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

Abstract

The convergence of the Gaussian mixture extended-target probability hypothesis density (GM-EPHD) filter and its extended Kalman (EK) 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.

1. 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.

Recently, the random-finite-set- (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 extended-target PHD (EPHD) filter was proposed by Granström et al. [16], in 2010. Similar to the point-target GM-PHD filter, the GM-EPHD filter can also be extended to accommodate mildly nonlinear target dynamics using the extended Kalman (EK) filtering [17] approximation. The extension is called EK-GM-EPHD filter. Experimental results showed the EK-GM-EPHD 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 GM-EPHD and EK-GM-EPHD filters have been successfully used for many real-world problems, there have been no results showing the convergence for the two filters. The convergence results on point-target particle-PHD and GM-PHD filters [18, 19] do not apply directly for the GM-EPHD and EK-GM-EPHD 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 “real-life” problems, it is necessary to answer the following question: do the GM-EPHD and EK-GM-EPHD 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 GM-EPHD filter. In other words, given simple sufficient conditions, the approximation error of the measurement-updated 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 EK-GM-EPHD filter are derived from Proposition 3.3.

2. EPHD and GM-EPHD Filters

At time 𝑘, let 𝐱_𝑘 be the state vector of a single extended target, and 𝐳_𝑘 a single measurement vector received by sensor. Multiple extended-target states and sensor measurements can, respectively, be represented as finite sets 𝑋_𝑘={𝐱_𝑖,𝑘}^𝑛𝑘_𝑖=1 and 𝑍_𝑘={𝐳_𝑖,𝑘}^𝑚𝑘_𝑖=1, where 𝑛_𝑘 and 𝑚_𝑘 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]: 𝑙_𝑍𝑘𝐱_𝑘=𝑒^{−𝛾(𝐱𝑘)}_{𝐳𝑘∈𝑍𝑘}𝛾𝐱_𝑘𝜙_𝐳𝑘𝐱_𝑘, (2.1) where 𝜙_𝐳𝑘(𝐱_𝑘) denotes the single-measurement single-target likelihood density; 𝛾(𝐱_𝑘) denotes the expected number of measurements arising from an extended target.

The clutter is modeled as a Poisson RFS with the intensity 𝜅_𝑘(𝐳_𝑘)=𝜆_𝑘𝑐_𝑘(𝐳_𝑘), where 𝜆_𝑘 is the average clutter number per scan and 𝑐_𝑘(𝐳_𝑘) is the density of clutter spatial distribution.

Given the Poisson likelihood model for the extended targets, Mahler derived the EPHD filter using finite-set statistics [15, 23]. The prediction equations of the EPHD filter are identical to those of the point-target PHD filter [10]. The EPHD measurement update equations are 𝜐_𝑘𝐱_𝑘=1−𝑝_𝐷,𝑘𝐱_𝑘+𝑒^{−𝛾(𝐱𝑘)}𝑝_𝐷,𝑘𝐱_𝑘+_{℘𝑘∠𝑍𝑘}𝜔_{℘𝑘𝑊𝑘∈℘𝑘}𝜂_𝑊𝑘𝐱_𝑘𝑑_𝑊𝑘𝜐_{𝑘|𝑘−1}𝐱_𝑘,𝜂_𝑊𝑘𝐱_𝑘=𝑝_𝐷,𝑘𝐱_𝑘𝑒^{−𝛾(𝐱𝑘)}_{𝐳𝑘∈𝑊𝑘}𝛾𝐱_𝑘𝜙_𝐳𝑘𝐱_𝑘𝜆_𝑘𝑐_𝑘𝐳_𝑘,𝜔_℘𝑘=∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊′𝑘;𝑑_𝑊𝑘=𝛿_|𝑊𝑘|,1+𝜐_{𝑘∣𝑘−1},𝜂_𝑊𝑘, (2.2)(2.3)(2.4) where 𝑝_𝐷,𝑘(𝐱_𝑘) denotes the probability that the set of observations from the extended target will be detected at time 𝑘; ℘_𝑘∠𝑍_𝑘 denotes that ℘_𝑘 partitions set 𝑍_𝑘 [15], for example, let 𝑍={𝐳₁,𝐳₂,𝐳₃}, then the partitions of 𝑍 are ℘₁={{𝐳₁,𝐳₂,𝐳₃}}, ℘₂={{𝐳₁},{𝐳₂},{𝐳₃}}, ℘₃={{𝐳₁,𝐳₂},{𝐳₃}}, ℘₄={{𝐳₁,𝐳₃},{𝐳₂}}, and ℘₅={{𝐳₁},{𝐳₂,𝐳₃}}; |𝑊_𝑘| denotes the cardinality of the set 𝑊_𝑘; 𝛿_|𝑊𝑘|,1=1 if |𝑊_𝑘|=1, and 𝛿_|𝑊𝑘|,1=0 otherwise; the notation ⟨⋅,⋅⟩ is the usual inner product. The measure in ⟨⋅,⋅⟩ of (2.4) is continuous, it defines the integral inner product 𝜐_{𝑘∣𝑘−1},𝜂_𝑊𝑘=𝜐_{𝑘∣𝑘−1}𝐱_𝑘𝜂_𝑊𝑘𝐱_𝑘𝑑𝐱_𝑘. (2.5)

