*2.2. The Multi-Object Transition Kernel*

In standard tracking scenario, an existing object can either survive or die in the next time step. The surviving objects are modeled as an LMB RFS with a survival probability of *pS*(*x*, *l*), a disappearance probability of *qS*(*x*, *l*) = 1 − *pS*(*x*, *l*), and a spatial distribution of *fS*+(*x*+|*x*, *l*). The model for such surviving objects is given as follows [13,15,16]:

$$\mathbf{f}\_{\mathbb{S}+}(\mathbb{X}\_{\mathbb{S}+}|\mathbb{X}) = \Delta(\mathbb{X})\Delta(\mathbb{X}\_{\mathbb{S}+})\mathbf{1}\_{\mathcal{L}(\mathbb{X})}(\mathcal{L}(\mathbb{X}\_{\mathbb{S}+})) [\Phi\_{\mathbb{S}+}(\mathbb{X}\_{\mathbb{S}+}|\cdot)]^{\mathbb{X}} \tag{2}$$

where

$$\Phi\_{\mathbb{S}+}(\mathbb{X}\_{\mathbb{S}+}|\mathbf{x},l) = \sum\_{(\mathbf{x}\_{+},l\_{+}) \in \mathbb{X}\_{\mathbb{S}+}} \delta\_{l}(l\_{+})p\_{\mathbb{S}}(\mathbf{x},l)f\_{\mathbb{S}+}(\mathbf{x}\_{+}|\mathbf{x},l) + [1-\mathbf{1}\_{\mathcal{L}(\mathbf{X}\_{\mathbb{S}+})}(l)]q\_{\mathbb{S}}(\mathbf{x},l)$$

In addition, the new birth objects can instantaneously appear at each time steps and they are modeled with LMB RFS as follows [13,15,16]:

$$\mathbf{f}\_{\mathcal{B}+} (\mathbf{X}\_{\mathcal{B}+}) = \Delta(\mathbf{X}\_{\mathcal{B}+}) w\_{\mathcal{B}}(\mathcal{L}(\mathbf{X}\_{\mathcal{B}+})) [p\_{\mathcal{B}+}]^{\mathbf{X}\_{\mathcal{B}+}} \tag{3}$$

$$w\_B(\mathcal{L}(\mathsf{X}\_{B+})) = \mathbf{1}\_{\mathbb{B}+} (\mathcal{L}(\mathsf{X}\_{B+})) [1 - r\_{B+}]^{\mathbb{B}\_+ - \mathcal{L}(\mathsf{X}\_{B+})} [r\_{B+}]^{\mathcal{L}(\mathsf{X}\_{B+})}$$

Furthermore, in certain scenarios, new objects can also be generated from existing objects, which leads to the need of a spawning model in order to correctly predict the state of the system at the next time step. Recently, a spawning model for GLMB filter has been proposed in Reference [30]; we introduce this model again here as follows for the sake of completeness.

For spawned objects, the naming convention is given as follows: if at time step *k* the label of an object is *l*, then the spawned labels from *l* at the next time step is *lspawn* = (*l*, *k* + 1, *i*), where *i* is the index to distinguish between different spawned objects from the same parent. Following this convention, the set of all spawned labels in the next time step is <sup>S</sup><sup>+</sup> <sup>=</sup> <sup>L</sup> × {*<sup>k</sup>* <sup>+</sup> <sup>1</sup>} × <sup>N</sup>, where <sup>N</sup> is the set of positive natural numbers [30].

For each spawned object with the label *lspawn* <sup>∈</sup> <sup>S</sup>+(L(**x**)), it will either exist with the probability *pT*(**x**; *lspawn*) and a spatial distribution *fT*+(*x*+|**x**; *lspawn*) or not with the probability *qT*(**x**; *lspawn*) = 1 − *pT*(**x**; *lspawn*).

The density of the set **P** of new spawned objects from **x** is formulated as follows [30]:

$$\mathbf{f}\_{T+}(\mathbf{P}|\mathbf{x}, l\_{\text{span}}) = \Delta(\mathbf{P}) \mathbf{1}\_{\mathbb{S}\_{+}(\mathcal{L}(\mathbf{x}))}(\mathcal{L}(\mathbf{P})) [\Phi\_{T+}(\mathbf{P}|\mathbf{x}; \cdot)]^{\mathbb{S}\_{+}(\mathcal{L}(\mathbf{x}))} \tag{4}$$

where

$$\Phi\_{\mathbf{T}+}(\mathbf{P}|\mathbf{x};l\_{\mathrm{spawn}}) = \sum\_{(\mathbf{x},l\_{+})} \delta\_{l\_{\mathrm{spawn}}}(l\_{+}) p\_{\mathrm{T}}(\mathbf{x},l\_{\mathrm{spawn}}) f\_{\mathrm{T}+}(\mathbf{x}+|\mathbf{x},l\_{\mathrm{spawn}}) + [1-1\_{\mathcal{L}(\mathbf{P})}(l\_{\mathrm{spawn}})] q\_{\mathrm{T}}(\mathbf{x},l\_{\mathrm{spawn}}) $$

Let **<sup>Q</sup>** be a labeled set of objects spawned from **<sup>X</sup>** with <sup>L</sup>(**Q**) <sup>⊆</sup> <sup>S</sup>+(L(**X**)). As all labels sets are disjoint, the FISST convolution theorem [54] can be applied.

$$\mathbf{f}\_{T+}(\mathbf{Q}|\mathbf{X}) = \Lambda(\mathbf{Q})\mathbf{1}\_{\mathbb{S}\_{+}(\mathcal{L}(\mathbf{X}))}(\mathcal{L}(\mathbf{Q}))[\Phi\_{T+}(\mathbf{Q}|\cdot)]^{\mathbf{X}} \tag{5}$$

where

$$\Phi\_{T+}(\mathbf{Q}|\mathbf{x}) = [\Phi\_{T+}(\mathbf{Q}\cap(\mathbb{X}\times\mathbb{B}\_{+}(\mathcal{L}(\mathbf{x}))|\mathbf{x};\cdot)]^{\mathbb{B}\_{+}(\mathcal{L}(\mathbf{x}))}$$

As new birth objects (given in Equation (3)) are independent of the previous time step objects, the overall transition model is given as follows:

$$\mathbf{f}(\mathbf{X}\_{+}|\mathbf{X}) = \mathbf{f}\_{\mathbb{S}+}(\mathbf{X}\_{\mathbb{S}+}|\mathbf{X})\mathbf{f}\_{T+}(\mathbf{Q}|\mathbf{X})\mathbf{f}\_{\mathbb{B}+}(\mathbf{X}\_{\mathbb{B}+}) \tag{6}$$

As the spawned objects depend upon the objects from previous time steps, the prediction step of the filtering stage needs to be done in a joint manner to capture the objects' dependency. As a result, approximation is needed to convert the joint object distribution to a standard GLMB density for each time step in order to keep the algorithm tractable.

In the scenario where the spawning process is not present, the multi-object transition kernel is reduced to the following:

$$\mathbf{f}(\mathbf{X}\_{+}|\mathbf{X}) = \mathbf{f}\_{\mathbb{S}+}(\mathbf{X}\_{\mathbb{S}+}|\mathbf{X})\mathbf{f}\_{\mathbb{B}+}(\mathbf{X}\_{\mathbb{B}+}) \tag{7}$$
