## 3 Adaptive TPHD Filter with Unknown Measurement Noise
### Tphd Using VB Approximation  
The development of robust PHD gradually solves the problem of multi-target tracking in the case of unknown detection contour and unknown clutter rate. Comparatively speaking, the processing of unknown measurement noise covariance is more difficult. TPHD provides an excellent framework for us to employ variational means and can also be used to spread the posterior probability of the extended trajectory state \(\overline{X}\).  
The multi-trajectory density propagated by TPHD filter in the trajectory space is Poisson distribution, which is no longer established in extended state space from Section 3.1.  
VB-TPHD minimizes KL divergence to achieve the best Poisson approximation above all. It is assumed that the dynamic models and the measurement noise covariance are independent of each other, and therefore, \(p_{s,k}\big{(}\overline{X}\big{)}=\ p_{s,k}\big{(}x^{i-1}\big{)}\), \(p_{D,k}\big{(}\overline{X}\big{)}=\ p_{D,k}\big{(}x^{i}\big{)}\).  
**Proposition 1**.: _Given a posterior PHD \(D_{\gamma_{k-1}}(\cdot)\), VB-TPHD can be derived from the following equations:_  
\[D_{\omega_{k}}\big{(}\overline{X}\big{)}=D_{\xi_{k}}\big{(}\overline{X}\big{)} +D_{\gamma_{k}}\big{(}\overline{X}\big{)}, \tag{17}\]  
_where_  
\[D_{\gamma_{k}}\Big{(}t,x^{1:i},R^{1:i}\Big{)}=D_{\gamma}\Big{(}t,x^{1:i},R^{1: i}\Big{)}\delta_{k}[t], \tag{18}\]  
\[D_{\xi_{k}}\Big{(}t,x^{1:i},R^{1:i}\Big{)}=p_{s,k}\Big{(}x^{i-1}\Big{)}\cdot f \Big{(}x^{i}|x^{i-1}\Big{)}g\Big{(}R^{i}|R^{i-1}\Big{)}D_{\gamma_{k-1}}\Big{(} t,x^{1:i-1},R^{1:i-1}\Big{)}\delta_{N_{k}}[t]. \tag{19}\]  
Similar to the background part, VB-TPHD also only considers the survival trajectory. We put the proof of Proposition 1 in Appendix A. When \(t+i-1\
eq k\) at the end of the trajectory, the disappearing trajectory and unknown covariance sequence are not considered.  
**Proposition 2**.: _Assume that the TPHD can predict the trajectory density\(D_{\omega_{k}}\big{(}\overline{X}\big{)}\) at time \(k\), then the VB-TPHD equation can be given as follows:_  
\[D_{\gamma_{k}}\big{(}\overline{X}\big{)}=D_{\omega_{k}}\big{(} \overline{X}\big{)}\Big{(}1-\ p_{D,k}\big{(}x^{i}\big{)}\Big{)}\] (20) \[+\ p_{D,k}\big{(}x^{i}\big{)}\times\sum\limits_{z\in Z_{k}}\frac{ l_{k}\Big{(}  
Considering the \(D_{X_{k}}(x^{1:i})\) and \(D_{R_{k}}(R^{1:i})\) are both coupling, fix \(D_{R_{k}}(R^{1:i})\) to obtain \(D_{X_{k}}\big{(}x^{1:i}\big{)}\) and vice versa.  
\[D_{X_{k}}\Big{(}x^{1:i}\Big{)}\propto\exp\bigg{(}\int logD_{D_{k}}\Big{(}x^{1: i},R^{1:i},z_{k}|z_{1:k-1}\Big{)}D_{R_{k}}\Big{(}R^{1:i}\Big{)}dR\bigg{)},\]  
\[D_{R_{k}}\Big{(}R^{1:i}\Big{)}\propto\exp\bigg{(}\int logD_{D_{k}}\Big{(}x^{1: i},R^{1:i},z_{k}|z_{1:k-1}\Big{)}D_{X_{k}}\Big{(}x^{1:i}\Big{)}dX\bigg{)} \tag{25}\]  
VB-TPHD not only contains the TPHD key factors including starting time and duration namely time factors as well as trajectory state factor, but also includes track and the corresponding noise covariance sequence. For the trajectory state and covariance matrix coupled in the likelihood function, the accuracy of the estimated covariance matrix increases as the accuracy of the trajectory state estimation is improved at the same time, which is specified in the next section.