By making the same six assumptions that are made in [12] and the additional assumption that 𝛾(𝐱_𝑘) can be approximated as functions of the mean of the individual Gaussian components, Granström et al. proposed the GM-EPHD filter [16]. At time 𝑘, let 𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1} denote the GM approximation to the predicted EPHD 𝜐_{𝑘∣𝑘−1} with 𝐽_{𝑘∣𝑘−1} Gaussian components, and 𝜐^𝐽𝑘_𝑘 the GM approximation to the measurement-updated EPHD 𝜐_𝑘 with 𝐽_𝑘 Gaussian components. The prediction equations of the GM-EPHD filter are identical to those of point-target GM-PHD filter [12]. The GM-EPHD measurement update equations are as follows.

Let the predicted EPHD be a GM of the form 𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝐱_𝑘=^{𝐽𝑘|𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘|𝑘−1}𝒩𝐱_𝑘∣𝐦^(𝑖)_{𝑘∣𝑘−1},𝐏^(𝑖)_{𝑘∣𝑘−1}, (2.6) 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 𝜐^𝐽𝑘_𝑘𝐱_𝑘=𝜐^{𝑁𝐷,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘+_{℘𝑘∠𝑍𝑘𝑊𝑘∈℘𝑘}𝜐^{𝐷,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘,𝑊_𝑘, (2.7) where 𝜐^{𝑁𝐷,𝐽𝑘|𝑘−1}_𝑘(𝐱_𝑘) denotes the Gaussian components handling no detections, 𝜐^{𝑁𝐷,𝐽𝑘|𝑘−1}_𝑘𝐱_𝑘=^{𝐽𝑘|𝑘−1}_𝑗=1𝑤_𝑘^(𝑗)𝒩𝐱_𝑘∣𝐦^(𝑗)_{𝑘|𝑘−1},𝐏^(𝑗)_{𝑘|𝑘−1},𝑤_𝑘^(𝑗)=1−1−𝑒^{−𝛾(𝐦(𝑗)𝑘∣𝑘−1)}𝑝_𝐷,𝑘𝑤^(𝑗)_{𝑘∣𝑘−1},𝐦_𝑘^(𝑗)=𝐦^(𝑗)_{𝑘∣𝑘−1};𝐏_𝑘^(𝑗)=𝐏^(𝑗)_{𝑘∣𝑘−1}, (2.8) and 𝜐^{𝐷,𝐽𝑘|𝑘−1}_𝑘(𝐱_𝑘,𝑊_𝑘) denotes the Gaussian components handling detected targets 𝜐^{𝐷,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘,𝑊_𝑘=^{𝐽𝑘∣𝑘−1}_𝑗=1𝑤_𝑘^(𝑗)𝒩𝐱_𝑘∣𝐦_𝑘^(𝑗),𝐏_𝑘^(𝑗),𝑤_𝑘^(𝑗)=𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜂_𝑊𝑘𝐦^(𝑗)_{𝑘∣𝑘−1}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝑤^(𝑗)_{𝑘∣𝑘−1},𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘=∏_{𝑊𝑘∈℘𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘;𝑑^{𝐽𝑘|𝑘−1}_𝑊𝑘=𝛿_|𝑊𝑘|,1+𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘,𝜂_𝑊𝑘𝐦^(𝑗)_{𝑘∣𝑘−1}=𝑝_𝐷,𝑘𝑒^{−𝛾(𝐦(𝑗)𝑘∣𝑘−1)}_{𝐳𝑘∈𝑊𝑘}𝛾𝐦^(𝑗)_{𝑘∣𝑘−1}𝜙_𝐳𝑘𝐦^(𝑗)_{𝑘∣𝑘−1}𝜆_𝑘𝑐_𝑘𝐳_𝑘,𝜙_𝐳𝑘𝐦^(𝑗)_{𝑘∣𝑘−1}=𝒩𝐳_𝑘∣𝐇_𝑘𝐦^(𝑗)_{𝑘∣𝑘−1},𝐑_𝑘+𝐇_𝑘𝐏^(𝑗)_{𝑘∣𝑘−1}𝐇^𝑇_𝑘𝐊_𝑘(𝑗),=𝐏^(𝑗)_{𝑘∣𝑘−1}𝐇^𝑇_𝑘𝐇_𝑘𝐏^(𝑗)_{𝑘∣𝑘−1}𝐇^𝑇_𝑘+𝐑_𝑘⁻¹,𝐦_𝑘^(𝑗)=𝐦^(𝑗)_{𝑘∣𝑘−1}+𝐊_𝑘^(𝑗)𝐳₁⋮𝐳_|𝑊𝑘|−𝐇_𝑘𝐦^(𝑗)_{𝑘∣𝑘−1};𝐏_𝑘^(𝑗)=𝐈−𝐊_𝑘^(𝑗)𝐇_𝑘𝐏^(𝑗)_{𝑘∣𝑘−1},𝐇_𝑘=𝐇_𝑘⋮𝐇_𝑘𝑊_𝑘;𝐑_𝑘=blkdiag𝐑_𝑘,…^|𝑊𝑘|,𝐑_𝑘, (2.9)(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)(2.16) 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; blkdiag(⋅) denotes block diagonal matrix, the measure in ⟨⋅,⋅⟩ of (2.11) is discrete, and it defines the summation inner product 𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘=^{𝐽𝑘|𝑘−1}_𝑙=1𝜂_𝑊𝑘𝐦^(𝑙)_{𝑘∣𝑘−1}𝑤^(𝑙)_{𝑘∣𝑘−1}. (2.17)

