Generalized Labeled Multi-Bernoulli Extended Target Tracking Based on Gaussian Process Regression

For the problems that Gamma Gaussian Inverse Wishart Cardinalized Probability Hypothesis Density (GGIW-CPHD) filter cannot accurately estimate the extended target shape and has a bad tracking performance under the condition of low SNR, a new generalized labeled multi-Bernoulli algorithm based on Gaussian process regression is proposed. The algorithm adopts the star convex to model the extended target, and realizes the online learning of the Gaussian process by constructing the state space model to complete the estimation of the extended target shape. At the same time, in the low SNR environment , the target motion state is tracked by the good tracking performance of the generalized label Bernoulli filter. Simulation results show that for any target with unknown shape, the proposed algorithm can well offer its extended shape and in the low SNR environment it can greatly improve the accuracy and stability of target tracking.


Introduction
In an extended target tracking problem, target measurements collected by the sensor are no longer one point for one target, but a set of points. By obtaining the extended form of the target from measurement sets, we can achieve better tracking and recognition performance. The extended target form can be modelled as circle, rectangle, ellipse, star convex and other geometric shapes. In 2008, Koch first proposed the random matrix [1] which modelled the target as ellipse and estimated the motion state and extended form of the target. In 2012, Granstrom applied the random matrix to the extended target probability hypothesis density(PHD) filter and proposed the Gaussian Inverse Wishart PHD (GIW-PHD)filter [2] , then he applied it to the Cardinalized probability hypothesis density (CPHD) filter and got Gamma Gaussian inverse Wishart CPHD (GGIW-CPHD)filter [3] . However, in low SNR environment, the tracking performance of above RFS algorithm is greatly reduced. In 2008, Baum proposed an extended target tracking method based on random hypersurfaces [4] . Then he modelled the targets as star-convex [5] and realized the tracking of irregular shape targets. In 2015, Gaussian process regression of machine learning is proposed to estimate extended target shape [6] , avoiding complex calculation problems of the above algorithm due to excessive parameters, and achieving the accurate estimation of the target of any unknown shape.
A new generalized labeled multi-Bernoulli algorithm based on Gaussian process regression is proposed to solve the problem that GGIW-CPHD filter cannot accurately estimate the extended target shape and has a bad tracking performance under the condition of low SNR. The algorithm improves the accuracy and stability of target tracking in low SNR environment and realizes the accurate estimation of the unknown extended target shape using the Gaussian process regression.

Gaussian process regression
At present, Gaussian process is widely used in machine learning, statistics and communication processing to classify and identify signals [7][8][9] . In simple terms, the Gaussian process can be regarded as a distribution function ( ) f u and can be defined by means functions ( ) u  and covariance function ( , ') k u u .
The Gaussian process learns the unknown function through the input training data. The measurement model is given below: . According to literature 6, it is known that the measurement value k z and function value f are subject to the joint Gaussian distribution.
From the above joint Gaussian distribution, it is easy to obtain the likelihood function and the initial prior probability: Through (6) the likelihood function is obtained, then establish the state space model, and the Gaussian process regression can be realized using the Kalman filter.

Star convex model for extended target based on Gaussian process
In this paper, the extended target shape is modeled by star convex, and the estimation of it is realized by combining the learning characteristics of Gaussian process regression. According to the definition of star convex model, the shape of the target can be represented by the radius function The radius function of the target shape can be learned through the Gaussian process Put the formula (11) and (16) in (14), the standard measurement equation can be obtained, and its expression is: , , , , 3 Generalized labeled multi-Bernoulli algorithm based on Gaussian process regression

Generalized labeled Multi-Bernoulli filter
In multiple target tracking problem, filtering methods based on traditional RFS theory cannot provide target track, therefore, Vo introduced label into RFS and put forward the general labeled multi-Bernoulli filter (GLMB) [10] [11] . In the process of Bayesian recursion, the posterior probability density of GLMB RFS has the characteristics of maintaining the same form, which can avoid the complicated integral solving process in , ; , ) (, ; ), , ; ( , ), ( ) 0 (2) tracking performance: whether it has good tracking accuracy and stability under low SNR environment.
Our experiment assumes that the extended target is shaped like a star. The new target random set can be represented by the labeled Bernoulli parameter set   . Target position and rotation direction are modeled as the uniform velocity model: Assuming the position process noise 2 1 q   , rotation direction process noise 2 0.1 The covariance function of Gaussian process [6]  The whole tracking process continuous 100s and 100 Monte Carlo experiment is conducted. The initial state of the target is shown in table 1.

Analysis of simulation result
In this paper, the simulation results are shown in figure 1-4. Fig.1 shows the extended target real tracks and filtering results of GGIW-CPHD and our proposed algorithm.  The black line , blue line and red line are respectively stand for target real tracks and real shape of the extension 、 GGIW-CPHD filtering results and shape estimation 、our proposed algorithm filtering results and shape estimation. We can see intuitively from Fig.1、 2 that in a low SNR environment, our proposed algorithm can effectively suppress clutter measurement interference and get better and sustained tracking results. Meanwhile, the proposed algorithm is more precise in shape estimation than GGIW-CPHD.  Fig.3 shows target estimation number of GGIW-CPHD and the proposed algorithm. Yellow, green shadows respectively stand for target number statistical standard deviation of GGIW -CPHD and the proposed algorithm. We can see that GGIW -CPHD and the proposed algorithm both carry on the unbiased estimation, but the standard deviation of GGIW-CPHD is bigger than the proposed algorithm which verifies that in low SNR environment, our proposed algorithm has better tracking accuracy and stability.    4 shows OSPA distance contrast of GGIW-CPHD and the proposed algorithm. It can be seen from the figure, when target number changes, the filtering effect of both algorithm has varying degrees of decline, but the proposed algorithm is smaller on the whole and proves to have a higher filtering accuracy and stability.

Conclusions
This paper presents a generalized labeled Bernoulli algorithm under Gaussian process regression. The proposed algorithm uses Gaussian process of machine learning to learn extended target shape, avoiding the existing algorithms' excessive parameters calculation complex problems, then tracking the targets based on the generalized labeled Bernoulli filter good tracking performance under low SNR environment. The experimental results show that the proposed improved algorithm can effectively track the extended target in the low SNR environment, which also has better stability and more accurate shape estimation.