Gaussian-Student’s t mixture distribution PHD robust filtering algorithm based on variational Bayesian inference①

HU Zhentao(胡振涛), YANG Linlin,HU Yumei②, YANG Shibo

(∗School of Artificial Intelligence, Henan University, Zhengzhou 450046, P.R.China)(∗∗School of Automation, Northwestern Polytechnical University, Xi’an 710029, P.R.China)

Abstract Aiming at the problem of filtering precision degradation caused by the random outliers of process noise and measurement noise in multi-target tracking (MTT) system, a new Gaussian-Student’s t mixture distribution probability hypothesis density (PHD) robust filtering algorithm based on variational Bayesian inference (GST-vbPHD) is proposed. Firstly, since it can accurately describe the heavy-tailed characteristics of noise with outliers, Gaussian-Student’s t mixture distribution is employed to model process noise and measurement noise respectively. Then Bernoulli random variable is introduced to correct the likelihood distribution of the mixture probability, leading hierarchical Gaussian distribution constructed by the Gaussian-Student’s t mixture distribution suitable to model non-stationary noise. Finally, the approximate solutions including target weights,measurement noise covariance and state estimation error covariance are obtained according to variational Bayesian inference approach. The simulation results show that, in the heavy-tailed noise environment, the proposed algorithm leads to strong improvements over the traditional PHD filter and the Student’s t distribution PHD filter.

Key words: multi-target tracking (MTT), variational Bayesian inference, Gaussian-Student’s t mixture distribution, heavy-tailed noise

0 Introduction

Multi-target tracking (MTT) technique based on point measurements is used to real-time estimate the number of targets, status, trajectory, and other attribute information with the processing of measurement information. The traditional implementation of MTT generally adopts the data association strategies, such as joint probabilistic data association (JPDA)[1], multihypothesis tracking ( MHT)[2], and probabilistic multi-hypothesis tracking (PMHT)[3]. However,these above methods cannot deal well with the time-varying characteristics of the target state, i.e. the time-varying number of targets makes it difficult to achieve an effective correlation between the state set and the measurement set of the target. Recently, since bypassing the complex data association,the MTT based on random finite set (RFS) theory and its improvements have attracted extensive attention[4]. Specifically, their complexity and track ability are better than those methods using data association strategy. A typical implementation mentioned above is the probability hypothesis density (PHD) filter which recursively solves the state posterior first-order statistical moments, thus gives the first engineering implementation of RFS[5]. The existing PHD filter implementation strategies mainly include sequential Monte Carlo PHD (SMC-PHD)[6]and Gaussian mixture PHD (GM-PHD)[7-8].

In practical engineering applications, the noise outlier induced by electromagnetic interference, aging of the sensor, and uncertainty of the dynamic model will deteriorate PHD filter tracking accuracy. Besides,the outlier-containing noise usually exhibits heavytailed characteristics. However, traditional GM-PHD suffers poor robustness at heavy-tailed process noise and measurement noise existing[9]. Under the condition of Gaussian distribution, the SMC-PHD filter may partially relieve the above problem with high computational cost. Huber’s M-estimation theory can be used to improve the GM-PHD filter’s performance when outliers exist in the measurement model, but it cannot deal with outliers in process noisy. Moreover, since it is based on Gaussian distribution approximation, GMPHD filter may induce biased estimates on the state and number of targets, thus is unsuitable to handle non-Gaussian noise system model with noisy outliers[10]. Existing literatures show that heavy-tailed noise may not be efficiently tackled in Gaussian noise hypothetical scenario, so heavy-tailed noise modeling becomes the key to deal with multi-target tracking problem with noise outliers[11].

