*2.2. SMC-PHD Filter*

Up to now, PHD filters can be realized by SMC approximation or GM approximation. Compared with the GM-PHD filter, the SMC-PHD filter is suitable for problems involving non-linear non-Gaussian dynamics. Regardless of spawned targets, the following sequentially describes each of the SMC-PHD filter processing steps: initialization, prediction, correction, and state estimation.

Initialization: Suppose prior PHD at time 0 is

$$D\_{0|0}(\mathbf{x}) \approx \sum\_{i=1}^{v\_{0|0}} w\_{0|0}^i \delta(\mathbf{x} - \mathbf{x}\_{0|0}^i),\tag{3}$$

where <sup>δ</sup>(*x*) is Dirac delta function, *<sup>v</sup>*0|<sup>0</sup> is the number of particles, *<sup>x</sup><sup>i</sup>* <sup>0</sup>|<sup>0</sup> is the *<sup>i</sup>*th particle and *<sup>w</sup><sup>i</sup>* <sup>0</sup>|<sup>0</sup> is the corresponding weight.

Prediction: Suppose PHD at time *k* − 1 can be approximated using a group of particles

$$D\_{k-1|k-1}(x) \approx \sum\_{i=1}^{v\_{k-1|k-1}} w\_{k-1|k-1}^i \delta(x - x\_{k-1|k-1}^i). \tag{4}$$

The meaning of the variables in the above formula is similar to that of Equation (3). Then the predicted PHD at time *k* is

$$D\_{k|k-1}(\mathbf{x}) \approx \sum\_{i=1}^{\mathcal{V}\_{k|k-1}} w\_{k|k-1}^{i} \delta(\mathbf{x} - \mathbf{x}\_{k|k-1}^{i}),\tag{5}$$

where *vk*|*k*−<sup>1</sup> <sup>=</sup> *vk*−1|*k*−<sup>1</sup> <sup>+</sup> *<sup>v</sup>birth <sup>k</sup>*|*k*−<sup>1</sup> is the number of predicted particles, *vbirth <sup>k</sup>*|*k*−<sup>1</sup> is the number of appearing particles, *xi k*|*k*−1 , *<sup>i</sup>* <sup>=</sup> 1, ··· , *vk*−1|*k*−<sup>1</sup> is obtained by the single-target Markov transition density, *<sup>x</sup><sup>i</sup> k*|*k*−1 , *i* = *vk*−1|*k*−<sup>1</sup> + 1, ··· , *vk*|*k*−<sup>1</sup> is sampled from the probability density of the spontaneously appearing targets, *wi <sup>k</sup>*|*k*−<sup>1</sup> <sup>=</sup> *pS*(*xi <sup>k</sup>*−1|*k*−1) · *<sup>w</sup><sup>i</sup> k*−1|*k*−1 , *i* = 1, ··· , *vk*−1|*k*−<sup>1</sup> is the weight corresponding to persisting particles, *wi <sup>k</sup>*|*k*−<sup>1</sup> <sup>=</sup> 1/ρ, *<sup>i</sup>* <sup>=</sup> *vk*−1|*k*−<sup>1</sup> <sup>+</sup> 1, ··· , *vk*|*k*−<sup>1</sup> is the weight corresponding to appearing particles, and the PHD filter requires ρ particles to adequately maintain track on any individual target.

Correction: After receiving the multi-target measurement set, the posterior PHD at time *k* can be approximated as

$$D\_{k|k}(\mathbf{x}) \approx \sum\_{i=1}^{v\_{k|k}} w\_{k|k}^{i} \delta(\mathbf{x} - \mathbf{x}\_{k|k}^{i}) \, \tag{6}$$

where *vk*|*<sup>k</sup>* <sup>=</sup> *vk*|*k*−<sup>1</sup> and *<sup>x</sup><sup>i</sup> <sup>k</sup>*|*<sup>k</sup>* <sup>=</sup> *xi k*|*k*−1 , *i* = 1, ··· , *vk*|*<sup>k</sup>* are the same as those of the predicted PHD, and *i*th particle weight can be calculated by

$$w\_{k\mid k}^i = w\_{k\mid k-1}^i p\_D(\mathbf{x}\_{k\mid k-1}^i) \sum\_{z \in \mathbb{Z}\_k} \frac{L\_z(\mathbf{x}\_{k\mid k-1}^i)}{\lambda c(z) + \sum\_{\varepsilon=1}^{\varpi\_{k\mid k-1}} w\_{k\mid k-1}^\varepsilon p\_D(\mathbf{x}\_{k\mid k-1}^\varepsilon) L\_z(\mathbf{x}\_{k\mid k-1}^\varepsilon)} + w\_{k\mid k-1}^i \left[1 - p\_D(\mathbf{x}\_{k\mid k-1}^i)\right]. \tag{7}$$

The above particle weights are not equal, and the resampling technique can be utilized to replace them with new, equal weights.

State estimation: The expected number of targets at time *<sup>k</sup>* is *<sup>N</sup>*<sup>ˆ</sup> *<sup>k</sup>*|*<sup>k</sup>* <sup>≈</sup> *round<sup>v</sup>* \$*k*|*k i*=1 *wi k*|*k* , and the state-estimates of the targets can be obtained by clustering [26,27], data-driven methods [28–31], and so on.
