Convergence Results for the Gaussian Mixture Implementation of the Extended-Target PHD Filter and Its Extended Kalman Filtering Approximation

The convergence of the Gaussian mixture extended-target probability hypothesis density (cid:2) GM-EPHD (cid:3) ﬁlter and its extended Kalman (cid:2) EK (cid:3) ﬁltering approximation in mildly nonlinear condition, namely, the EK-GM-EPHD ﬁlter, is studied here. This paper proves that both the GM-EPHD ﬁlter and the EK-GM-EPHD ﬁlter converge uniformly to the true EPHD ﬁlter. The signiﬁcance of this paper is in theory to present the convergence results of the GM-EPHD and EK-GM-EPHD ﬁlters and the conditions under which the two ﬁlters satisfy uniform convergence.


Introduction
The problem of extended-target tracking ETT 1, 2 arises because of the sensor resolution capacities 3 , the high density of targets, the sensor-to-target geometry, and so forth. For targets in near field of a high-resolution 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 association-based 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. where N ·|m, P denotes the density of Gaussian distribution with the mean m and covariance P. Then, the measurement-updated EPHD is a GM given by where I denotes the identity matrix; p D,k has been assumed to be state independent; H k and R k denote the observation matrix and the observation noise covariance, respectively; Journal of Applied Mathematics 5 blkdiag · denotes block diagonal matrix, the measure in ·, · of 2.11 is discrete, and it defines the summation inner product

Convergence of the GM-EPHD and EK-GM-EPHD Filters
The convergence properties and corresponding proof of the initialization step, prediction step, and pruning and merging step for the GM-EPHD filter are identical to those for point-target GM-PHD 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 GM-EPHD filter, the following lemma is first presented.
Consider the following assumptions.
B1: After the prediction step at time k, υ J k|k−1 k|k−1 converges uniformly to υ k|k−1 . In other words, for any given ε k|k−1 > 0 and any bounded measurable function ϕ ∈ B R d , where B R d is the set of bounded Borel measurable functions on R d , there is a positive integer J such that for J k|k−1 ≥ J, where · ∞ denotes ∞-norm. ϕ ∞ sup |ϕ| , sup · denotes the supremum.
B2: The clutter intensity κ k z k λ k c k z k is known a priori.
The proof of Lemma 3.1 can be found in Appendix A. The uniform convergence of the measurement-updated GM-EPHD is now established by Proposition 3.2.
where a k is defined by B.10 .

Journal of Applied Mathematics
The proof of the Proposition 3.2 can be found in Appendix B. Proposition 3.2 shows that the error for the GM-EPHD 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 GM-EPHD 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 EK-GM-EPHD filter, which is the nonlinear extension of the GM-EPHD filter. Due to the nonlinearity of the extended-target state and observation processes, the EPHD can no longer be represented as a GM. However, the EK-GM-EPHD filter can be adapted to accommodate models with mild nonlinearities. The convergence property and corresponding proof of the prediction step for the EK-GM-EPHD filter are identical to those for point-target EK-GM-PHD filter 19 . We now establish the conditions for uniform convergence of the measurement update step for the EK-GM-EPHD filter.

Proposition 3.3. Suppose that the predicted EK-EPHD is given by the sum of Gaussians
and the φ z k x k in 2.1 is given by the nonlinear single-measurement single-target equation z k h k x k , v k , where h k is known nonlinear functions and v k is zero-mean Gaussian measurement noise with covariance R k , then the measurement-updated EK-EPHD approaches the Gaussian sum The proof of Proposition 3.3 can be found in Appendix C. From Propositions 3.2 and 3.3, we can obtain that the EK-GM-EPHD corrector uniformly converges to the true EPHD corrector in x k and Z k under the assumptions that P i k|k−1 → 0 for i 1, . . . , J k|k−1 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 EK-GM-EPHD filter have demonstrated its potential for real-world applications. However, Propositions 3.2 and 3.3 give further theoretical justification for the use of the GM-EPHD and EK-GM-EPHD filters in ETT problem.