Since Student’s t distribution exhibits heavier tail than Gaussian distribution and converges to Gaussian distribution as its freedom increasing, it may be suitable for modeling non-Gaussian noise with significant heavy-tailed. Assuming the measurement noise follows Student’s t distribution, Li et al.[12]proposed a robust PHD filter which used VB to update the posterior likelihood function, but the method is unsuitable for noisy outliers. Liu et al.[13]presented a robust Student’s t mixture PHD filter by recursively propagating the intensity as a mixture of Student’s t components in PHD filtering framework. In addition, to alleviate the unfavorable effects on filtering performance induced by heavytailed noise, Liu re-weighted on true measurement,outliers and clutter according to their value, and proposed M-estimation based dual-gating strategy to construct a Student’s t mixture distribution. With approximately regarding the process noise and measurement noise as the Student’s t distribution, Hong et al.[14]proposed a Student’s t mixture particle PHD (STMPPHD) filter. They argued that the intensity of the multi-target may be approximated by using a Student’s t mixture model,while Monte Carlo is utilized to calculate the Student’ s t function integral, leading to a closed Student’s t hybrid recursive framework. However, few literatures above focus on improving the filtering robustness by using variational Bayesian inference. Zhang et al.[15]designed a robust Student’s t based labeled multi-Bernoulli ( RSTLMB ) filter through modeling the Student’s t distribution with the state prediction probability density and the measurement likelihood function of individual targets. Moreover, a closed recursion filter is proposed to jointly estimate the target state and the parameters of the Student’s t distribution. Due to the random occurrence of the outliers in noise, RSTLMB hardly model nonstationarity of noise by using one single Student’s t distribution.

Obviously, using fixed inverse scale matrix or Student’s t distribution can hardly model random noise with heavy-tailed outliers. To address the above problem, a new Gaussian-Student’s t mixture distribution PHD robust filtering algorithm based on variational Bayesian inference (GST-vbPHD) is proposed here.The main contributions are summarized as follows.

(1) Random outliers existing in process noise and measurement noise are modeled as Gaussian-Student’s t mixture distribution, in addition, the parameters of the mixture distribution and kinematic state are integrated in the augmentation matrix.

(2) Bernoulli random variables are introduced to transform the mixture distribution model of noise outliers into a hierarchical Gaussian form in which parameters including targets states and weights are updated by variational Bayesian inference.

(3) In different experiment scenarios, two types of performance indicators, the optimal subpattern assignment (OSPA) distance and the accuracy of the target number estimation, are used to verify the feasibility and validity of the proposed algorithm. The experiment results demonstrate that the proposed algorithm outperforms the comparison methods on tracking accuracy.

1 Gaussian mixed PHD filter

whereSk∣k-1(x) andBk|k-1(x) denote the random finite sets of survival targets and spawned targets fromXk-1at timek,respectively.Γkis the random finite sets of birth targets at timek. Θ(x) andκkdenote respectively the observed random sets generated by targets and clutters at timek.

The PHD filter estimates the states of targets and its number by iteratively propagating the posterior intensity, which is a first order statistic of the random finite set[7]. The linear Gaussian MTT system develops Gaussian mixture implementation in finding the analytic solution of the Bayesian integral, the process of which clearly demonstrates how the Gaussian components propagate analytically to the next moment. Assume that the prior intensity functionvk-1at timek- 1 obeys the Gaussian distribution

2 GST-vbPHD filter

2.1 Gaussian-Student’s t mixture distribution

The outliers of the process noise and measurement noise may appear at different moments in practical engineering application, resulting in the non-stationarity characteristic of the non-Gaussian noise. The modeling of noise with outliers, by adopting the mixed probabilityηas a mixed Gaussian-Student’s t distribution, is as follows

The auxiliary variableλis introduced to transform the mixed Gaussian-Student’s t distribution into the following hierarchical Gaussian form[16]

2.2 GST-vbPHD filting

2.2.1 Predict

Combined with the model constructed by Eq.(9)and Eq.(10), the implementation of GST-vbPHD is derived for linear multi-target systems. According to Bayesian probability theory, a beta distribution is selected as the conjugate prior distribution of unknown mixing probabilityηk[16].The likelihood distribution of theηkis expressed as Bernoulli distribution, and Bernoulli componentεkis introduced to select Gaussian or Student’ s t distribution. It is well known that the Student’s t distribution can be expressed as the product of gamma distribution and Gaussian distribution after introducing auxiliary variablesλk.

Suppose the augmented stateϑkof one single target, which contains one single target state and a set of parameters for constructing the distribution, can be represented asϑk≜(xk,ηk,εk,λk),whereηk,εkandλkrespectively refer to the mixed probability, Bernoulli random variables and auxiliary variables, and they are mutually independent ofxk.The components of the predicted intensity are the same as Eq.(1),so the mixed distribution model of joint probability density is expressed as

