## 3 Required Data Rate in V2V for Safe Overtaking
### Derivation of the Required Data Rate  
In this section, the details of the recognition process are introduced. In general, object recognition can be classified into two cases. One is specific object recognition. This recognition tries to classify an object as a specific object. The other is general object recognition. In contrast to the former recognition, this recognition tries to classify an object as a generic object. Since we focused on the recognition of a vehicle, specific object recognition was adopted, and we refer to the recognition target as the target vehicle. In this case, the ego vehicle wants to prevent a collision with the oncoming vehicle so that the oncoming vehicle becomes the target vehicle. The recognition part consists of three phases. The first phase is the simulation of LiDAR sensor data in the virtual environment and clustering point cloud about the target vehicle. The second phase is the extraction of feature points from the clustered points. The final phase is the decision about recognition. The following paragraphs explain the details of each phase. In the first phase, regarding lasers from a LiDAR sensor as geometric optics, ray-tracing simulation of LiDAR sensor data was adopted. In order to implement ray tracing easily, objects such as vehicles, buildings, and roads consist of triangle meshes. From this setting, a point \(\mathbf{p}\) on a triangle mesh can be described with three position vectors \(\mathbf{p}_{1}\), \(\mathbf{p}_{2}\), and \(\mathbf{p}_{3}\) and two parameters \(u,\ v\) as the following formula. \[\mathbf{p}=(1-u-v)\mathbf{p}_{1}+u\mathbf{p}_{2}+v\mathbf{p}_{3} \tag{6}\]  
Furthermore, the point \(\mathbf{p}\) can be also described by a normalized direction vector \(\mathbf{d}\) departing from the laser source \(\mathbf{O}\) to \(\mathbf{p}\). \[\mathbf{p}=\mathbf{O}+t\mathbf{d} \tag{7}\]  
Since the laser propagates in three-dimensional space, the departure angle can be described by azimuth angle \(\phi\) and elevation angle \(\theta\). When a point is on the mesh, parameters \(u\) and \(v\) have to meet \(0\leq u,\ v\leq 1\) and \(0\leq u+v\leq 1\). On the other hand, parameter \(t\) has to meet \(0\leq t\). In order to confirm whether these conditions are met or not, we solved these parameters by combining Equations (6) and (7) and adopting Cramer's rule. As mentioned above, before extracting the feature points only from the target vehicle points for recognition, clustering is needed to remove irrelevant points. In this simple ray tracing algorithm, the function of linking the hit object to the laser is implemented. As a result, the LiDAR sensor in our simulation knows which object the laser is reflected from so that we can only select the points of the target vehicle and perform clustering easily. In the second phase, we extracted feature points from the clustered points. When we want to describe features of point cloud data, or LiDAR sensor data, a feature descriptor is often used. SHOT (Signature of Histogram of Orientation) and PFH (Point Feature Histogram) are the typical feature descriptors. These descriptors use a histogram to describe features around a point [26; 27]. In general, the calculation time of a feature descriptor depends on the dimension of the descriptor. In order to avoid this complicated discussion, we used edge points, which are basic features. Extracting edges was performed by PCA (Principal Component Analysis) [28]. This PCA method is faster and more robust to noise than using a Gauss map. The key point of this process is that edge points are extracted by the eigenvalues of a covariance matrix. The quantity made of the eigenvalues is called the surface curvature, and it is calculated for each point. When the surface curvature exceeds a threshold, the point is regarded as an edge point. The threshold is tuned by observing the distribution of the surface curvature. The final phase is the decision about recognition. In this simulation, we adopted model-based recognition.