3. 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^{𝐽𝑘∣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 𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝜐_{𝑘∣𝑘−1},𝜑≤𝜀_{𝑘∣𝑘−1}𝜑_∞, (3.1) 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_𝑘𝑐_𝑘(𝐳_𝑘) is known a priori.B3: 𝛾(𝐱_𝑘)∈𝐶_𝑏(ℝ^𝑑), where 𝐶_𝑏(ℝ^𝑑) denotes the set of the continuous bounded functions on ℝ^𝑑.

Lemma 3.1. Given a partition℘_𝑘={𝑊_1,𝑘,𝑊_2,𝑘,…,𝑊_𝑛,𝑘} and suppose that assumptions B1–B3 hold, then _{𝑖=1,…,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}−_{𝑖=1,…,𝑛}𝑑_{𝑊𝑖,𝑘}≤𝜀_{𝑘∣𝑘−1𝑗=1,…,𝑛}𝜂_{𝑊𝑗,𝑘∞𝑖=1,…,𝑛;𝑖≠𝑗}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+𝑑_{𝑊𝑖,𝑘}. (3.2)

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.

Proposition 3.2. After the measurement update step of the GM-EPHD filter, there exists a real number 𝑎_𝑘, dependent on the number of measurements such that 𝜐^𝐽𝑘_𝑘−𝜐_𝑘,𝜑≤𝑎_𝑘𝜀_{𝑘∣𝑘−1}𝜑_∞, (3.3) where 𝑎_𝑘 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 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 𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝐱_𝑘=^{𝐽𝑘∣𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦^(𝑖)_{𝑘∣𝑘−1},𝐏^(𝑖)_{𝑘∣𝑘−1}, (3.4) and the 𝜙_𝐳𝑘(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 𝜐_𝑘𝐱_𝑘→𝜐^{𝐸𝐾,𝐽𝑘}_𝑘𝐱_𝑘=𝜐^{𝑁𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘+_{℘𝑘∠𝑍𝑘𝑊𝑘∈℘𝑘}𝜐^{𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘,𝑊_𝑘, (3.5) uniformly in 𝐱_𝑘 and 𝑍_𝑘 as 𝐏^(𝑖)_{𝑘|𝑘−1}→0 for 𝑖=1,…,𝐽_{𝑘∣𝑘−1}, and where 𝜐^{𝑁𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘=^{𝐽𝑘∣𝑘−1}_𝑖=11−𝑝_𝐷,𝑘+𝑒^{−𝛾(𝐦(𝑖)𝑘∣𝑘−1)}𝑝_𝐷,𝑘𝑤^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦^(𝑖)_{𝑘∣𝑘−1},𝐏^(𝑖)_{𝑘∣𝑘−1},𝜐^{𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘,𝑊_𝑘=^{𝐽𝑘∣𝑘−1}_𝑖=1𝜔^{𝐸𝐾,𝐽𝑘∣𝑘−1}_℘𝑘𝜂_𝑊^𝐸𝐾_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1}𝑑^{𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑊𝑘𝑤^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦_𝑘^(𝑖),𝐏_𝑘^(𝑖),𝜔^{𝐸𝐾,𝐽𝑘∣𝑘−1}_℘𝑘=∏_{𝑊𝑘∈℘𝑘}𝑑^{𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊^𝐸𝐾_′𝑘;𝑑^{𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑊𝑘=𝛿_|𝑊𝑘|,1+^{𝐽𝑘∣𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝜂_𝑊^𝐸𝐾_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1},𝜂_𝑊^𝐸𝐾_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1}=𝑝_𝐷,𝑘𝑒^{−𝛾(𝐦(𝑖)𝑘∣𝑘−1)}×_{𝐳𝑘∈𝑊𝑘}𝛾𝐦^(𝑖)_{𝑘∣𝑘−1}𝒩𝐳_𝑘∣ℎ_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1},𝟎,𝐔_𝑘^(𝑖)𝐑_𝑘𝐔_𝑘^(𝑖)𝑇+𝐇_𝑘^(𝑖)𝐏^(𝑖)_{𝑘∣𝑘−1}𝐇_𝑘^(𝑖)𝑇𝜆_𝑘𝑐_𝑘𝐳_𝑘,𝐇_𝑘^(𝑖)=𝜕ℎ_𝑘𝐱_𝑘,𝟎𝜕𝐱_{𝑘𝐱𝑘=𝐦(𝑖)𝑘∣𝑘−1};𝐔_𝑘^(𝑖)=𝜕ℎ_𝑘𝐦^(𝑖)𝑛_{𝑘∣𝑘−1},𝐯_𝑘𝜕𝐯_{𝑘𝐯𝑘=0},𝐊_𝑘^(𝑖)=𝐏^(𝑖)_{𝑘∣𝑘−1}𝐇_𝑘^(𝑖)𝑇𝐇_𝑘^(𝑖)𝐏^(𝑖)_{𝑘∣𝑘−1}𝐇_𝑘^(𝑖)𝑇+𝐔_𝑘^(𝑖)𝐑_𝑘𝐔_𝑘^{(𝑖)𝑇−1},𝐦_𝑘^(𝑖)=𝐦^(𝑖)_{𝑘∣𝑘−1}+𝐊_𝑘^(𝑖)𝐳₁⋮𝐳_|𝑊𝑘|−ℎ𝐦^(𝑖)_{𝑘∣𝑘−1}⋮ℎ𝐦,𝟎_{(𝑖)𝑘∣𝑘−1},𝟎;𝐏_𝑘^(𝑖)=𝐈−𝐊_𝑘^(𝑖)𝐇_𝑘^(𝑖)𝐏^(𝑖)_{𝑘∣𝑘−1},𝐇_𝑘^(𝑖)=𝐇_𝑘^(𝑖)⋮𝐇_𝑘^(𝑖)𝑊_𝑘;𝐔_𝑘^(𝑖)=𝐔_𝑘^(𝑖)⋮𝐔_𝑘^(𝑖)𝑊_𝑘, (3.6)(3.7)(3.8)(3.9)(3.10)(3.11)(3.12)(3.13)

