Abstract
The Probability Hypothesis Density (PHD) and Cardinalized PHD (CPHD) filters are popular solutions to the multi-target tracking problem due to their low complexity and ability to estimate the number and states of targets in cluttered environments. The PHD filter propagates the first-order moment (i.e. mean) of the number of targets while the CPHD propagates the cardinality distribution in the number of targets, albeit for a greater computational cost. Introducing the Panjer point process, this paper proposes a second-order PHD filter, propagating the second-order moment (i.e. variance) of the number of targets alongside its mean. The resulting algorithm is more versatile in the modelling choices than the PHD filter, and its computational cost is significantly lower compared to the CPHD filter. The paper compares the three filters in statistical simulations which demonstrate that the proposed filter reacts more quickly to changes in the number of targets, i.e., target births and target deaths, than the CPHD filter. In addition, a new statistic for multi-object filters is introduced in order to study the correlation between the estimated number of targets in different regions of the state space, and propose a quantitative analysis of the spooky effect for the three filters.
Highlights
In the context of multi-target detection and tracking problems, methods based on the Random Finite Set (RFS) framework have recently attracted a lot of attention due to the development of low-complexity algorithms within this methodology [1]
The best-known algorithm is perhaps the Probability Hypothesis Density (PHD) filter that jointly estimates the number of targets and their states by propagating the firstorder moment of a RFS [2]; a Gaussian Mixture (GM) and a Sequential Monte Carlo (SMC) implementation have been presented in [3] and [4]
In 2007, Vo et al showed that the Cardinalized PHD (CPHD) filter can be overconfident in some cases [8], and in 2009, Franken et al identified a counter-intuitive property of the CPHD filter that occurs with the weights of the targets when they are miss-detected which they called the spooky effect [9]
Summary
In the context of multi-target detection and tracking problems, methods based on the Random Finite Set (RFS) framework have recently attracted a lot of attention due to the development of low-complexity algorithms within this methodology [1]. Mahler derived the Cardinalized PHD (CPHD) filter which propagates the cardinality distribution of the target point process alongside its first-order moment [6] It provides higher-order information on the number of targets, but to the expense of a higher computational cost. The proposed solution complements the original PHD filter with the variance in the estimated number of targets; it propagates less information than the CPHD filter but has a lower computational cost. We exploit the statistical tools introduced in this paper in order to study the correlation in the estimated number of targets in disjoint regions of the state space, and provide a quantitative analysis of the wellknown spooky effect [9] for the PHD filter, CPHD filter, and the proposed second-order PHD filter. Pseudo-code and detailed proofs for the proposed algorithms are given in the appendix
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