Algorithm 1 The variational iteration process for each Gaussian Student’s t mixture components Inputs:m(j)k|k-1, P(j)k|k-1, w(j)k|k-1, Hk, Rk, zk, PD,k, r, e1, e2,ω1, ω2, N 1. Initialization:E(0)[γ] = 1,E(0)[logγ] = 0,E(0)[ε] = 1,E(0)[log(1 - σ)] = nψ(1 - e) - ψ(1)E(0)[logσ] = ψ(e) - ψ(1),2. m(j)(0)k = m(j)k∣k-1, P(j)(0)k = P(j)k∣k-1 3. for n = 0: N -1 do 4. Calculate ～P(j)(n)k|k-1 and ～R(j)(n)k using Eqs(48) and (49)5. Calculate m(j)(n+1)k and P(j)(n+1)k using Eqs(45) -(47)6. Calculate w(j)(n+1)k (z) using Eqs(43) -(44)7. Calculate A(n+1)k , B(n+1)k using Eqs(39) - (40) Update qn+1(ε1,k), qn+1(ε2,k) as Bernoulli distributions

8. Calculate Prn+1(ε1,k = 1), Prn+1(ε1,k = 0) and Prn+1(ε2,k = 1),Prn+1(ε2,k = 0) using Eqs(33) -(36)9. Calculate En+1[ε1,k], En+1[ε2,k] using Eq.(50)Update qn+1(γ1,k), qn+1(γ2,k) as Gamma distributions 10. Calculate wn+1 1,k, hn+1 1,k and wn+12,k, hn+1 2,k using Eqs(29) -(32)11. Calculate En+1[γ1,k], En+1[γ2,k] and En+1[logγ1,k], En+1[logγ2,k] using Eqs(51) -(52)Update qn+1(σ1,k), qn+1(σ2,k) as Bate distributions 12. Calculate en+1 1,k, tn+1 1,k and en+12,k, tn+1 2,k using Eqs(25) -(28)13. Calculate En+1[logσ1,k], En+1[logσ2,k] and En+1[log(1 - σ1,k)], En+1[log(1 - σ2,k)] using Eqs(39) -(40)14. if m(j+1)(n+1)k - m(j+1)(n)k ≤ε then 15. Stop the iteration 16. end if 17. end for Outputs:m(j)k 、P(j)k and w(j)k

3 Simulation results and analysis

3.1 Scenario design

To verify the tracking performance of the proposed algorithm, two simulation scenarios are designed in the 2-D plane, i.e. the scenario of the measurement noise with outliers and the scenario of outliers in both process noise and measurement noise. In addition, the different probabilities of generating outliers are compared in the two scenarios. For comparison, the tracking performance of the Gaussian mixture PHD filter (GMPHD), the robust Student’s t based PHD filter (RSTPHD) and the GST-vbPHD are employed.

Assuming that there are four targets in the surveillance range,they are present at time [1 8 12 26](s)until time [20 25 25 40](s) in turn disappears,with uniform motion during the survival period. The real trajectories of all targets are plotted in Fig.1.

Fig.1 The true trajectory of multiple targets

3.2 Simulation parameters

A total of 40 steps are running in the simulation process, and the simulation results are the average after 300 Monte Carlo (MC) trials. What is more, the objective survival probabilityPS,k= 0.99, detection probabilityPD,k= 0.98 and the clutter rateλ= 3.The state and measurement equations of the targets are modeled as the following form

3.3 Results and analysis

3.3.1 Scenario 1

To observe the performance of MTT at measurement outliers existing, the measurement noise covariance is constructed with outlier according to Ref.[21].

where,p1 is the probability of the measurement noise without outliers and is set to be in a range of 5 -30 s during the multi-target motion.

Fig.2 shows the OSPA distance errors for the three filters with probabilityp1 = 0. 98. Due to the measurement outliers, it can be observed that the GMPHD has significantly inferior tracking performance to the other two filters. Specifically, when there are outliers in the measurement,because of the light-weight tail property of Gaussian distribution, the weight of Gaussian components tends to be a small value or even zero in some cases, which leads to a larger OSPA distance. In contrast, although RST-PHD takes into account the heavy-tailed feature of noise, it uses a fixed Student’s t distribution for modeling, which lacks robustness to randomly occurring outliers. The OSPA distance curve of GST-vbPHD is lower than that of GM-PHD and RSTPHD, which demonstrates that the tracking performance of GST-vbPHD surpasses the other two algorithms. Due to the random characteristic of the measurement noise outliers, the noise cannot always remain in a heavy-tailed or Gaussian distribution state. GSTvbPHD employs the model with mixture distribution to better estimate the target weights, which helps to track the target without loss. The results of three algorithms for estimating the number of targets are given in Fig.3,and it can be seen that GST-vbPHD significantly outperforms GM-PHD and RST-PHD.