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.

4. 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.

Example 4.1 (GM-EPHD filter to linear ETT problem). Consider a two-dimensional scenario with an unknown and time varying number of the extended targets observed over the region [−1000,1000]×[−1000,1000] (in m) for a period of 𝑇=45 time steps. The sampling interval is Δ𝑡=1s. At time 𝑘, the actual number of the existing extended targets is 𝑛_𝑘 and the state of the 𝑖th target is 𝐱_𝑖,𝑘=[𝑥_𝑖,𝑘,𝑦_𝑖,𝑘,𝑥̇_𝑖,𝑘,𝑦̇_𝑖,𝑘,𝑥̈_𝑖,𝑘,𝑦̈_𝑖,𝑘]^𝑇 (𝑖=1,…,𝑛_𝑘). Assume that the process noise 𝝕_𝑖,𝑘 of the 𝑖th extended target is independent and identically distributed (IID) zero-mean Gaussian white noise with the covariance matrix 𝐐_𝑖,𝑘. Then the Markovian transition density of 𝐱_𝑖,𝑘 could be modeled as 𝑓_{𝑘|𝑘−1}𝐱_𝑖,𝑘∣𝐱_{𝑖,𝑘−1}=𝒩𝐱_𝑖,𝑘∣𝚽_𝑖,𝑘𝐱_{𝑖,𝑘−1},𝐐_𝑖,𝑘, (4.1) 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 𝚽_𝑖,𝑘=1Δ𝑡Δ𝑡²211Δ𝑡⊗𝐈₂;𝐐_𝑖,𝑘=𝜎²_𝜛Δ𝑡⁴4Δ𝑡³2Δ𝑡²2Δ𝑡³2Δ𝑡²2Δ𝑡Δ𝑡²2Δ𝑡1⊗𝐈₂;𝐈₂=11, (4.2) where “⊗” denotes the Kronecker product. The parameter 𝜎_𝜛 is the instantaneous standard deviation of the acceleration, given by 𝜎_𝜛=0.05m/s².
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(𝜎²_𝑥,𝜎²_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=25m. The single-measurement single-target likelihood density 𝜙_𝐳𝑘(𝐱_𝑖,𝑘) is 𝜙_𝐳𝑘𝐱_𝑖,𝑘=𝒩𝐳_𝑘∣𝐇_𝑘𝐱_𝑖,𝑘,𝐑_𝑘, (4.3) where 𝐇_𝑘=100000010000. (4.4)
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)=3 (𝑖=1,…,𝑛_𝑘).
The clutter is modeled as a Poisson RFS with the intensity 𝜅_𝑘(𝐳_𝑘)=𝜆_𝑘𝑐_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.

Figure 1 

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 𝛽_𝑘𝐱_𝑘=𝜆_𝛽𝑓_𝛽𝐱_𝑘∣𝜓_𝛽, (4.5) 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, 𝑓_𝛽(𝐱_𝑘|𝜓_𝛽)=𝜋¹_𝛽𝒩(𝐱_𝑘|𝝁¹_𝛽,𝚺¹_𝛽)+𝜋²_𝛽𝒩(𝐱_𝑘|𝝁²_𝛽,𝚺²_𝛽), where 𝜓_𝛽={𝜋¹_𝛽,𝜋²_𝛽,𝝁¹_𝛽,𝝁²_𝛽,𝚺¹_𝛽,𝚺²_𝛽}, 𝜋¹_𝛽=𝜋²_𝛽=0.5, 𝝁¹_𝛽=[−600,750,0,0,0,0]^𝑇, 𝝁²_𝛽=[−650,−800,0,0,0,0]^𝑇, and 𝚺¹_𝛽=𝚺²_𝛽=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=100m, 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 

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the GM-EPHD filter in various 𝐽 given 𝜆=50.

