## 3 Proposed Real-Time Data Fusion Method Based on a New Distance Metric
### Data Fusion for Object Information
#### 3.1.5 Proposed Data Fusion for Object Classification  
Next, we consider the object classification process. The frame of discernment for classification is given by the set of objects in the roads, e.g., \(\Theta_{C}=\{\text{Sedan, SUV, Pedestrian, Bicycle,},\cdots\}\). Let \(c_{i}^{k}(X)\) be the output from object detectors of vehicle i using about class \(X\) for object \(k\). We define the set of outputs as \(C_{k}^{i}=\{c_{k}^{i}(X),\forall X\subseteq\Theta_{C}\}\). Any object detector providing confidence value for each element in \(C_{k}^{i}\) can be applied, e.g., deep neural networks [48], classified vector quantization [49], and \(k\)-nearest neighbor [50]. In general, the outputs from the object detection system are not suitable for direct use as confidences [51]. Instead, we apply temperature scaling technique [51] to classifier output, and the calibrated result \(\bar{c}^{i}_{k}(\cdot)\) is defined by  
\[\bar{c}^{i}_{k}(X)=\frac{\exp(c^{i}_{k}(X)/T)}{\sum_{Y\in 2^{\Theta_{C}}}\exp(c^{i }_{k}(Y)/T)},\ \forall X\in 2^{\Theta_{C}}, \tag{24}\]  
where temperature \(T\) is a control parameter. We can tune the deviation among confidences by adjusting the parameter \(T\). This adjustment allows the result to be \"soft\" while preserving the existing confidence [52]. The classifier assigns confidence (BPA) to each element of \(\Theta_{C}\), and it satisfies the conditions in (1) and (2). It is reasonable for a classifier to produce distinct probabilities for each candidate class instead of a single output across multiple classes. Therefore, we have the same number of focal elements to \(|\Theta_{C}|\). Next, we will calculate the distance of confidences as in (10) using calibrated confidences. Assume that vehicle \(i\) receives \(n\)-th data point. Note that only the vehicles who identify \"existence\" with respect to object \(k\) can have the corresponding class confidences, which is defined by \(E^{i}_{k}=\{j|m^{j}_{k}(E)>0,\forall j\in\mathcal{M}_{i(k)}(n)\}\). The same procedure in (15)-(18) is applied to obtain the weighted confidences, and the DS combination is performed with respect to the weighted confidences for \(|E^{i}_{k}|-1\) times. Let us denote the obtained DS combination results for classification as \(W^{i}_{k}(X)\), \(\forall X\in 2^{\Theta_{C}}\). Finally, we choose the class with the highest confidence in the estimated class given by  
\[CL^{i}_{k}=\operatorname*{arg\,max}_{X\in 2^{\Theta_{C}}}W^{i}_{k}(X) \tag{25}\]  
where \(CL^{i}_{k}\) is the class decided at vehicle \(i\) about object \(k\). The classification process is presented in Algorithm 2. ```
1:function\([\mathcal{C}]=\textsc{Class}\ (C^{i}_{k},\forall i\in E^{i}_{k})\)
2:Initialize : \(\mathcal{C},\mathcal{W}=\emptyset\),
3:for\(i\in E^{i}_{k}\)do
4:\(\bar{c}^{i}_{k}(X)\leftarrow\frac{exp(c^{i}_{k}(X)/T)}{\sum_{j=1}^{|\Theta_ {C}|}exp(c^{i}_{k}(X)/T)},\ \forall X\in 2^{\Theta_{C}}\)
5:endfor
6:\(\bar{c}^{i}_{k}(X)\leftarrow\) Calculate the weighted confidences according to (10) and (15)-(18) based on \(\bar{c}^{i}_{k}(X)\), \(\forall X\in 2^{\Theta_{C}}\)
7:for\(j=1,\cdots,|E^{i}_{k}|-1\)do
8:\(W^{i}_{k}(X)\gets W^{i}_{k}(X)\oplus\bar{c}^{i}_{k}(X)\), \(\forall X\in 2^{\Theta_{C}}\)
9:endfor
10:\(\mathcal{C}\leftarrow\) arg max\({}_{X\in 2^{\Theta_{C}}}\ W^{i}_{k}(X)\)
11:return\(\bar{\mathcal{C}}\)
12:endfunction
```  
**Algorithm 2** Classification fusion for vehicle \(i\) about object \(k\)  
We can show that the data fusion for classification also has polynomial time complexity as follows. Under the classification fusion, as explained previously, we have independent confidences for each class. Consequently, the valid power set of \(\Theta_{C}\) is equal to \(\Theta_{C}\), which results in \(O(|\Theta_{C}|^{2})\) complexity to calculate the distance between confidences. Therefore, in total, the complexity of the class fusion process is \(O(|E^{i}_{k}|\lvert\Theta_{C}\rvert^{2})\). The integrated decision-making is outlined in Algorithm 3. The algorithm uses functions of Existence and Class from Algorithms 1 and 2.