## 3 The Proposed Fusion Method
### Multi-Sensor Fusion  
This section illustrates a three-sensor example for multi-sensor fusion in the presence of limited FOVs, and the relevant conclusions can be extended to more multi-sensor fusion scenarios. As shown in Figure 2, the posterior densities from sensors \(a\), \(b\), and \(c\) can be expressed as follows:  
\[\mathbf{X}_{a} =\mathbf{X}_{1}\cup\mathbf{X}_{2}\cup\mathbf{X}_{4}\cup\mathbf{X}_{5}, \tag{41}\] \[\mathbf{X}_{b} =\mathbf{X}_{2}\cup\mathbf{X}_{3}\cup\mathbf{X}_{5}\cup\mathbf{X}_ {6},\] (42) \[\mathbf{X}_{c} =\mathbf{X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\cup\mathbf{X}_ {7}. \tag{43}\]  
Then, when merging the three posterior densities, the process can be expressed as follows:  
\[\mathbf{X}_{f} =\mathbf{X}_{1}\cup\mathbf{X}_{2}\cup\mathbf{X}_{3}\cup\mathbf{X}_ {4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\cup\mathbf{X}_{7} \tag{44}\] \[=\{\mathbf{X}_{1}\cup\mathbf{X}_{2}\cup\mathbf{X}_{3}\cup\mathbf{ X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\}\] (45) \[\cup\{\mathbf{X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\cup \mathbf{X}_{7}\}\backslash\{\mathbf{X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\}\] \[=\{\mathbf{X}_{1}\cup\mathbf{X}_{2}\cup\mathbf{X}_{4}\cup\mathbf{ X}_{5}\}\cup\{\mathbf{X}_{2}\cup\mathbf{X}_{3}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\}\] (46) \[\backslash\{\mathbf{X}_{2}\cup\mathbf{X}_{5}\}\cup\{\mathbf{X}_{4} \cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\cup\mathbf{X}_{7}\}\] \[\backslash\{\mathbf{X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\},\]  
where \(\{\mathbf{X}_{1}\cup\mathbf{X}_{2}\cup\mathbf{X}_{4}\cup\mathbf{X}_{5}\}\cup\{ \mathbf{X}_{2}\cup\mathbf{X}_{3}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\}\backslash \{\mathbf{X}_{2}\cup\mathbf{X}_{5}\}\) is the fusion result of sensors \(a\) and \(b\), and it is denoted as \(\mathbf{X}_{ab}\). The result \(\mathbf{X}_{ab}\cup\{\mathbf{X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\cup \mathbf{X}_{7}\}\backslash\{\mathbf{X}_{4}\cup\mathbf{X}_{5}\cup\mathbf{X}_{6}\}\) in (46) is the fusion result of \(\mathbf{X}_{ab}\) and sensor \(c\); thus, it is the fusion result of sensors \(a\), \(b\), and \(c\) and is denoted as \(\mathbf{X}_{f}\).  
The above process is the multi-sensor sequential fusion method proposed in this study; it fuses the results of two sensors at a time and then continues to fuse that result with the results of the next unfused sensor. The process is repeated until all sensors are involved in the fusion process. In summary, the pseudocode of the distributed multi-sensor fusion process is shown in Algorithm 2.  
Figure 2: Schematic of the limited FOVs for three-sensor fusion.  
```
Input: Parameterized posterior densities \(\{\omega_{i_{1}^{\texttt{T}}}^{\texttt{T}},\{r,f\}_{i_{1}}^{\texttt{T}}\},\{ \omega_{i_{2}^{\texttt{T}}}^{\texttt{T}},\{r,f\}_{i_{2}}^{\texttt{T}}\},\ldots\), \(\{\omega_{i_{l^{\texttt{N}}}}^{\texttt{N}},\{r,f\}_{i_{l}}^{\texttt{N}}\}\) obtained from \(\mathbb{N}\) sensors at time \(k\). Output: Fused posterior \(\omega_{i_{p}^{\texttt{F}}}^{\texttt{F}},\{r,f\}_{i_{p}}^{\texttt{F}}\). Perform a two-sensor fusion process to \(\{\omega_{i_{1}^{\texttt{T}}}^{\texttt{T}},\{r,f\}_{i_{1}}^{\texttt{T}}\}\) and \(\{\omega_{i_{2}^{\texttt{T}}}^{\texttt{T}},\{r,f\}_{i_{2}}^{\texttt{T}}\}\) to obtain \(\omega_{i_{p}^{\texttt{F}}}^{\texttt{F}},\{r,f\}_{i_{p}}^{\texttt{F}}\) (Algorithm 1). for\(j=3:\mathbb{N}\)do  Choose \(\{\omega_{i_{p}^{\texttt{T}}}^{j},\{r,f\}_{i_{j}}^{j}\}\) for fusion;  Perform a two-sensor fusion process on \(\omega_{i_{p}^{\texttt{F}}}^{\texttt{F}},\{r,f\}_{i_{p}}^{\texttt{F}}\) and \(\{\omega_{i_{p}^{\texttt{T}}}^{j},\{r,f\}_{i_{j}}^{j}\}\) to obtain a new \(\omega_{i_{p}^{\texttt{F}}}^{\texttt{F}},\{r,f\}_{i_{p}}^{\texttt{F}}\) (Algorithm 1).  end Output the fused posterior \(\omega_{i_{p}^{\texttt{F}}}^{\texttt{F}},\{r,f\}_{i_{p}}^{\texttt{F}}\).
```  
**Algorithm 2** Multi-sensor fusion pseudocode