## 3 The Proposed Fusion Method
### Two-Sensor Fusion  
As shown in Figure 1, the targets detected by the two sensors \(a\) and \(b\) are denoted as \(\mathbf{X}_{a}\), \(\mathbf{X}_{b}\). The overlapping field of view of the sensors \(a\) and \(b\) is shown in the green part, and the unique FOVs are shown in blue and yellow, respectively. \(\{\mathbf{X}_{a}\diagdown\mathbf{X}_{b}\}\) denotes the removal of the targets in \(\mathbf{X}_{b}\) from the targets in \(\mathbf{X}_{a}\), and \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\) indicates the targets that are common to both \(\mathbf{X}_{a}\) and \(\mathbf{X}_{b}\). For fusion, it is essential to ensure that the fusion approach does not introduce false targets, miss real targets, or cause targets to be counted twice and that the approach can be independent of any a priori information about the targets' fields of view. According to Figure 1, the fused targets should be represented as follows:  
\[\mathbf{X} =\{\mathbf{X}_{a}\cup\mathbf{X}_{b}\}\diagdown\{\mathbf{X}_{a} \cap\mathbf{X}_{b}\} \tag{22}\] \[=\{\mathbf{X}_{a}\diagdown\mathbf{X}_{b}\}\cup\{\mathbf{X}_{a} \cap\mathbf{X}_{b}\}\cup\{\mathbf{X}_{b}\diagdown\mathbf{X}_{a}\}.\]  
Therefore, for the final fusion result, it is only necessary to merge the targets represented by each of the three parts in (22). Combining the three parts may be called a particular WAA procedure in which we attach a weight of 1 to each part. First, look at the \(\{\mathbf{X}_{a}\cap\mathbf{X}_{b}\}\) part. The targets in this part belong to the common FOV of the sensors \(a\) and \(b\). To obtain these targets, we can use the properties of the WGA method to suppress the targets in the sensors' different FOVs and keep only the targets in the common FOV. In fusing the local MBM a posteriori densities from different sensors, we perform MB one-to-one pairing fusion between different sensors, since the MBM is the result of combining multiple MBs. In the MB fusion process, we perform the Bernoulli one  
Figure 1: Schematic of the limited FOVs for two-sensor fusion. to-one pairing fusion process due to the non-closure of the results when fusing MBs directly. Combined with the locally obtained filter results in GGIW form, our final fusion results remain in GGIW form when performing fusion between different Bernoulli terms. The specific fusion results are illustrated below with one MB, \(\{\omega_{i_{a}}^{a},\{r,f\}_{i_{a}}^{a}\},i_{a}=1,2,\cdots,I_{a}\) and \(\{\omega_{i_{b}}^{b},\{r,f\}_{i_{b}}^{b}\},i_{b}=1,2,\cdots,I_{b}\) for sensors \(a\) and \(b\); \(I_{a}\) and \(I_{b}\) are the numbers of Bernoulli components. Assume that one Bernoulli term \(\{r^{a},f(\xi^{a})\}\), \(\{r^{b},f(\xi^{b})\}\) from sensors \(a\) and \(b\) can be represented as follows:  
\[f(\xi^{a}) =\mathcal{GAM}(\gamma;\alpha^{a},\beta^{a})\mathcal{N}(\mathbf{x} ;\mathbf{m}^{a},\mathbf{P}^{a})\mathcal{IW}_{d}(\mathbf{\Sigma};v^{a}, \mathbf{V}^{a}), \tag{23}\] \[f(\xi^{b}) =\mathcal{GAM}\Big{(}\gamma;\alpha^{b},\beta^{b}\Big{)}\mathcal{ N}\Big{(}\mathbf{x};\mathbf{m}^{b},\mathbf{P}^{b}\Big{)}\mathcal{IW}_{d} \Big{(}\mathbf{\Sigma};v^{b},\mathbf{V}^{b}\Big{)}. \tag{24}\]  
The weight \(\omega^{a,b}\) of the new MB formed by the fusion of the MB terms of sensors \(a\) and \(b\) is denoted as follows:  
\[\omega^{a,b}=\omega_{i_{a}}^{a}\omega_{i_{b}}^{b}. \tag{25}\]  
The fused GGIW parameters can be expressed as  
The existence probability obtained using (38) has not been normalized. The normalization process is described in (39), where the normalized denominator is the sum of all presence probabilities \(r^{a,ib}\) obtained from the sensors \(b\) and \(a\), while the numerator remains unchanged. After completing the one-to-one pairwise fusion of the Bernoulli terms from \(\{\omega_{i_{a}}^{a},\{r,f\}_{i_{b}}^{a}\}\) and \(\{\omega_{i_{b}}^{b},\{r,f\}_{i_{b}}^{b}\}\), we perform a pruning and merging process on the results obtained, which is similar to that described in [12], and it should be noted that the merging process is for the GGIN parameters.