^{1}

As far as the increasing number of mixture components in the Gaussian mixture PHD filter is concerned, an iterative mixture component pruning algorithm is proposed. The pruning algorithm is based on maximizing the posterior probability density of the mixture weights. The entropy distribution of the mixture weights is adopted as the prior distribution of mixture component parameters. The iterative update formulations of the mixture weights are derived by Lagrange multiplier and Lambert

The objective of multitarget tracking is to estimate target number and target states from a sequence of noisy and cluttered measurement sets. The tracked target is generally simplified as a point [

As far as the Gaussian mixture implementation of the PHD filter is concerned, it approximates the PHD by the summation of weighted Gaussian components under the multitarget linear Gaussian assumptions [

The remaining parts of this paper are organized as follows. Section

The predictor and connector of PHD filter [

Under the linear Gaussian assumptions, the Gaussian mixture PHD filter is derived in [

It can be seen from formula (

For simplicity, the time index

The entropy distribution of the mixture weights is adopted as the prior of

Formula (

In formula (

Formula (

At the normalization step of the iteration procedure, if a mixture weight becomes negative, the corresponding component is removed from the mixture components by setting its weight to zero. The removed mixture component will not be considered when the log-posterior is computed in the following iterations. The mixing weights of survival mixture components are normalized at the end of this step.

The effect of entropy distribution of mixing weights is taken during the iterative procedure. The mixture weights of components negligible to the PHD become smaller and smaller iteration by iteration, since the parameter estimates are driven into low-entropy direction by entropy distribution. The low-entropy tendency can also promote competition among the mixture components with similar parameters which can then be merged into one mixture component with larger weight.

For the mean

The main steps of iterative mixture component pruning algorithm are summarized in Algorithm

(1) normalize

(2)

(3)

(4)

(5)

(6) compute

(7)

(8)

(9)

(10) compute

(11)

(12)

(13) compute

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25) compute

(26) compute

(27)

(28)

(29) normalize

(30) compute

(31)

(32) goto step 3;

(33)

A two-dimensional scenario with unknown and time-varying target number is considered to test the proposed iterative mixture component pruning algorithm. The surveillance region is [−1000, 1000]

Each target is detected with probability

The means of the Gaussian mixture components with mixing weights greater than 0.5 are chosen as the estimates of multitarget states after the mixture reduction.

The tracking results in one Monte Carlo trial are presented in Figures

True traces and estimates of

True traces and estimates of

The mixture components with weights larger than 0.0005 at the 86th time step before pruning operation in the above Monte Carlo simulation trial are presented in Figure

Components before pruning operation.

Components after pruning operation.

The typical mixture component pruning algorithm based on thresholds in [

The averaged Wasserstein distances.

Figure

Estimates of target numbers.

Figure

The averaged component numbers.

The case of low signal-to-noise rate (SNR) is yet considered for the further comparison of two algorithms.

The averaged Wasserstein distances under low SNR.

Estimates of target numbers under low SNR.

The averaged component numbers under low SNR.

An iterative mixture component pruning algorithm is proposed for the Gaussian mixture PHD filter. The entropy distribution of the mixture weights is used as the prior distribution of mixture parameters. The update formula of the mixture weight is derived by Lagrange multiplier and Lambert

The author declares that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (Grant no. 61304261), the Senior Professionals Scientific Research Foundation of Jiangsu University (Grant no. 12JDG076), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).