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.

Table 2 

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the GM-EPHD filter in various 𝜆 given 𝐽=700.

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_𝐳𝑘(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(𝜎²_𝑟,𝜎²_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=25m and 𝜎_𝜃=0.025rad. The 𝜙_𝐳𝑘(𝐱_𝑖,𝑘) becomes 𝜙_𝐳𝑘𝐱_𝑖,𝑘=𝒩𝐳_𝑘∣ℎ_𝑘𝐱_𝑖,𝑘,𝐑_𝑘, (4.6) where ℎ_𝑘𝐱_𝑖,𝑘=𝑥²_𝑖,𝑘+𝑦²_𝑖,𝑘𝑦arctan^𝑖,𝑘𝑥_𝑖,𝑘^𝑇. (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.

Table 3 

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the EK-GM-EPHD filter in various 𝐽 given 𝜆=50.

Table 4 

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the EK-GM-EPHD filter in various 𝜆 given 𝐽=700.

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.

5. 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 tractable approximation for it and providing the convergence results and error bounds for the approximate GM-EPHD corrector.

Appendices

A. Proof of Lemma 3.1

We have known that 𝛾(𝐱_𝑘)∈𝐶_𝑏(ℝ^𝑑), 𝜙_𝐳𝑘(𝐱_𝑘)=𝒩(𝐳_𝑘|𝐇_𝑘𝐱_𝑘,𝐑_𝑘)∈𝐶_𝑏(ℝ^𝑑), 𝜆_𝑘𝑐_𝑘(𝐳_𝑘) is known a priori and 0≤𝑝_𝐷,𝑘≤1 according to assumptions A1–A6 in [12] and assumptions B1–B3. So, from (2.3) we get 𝜂_𝑊𝑘(𝐱_𝑘)∈𝐶_𝑏(ℝ^𝑑). In addition, by (2.4) and (2.11) and the definition of ∞-norm, we have 𝑑_𝑊𝑘≥0, 𝑑^{𝐽𝑘|𝑘−1}_𝑊𝑘≥0, and ‖𝜂_𝑊𝑘‖_∞≥0 because of the facts that 𝜐_{𝑘∣𝑘−1}(𝐱_𝑘)≥0, 𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}(𝐱_𝑘)≥0 and 𝜂_𝑊𝑘(𝐱_𝑘)≥0.

For the initial induction step, assume 𝑛=1. In this case, from (3.1) we get 𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘−𝑑_𝑊1,𝑘=𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊1,𝑘−𝜐_{𝑘∣𝑘−1},𝜂_𝑊1,𝑘=𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝜐_{𝑘∣𝑘−1},𝜂_𝑊1,𝑘≤𝜀_{𝑘∣𝑘−1}𝜂_{𝑊1,𝑘∞}. (A.1)

In the case of 𝑛=2, by the triangle inequality and (A.1), we have 𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘−𝑑_𝑊1,𝑘𝑑_𝑊2,𝑘=𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘+𝑑_𝑊2,𝑘2𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘−𝑑_𝑊1,𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘−𝑑_𝑊2,𝑘2𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘+𝑑_𝑊1,𝑘≤𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘+𝑑_𝑊2,𝑘2𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘−𝑑_𝑊1,𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘−𝑑_𝑊2,𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘+𝑑_𝑊1,𝑘2≤𝜀_{𝑘∣𝑘−1}𝜂_{𝑊1,𝑘∞}𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘+𝑑_𝑊2,𝑘2+𝜀_{𝑘∣𝑘−1}𝜂_{𝑊2,𝑘∞}𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘+𝑑_𝑊1,𝑘2. (A.2) Since 𝜀_{𝑘∣𝑘−1}>0, 𝑑_𝑊𝑘≥0, 𝑑^{𝐽𝑘|𝑘−1}_𝑊𝑘≥0 and ‖𝜂_𝑊𝑘‖_∞≥0, (A.2) becomes 𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘−𝑑_𝑊1,𝑘𝑑_𝑊2,𝑘≤𝜀_{𝑘∣𝑘−1}𝜂_{𝑊1,𝑘∞}𝑑_𝑊2,𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊2,𝑘+𝜂_{𝑊2,𝑘∞}𝑑_𝑊1,𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊1,𝑘. (A.3)

