## 2 Trajectory Initialization Based on MDF-Clustering
### Fuzzy Clustering Based on Multidimensional Features
Then, the total distance is:  
\[d_{ii^{\prime}}^{2}=\left(w_{1}\cdot_{1}d_{ii^{\prime}}\right)^{2}+\left(w_{2} \cdot_{2}d_{ii^{\prime}}\right)^{2} \tag{11}\]  
According to the fuzzy association matrix in the fuzzy framework, suppose \(\widetilde{\chi}_{s}=\left\{\widetilde{\chi}_{1s^{\prime}},\ldots,\widetilde{ \chi}_{cs^{\prime}},\ldots,\widetilde{\chi}_{C^{\prime}s}\right\}\) is a subset of \(\chi_{s}\), the clustered clusters are \(C\), and \(\widetilde{\chi}_{C^{\prime}s}\in\widetilde{\chi}_{s}\) is the eigenvalue in the \(C^{\prime}\)th cluster in \(\widetilde{\chi}_{s}\). Then, \(\widetilde{\chi}_{s}=\left\{\widetilde{\chi}_{1s^{\prime}},\ldots,\widetilde{ \chi}_{cs^{\prime}},\ldots,\widetilde{\chi}_{C^{\prime}s}\right\}\) is a subset of \(\chi\) representing the centroid of the current cluster, and its number of clusters after clustering is \(C\). where \(u_{i}=(u_{i1},\ldots,u_{ic},\ldots,u_{iC})^{\prime}\) and \(\lambda\) is the Lagrange multiplier. Therefore, setting the first order derivatives of Equation (15) with respect to \(u_{ic}\) and \(\lambda\) to zero yields:  
\[\frac{\partial L_{m}(u_{i},\lambda)}{\partial u_{ic}}=0\Leftrightarrow m\mathbf{u} _{ic}^{m-1}\sum_{s=1}^{S}\left(w_{s}{}_{s}d_{ic}\right)^{2}-\lambda=0 \tag{16}\]  
\[\frac{\partial L_{m}(u_{i},\lambda)}{\partial\lambda}=0\Leftrightarrow\sum_{ c=1}^{C}u_{ic}-1=0 \tag{17}\]  
According to Equation (16) it can be concluded that  
\[u_{ic}=\left[\frac{\lambda}{m\sum_{s=1}^{S}\left(w_{s}{}_{s}d_{ic}\right)^{2}} \right]^{\frac{1}{m-1}} \tag{18}\]  
It can be derived by also considering Equation (17):  
\[\frac{\lambda}{m}=\left\{\frac{1}{\sum_{c=1}^{C}\left[\frac{1}{\sum_{s=1}^{S} \left(w_{s}{}_{s}d_{ic}\right)^{2}}\right]}\right\}^{\frac{1}{m-1}} \tag{19}\]  
Finally, Equation (19) is substituted into Equation (18) to obtain \(u_{ic}\) as shown in Equation (13). \(u_{ic}\) is then fixed to yield \(w_{s}\). The Lagrangian function is  
\[L_{w}(\mathbf{w},\xi)=\sum_{i=1}^{n}\sum_{c=1}^{C}u_{ic}^{m}\sum_{s=1}^{S}\left(w _{s}{}_{s}d_{ic}\right)^{2}-\xi\left(\sum_{c=1}^{C}u_{ic}-1\right) \tag{20}\]  
where \(\mathbf{w}=\left(w_{1},\ldots,w_{s},\ldots,w_{S}\right)^{\prime}\) and \(\xi\) are Lagrange multipliers. Setting the first order derivatives of Equation (20) with respect to \(w_{s}\) and \(\xi\) equal to zero, one obtains respectively  
\[\frac{\partial L_{m}(\mathbf{w},\lambda)}{\partial w_{s}}=0\Leftrightarrow 2w_{s} \sum_{i=1}^{n}\sum_{c=1}^{C}u_{ic}^{m}{}_{s}d_{ic}^{2}-\xi=0 \tag{21}\]  
\[{}_{s}d_{ii^{\prime}}=d(\chi_{is^{\prime}},\chi_{i^{\prime}s}) \tag{9}\]0  
It can be derived from Equation (21):  
\[{}_{s}d_{ii^{\prime}}=d(\chi_{is^{\prime}},\chi_{i^{\prime}s}) \tag{9}\]1  
After that, it is derived using Equation (21):  
\[{}_{s}d_{ii^{\prime}}=d(\chi_{is^{\prime}},\chi_{i^{\prime}s}) \tag{9}\]2  
Then, the weight formula in Equation (14) is obtained by replacing Equation (24) with Equation (23). In this way, using the combination of the weight information and the affiliation degree information of fuzzy clustering, the multi-dimensional fuzzy clustering is achieved to differentiate the target and background clutter. The detailed step-by-step process can be found in Algorithm 1, which illustrates the initialization membership update, weight update, and centroid recalculation steps of the MDF-clustering algorithm.