Fig.2 The OSPA distance

Fig.3 The number of targets

To further analyze the impact of the probabilityp1on the filter performance, the statistical analysis on the average OSPA distance is shown in Fig.4. When the probability of the measurement noise without outliers is 0.98,0.96,0.94,0.92 and 0.9 respectively, OSPA average distance all decrease with increasing of the probability of the measurement noise, and that is because the effect of measurement noise outliers on the system significantly weakens. GST-vbPHD has a lower OSPA average distance than GM-PHD and RST-PHD,and has better tracking accuracy under lighter-tailed measurements or even heavy-tailed measurements.

Fig.4 Average OSPA with different p1

3.3.2 Scenario 2

To evaluate the performance of MTT at outlier both existing in process noise and measurement noise,a new experiment Scenario 2 is constructed. The measurement noise outliers can be generated according to Scenario 1, and the process noise covariance with outliers is shown as follows.

wherep2is the probability of the process noise without outliers. Assume that the time period of outliers in the noise is the same as Scenario 1.

For outliers of the process noise and the measurement noise with the same probability, Fig.5 shows the OSPA distance comparison of three filters. From Fig.5 and Fig.6, it can be found that GST-vbPHD shows a better tracking result for both the tracking accuracy and the estimation of target number. The process noise outliers may be induced by target maneuvers, while the GM-PHD filter cannot capture the target due to the light tail of the Gaussian distribution, and RST-PHD lacks adaptability to random outliers. The proposed algorithm utilizes the mixture distribution model of noise to correct state error covariance, effectively eliminates the adverse effects induced by process noise outliers.Overall, if there are outliers in both process noise and measurement noise, the GST-vbPHD can achieve reliable and effective performance in MTT.

Fig.5 The OSPA distance

Fig.6 The number of targets

In order to deeply analyze the tracking performance of the filters under different probabilities of outliers, the additional experiments are executed. First,p2is fixed, whilep1is 0. 9, 0. 92, 0. 94, 0. 96, and 0.98 respectively. After that, the average OSPA distance is given in Fig.7. On the contrary, whenp1is fixed,p2changes and the corresponding simulation results are shown in Fig.8. The average OSPA distance of the three filters gradually decreases, which means that the poor tracking performance with the occurrence probability of outliers increases. In Fig.8, the average OSPA distance of the three filters decreases less obviously than that in Fig.7. The results can be attributed to the different multiples of setting outliers in the dynamic model, and the filter is more sensitive to the process noise. As shown in Fig.7 and Fig.8, the proposed algorithm achieves relatively stable tracking performance for different probabilities of outliers in process noise and measurement noise.

Fig.7 Average OSPA with different p1

Fig.8 Average OSPA with different p2

The computation time of the RST-PHD and the GST-vbPHD in this paper is 1.4 s and 3.5 s respectively when both the variational approximation iterations are used. The proposed algorithm not only improves the accuracy of tracking and estimating the number of targets, but also increases the operation time. This is because we consider that both process noise and measurement noise may have outliers. Two sets of parameters are used to modify the measurement noise covariance and state error covariance respectively, and participate in the variational iteration. The contrast algorithm only considers the heavy tail characteristics of noise outliers, but ignores the associate the nonstationarity. It simply uses student t distribution to model the noise.Therefore, the proposed algorithm in this paper does not perform well on the evaluation index of operation time.

4 Conclusions

In this paper, a new Gaussian-Student’s t mixture distribution PHD robust filtering algorithm is proposed based on variational Bayesian inference, which models the one-step state prediction PDF and the measurement likelihood PDF as the hierarchical Gaussian forms. Concretely, the hierarchical Gaussian form is employed to correct state error covariance matrix and measurement noise covariance matrix, eliminating the adverse effects of process noise and measurement noise both with outliers on the tracking performance. In addition, the parameters in the mixed distribution term are iteratively optimized by variational inference to obtain the target posterior probability density. The simulation results show that the proposed algorithm can achieve competitive performance with the traditional Gaussian hybrid PHD filter and the Student’s t PHD filter on tracking accuracy in MTT. Future work will focus on how to construct a hierarchical Gaussian noise distribution for nonlinear systems to effectively solve the influence of noise outliers.