*4.4. GLMB Filter*

The GLMB filter is a Bayesian recursion from the multi-Bernoulli distribution, which satisfies the following formula [34]:

$$\mathbb{C} = \mathcal{F}\left(\mathbb{L}\right) \times \Xi$$

$$\omega^{(\varepsilon)}\left(L\right) = \omega^{\left(I, \mathbb{\tilde{x}}\right)}\left(L\right) = \omega^{\left(I, \mathbb{\tilde{x}}\right)}\delta\_I\left(L\right)\tag{28}$$

$$p^{\left(c\right)} = p^{\left(I, \mathbb{\tilde{x}}\right)} = p^{\left(\mathbb{\tilde{x}}\right)}.$$

The forward propagation expression of GLMB Filter is as follows:

$$
\pi \left( \mathbf{X} \right) = \Delta \left( \mathbf{X} \right) \sum\_{\left( I, \underline{\varepsilon} \right) \in \mathcal{F}\left( \mathbb{L} \right) \times \boldsymbol{\Sigma}} \omega^{\left( I, \underline{\varepsilon} \right)} \delta\_{I} \left( \mathcal{L} \left( \mathbf{X} \right) \right) \left[ p^{\left( \underline{\varepsilon} \right)} \right]^{\mathbf{X}}.\tag{29}
$$

The distribution of multi-target prior probability is given by the Equation (29), thus, the multi-target prediction is still the multi-Bernoulli distribution and the prediction step can be expressed as:

$$
\boldsymbol{\pi}\_{+}\left(\boldsymbol{\mathsf{X}}\_{+}\right) = \boldsymbol{\Delta}\left(\boldsymbol{\mathsf{X}}\_{+}\right) \sum\_{\left(\boldsymbol{I}\_{+},\boldsymbol{\mathsf{X}}\_{\boldsymbol{\mathsf{X}}}^{\boldsymbol{\mathsf{X}}}\right) \in \boldsymbol{\mathcal{F}}\left(\boldsymbol{\mathsf{L}}\right) \times \boldsymbol{\Xi}} \boldsymbol{\omega}\_{+}\left(\boldsymbol{I}\_{+},\boldsymbol{\mathsf{X}}\_{+}^{\boldsymbol{\mathsf{X}}}\right) \delta\_{\boldsymbol{I}\_{+}}\left(\boldsymbol{\mathsf{L}}\left(\boldsymbol{\mathsf{X}}\_{+}\right)\right) \left[\boldsymbol{\mathsf{P}}\_{+}\left(\boldsymbol{\mathsf{S}}\_{+}\right)\right]^{\boldsymbol{\mathsf{X}}^{+}} \tag{30}
$$

where

$$
\omega\_{+}^{\left(l\_{+},\mathbb{Z}\right)} = \omega\_{\mathbb{s}}^{\left(\frac{\mathbb{Z}}{\mathbb{S}}\right)} \left(I\_{+} \cap \mathbb{L}\right) \omega\_{\mathbb{B}} \left(I\_{+} \cap \mathbb{B}\right) \tag{31}
$$

$$p\_{+}\left(^{\langle\xi\rangle}\langle\mathbf{x},l\rangle=1\_{\mathbb{L}}\left(l\right)p\_{s}^{\langle\xi\rangle}\left(\mathbf{x},l\right)+\left(1-1\_{\mathbb{L}}\left(l\right)\right)p\_{\mathbb{B}}\left(\mathbf{x},l\right)\tag{32}$$

$$p\_s^{(\xi)}\left(\mathbf{x},l\right) = \frac{\left\langle p\_s\left(\cdot,l\right)f\left(\mathbf{x}|\cdot,l\right), p^{(\xi)}\left(\cdot,l\right)\right\rangle}{\eta\_s^{(\xi)}\left(l\right)}\tag{33}$$

$$\eta\_s^{(\xi)}\left(l\right) = \int \left\langle p\_s\left(\cdot, l\right) f\left(\mathbf{x}\middle|\cdot, l\right), p^{(\xi)}\left(\cdot, l\right) \right\rangle d\mathbf{x} \tag{34}$$

$$
\omega\_s^{\left(\xi\right)}\left(L\right) = \left[\eta\_s^{\left(\xi\right)}\right]^L \sum\_{I \in L} \mathbf{1}\_I\left(L\right) \left[\eta\_s^{\left(\xi\right)}\right]^{I-L} \omega^{\left(I,\xi\right)}\tag{35}
$$

$$q\_s^{\left(\xi\right)}\left(l\right) = \left\langle q\_s\left(\cdot, l\right), p\_s^{\left(\xi\right)}\left(\cdot, l\right) \right\rangle \tag{36}$$