Assume that we have established (3.2) for 𝑖=1,…,𝑛. We are to establish (3.2) for 𝑖=1,…,𝑛+1. Using the triangle inequality and (A.1), we get _{𝑖=1,...,𝑛+1}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}−_{𝑖=1,...,𝑛+1}𝑑_{𝑊𝑖,𝑘}=𝑑_{𝑊𝑛+1,𝑘}+𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑛+1,𝑘}2_{𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}−_{𝑖=1,...,𝑛}𝑑_{𝑊𝑖,𝑘}+𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑛+1,𝑘}−𝑑_{𝑊𝑛+1,𝑘}2_{𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+_{𝑖=1,...,𝑛}𝑑_{𝑊𝑖,𝑘}≤𝑑_{𝑊𝑛+1,𝑘}+𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑛+1,𝑘}2_{𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}−_{𝑖=1,...,𝑛}𝑑_{𝑊𝑖,𝑘}+𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑛+1,𝑘}−𝑑_{𝑊𝑛+1,𝑘}2_{𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+_{𝑖=1,...,𝑛}𝑑_{𝑊𝑖,𝑘}≤𝜀_{𝑘|𝑘−1}𝑑_{𝑊𝑛+1,𝑘}+𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑛+1,𝑘}2_{𝑗=1,...,𝑛}𝜂_{𝑊𝑗,𝑘∞𝑖=1,...,𝑛;𝑖≠𝑗}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+𝑑_{𝑊𝑖,𝑘}+𝜀_{𝑘∣𝑘−1}𝜂_{𝑊𝑛+1,𝑘∞}2_{𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+_{𝑖=1,...,𝑛}𝑑_{𝑊𝑖,𝑘}. (A.4)

Since 𝜀_{𝑘|𝑘−1}>0, 𝑑_𝑊𝑘≥0, 𝑑^{𝐽𝑘|𝑘−1}_𝑊𝑘≥0 and ‖𝜂_𝑊𝑘‖_∞≥0, (A.4) becomes _{𝑖=1,...,𝑛+1}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}−_{𝑖=1,...,𝑛+1}𝑑_{𝑊𝑖,𝑘}≤𝜀_{𝑘∣𝑘−1𝑗=1,...,𝑛}𝜂_{𝑊𝑗,𝑘∞𝑖=1,...,𝑛+1;𝑖≠𝑗}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+𝑑_{𝑊𝑖,𝑘}+𝜀_{𝑘∣𝑘−1}𝜂_{𝑊𝑛+1,𝑘∞𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+_{𝑖=1,...,𝑛}𝑑_{𝑊𝑖,𝑘}≤𝜀_{𝑘∣𝑘−1𝑗=1,...,𝑛}𝜂_{𝑊𝑗,𝑘∞𝑖=1,...,𝑛+1;𝑖≠𝑗}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+𝑑_{𝑊𝑖,𝑘}+𝜀_{𝑘∣𝑘−1}𝜂_{𝑊𝑛+1,𝑘∞𝑖=1,...,𝑛}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+𝑑_{𝑊𝑖,𝑘}=𝜀_{𝑘∣𝑘−1𝑗=1,...,𝑛+1}𝜂_{𝑊𝑗,𝑘∞𝑖=1,...,𝑛+1;𝑖≠𝑗}𝑑^{𝐽𝑘∣𝑘−1}_{𝑊𝑖,𝑘}+𝑑_{𝑊𝑖,𝑘}, (A.5) 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 𝜐^𝐽𝑘_𝑘−𝜐_𝑘=,𝜑1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘+_{℘𝑘∠𝑍𝑘}𝜔^{𝐽𝑘∣𝑘−1}_{℘𝑘𝑊𝑘∈℘𝑘}𝜂_𝑊𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}−,𝜑1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘+_{℘𝑘∠𝑍𝑘}𝜔_{℘𝑘𝑊𝑘∈℘𝑘}𝜂_𝑊𝑘𝑑_𝑊𝑘𝜐_{𝑘|𝑘−1}=𝜐,𝜑_{𝐽𝑘∣𝑘−1𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘−𝜐_{𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘+∑_{℘𝑘∠𝑍𝑘}∑_{𝑊𝑘∈℘𝑘}𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘−𝜔_℘𝑘𝜐_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘≤𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝜐_{𝑘∣𝑘−1},𝜑1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘+_{℘𝑘∠𝑍𝑘𝑊𝑘∈℘𝑘}𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘−𝜔_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘+𝜔_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘−𝜔_℘𝑘𝜐_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘. (B.1) By (3.1), the second term in the summation of (B.1) is 𝜔_℘𝑘𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘−𝜔_℘𝑘𝜐_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘=𝜔_℘𝑘𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1}−𝜐_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘≤𝜔_℘𝑘𝜀_{𝑘|𝑘−1}𝜑𝜂_𝑊𝑘∞𝑑_𝑊𝑘. (B.2)

Using the triangle inequality, the first term in the summation of (B.1) is 𝜔^{𝐽𝑘|𝑘−1}_℘𝑘𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑^{𝐽𝑘|𝑘−1}_𝑊𝑘−𝜔_℘𝑘𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘=𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘−𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜔_℘𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝑑_𝑊𝑘=𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘𝜔^{𝐽𝑘|𝑘−1}_℘𝑘−𝑑^{𝐽𝑘|𝑘−1}_𝑊𝑘𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘−𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜔_℘𝑘𝛿_|𝑊𝑘|,1+𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘𝑑_𝑊𝑘≤𝜐^{𝐽𝑘|𝑘−1}_{𝑘|𝑘−1},𝜂_𝑊𝑘𝜑_∞𝜔^{𝐽𝑘|𝑘−1}_℘𝑘𝑑_𝑊𝑘−𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘−𝜔_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘𝑑_𝑊𝑘≤𝜑_∞𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝑑_𝑊𝑘−𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘−𝜔_℘𝑘𝑑_𝑊𝑘. (B.3)

