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.

The problem of extended-target tracking (ETT) [

The ETT problem has attracted great interest in recent years. Some approaches [

Recently, the random-finite-set- (RFS-) based tracking approaches [

Given the Poisson likelihood model for the extended target [

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 [

The answer can actually be derived from Propositions

At time

The clutter is modeled as a Poisson RFS with the intensity

Given the Poisson likelihood model for the extended targets, Mahler derived the EPHD filter using finite-set statistics [

By making the same six assumptions that are made in [

Let the predicted EPHD be a GM of the form

Then, the measurement-updated EPHD is a GM given by

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 [

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.

: After the prediction step at time

: The clutter intensity

:

Given a partition

The proof of Lemma

The uniform convergence of the measurement-updated GM-EPHD is now established by Proposition

After the measurement update step of the GM-EPHD filter, there exists a real number

The proof of the Proposition

Proposition

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 [

Suppose that the predicted EK-EPHD is given by the sum of Gaussians

The proof of Proposition

From Propositions

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.

Consider a two-dimensional scenario with an unknown and time varying number of the extended targets observed over the region

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 [

At time

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 clutter is modeled as a Poisson RFS with the intensity

Figure

The true trajectories for extended targets and sensor location.

In Figure

The intensity of the extended-target birth at time

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

The standard deviation of the estimated cardinality distribution and the optimal subpattern assignment (OSPA) multitarget miss distance [

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the GM-EPHD filter in various

Gaussian number |
50 | 100 | 300 | 500 | 700 |
---|---|---|---|---|---|

Time averaged standard deviation of the estimated cardinality distribution from the GM-EPHD filter | 2.12 | 1.39 | 0.97 | 0.71 | 0.58 |

OSPA (m) from the GM-EPHD filter | 83.5 | 58.7 | 49.6 | 43.1 | 39.5 |

Table

Given

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the GM-EPHD filter in various

Clutter rate |
50 | 100 | 200 | 300 | 400 |
---|---|---|---|---|---|

Time averaged standard deviation of the estimated cardinality distribution from the GM-EPHD filter | 0.58 | 0.70 | 0.95 | 1.23 | 1.48 |

OSPA (m) from the GM-EPHD filter | 39.5 | 42.9 | 49.0 | 56.1 | 61.7 |

From Table

The experiment settings are the same as those of Example

The EK-GM-EPHD filter is used to estimate the number and states of the extended targets in the nonlinear ETT problem. Given

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the EK-GM-EPHD filter in various

Gaussian number |
50 | 100 | 300 | 500 | 700 |
---|---|---|---|---|---|

Time averaged standard deviation of the estimated cardinality distribution from the EK-GM-EPHD filter | 3.15 | 2.29 | 1.77 | 1.21 | 0.76 |

OSPA (m) from the EK-GM-EPHD filter | 93.2 | 86.7 | 75.3 | 54.6 | 43.9 |

Time averaged standard deviation of the estimated cardinality distribution and time averaged OSPA (m) from the EK-GM-EPHD filter in various

Clutter rate |
50 | 100 | 200 | 300 | 400 |
---|---|---|---|---|---|

Time averaged standard deviation of the estimated cardinality distribution from the EK-GM-EPHD filter | 0.76 | 0.92 | 1.29 | 1.61 | 1.92 |

OSPA (m) from the EK-GM-EPHD filter | 43.9 | 48.1 | 55.8 | 67.8 | 79.5 |

As expected, Tables

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.

We have known that

For the initial induction step, assume

In the case of

Assume that we have established (

Since

By the EPHD corrector equations, (

Using the triangle inequality, the first term in the summation of (

Using the triangle inequality again for the term

Using Lemma

Then, (

Substitute (

Substituting (

So that Proposition

This completes the proof.

Clearly, by the EPHD corrector equations, (

And by the result for the EK Gaussian sum filter [

Now consider the terms

And by the result for the EK Gaussian sum filter [

Changing the order of the summation and integral, (

Then, the expressions of

Finally, (

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.