Here, the *<sup>ω</sup><sup>B</sup>* (*I*<sup>+</sup> <sup>∩</sup> <sup>B</sup>) and *<sup>ω</sup><sup>s</sup>* (*ξ*) (*I*<sup>+</sup> <sup>∩</sup> <sup>L</sup>) are weights of the birth labels (*I*<sup>+</sup> <sup>∩</sup> <sup>B</sup>) and surviving labels (*I*<sup>+</sup> <sup>∩</sup> <sup>L</sup>), respectively. *<sup>p</sup>*<sup>B</sup> (*x*, *<sup>l</sup>*) is density of a new-born target, *<sup>p</sup>* (*ξ*) *<sup>s</sup>* (*x*, *l*) is the probability density of the surviving target obtained from the prior probability *<sup>p</sup>*(*ξ*) (·, *<sup>l</sup>*). *<sup>f</sup>* (*x*|·, *<sup>l</sup>*) means density weighted by the probability of survival *ps* (·, *l*).

Given the predicted density as Equation (29), the update step can be expressed in the form of a truncated estimate:

$$\pi\left(X\left|Z\right>\approx\Lambda\left(X\right)\sum\_{\left(l,\xi\right)\in\mathcal{F}\left(\mathbb{L}\right)\times\Xi}\sum\_{\theta\in\Theta^{\{M\}}}\tilde{w}^{\left(l,\xi,\theta\right)}\left(Z\right)\delta\_{l}\left(\mathcal{L}\left(\mathbb{X}\right)\right)\left[p^{\left(\xi,\theta\right)}\left(\cdot\mid Z\right)\right]^{\lambda},\tag{37}$$

where (*I*, *ξ*) is fixed parameter. For *M* elements set Θ(*M*) = *<sup>ξ</sup>*(1), ··· , *<sup>ξ</sup>*(*M*) there is the highest weight *<sup>ω</sup>*(*I*,*ξ*,*θ*), *<sup>ω</sup>*,(*I*,*ξ*,*θ*) is the re-normalized weight after truncation and

$$
\hat{\omega}^{(I,\underline{\xi},\theta)}\left(\mathcal{Z}\right) = \frac{\delta\_{\theta^{-1}\left(\{0;|\mathcal{Z}\,\,\|\,\,\|\,\,\|\,\,\theta\}\right)}\left(I\right)\omega^{(I,\underline{\xi})}\left[\eta\_{\underline{Z}}^{\left(\underline{\xi},\theta\right)}\right]^{I}}{\sum\limits\_{(I,\underline{\xi})\in\mathcal{F}(\mathcal{L})\times\mathcal{Z}}\sum\_{\theta\in\Theta}\delta\_{\theta^{-1}\left(\{0;|\mathcal{Z}\,\,\|\,\,\|\,\,\|\,\,\theta\}\right)}\left(I\right)\omega^{(I,\underline{\xi})}\left[\eta\_{\underline{Z}}^{\left(\underline{\xi},\theta\right)}\right]^{I}}\tag{38}
$$

$$p^{\left(\xi,\theta\right)}\left(\mathbf{x},l\,\vert\,\mathbf{Z}\right) = \frac{p^{\left(\xi\right)}\left(\mathbf{x},l\right)\psi\_{\mathbf{Z}}\left(\mathbf{x},l\,\vert\theta\right)}{\eta\_{\mathbf{Z}}^{\left(\xi,\theta\right)}\left(l\right)}\tag{39}$$

$$
\eta\_{\mathbb{Z}}^{\left(\xi,\theta\right)}\left(l\right) = \left\langle p^{\left(\xi\right)}\left(\cdot,l\right)\Psi\_{\mathbb{Z}}\left(\cdot,l;\theta\right)\right\rangle\tag{40}
$$

$$\psi\_{\rm Z}\left(\mathbf{x},l;\theta\right) = \delta\_{0}\left(\theta\left(l\right)\right)q\_{D}\left(\mathbf{x},l\right) + \left(1-\delta\_{0}\left(\theta\left(l\right)\right)\right)\frac{p\_{D}\left(\mathbf{x},l\right)g\left(z\_{\theta\left(l\right)}|\mathbf{x},l\right)}{\mathcal{K}\left(z\_{\theta\left(l\right)}\right)}.\tag{41}$$