Using the triangle inequality again for the term |𝜔^{𝐽𝑘∣𝑘−1}_℘k−𝜔_℘k| in the numerator of (B.3), we get 𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘−𝜔_℘𝑘=∏_{𝑊𝑘∈℘𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘−∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊′𝑘=∏_{𝑊𝑘∈℘𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘−∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘+∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘−∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊′𝑘≤∏_{𝑊𝑘∈℘𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘−∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘+∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊′𝑘⋅∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊′𝑘−∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘≤∏_{𝑊𝑘∈℘𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘−∏_{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘+𝜔_℘𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑_𝑊′𝑘−∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′𝑘. (B.4)

Using Lemma 3.1, we get _{𝑊𝑘∈℘𝑘}𝑑_𝑊𝑘−_{𝑊𝑘∈℘𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘≤𝜀_{𝑘∣𝑘−1𝑊𝑘∈℘𝑘}𝜂_{𝑊𝑘∞𝑉𝑘∈℘𝑘−𝑊𝑘}𝑑_𝑉𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑉𝑘, (B.5) where ℘_𝑘−𝑊_𝑘 denotes the complement of 𝑊_𝑘 in ℘_𝑘.

Then, (B.4) can be rewritten as 𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘−𝜔_℘𝑘≤𝜀_{𝑘∣𝑘−1}𝜌_𝑘, (B.6) where 𝜌_𝑘=∑_{𝑊𝑘∈℘𝑘}𝜂_𝑊𝑘∞∏_{𝑉𝑘∈℘𝑘−𝑊𝑘}𝑑_𝑉𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑉𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′+𝜔_℘𝑘∑_{℘′𝑘∠𝑍𝑘}∑_{𝑊′𝑘∈℘′𝑘}𝜂_{𝑊′𝑘∞}∏_{𝑉′𝑘∈℘′𝑘−𝑊′𝑘}𝑑_𝑉′𝑘+𝑑^{𝐽𝑘∣𝑘−1}_𝑉′𝑘∑_{℘′𝑘∠𝑍𝑘}∏_{𝑊′𝑘∈℘′𝑘}𝑑^{𝐽𝑘∣𝑘−1}_𝑊′. (B.7)

Substitute (A.1) and (B.6) into (B.3), 𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑^𝐽𝑘_𝑊𝑘−𝜔_℘𝑘𝜐^{𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜑𝜂_𝑊𝑘𝑑_𝑊𝑘≤𝜀_{𝑘∣𝑘−1}𝜑_∞𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜂_𝑊𝑘∞+𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜌_𝑘𝑑_𝑊𝑘. (B.8)

Substituting (3.1), (B.2), and (B.8) into (B.1), we have 𝜐^{𝐽𝑘∣𝑘−1}_𝑘−𝜐_𝑘,𝜑≤𝜀_{𝑘∣𝑘−1}𝜑_∞1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘∞+_{℘𝑘∠𝑍𝑘𝑊𝑘∈℘𝑘}𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜂_𝑊𝑘∞+𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜌_𝑘𝑑_𝑊𝑘+𝜔_℘𝑘𝜂_𝑊𝑘∞𝑑_𝑊𝑘. (B.9)

So that Proposition 3.2 is proved with 𝑎_𝑘=1−𝑝_𝐷,𝑘+𝑒^−𝛾𝑝_𝐷,𝑘∞+_{℘𝑘∠𝑍𝑘𝑊𝑘∈℘𝑘}𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜂_𝑊𝑘∞+𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜌_𝑘+𝜔_℘𝑘𝜂_𝑊𝑘∞𝑑_𝑊𝑘. (B.10)

This completes the proof.

C. Proof of Proposition 3.3

Clearly, by the EPHD corrector equations, (2.2)–(2.4), and the predicted EK-EPHD, (3.4), we obtain that the 𝜐^{𝑁𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘(𝐱_𝑘) in (3.5) is a Gaussian sum presented by (3.6). Now turn to the 𝜐^{𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘(𝐱_𝑘,𝑊_𝑘) in (3.5). From (2.2), we get 𝜐^{𝐷,𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑘𝐱_𝑘,𝑊_𝑘=𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘𝜂_𝑊𝑘𝐱_𝑘𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘|𝑘−1}𝐱_𝑘. (C.1) Consider the term 𝜂_𝑊𝑘(𝐱_𝑘)𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}(𝐱_𝑘) in (C.1). Using the predicted EK-EPHD, (3.4), 𝜂_𝑊𝑘𝐱_𝑘𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝐱_𝑘=𝑝_𝐷,𝑘𝑒^{−𝛾(𝐱𝑘)}_{𝐳𝑘∈𝑊𝑘}𝛾𝐱_𝑘𝜙_𝐳𝑘𝐱_𝑘𝜆_𝑘𝑐_𝑘𝐳_𝑘^{𝐽𝑘∣𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦^(𝑖)_{𝑘∣𝑘−1},𝐏^(𝑖)_{𝑘∣𝑘−1}. (C.2)

