## 3 The Proposed Fusion Method
### Two-Sensor Fusion
Then, we compute \(\{\mathbf{X}_{a}\backslash\mathbf{X}_{b}\}\). \(\{\mathbf{X}_{b}\backslash\mathbf{X}_{a}\}\) is obtained in a similar way. Because \(\{\mathbf{X}_{a}\backslash\mathbf{X}_{b}\}=\mathbf{X}_{a}\backslash\{\mathbf{ X}_{a}\cap\mathbf{X}_{b}\}\), \(\{\mathbf{X}_{a}\backslash\mathbf{X}_{b}\}\) can be obtained by eliminating the targets \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\) from \(\mathbf{X}_{a}\). Finding the targets of \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\) in \(\mathbf{X}_{a}\) can be solved with the following optimization problem:  
\[\operatorname*{arg\,max}_{a_{i}} \int\big{(}\sum_{i=1}^{l}a_{i}\mathcal{GGIN}(\tilde{s};\zeta^{i} )-\sum_{j=1}^{l}\mathcal{GGIN}(\tilde{s};\zeta^{j})\big{)}^{2}d\zeta \tag{40}\] \[s.t. \sum_{i=1}^{l}a_{i}=J,a_{i}=0\,or\,\,\,1.\]  
The implication of this optimization problem is finding several GGIW components in \(\mathbf{X}_{a}\) such that the GGIW components found minimize the error of the square of the L2 norm with respect to the \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\) part of the objective. Here, \(\sum_{j=1}^{l}\mathcal{GGIN}(\tilde{s};\zeta^{j})\) denotes the targets of \(\mathbf{X}_{a}\cap\mathbf{X}_{b}\), which contains a total of \(J\) GGIW components corresponding to \(J\) targets. \(a_{i}\) is the optimization variable; there are \(I\) in total. When \(a_{i}=1\), this indicates that the \(i\)-th objective in \(\sum_{i=1}^{l}a_{i}\mathcal{GGIN}(\tilde{s};\zeta^{i})\) should be included in \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\), and the opposite is not included. In addition, it is necessary to ensure that only \(J\) targets in \(\sum_{i=1}^{l}a_{i}\mathcal{GGIN}(\tilde{s};\zeta^{i})\) can be included in \(\mathbf{X}_{a}\cap\mathbf{X}_{b}\). See Appendix A for the simplification of the optimization objective function in (40), which can be solved using linear programming. If the three parts \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\), \(\{\mathbf{X}_{a}\backslash\mathbf{X}_{b}\}\), and \(\{\mathbf{X}_{b}\backslash\mathbf{X}_{a}\}\) are obtained separately, it is only necessary to merge these three parts as the final MB fusion result. Then, the remaining MB elements of sensors \(a\) and \(b\) are traversed until all MBs are fused one by one, and then these fused MBs are reorganized into a new MBM. Similarly, the new MBM formed after fusion also undergoes a pruning and merging process for MB components, which can be found in [40]. The pseudocode for the whole fusion process under the condition of two sensors is shown in Algorithm 1. ```
Input: Parameterized posterior densities \(\omega_{i_{a}}^{a},\{r,f\}_{i_{a}}^{a},i_{a}\in I_{a}\) and \(\omega_{i_{b}}^{b},\{r,f\}_{i_{b}}^{b},i_{b}\in I_{b}\) obtained from sensors \(a\) and \(b\) at time \(k\). Output: Fused posterior density \(\omega_{i_{a}}^{ab},\{r,f\}_{i_{ab}}^{ab},i_{ab}=I_{ab}\). for\(i_{a}=1:I_{a}\)do  Choose \(\omega_{i_{a}}^{a},\{r,f\}_{i_{a}}^{a}\) for fusion; for\(i_{b}=1:I_{b}\)do  Choose \(\omega_{i_{b}}^{b},\{r,f\}_{i_{b}}^{b}\) for fusion;  Calculate \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\) using (25)-(39);  Perform a pruning and merging process for Bernoulli terms in \(\mathbf{X}_{a}\cap\mathbf{X}_{b}\);  Calculate \(\{\mathbf{X}_{a}\backslash\mathbf{X}_{b}\}\) using (40);  Calculate \(\{\mathbf{X}_{b}\backslash\mathbf{X}_{a}\}\) using (40);  Merge \(\mathbf{X}_{a}\cap\mathbf{X}_{b}\), \(\{\mathbf{X}_{a}\backslash\mathbf{X}_{b}\}\), \(\{\mathbf{X}_{b}\backslash\mathbf{X}_{a}\}\) to form \(\omega_{i_{ab}}^{ab},\{r,f\}_{i_{ab}}^{ab}\);  end for  end for  Perform a pruning and merging process for fused MB terms and output the fused posterior \(\omega_{i_{ab}}^{ab},\{r,f\}_{i_{ab}}^{ab},i_{ab}=I_{ab}\). ```  
**Algorithm 1** Two-sensor fusion pseudocode