**2. ET-GIW-PHD Review**

In ET-GIW-PHD, both predicted PHD and corrected PHD can be approximated as an unnormalized mixture of Gaussian inverse Wishart distributions. Let *<sup>ζ</sup><sup>k</sup>* <sup>=</sup> {*xk*, **<sup>X</sup>***k*} be the sufficient statistics of the GIW components at time which contains kinematical state *x<sup>k</sup>* and extension state **X***<sup>k</sup>* which is mathematically described by a symmetric and positively definite (SPD) random matrix. The iterative formulae for *ξ<sup>k</sup>* are obtained in [39]. More implementation details, such as pruning and merging, can also be found in [39].

Prediction:

$$D\_{k+1|k}(\mathfrak{F}\_{k+1}) = \int p\_s(\mathfrak{F}\_k) p\_{k+1|k}(\mathfrak{F}\_{k+1}|\mathfrak{F}\_k) \times D\_{k|k}(\mathfrak{F}\_k) d\mathfrak{F}\_k + D\_{k+1}^b(\mathfrak{F}\_{k+1}),\tag{1}$$

where *ps*(·) is the probability of survival, *pk*<sup>+</sup>1|*k*(*ξk*+1|*ξk*) is the state transition density, *<sup>D</sup><sup>b</sup> <sup>k</sup>*+1(·) is the birth PHD, new target spawning is omitted [39].

Correction:

The corrected PHD *Dk*|*k*(*ξk*) can be summarized as:

$$D\_{k|k}(\mathfrak{F}\_k) = D\_{k|k}^{ND}(\mathfrak{F}\_k) + \sum\_{\mathbf{p} \in \mathbb{Z}\_k} \sum\_{\mathcal{W} \in p} D\_{k|k}^D(\mathfrak{F}\_{k'} \mathcal{W})\_{\mathbf{\vargamma}} \tag{2}$$

where p∠**Z***<sup>k</sup>* means that the measurement sets **Z***<sup>k</sup>* are partitioned into non-empty cells, *W* ∈ p means that the cell *W* is in the partition p.

*DND <sup>k</sup>*|*<sup>k</sup>* (*ξk*) handles the undetected target case, because *Dk*<sup>+</sup>1|*k*(*ξk*+1) is approximated as an unnormalized mixture of Gaussian inverse Wishart distributions, it is given by

$$D\_{k|k}^{ND}(\mathbf{f}\_k) = \sum\_{j=1}^{J\_{k|k-1}} w\_{k|k}^j \mathcal{N}(\mathbf{x}\_k; \mathfrak{m}\_{k|k'}^{(j)} \mathbf{P}\_{k|k}^{(j)} \otimes \mathbf{X}\_k) \mathcal{D}\mathcal{W}(\mathbf{X}\_k; \mathfrak{v}\_{k|k'}^{(j)} \mathbf{V}\_{k|k}^{(j)}),\tag{3}$$

where *Jk*|*k*−<sup>1</sup> is the number of components of predicted PHD, *<sup>w</sup>*(*j*) *<sup>k</sup>*|*<sup>k</sup>* is the weight of GIW component. <sup>N</sup> (*x*; *<sup>m</sup>*, **<sup>P</sup>**) means that a vector *<sup>x</sup>* is subject to Gaussian distribution with mean *<sup>m</sup>* and covariance **<sup>P</sup>**, IW(**X**; *v*, **V**) means that a matrix is subject to inverse Wishart distribution with degree of freedom *v* and inverse scale matrix **V**. ⊗ is the Kronecker product.

*D<sup>D</sup> <sup>k</sup>*|*k*(*ξk*, *<sup>W</sup>*) handles the detected target case, which is given by

$$D^D\_{k|k}(\mathbf{f}\_{k^\*}\mathcal{W}) = \sum\_{j=1}^{J\_{k|k-1}} w\_{k|k}^{(j,\mathcal{W})} \mathcal{N}(\mathbf{x}\_k; \mathfrak{m}^{(j,\mathcal{W})}\_{k|k}, \mathbf{P}^{(j,\mathcal{W})}\_{k|k} \otimes \mathbf{X}\_k) \mathcal{T}\mathcal{W}(\mathbf{X}\_k; \mathfrak{v}^{(j,\mathcal{W})}\_{k|k}, \mathbf{V}^{(j,\mathcal{W})}\_{k|k}).\tag{4}$$

*<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* can be obtained by

$$w\_{k|k}^{(j, \mathcal{W})} = \frac{\omega\_p}{d\_{\mathcal{W}}} e^{-\gamma^{(j)}} \left(\frac{\gamma^{(j)}}{\lambda\_c c\_k}\right)^{|\mathcal{W}|} p\_D^{(j)} \Lambda\_k^{(j, \mathcal{W})} w\_{k|k-1'}^{(j)} \tag{5}$$

where

$$d\_{\mathcal{W}} = \delta\_{|\mathcal{W}|,1} + \sum\_{j=1}^{I\_{k|k-1}} e^{-\gamma(j)} \left(\frac{\gamma^{(j)}}{\lambda\_c c\_k}\right)^{|\mathcal{W}|} p\_D^{(j)} \Lambda\_k^{(j, \mathcal{W})} w\_{k|k-1}^{(j)} \tag{6}$$

$$\Lambda\_k^{(j,\mathcal{W})} = \frac{1}{\left(\pi^{|\mathcal{W}|} |\mathcal{W}| \mathbf{S}\_{k|k-1}^{(j,\mathcal{W})}\right)^{\frac{\nu\_k^{(j)}}{2}}} \frac{|\mathbf{V}\_{k|k-1}^{(j)}|^{\frac{\nu\_{k|k-1}^{(j,\mathcal{W})}}{2}} \Gamma\_d(\frac{\nu\_{k|k}^{(j,\mathcal{W})}}{2})}{|\mathbf{V}\_{k|k-1}^{(j,\mathcal{W})}|^{\frac{\nu\_{k|k}^{(j,\mathcal{W})}}{2}} \Gamma\_d(\frac{\nu\_{k|k-1}^{(j)}}{2})} \tag{7}$$

$$
\omega\_p = \frac{\prod\_{W \in P} d\_W}{\sum\_{P' \angle Z\_k} \prod\_{W' \in P'} d\_{W'}}.\tag{8}
$$

Λ(*j*,*W*) *<sup>k</sup>* presents the likelihood of the jth GIW component given the measurements of the Wth cell, *ω<sup>p</sup>* is the weight of pth partition, *p* (*j*) *<sup>D</sup>* is the detection probability of jth GIW component, *<sup>γ</sup>*(*j*) is the expected number of measurements generated by jth GIW component, *λ<sup>c</sup>* is the mean number of clutter measurements, *ck* is the spatial distribution of the clutter over the surveillance volume, *δi*,*<sup>j</sup>* is the Kronecker delta, <sup>|</sup>*W*<sup>|</sup> is the the number of measurements in the Wth cell, **<sup>S</sup>**(*j*,*W*) *<sup>k</sup>*|*k*−<sup>1</sup> is innovation factor, Γ*d*(·) is the multivariate Gamma function.
