
The proof of Lemma 3.1 can be found in Appendix A.
The uniform convergence of the measurementupdated GMEPHD is now established by Proposition 3.2.
Proposition 3.2. After the measurement update step of the GMEPHD filter, there exists a real number , dependent on the number of measurements such that where is defined by (B.10).
The proof of the Proposition 3.2 can be found in Appendix B.
Proposition 3.2 shows that the error for the GMEPHD corrector converges uniformly to the true EPHD corrector at each stage of the algorithm and the corresponding error bound is also provided. The error tends to zero as the number of Gaussians in the mixture tends to infinity. However, from (B.10), it can be seen that the error bound for the GMEPHD corrector depends on the number of all partitions of the measurement set. It is quickly realized that as the size of the measurement set increases, the number of possible partitions grows very large. Therefore, the number of Gaussians in the mixture to ensure the asymptotic convergence of the error to a given bound would grow very quickly with the increase of the measurement number.
Now turn to the convergence for the EKGMEPHD filter, which is the nonlinear extension of the GMEPHD filter. Due to the nonlinearity of the extendedtarget state and observation processes, the EPHD can no longer be represented as a GM. However, the EKGMEPHD filter can be adapted to accommodate models with mild nonlinearities. The convergence property and corresponding proof of the prediction step for the EKGMEPHD filter are identical to those for pointtarget EKGMPHD filter [19]. We now establish the conditions for uniform convergence of the measurement update step for the EKGMEPHD filter.
Proposition 3.3. Suppose that the predicted EKEPHD is given by the sum of Gaussians and the in (2.1) is given by the nonlinear singlemeasurement singletarget equation , where is known nonlinear functions and is zeromean Gaussian measurement noise with covariance , then the measurementupdated EKEPHD approaches the Gaussian sum uniformly in and as for , and where
The proof of Proposition 3.3 can be found in Appendix C.
From Propositions 3.2 and 3.3, we can obtain that the EKGMEPHD corrector uniformly converges to the true EPHD corrector in and under the assumptions that for and the number of Gaussians in the mixture tends to infinity. These assumptions may be too restrictive or be unrealistic for practical problems, although the EKGMEPHD filter have demonstrated its potential for realworld applications. However, Propositions 3.2 and 3.3 give further theoretical justification for the use of the GMEPHD and EKGMEPHD filters in ETT problem.
4. Simulations
Here we briefly describe the application of the convergence results for the GMEPHD and EKGMEPHD filters to the linear and nonlinear ETT examples.
Example 4.1 (GMEPHD filter to linear ETT problem). Consider a twodimensional scenario with an unknown and time varying number of the extended targets observed over the region (in m) for a period of time steps. The sampling interval is s. At time , the actual number of the existing extended targets is and the state of the th target is (). Assume that the process noise of the th extended target is independent and identically distributed (IID) zeromean Gaussian white noise with the covariance matrix . Then the Markovian transition density of could be modeled as
where is discretetime evolution matrix. Here and are given by the constant acceleration model [24], as
where “” denotes the Kronecker product. The parameter is the instantaneous standard deviation of the acceleration, given by .
Note that the objective of this paper is to focus on the convergence analysis for the GMEPHD and EKGMEPHD filters, rather than the simulation of the extendedtarget motions. Therefore, although the proposed evolutions for the extended targets seem to be uncritical and oversimplifying, they will have little effect on the intention of the paper. Readers could be referred to [25] for further discussion on the extendedtarget motion models. The models proposed in [25] can also be accommodated within the EPHD filter straightforwardly.
At time , the xcoordinate and ycoordinate measurements of the extended targets are generated by a sensor located at . The measurement noise is IID zeromean Gaussian white noise with covariance matrix , where denotes the diagonal matrix, and are, respectively, standard deviations of the xcoordinate and ycoordinate measurements. In this simulation, they are given as m. The singlemeasurement singletarget likelihood density is
where
The detection probability of the sensor is .
In this simulation, it is assumed that the effect of the shape for each extended target is much smaller than that of the measurement noise. Hence, the shape estimation is not considered here.
At time , the number of the measurements arising from the th extended target satisfies Poisson distribution with the mean . In this simulation, it is given as ().
The clutter is modeled as a Poisson RFS with the intensity . In this example, the actual clutter density is . It means that the clutter is uniformly distributed over the observation region.
Figure 1 shows the true trajectories for extended targets and sensor location.
In Figure 1, “” denotes the sensor location, “” denotes the locations at which the extended targets are born, “” denotes the locations at which the extended targets die, and “+” denotes the measurements generated by the extended targets. Extended target 1 is born at 1 s and dies at 25 s. Extended target 2 is born at 1 s and dies at 30 s. Extended target 3 is born at 10 s and dies at 35 s. Extended target 4 is born at 20s and dies at 45 s.
The intensity of the extendedtarget birth at time is modeled as where is the average number of the extendedtarget birth per scan, is the probability density of the new born extendedtarget state, and is the set of the density parameters. In this example, they are taken as , , where , , , , and .
The GMEPHD filter is used to estimate the number and states of the extended targets in the linear ETT problem. We now conduct Monte Carlo (MC) simulation experiments on the same clutter intensity and target trajectories but with independently generated clutter and targetgenerated measurements in each trial. Via comparing the tracking performance of the GMEPHD filter in the various number of Gaussians in the mixture and in various clutter rate , the convergence results for the algorithm can be verified to a great extent. For convenience, we assume and at each time step. Assumptions B2–B3 are satisfied in this example. So, the GMEPHD filter uniformly converges to the ground truth.
The standard deviation of the estimated cardinality distribution and the optimal subpattern assignment (OSPA) multitarget miss distance [26] of order with cutoff m, which jointly captures differences in cardinality and individual elements between two finite sets, are used to evaluate the performance of the method. Given the clutter rate , Table 1 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the GMEPHD filter in various via 200 MC simulation experiments.

