# Tracking Extended Targets: Novel TPMB Filter Driven by Model and Data Collaboration
## 2 Background
### Trajectory PMB Approximation  
According to reference [16], the TPMBM approximates the TPMB method, and this paper also utilizes special multi-trajectory states and measurement models. According to Equation (10), we derive the TPMB prediction equation. Given the multi-trajectory set prediction density \(\widetilde{f}_{k^{\prime}|k}\big{(}\widetilde{\mathbf{X}}_{k^{\prime}}\big{)}\) and measurement set \(\mathbf{z}_{k}=\big{\{}z_{k}^{1},\ldots,z_{k}^{m_{k}}\big{\}}\) at time step \(k\), from Equation (5) to Equation (8), it is known that the number of updated Bernoulli components \(n_{k|k}=n_{k|k-1}+m_{k}\), and the updated Poisson intensity is as follows:  
\[\lambda_{k|k}(X)=\mathscr{L}_{k}(\varnothing|X)\lambda_{k|k-1}(X) \tag{9}\]  
For each Bernoulli component predicted at the previous measurements \(i\in\{1,\ldots,n_{k|k^{\prime}}\}\), the current moment's new measurements generate \(m_{k}\) Bernoulli components. At the same time, each Bernoulli component predicted from the previous moment generates \(m_{k}\) new local hypotheses at the current moment. These newly generated local hypotheses may be false detection or updates for non-empty subsets of measurements \(z^{k}\). After augmenting the model using auxiliary variables, it is necessary to define the global hypothesis set of generalized measurement model parameter variables. We give the augmented representation method of measurement \(\mathcal{M}(k,j)=j\) for \(z_{k}^{j}\). For the Bernoulli hypothesis component \(i\), a set of measurement collections represented by a local hypothesis \(a^{i}\) is \(\mathcal{M}_{k}^{i,a^{i}}\subseteq\mathcal{M}_{k}\):  
next moment and is associated with measurement \(z_{k}^{j}\), which yields, \(j\in\{1,\ldots,2^{m_{k}}-1\}\), \(\mathcal{M}(i,j)=1\), \(r_{k|k}^{i,j+1}=1\). \[\epsilon_{k|k}^{i,j+1}=\left\langle f_{k|k-1}^{i},\ell_{k}\!\left(w_{k}^{j}|\cdot \right)\right\rangle \tag{16}\]  
\[w_{k|k}^{i,j+1}=w_{k|k-1}^{i}r_{k|k-1}^{i}\epsilon_{k|k}^{i,j+1} \tag{17}\]  
\[f_{k|k}^{i,j+1}(X)=\frac{\ell_{k}\!\left(v_{k}^{j}|X\right)f_{k|k-1}^{i}(X)}{ \epsilon_{k|k}^{i,j+1}} \tag{18}\]  
Then, we construct the set \(S_{i}\) to associate with the new Bernoulli component \(i\in\{1,\ldots,m_{k}\}\), with \(S_{i}=\left\{\left\{z_{k}^{i}\right\}\right\}\cup\left(\cup_{\
u\in\cup_{j=1}^ {i-1}S_{i}}\!\left\{\left\{z_{k}^{i}\right\}\cup\mathrm{w}\right\}\right)\), and use \(w_{k}^{i,\iota}\) to represent the Bernoulli component of the measurement set \(\iota\), where \(i=n_{k|k-1}+j\), \(j\in\{1,\ldots,m_{k}\}\), \(\iota\in\{1,\ldots,2^{j-1}\}\). For the new Bernoulli components \(i=n_{k|k-1}+j\) generated by new measurements \(z_{k}^{j}\) at time step \(k\), there are two single-target hypotheses for each Bernoulli component: one corresponds to a previously undetected and non-existent Bernoulli component, represented as \(\mathcal{M}(i,1)=\varnothing\), \(r_{k|k}^{i,1}=0\), \(w_{k|k}^{i,1}=1\); the other corresponds to a Bernoulli component that existed in the previous moment. \[\epsilon_{k|k}^{i,2}=\left\langle\lambda_{k|k-1},\ell_{k}\!\left(w_{k}^{i, \iota}|\cdot\right)\right\rangle \tag{19}\]  
\[w_{k|k}^{i,2}=\lambda^{\mathcal{C}}\!\left(w_{k}^{i}\right)+\left\langle \lambda_{k|k-1},l\!\left(w_{k}^{i,\iota}|\cdot\right)\right\rangle \tag{20}\]  
\[r_{k|k}^{i,2}=\frac{\left\langle\lambda_{k|k-1},l\!\left(w_{k}^{i,\iota}| \cdot\right)\right\rangle}{\lambda^{\mathcal{C}}\!\left(w_{k}^{i,\iota}\right) +\left\langle\lambda_{k|k-1},l\!\left(w_{k}^{i,\iota}|\cdot\right)\right\rangle} \tag{21}\]  
\[f_{k|k}^{i,a^{j}}(X)=\frac{\ell_{k}\!\left(w_{k}^{i,\iota}|X\right)\!\lambda_{ k|k-1}(X)}{\ell_{k|k}^{i}} \tag{22}\]  
The set \(\mathcal{M}_{k}^{i,j}\) represents the local hypotheses \(a^{i}\) related to the Bernoulli component \(i\) measurement indicators. After the Bayesian update, the updated density of PMBM is obtained through the aforementioned process.