Simulations
Here we briefly describe the application of the convergence results for the GM-EPHD and EK-GM-EPHD filters to the linear and nonlinear ETT examples. Gaussian white noise with the covariance matrix Q i,k . Then the Markovian transition density of x i,k could be modeled as where Φ i,k is discrete-time evolution matrix. Here Φ i,k and Q i,k are given by the constant acceleration model 24 , as where "⊗" denotes the Kronecker product. The parameter σ is the instantaneous standard deviation of the acceleration, given by σ 0.05 m/s 2 . Note that the objective of this paper is to focus on the convergence analysis for the GM-EPHD and EK-GM-EPHD filters, rather than the simulation of the extended-target 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 extended-target motion models. The models proposed in 25 can also be accommodated within the EPHD filter straightforwardly.
At time k, the x-coordinate and y-coordinate measurements of the extended targets are generated by a sensor located at 0, 0 T . The measurement noise v k is IID zero-mean Gaussian white noise with covariance matrix R k diag σ 2 x , σ 2 y , where diag · denotes the diagonal matrix, σ x and σ y are, respectively, standard deviations of the x-coordinate and y-coordinate measurements. In this simulation, they are given as σ x σ y 25 m. The single-measurement single-target likelihood density φ z k x i,k is The detection probability of the sensor is p D,k x k 0.95. 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 k, the number of the measurements arising from the ith extended target satisfies Poisson distribution with the mean γ x i,k . In this simulation, it is given as γ x i,k The clutter is modeled as a Poisson RFS with the intensity κ k z k λ k c k z k . In this example, the actual clutter density is c k z k U z k . 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, "Δ" 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 extended-target birth at time k is modeled as where λ β is the average number of the extended-target birth per scan, f β x k |ψ β is the probability density of the new born extended-target state, and ψ β is the set of the density parameters. In this example, they are taken as λ β 0.05, diag 400, 400, 100, 100, 9, 9 . The GM-EPHD 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 target-generated measurements in each trial. Via comparing the tracking performance of the GM-EPHD filter in the various number J k of Gaussians in the mixture and in various clutter rate λ k , the convergence results for the algorithm can be verified to a great extent. For convenience, we assume J k J and λ k λ at each time step. Assumptions B2-B3 are satisfied in this example. So, the GM-EPHD 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 p 2 with cutoff  c 100 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 λ 50, Table 1 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the GM-EPHD filter in various J 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 J 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 GM-EPHD decreases as J increases; then, the more precise estimates of the multitarget number and states can be derived from the more precise GM-EPHD, which eventually leads to the results presented in Table 1.
Given J 700, Table 2 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the GM-EPHD filter in various clutter rate λ via 200 MC simulation experiments. Obviously, the number of the measurements collected at each time step increases with the increase of λ.
From Table 2, it can be seen that the errors of the multitarget number and state estimates from the GM-EPHD filter grow significantly with the increase of λ. A reasonable explanation for this is that the partition operation included in B.10 leads that the error bound of the GM-EPHD 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 EK-GM-EPHD filter to nonlinear ETT problem . The experiment settings are the same as those of Example 4.1 except the single-measurement single-target likelihood density φ z k x i,k . The range r k and bearing θ k measurements of the extended targets are generated with the noise covariance matrix R k diag σ 2 r , σ 2 θ , where σ r and σ θ are, respectively, standard deviations of the range and bearing measurements. In this simulation, they are given as σ r 25 m and σ θ 0.025 rad. The φ z k x i,k becomes  where

4.7
The EK-GM-EPHD filter is used to estimate the number and states of the extended targets in the nonlinear ETT problem. Given λ 50, Table 3 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the EK-GM-EPHD filter in various J via 200 MC simulation experiments while, given J 700, Table 4 shows the time averaged standard deviation of the estimated cardinality distribution and the time averaged OSPA from the EK-GM-EPHD filter in various λ 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 EK-GM-EPHD filter decrease with the increase of J and increase with the increase of λ. 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 EK-GM-EPHD filter are obviously larger than the errors from the GM-EPHD filter given the same J and λ. The additional errors from the EK-GM-EPHD filter are caused by the reason that the condition P i k|k−1 → 0 for i 1, . . . , J k|k−1 in Proposition 3.3 is very difficult to approach in this example.

Conclusions and Future Work
This paper shows that the recently proposed GM-EPHD 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 EK-GM-EPHD filter. Since the GM-EPHD 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