Table 1 shows that both the standard deviation of the estimated cardinality distribution and OSPA decrease with the increase of the Gaussian number in the mixture. This phenomenon can be reasonably explained by the convergence results derived in this paper. First, according to Proposition 3.2, the error of the GMEPHD decreases as increases; then, the more precise estimates of the multitarget number and states can be derived from the more precise GMEPHD, which eventually leads to the results presented in Table 1.
Given , Table 2 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the GMEPHD filter in various clutter rate via 200 MC simulation experiments. Obviously, the number of the measurements collected at each time step increases with the increase of .

From Table 2, it can be seen that the errors of the multitarget number and state estimates from the GMEPHD filter grow significantly with the increase of . A reasonable explanation for this is that the partition operation included in (B.10) leads that the error bound of the GMEPHD corrector grows very quickly with the increase of the measurement number. Therefore, Table 2 consists with the convergence results established by Proposition 3.2, too.
Example 4.2 (EKGMEPHD filter to nonlinear ETT problem). The experiment settings are the same as those of Example 4.1 except the singlemeasurement singletarget likelihood density . The range and bearing measurements of the extended targets are generated with the noise covariance matrix , where and are, respectively, standard deviations of the range and bearing measurements. In this simulation, they are given as m and rad. The becomes
where
The EKGMEPHD filter is used to estimate the number and states of the extended targets in the nonlinear ETT problem. Given , Table 3 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the EKGMEPHD filter in various via 200 MC simulation experiments while, given , Table 4 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the EKGMEPHD filter in various via 200 MC simulation experiments.


As expected, Tables 3 and 4, respectively, show that the errors of the multitarget number and state estimates from the EKGMEPHD filter decrease with the increase of and increase with the increase of . These consist with the convergence results established by Propositions 3.2 and 3.3. In addition, comparing Tables 1 and 2 with Tables 3 and 4, it can be seen that the errors from the EKGMEPHD filter are obviously larger than the errors from the GMEPHD filter given the same and . The additional errors from the EKGMEPHD filter are caused by the reason that the condition for in Proposition 3.3 is very difficult to approach in this example.
5. Conclusions and Future Work
This paper shows that the recently proposed GMEPHD filter converges uniformly to the true EPHD filter as the number of Gaussians in the mixture tends to infinity. Proofs of uniform convergence are also derived for the EKGMEPHD filter. Since the GMEPHD corrector equations involve with the partition operation that grows very quickly with the increase of the measurement number, the future work is focused on studying the computationally tractable approximation for it and providing the convergence results and error bounds for the approximate GMEPHD corrector.
Appendices
A. Proof of Lemma 3.1
We have known that , , is known a priori and according to assumptions A1–A6 in [12] and assumptions B1–B3. So, from (2.3) we get . In addition, by (2.4) and (2.11) and the definition of norm, we have , , and because of the facts that , and .
For the initial induction step, assume . In this case, from (3.1) we get
In the case of , by the triangle inequality and (A.1), we have Since , , and , (A.2) becomes
Assume that we have established (3.2) for . We are to establish (3.2) for . Using the triangle inequality and (A.1), we get
Since , , and , (A.4) becomes and this closes the inductive step. This completes the proof.
B. Proof of Proposition 3.2
By the EPHD corrector equations, (2.2), and the triangle inequality, we get By (3.1), the second term in the summation of (B.1) is
Using the triangle inequality, the first term in the summation of (B.1) is
Using the triangle inequality again for the term in the numerator of (B.3), we get
Using Lemma 3.1, we get where denotes the complement of in .
Then, (B.4) can be rewritten as where
Substitute (A.1) and (B.6) into (B.3),
Substituting (3.1), (B.2), and (B.8) into (B.1), we have
So that Proposition 3.2 is proved with
This completes the proof.
C. Proof of Proposition 3.3
Clearly, by the EPHD corrector equations, (2.2)–(2.4), and the predicted EKEPHD, (3.4), we obtain that the in (3.5) is a Gaussian sum presented by (3.6). Now turn to the in (3.5). From (2.2), we get Consider the term in (C.1). Using the predicted EKEPHD, (3.4),
And by the result for the EK Gaussian sum filter [17], we get uniformly as for all , and , , , , are given by (3.9)–(3.13), respectively.
Now consider the terms and in (C.1). First, using the predicted EKEPHD, (3.4), the inner product is given by
And by the result for the EK Gaussian sum filter [17], we get uniformly as for all .
Changing the order of the summation and integral, (C.5) is equal to
Then, the expressions of and (see (3.8)) are derived by (2.4) and (C.6).
Finally, (3.7) is obtained by substituting (3.8) and (C.3) into (C.1). This completes the proof.
Acknowledgments
This research work was supported by Natural Science Foundation of China (61004087, 61104051, 61104214, and 61005026), China Postdoctoral Science Foundation (20100481338 and 2011M501443), and Fundamental Research Funds for the Central University.
References
 M. J. Waxmann and O. E. Drummond, “A bibliography of cluster (group) tracking,” in Signal and Data Processing of Small Targets, vol. 5428 of Proceedings of SPIE, pp. 551–560, Orlando, Fla, USA. View at: