# 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 Two Consecutive Time Points
Since the posterior probability density \(\mu_{k}^{\xi}\) changes over time, \(\pi_{k}^{\xi^{\prime}|\xi^{\prime}}\) can be represented by the model transition probability from the previous moment \(\pi_{k-1}^{\xi^{\prime}|\xi^{\prime}}\) and the relative rate of change of probability \(\Delta_{k}^{\xi^{\prime}|\xi^{\prime}}\). Assuming a normalization constant \(\rho_{k}^{\xi^{\prime}|\xi^{\prime}}=\frac{\Delta_{k}^{\xi^{\prime}|\xi^{ \prime}}}{\sum\limits_{\xi^{\prime}=1}^{M}\Delta_{k}^{\xi^{\prime}|\xi^{\prime }}\pi_{k-1}^{\xi^{\prime}|\xi^{\prime}}}\), the update of the transition probability \(\pi_{k}^{\xi^{\prime}|\xi^{\prime}}\) can be represented as follows:  
\[\pi_{k}^{\xi^{\prime}|\xi^{\prime}}=\frac{\Delta_{k}^{\xi^{\prime}}\pi_{k-1}^{ \xi^{\prime}}}{\sum\limits_{\xi^{\prime}=1}^{M}\Delta_{k}^{\xi^{\prime}|\xi^{ \prime}}\pi_{k-1}^{\xi^{\prime}|\xi^{\prime}}}=\rho_{k}^{\xi^{\prime}|\xi^{ \prime}}\pi_{k-1}^{\xi^{\prime}|\xi^{\prime}} \tag{34}\]  
According to Equation (34), \(\pi_{k}^{\xi^{\prime}|\xi^{\prime}}\) is represented as the product of \(\rho_{k}^{\xi^{\prime}|\xi^{\prime}}\) and \(\pi_{k-1}^{\xi^{\prime}|\xi^{\prime}}\), and there exists \(\sum\limits_{\xi^{\prime}=1}^{M}\Delta_{k}^{\xi^{\prime}|\xi^{\prime}}=1\), thus satisfying the Markov transition matrix conditions. Obviously, the above parameters can inherently match the maneuvering dynamics, including the CT motion and the motion switching. As a result, the maneuvering targets with different dynamics can be detected effectively. On the other hand, the intensity of target maneuvers directly determines the magnitude of process noise. The standard IMM algorithm considers process noise as a constant, which affects the accuracy of the filtering algorithm. Moreover, the process noise changes with unknown noise influences. Therefore, it is necessary to adapt the IMM algorithm to adjust the process noise to accommodate scenarios where the target is maneuvering or non-maneuvering and the noise is unknown. When the target is maneuvering, increase the process noise \(V_{k}\) to compensate for modeling errors, resulting in smaller filtering estimation errors; when the target is non-maneuvering, reduce \(V_{k}\) to maintain tracking accuracy. As suming \(\mu^{\zeta_{t}^{\prime}}_{\max k-1}\) is the maximum value of the posterior probability density of each model at time step \(k-1\), then the covariance matrix \(Q_{k}^{\xi}\) of the process noise \(V_{k}\) at time step \(k\) can be represented as follows:  
\[Q_{k}^{\xi}= \begin{cases}\left(1-\mu^{\zeta_{t}}_{\max k-1}\right)^{2}Q_{0}^{ \zeta_{t}^{\prime}},&\xi=\xi^{\prime}\\ \left(1+\mu^{\zeta_{t}}_{\max k-1}\right)^{2}Q_{0}^{\zeta_{t}^{\prime}},&\xi \
eq\xi^{\prime}\end{cases}  \tag{35}\]  
where \(Q_{0}^{\zeta_{t}^{\prime}}\) is the initial value of the process noise covariance matrix for model \(M_{k-1}^{\zeta_{t}^{\prime}}\). According to Equation (35), when the models described by \(M_{k-1}^{\zeta_{t}^{\prime}}\) and \(M_{k}^{\xi}\) are the same, \(M_{k-1}^{\zeta_{t}^{\prime}}\) matches the current target motion model, and \(Q_{k}^{\xi}\) takes the minimum value; when \(M_{k-1}^{\zeta_{t}^{\prime}}\) and \(M_{k}^{\xi}\) describe different models, \(M_{k-1}^{\zeta_{t}^{\prime}}\) does not match the current target motion model, and \(Q_{k}^{\xi_{t}}\) takes the maximum value. Considering the impact from the process noise, the adaptive adjustment of \(Q_{k}^{\xi}\) ensures consistency with the varying intensity of target maneuvers.