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

Feng Lian,Chongzhao Han,Hui Chen,Jing Liu

doi:10.1155/2012/141727

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.

Highlights

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
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
The additional errors from the EK-GM-EPHD filter are caused by the reason that the condition Pki|k−1 → 0 for i 1, . . . , Jk|k−1 in Proposition 3.3 is very difficult to approach in this example

Summary

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. 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. It is especially suitable for the ETT problem. The uniform convergence results for the measurement update step of the EK-GM-EPHD filter are derived from Proposition 3.3

EPHD and GM-EPHD Filters

Convergence of the GM-EPHD and EK-GM-EPHD Filters

Simulations

Conclusions and Future Work