# Tracking Extended Targets: Novel TPMB Filter Driven by Model and Data Collaboration
## 3 Model-Driven IMM-TPMB Filter
### Implementation of an IMM-TPMB Filter Based on Gate-Control
#### 3.2.2 Update  
In the BP iterative propagation algorithm, the optimization of the propagated information is achieved through \(P\) iterations, leading to the convergence of belief for all nodes. At this point, the label of each node aligns with the label of the node with the highest posterior probability density among the related nodes, representing the optimal label. In the update step, we improve the way of information transmission and reduce the amount of data association calculation by using the GBP. The specific improvement method is in step 3. The update step in model-driven scenarios includes six stages: particle initialization, importance sampling of particles, iterative information transfer, belief calculation, pruning and resampling, and target state estimation. 1. Particle initialization: The Poisson intensity is represented as a set of particles denoted by \(\left\{\left(t_{k|k-1}^{0,\zeta^{\prime},(l)},X_{k|k-1}^{0,\zeta^{\prime},(l) }w_{k|k-1}^{0,\zeta^{\prime},(l)}e^{-\gamma_{k}(\overline{X}_{k|k-1}^{0,\zeta ^{\prime},(l)})}\right)\right\}_{l=1}^{L_{k|k-1}^{k}}\). Updating for new, missed, and surviving targets requires considering whether targets are alive at the current moment. 2. Particle importance sampling: Utilize the ASCKF for importance sampling of particles. After iterating the prediction and updating process of prior and posterior information at time step \(k-1\), the particles at time step \(k\) follow the distribution function \[x_{k}^{i,\zeta^{\prime},(l)}\sim q(\overline{X}_{k}^{i,\zeta^{\prime},(l)},Z_ {k})=N(\hat{x}_{k}^{i,\zeta^{\prime},(l)},P_{k}^{i,\zeta^{\prime},(l)})\] (46) Here, \(N(\hat{x}_{k}^{i,\zeta^{\prime},(l)},P_{k}^{i,\zeta^{\prime},(l)})\) denotes a Gaussian distribution with mean \(\hat{x}_{k}^{i,\zeta^{\prime},(l)}\) and variance \(P_{k}^{i,\zeta^{\prime},(l)}\), as determined by the ASCKF. 3. Iterative information transfer: Measurement assessment requires calculating the information \(m_{k}^{j}\) transferred from each factor node \(\underline{p}_{k}\) to variable node \(\beta_{k}^{j}\) in the vector measurement. Factor node \(\underline{p}_{k}\) represents the process of information transfer between surviving trajectories and both sides. \[m_{i,j}^{(p,l)}(i)=\frac{1}{\lambda_{k}^{\mathbb{C}}(w_{k}^{j})u_{i,j}^{(p)}} \sum_{l=1}^{L_{i}}w_{i,j}^{(p,l)}\delta_{k-k^{i,\mathbb{C}}_{k|k}}[v_{k|k}^{i, \zeta^{\prime},(l)}]\times\gamma_{k}(\overline{X}_{k|k}^{i,\zeta^{\prime},(l) })l_{k}(w_{k}^{\xi}\Big{|}\overline{X}_{k|k}^{i,\zeta^{\prime},(l)})\] (47)  
4. Belief calculation: After multiple iterations of the above steps, the information and example weights converge to the true posterior probability density of the target. The belief \(\widehat{f}^{p}_{k|k}(\cdot)\) for the target state in each MBM component is represented as a set \(\left(r^{i}_{k|k^{\prime}}\left\{\left(w^{i,j,(l)}_{k|k}\,,\,r^{i,j,(l)}_{k|k} \,,\,X^{i,j,(l)}_{k|k}\right)\right\}_{l=1}^{L_{i}}\right)\) of particles, and its calculation method is similar to information \(u^{(p+1)}_{i,j}\). Additionally, subsequent iterations still require calculating particle weights and the existence probability of Bernoulli to ensure the validity of trajectory belief, followed by weight normalization. For new targets \(i\) in the detection area at time step \(k\), when \(i\in\{1,\ldots,n_{k|k-1}\}\), then the updated existence probability of the target is as follows:  
\[r^{(i,l)}_{k|k}=\frac{r^{i}_{k|k}\sum\limits_{l=1}^{L_{l}}w^{(p+1,l)}_{i}}{r^{ i}_{k|k-1}\sum\limits_{l=1}^{L_{l}}w^{(p,l)}_{i}+1-r^{i}_{k|k-1}} \tag{54}\]  
For undetected potential targets \(i\), when \(i\in\{n_{k|k-1}+1,\ldots,n_{k|k}\}\), then the updated existence probability of the target is the following:  
\[r^{(i,l)}_{k|k}=\frac{\sum\limits_{l=1}^{L}w^{(p+1,l)}_{i}}{r^{i}_{k|k-1}\sum \limits_{l=1}^{L}w^{(p,l)}_{i}+1} \tag{55}\]  
The updated particle weight is normalized as follows:  
\[w^{(i,j,l)}_{k|k}=\frac{w^{(p+1,l)}_{i}}{\sum\limits_{l=1}^{L}w^{(p+1,l)}_{i}} \tag{56}\]
5.