3.1.1. GLMB Filter without Objects Spawning

The procedure to estimate the state of a set of objects with the standard GLMB filter without including the spawning model in the transition kernel is given as follows.

Given a GLMB prior [16]

$$
\pi(\mathbf{X}) = \Delta(\mathbf{X}) \sum\_{(I,\tilde{\mathbf{J}}) \in \mathcal{F}(\mathbb{L}) \times \boldsymbol{\Xi}} \omega^{(I,\tilde{\mathbf{J}})} \delta\_I(\mathcal{L}(\mathbf{X})) [p^{(\tilde{\mathbf{J}})}]^{\mathbf{X}} \tag{17}
$$

and the standard observation model as in Equation (8), the filtering density in the next time step is given by the following [16]:

$$
\pi\_{\mathbb{Z}\_+}(\mathbb{X}) \approx \Delta(\mathbb{X}) \sum\_{\mathbb{I}\_{\mathbb{S}}^{\mathbb{Z}}, \mathbb{I}\_+, \theta\_+} \omega^{(\mathbb{I}\_{\mathbb{S}}^{\mathbb{Z}})} \omega^{(\mathbb{I}\_{\mathbb{S}}^{\mathbb{Z}})}\_{\mathbb{Z}\_+} \omega^{(\mathbb{I}\_{\mathbb{S}}^{\mathbb{Z}}, \mathbb{I}\_+, \theta\_+)}\_{\mathbb{I}\_+} \delta\_{\mathbb{I}\_+}(\mathcal{L}(\mathbb{X})) [p\_{\mathbb{Z}\_+}^{(\mathbb{Z}, \theta\_+)}]^{\mathbb{X}} \tag{18}
$$

where *<sup>I</sup>* ∈ F(L), *<sup>ξ</sup>* <sup>∈</sup> <sup>Ξ</sup>, *<sup>I</sup>*<sup>+</sup> ∈ F(L+), *<sup>θ</sup>*<sup>+</sup> <sup>∈</sup> <sup>Θ</sup><sup>+</sup> where *<sup>ξ</sup>* is the tracks to measurement association history and Ξ is the entire space of *ξ*.

$$
\omega\_{Z\_{+}}^{(l,\xi,l\_{+},\theta\_{+})} = 1\_{\Theta\_{+}(I\_{+})}(\theta\_{+})[1 - P\_{S}^{(\xi)}]^{I-I\_{+}}[P\_{S}^{(\xi)}]^{I \cap I\_{+}}[1 - r\_{B+}]^{\mathbb{B}\_{+}-I\_{+}}[r\_{B+}]^{\mathbb{B}\_{+}\cap I\_{+}}[\Psi\_{Z\_{+}}^{(\xi,\theta\_{+})}]^{I\_{+}}
$$

$$
P\_{S}^{(\xi)}(l) = \langle p^{(\xi)}(\cdot,l), p\_{S}(\cdot,l) \rangle
$$

$$
\psi\_{Z\_+}^{(\xi,\theta\_+)}(l\_+) = \langle \mathfrak{p}\_+^{(\xi)}(\cdot,l\_+), \mathfrak{p}\_{Z\_+}^{(\theta\_+(l\_+))}(\cdot,l\_+) \rangle.
$$

$$p\_{Z\_+}^{(\xi,\theta\_+)}(\mathbf{x}\_+,l\_+) = \frac{p\_+^{(\xi)}(\mathbf{x}\_+,l\_+)\psi\_{Z\_+}^{(\theta\_+,(l\_+))}(\mathbf{x}\_+,l\_+)}{\psi\_{Z\_+}^{(\xi,\theta\_+)}(l\_+)}$$

$$p\_{+}^{(\underline{\xi})}(\mathbf{x}\_{+},l\_{+}) = \mathbf{1}\_{\mathcal{L}}(\{l\_{+}\}) \frac{\langle p\_{\mathcal{S}}(\cdot,l\_{+})f\_{\mathcal{S}+}(\mathbf{x}\_{+}|\cdot,l\_{+}), p^{(\underline{\xi})}(\cdot,l\_{+})\rangle}{P\_{\mathcal{S}}^{(\underline{\xi})}(l\_{+})} + \mathbf{1}\_{\mathbb{B}\_{+}}(\{l\_{+}\})p\_{\mathcal{B}+}(\mathbf{x}\_{+},l\_{+})$$

In tracking scenarios where raw spatial detection are also available, the hybrid model in Equation (11) can be used to replace the standard observation model with the probability of miss-detection being scaled by the spatial observation likelihood, i.e., given the GLMB prior as in Equation (17). The filtering density is then given as follows [27]:

$$\pi\_{\boldsymbol{y}\_{+}}(\mathbf{X}) \propto \boldsymbol{\Lambda}(\mathbf{X}) \sum\_{\boldsymbol{I}\_{\boldsymbol{\sigma}}^{\overline{\boldsymbol{s}}}, \boldsymbol{I}\_{+}, \boldsymbol{\theta}\_{+}} \boldsymbol{\omega}^{(\boldsymbol{I}\_{\boldsymbol{\sigma}}^{\overline{\boldsymbol{s}}})} \boldsymbol{\omega}^{(\boldsymbol{I}\_{\boldsymbol{\sigma}}^{\overline{\boldsymbol{s}}})}\_{\boldsymbol{y}\_{+}} \boldsymbol{\omega}^{(\boldsymbol{I}\_{\boldsymbol{\sigma}}^{\overline{\boldsymbol{s}}}, \boldsymbol{I}\_{+}, \boldsymbol{\theta}\_{+})} \delta\_{\boldsymbol{I}\_{+}}(\mathcal{L}(\mathbf{X})) [p\_{\boldsymbol{y}\_{+}}^{(\overline{\boldsymbol{s}}, \boldsymbol{\theta}\_{+})}]^{\mathbf{X}} \tag{19}$$

where

$$\begin{aligned} \alpha\_{y\_+^{(\ell)}} \mu\_{+}^{(I\_{\ell}^{\xi}, I\_{+}, \theta\_{+})} &= \mathbf{1}\_{\Theta\_{+}(I\_{+})} (\theta\_{+}) [1 - \mathcal{P}\_{\mathcal{S}}^{(\xi)}]^{I - I\_{+}} [\mathcal{P}\_{\mathcal{S}}^{(\xi)}]^{I \cap I\_{+}} [1 - r\_{B+}]^{\mathbb{B}\_{+} - I\_{+}} [r\_{B+}]^{\mathbb{B}\_{+} \cap I\_{+}} [\mathfrak{d}\_{y\_{+}^{(\xi, \theta\_{+})}}^{(\xi, \theta\_{+})}]^{I\_{+}} \\\\ \mathfrak{d}\_{y\_{+}^{(\xi, \theta\_{+})}}^{(\xi, \theta\_{+})} (l\_{+}) &= \langle \mathcal{P}\_{+}^{(\xi)}(\cdot, l\_{+}), \mathfrak{e}\_{\mathcal{Y}\_{+}}^{(\theta\_{+}, (l\_{+}))}(\cdot, l\_{+}) \rangle \\\\ p\_{y\_{+}^{(\xi, \theta\_{+})}}^{(\xi, \theta\_{+})} (x\_{+}, l\_{+}) &= \frac{p\_{+}^{(\xi)}(x\_{+}, l\_{+}) \mathfrak{e}\_{\mathcal{Y}\_{+}}^{(\theta\_{+}, (l\_{+}))}(x\_{+}, l\_{+})}{\mathfrak{e}\_{\mathcal{Y}\_{+}}^{(\xi, \theta\_{+})}(l\_{+})} \end{aligned}$$
