2.2.2. Update Step

Assuming that *nk*,*<sup>z</sup>* measurements are collected as *<sup>Z</sup><sup>k</sup>* <sup>=</sup> *<sup>z</sup>k*,1, ··· , *<sup>z</sup>k*,*nk*,*<sup>z</sup>* and the predicted probability density is <sup>π</sup>*k*|*k*−1(*X*) = {(*<sup>r</sup>* (*i*) *k*|*k*−1 , *p* (*i*) *<sup>k</sup>*|*k*−1(*xk*))} *Mk*|*k*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> , then the posterior multi-target density at time *k* can be approximated by a multi-Bernoulli as

$$\pi\pi\_k \approx \left\{ (r\_{L,k'}^{(i)} p\_{L,k}^{(i)}(\mathbf{x}\_k)) \right\}\_{i=1}^{M\_{k|k-1}} \cup \left\{ (r\_{L|k}^\*(\mathbf{z}), p\_{L|k}^\*(\mathbf{x}\_k; \mathbf{z})) \right\}\_{\mathbf{z} \in \mathbf{Z}\_k} \tag{16}$$

*Sensors* **2020**, *20*, 1679

The first term in Equation (16) corresponds to the multi-Bernoulli density for the legacy tracks and it can be given by

$$r\_{L,k}^{(i)} = r\_{k|k-1}^{(i)} \frac{1 - \left< p\_{k|k-1}^{(i)}(\mathbf{x}\_k), p\_{D,k}(\mathbf{x}\_k) \right>}{1 - r\_{k|k-1}^{(i)} \left< p\_{k|k-1}^{(i)}(\mathbf{x}\_k), p\_{D,k}(\mathbf{x}\_k) \right>} \tag{17}$$

$$p\_{L,k}^{(i)}(\mathbf{x}\_k) = p\_{k|k-1}^{(i)}(\mathbf{x}\_k) \frac{1 - p\_{D,k}(\mathbf{x}\_k)}{1 - \left\langle p\_{k|k-1}^{(i)}(\mathbf{x}\_k), p\_{D,k}(\mathbf{x}\_k) \right\rangle} \tag{18}$$

where *pD*,*k*(*xk*) is the detection probability.

The second term in Equation (16) corresponds to the multi-Bernoulli density for measurement-corrected tracks and it can be given as

$$r\_{Ll,k}^{r}(\mathbf{z}) = \frac{r\_{l,k-1}^{M\_{\tilde{k}k-1}} \frac{r\_{l\tilde{k}l-1}^{(i)} (1 - r\_{l\tilde{k}l-1}^{(i)}) \left\langle p\_{l\tilde{k}l-1}^{(i)}(\mathbf{x}\_k), g\_k(\mathbf{z}|\mathbf{x}\_k) p\_{D,k}(\mathbf{x}\_k) \right\rangle}{\left(1 - r\_{l\tilde{k}l-1}^{(i)} \sqrt{p\_{l\tilde{k}l-1}^{(i)}}(\mathbf{x}\_k), p\_{D,k}(\mathbf{x}\_k) \right)^2} \tag{19}$$
 
$$\kappa(\mathbf{z}) + \sum\_{i=1}^{M\_{\tilde{k}l\tilde{k}-1}} \frac{r\_{l\tilde{k}l-1}^{(i)} \left\langle p\_{l\tilde{k}l-1}^{(i)}(\mathbf{x}\_k), g\_k(\mathbf{z}|\mathbf{x}\_k) p\_{D,k}(\mathbf{x}\_k) \right\rangle}{1 - r\_{l\tilde{k}l-1}^{(i)} \left\langle p\_{l\tilde{k}l-1}^{(i)}(\mathbf{x}\_k), p\_{D,k}(\mathbf{x}\_k) \right\rangle} $$
 
$$\frac{M\_{\tilde{k}l-1}}{\sum\_{i=1}^{M\_{\tilde{k}l-1}} \frac{r\_{l\tilde{k}l-1}^{(i)} p\_{l\tilde{k}l-1}^{(i)}(\mathbf{x}\_k) p\_{D,k} \xi\_l(\mathbf{z}|\mathbf{x}\_k)}{\sqrt{p\_{l\tilde{k}l}}^{(i)} \left\langle p\_{l\tilde{k}l-1}^{(i)}(\mathbf{x}\_k), p\_{D,k}(\mathbf{z}|\mathbf{x}\_k) \right\rangle}}$$

$$p^\*\_{II,k}(\mathbf{x}\_k; \mathbf{z}) = \frac{\sum\_{i=1}^r \frac{r\_{\frac{lk-1}{k}, k \parallel k-1} \sim k\_r^\*(\mathbf{D}\_k \mathbf{x}\_k) \rightarrow \mathbf{x}\_k}{1 - r\_{\frac{lk}{k} - 1}^{(i)}}}{\sum\_{i=1}^{r\_{\frac{lk-1}{k}}} \frac{r\_{\frac{lk-1}{k}}^{(i)}}{1 - r\_{\frac{lk}{k} - 1}^{(i)}} \left\langle p\_{k|k-1}^{(i)}(\mathbf{x}\_k), p\_{D,k}(\mathbf{x}\_k) \bigotimes\_k \mathbf{z} | \mathbf{x}\_k \right\rangle} \tag{20}$$

where *gk*(*z*|*xk*) is the likelihood function, <sup>κ</sup>(*z*) is the clutter intensity function. There are *Mk* <sup>=</sup> *Mk*|*k*−<sup>1</sup> + *nk*,*<sup>z</sup>* updated hypothesized tracks.