And by the result for the EK Gaussian sum filter [17], we get 𝜂_𝑊𝑘𝐱_𝑘𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝐱_𝑘⟶^{𝐽𝑘|𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝑒^{−𝛾(𝐦(𝑖)𝑘∣𝑘−1)}×_{𝐳𝑘∈𝑊𝑘}𝛾𝐦^(𝑖)_{𝑘∣𝑘−1}𝒩𝐳_𝑘∣ℎ_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1},𝟎,𝐔_𝑘^(𝑖)𝐑_𝑘𝐔_𝑘^(𝑖)𝑇+𝐇_𝑘^(𝑖)𝐏^(𝑖)_{𝑘∣𝑘−1}𝐇_𝑘^(𝑖)𝑇𝜆_𝑘𝑐_𝑘𝐳_𝑘×𝒩𝐱_𝑘∣𝐦_𝑘^(𝑖),𝐏_𝑘^(𝑖)=^{𝐽𝑘∣𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝜂_𝑊^𝐸𝐾_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦_𝑘^(𝑖),𝐏_𝑘^(𝑖), (C.3) uniformly as 𝐏^(𝑖)_{𝑘|𝑘−1}→0 for all 𝑖=1,…,𝐽_{𝑘∣𝑘−1}, and 𝜂_𝑊^𝐸𝐾_𝑘(𝐦^(𝑖)_{𝑘∣𝑘−1}), 𝐇_𝑘^(𝑖), 𝐔_𝑘^(𝑖), 𝐦_𝑘^(𝑖), 𝐏_𝑘^(𝑖) are given by (3.9)–(3.13), respectively.

Now consider the terms 𝜔^{𝐽𝑘∣𝑘−1}_℘𝑘 and 𝑑^{𝐽𝑘∣𝑘−1}_𝑊𝑘 in (C.1). First, using the predicted EK-EPHD, (3.4), the inner product ⟨𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘⟩ is given by 𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘=𝜂_𝑊𝑘𝐱_𝑘𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1}𝐱_𝑘𝑑𝐱_𝑘=𝑝_𝐷,𝑘𝑒^{−𝛾(𝐱𝑘)}_{𝐳𝑘∈𝑊𝑘}𝛾𝐱_𝑘𝜙_𝐳𝑘𝐱_𝑘𝜆_𝑘𝑐_𝑘𝐳_𝑘^{𝐽𝑘∣𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦^(𝑖)_{𝑘∣𝑘−1},𝐏^(𝑖)_{𝑘∣𝑘−1}𝑑𝐱_𝑘. (C.4)

And by the result for the EK Gaussian sum filter [17], we get 𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘⟶^{𝐽𝑘|𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝑒^{−𝛾(𝐱𝑘)}×_{𝐳𝑘∈𝑊𝑘}𝛾𝐦^(𝑖)_{𝑘∣𝑘−1}𝒩𝐳_𝑘∣ℎ_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1},𝟎,𝐔_𝑘^(𝑖)𝐑_𝑘𝐔_𝑘^(𝑖)𝑇+𝐇_𝑘^(𝑖)𝐏^(𝑖)_{𝑘∣𝑘−1}𝐇_𝑘^(𝑖)𝑇𝜆_𝑘𝑐_𝑘𝐳_𝑘×𝒩𝐱_𝑘𝐦_𝑘^(𝑖),𝐏_𝑘^(𝑖)𝑑𝐱_𝑘, (C.5) uniformly as 𝐏^(𝑖)_{𝑘|𝑘−1}→0 for all 𝑖=1,…,𝐽_{𝑘|𝑘−1}.

Changing the order of the summation and integral, (C.5) is equal to 𝜐^{𝐸𝐾,𝐽𝑘∣𝑘−1}_{𝑘∣𝑘−1},𝜂_𝑊𝑘⟶^{𝐽𝑘|𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘|𝑘−1}𝑝_𝐷,𝑘𝑒^{−𝛾(𝐦(𝑖)𝑘∣𝑘−1)}×_{𝐳𝑘∈𝑊𝑘}𝛾𝐦^(𝑖)_{𝑘∣𝑘−1}𝒩𝐳_𝑘∣ℎ_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1},𝟎,𝐔_𝑘^(𝑖)𝐑_𝑘𝐔_𝑘^(𝑖)𝑇+𝐇_𝑘^(𝑖)𝐏^(𝑖)_{𝑘∣𝑘−1}𝐇_𝑘^(𝑖)𝑇𝜆_𝑘𝑐_𝑘𝐳_𝑘×𝒩𝐱_𝑘∣𝐦_𝑘^(𝑖),𝐏_𝑘^(𝑖)𝑑𝐱_𝑘=^{𝐽𝑘|𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝜂_𝑊^𝐸𝐾_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1}𝒩𝐱_𝑘∣𝐦_𝑘^(𝑖),𝐏_𝑘^(𝑖)𝑑𝐱_𝑘=^{𝐽𝑘|𝑘−1}_𝑖=1𝑤^(𝑖)_{𝑘∣𝑘−1}𝜂_𝑊^𝐸𝐾_𝑘𝐦^(𝑖)_{𝑘∣𝑘−1}. (C.6)

Then, the expressions of 𝜔^{𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑊𝑘 and 𝑑^{𝐸𝐾,𝐽𝑘∣𝑘−1}_𝑊𝑘 (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.