## 3 Proposed Real-Time Data Fusion Method Based on a New Distance Metric
### Data Fusion for Object Information
#### 3.1.2 Data Fusion for the Existence of Objects  
For the problem of determining existence, the frame of discernment is given by \(\Theta_{E}=\{\text{existence, non-existence}\}\). In this paper, we use notation \(2^{\Theta_{E}}\) to represent the powerset of \(\Theta_{E}\), i.e., \(2^{\Theta_{E}}=\{\text{existence}\}\), \(\{\text{non-existence}\}\), \(\{\text{uncertainty (=existence or non-existence)}\}\}\) where the elements are denoted by E, N, and U, respectively. Let us define \(m_{k}^{i}(X)\) as the confidence at vehicle \(i\) about  
Figure 1: Determining the object type information.  
object \(k\) that is committed to \(X\in 2^{\Theta_{E}}\), and \(M_{k}^{i}=\{m_{k}^{i}(\mathrm{E}),m_{k}^{i}(\mathrm{N}),m_{k}^{i}(\mathrm{U})\}\) is defined to be the set of confidences. For two vehicles \(i\) and \(j\), we can calculate \(m_{k}^{i}(X)\oplus m_{k}^{i}(X)\) for each element \(X\in\{\mathrm{E},\mathrm{N},\mathrm{U}\}\) from (3) as follows,  
\[m_{k}^{i}(\mathrm{E})\oplus m_{k}^{j}(\mathrm{E})= \frac{m_{k}^{i}(\mathrm{E})m_{k}^{j}(\mathrm{E})+m_{k}^{i}(\mathrm{ E})m_{k}^{j}(\mathrm{U})+m_{k}^{i}(\mathrm{U})m_{k}^{j}(\mathrm{E})}{1-\kappa},\] \[m_{k}^{i}(\mathrm{N})\oplus m_{k}^{j}(\mathrm{N})= \frac{m_{k}^{i}(\mathrm{N})m_{k}^{j}(\mathrm{N})+m_{k}^{j}( \mathrm{N})m_{k}^{j}(\mathrm{U})+m_{k}^{i}(\mathrm{U})m_{k}^{j}(\mathrm{N})} {1-\kappa}, \tag{4}\] \[m_{k}^{i}(\mathrm{U})\oplus m_{k}^{j}(\mathrm{U})= \frac{m_{k}^{i}(\mathrm{U})m_{k}^{j}(\mathrm{U})}{1-\kappa},\]  
where \(\kappa=m_{k}^{i}(\mathrm{E})m_{k}^{j}(\mathrm{N})+m_{k}^{i}(\mathrm{N})m_{k}^ {j}(\mathrm{E})\). For instance, given \(M_{k}^{i}=\{0.88,0,0.12\}\) and \(M_{k}^{i}=\{0,0.7,0.3\}\), \(m_{k}^{i}(\mathrm{E})\oplus m_{k}^{j}(\mathrm{E})\) is given by \(\frac{0.88\otimes 0.3}{1-0.88\times 0.7}\approx 0.68\).  
Below we define \(\mathrm{U}\) or the \"uncertainty\" element, in a rigorous manner. There are two types of uncertainty. The first type is the uncertainty due to inherent limitation of a sensor. For example, a vision sensor cannot detect objects behind obstacles. As shown in Figure 2, the vehicle cannot recognize the existence of the completely occluded object unless a peer vehicle reports the object to that vehicle even if the object is within the sensing area. In this case, the confidence assignments by vehicle \(i\) about \(k\) is given by  
\[m_{k}^{i}(\mathrm{E})=m_{k}^{i}(\mathrm{N})=0,\ m_{k}^{i}(\mathrm{U})=1. \tag{5}\]  
The second type of uncertainty is from the limited accuracy of sensors. The accuracy of a sensor may be known in advance from its manufacturer [42]. In addition, the confidence can be adjusted for the environment such as weather condition or the distance to the object [16]. For example, we can set the BPAs for clear and rainy days to \(\{0.9,0,0.1\}\) and \(\{0.9-\delta,0.0.1+\delta\}\), respectively, where \(\delta\) is a system parameter.  
However, the classical DS combination may yield unreasonable outputs when evidences are highly conflicting. This problem is due to combining in (3) which ignores the conflicts in evidence. In order to illustrate this problem, let us consider the following example [19; 40; 43]. Given 3 events \(Z=\{A,B,C\}\) and 2 Observers, Observer 1 and 2 assign \(m_{1}(A)=0.9,m_{1}(B)=0.1,m_{1}(C)=0.0\) and \(m_{2}(A)=0.0,m_{2}(B)=0.1,m_{2}(C)=0.9\) as confidences in \(Z\), respectively. Although Observers are quite certain that either event \(A\) or \(C\) has occurred, the combination assigns the highest confidence to event \(B\) as \(m_{1}(B)\oplus m_{2}(B)=1\). Therefore, in order to deal with highly conflicting evidences, we consider the evidence of distance in order to measure the degree of conflict among evidences [24; 33].  
Figure 2: Definitions of sensing and uncertainty areas. The object in uncertainty area can be perceived through V2V communication.