# Radar Voxel Fusion for 3D Object Detection
## 3 Methodology
### Yaw Loss Parameterization  
While the original VoxelNet paper uses a simple yaw regression, we use a more complex parameterization to facilitate the learning process. [PERSON] [30] argues that a simple yaw representation is disadvantageous, as the optimizer needs to regress a smooth function over a discontinuity, e.g., from \(-\pi\,\text{rad}\) to \(+\pi\,\text{rad}\). Furthermore, the loss value for small positive angle differences is much lower than that of greater positive angle differences, while the absolute angle difference from the anchor orientation might be the same. Figure 2 visualizes this problem of an exemplary simple yaw regression.  
To account for this problem, the network estimates the yaw angle with a combination of a classification and a regression head. The classifier is inherently designed to deal with a discontinuous domain, enabling the regression of a continuous target. The regression head regresses the actual angle difference in the interval \([-\pi,\pi)\) with a smooth sine function, which is continuous even at the limits of the interval. The regression output of the yaw angle of a bounding box is  
\[\theta_{d} =\theta_{GT}-\theta_{A} \tag{1}\] \[e_{\theta} =sin(\theta_{d}),\]  
Figure 1: Network architecture of the proposed RVF-Net.  
Figure 2: Vehicle bounding boxes are visualized in a BEV. The heading of the vehicles is visualized with a line from the middle of the bounding box to the front. The relative angular deviations from the orange and blue ground truth boxes to the anchor box are equal. However, the resulting loss value of the orange bounding box is significantly higher than that of the blue one.  
where \(\theta_{GT}\) is the ground truth box yaw angle and \(\theta_{A}\) is the associated anchor box yaw angle.  
The classification head determines whether or not the angle difference between the predicted bounding box and the associated anchor lies inside or outside of the interval \([-\pi/2,\pi/2)\). The classification value of the yaw is modeled as  
\[c_{dir}=\begin{cases}1,&\text{if }-\frac{\pi}{2}\leq(\theta_{d}+\pi)\text{ mod }2\pi-\pi<\frac{\pi}{2}\\ 0,&\text{otherwise}\end{cases}. \tag{2}\]  
As seen above, the directional bin classification head splits the angle space into two equally spaced regions. The network uses two anchor angle parameterizations at 0 and \(\pi/2\). A vehicle driving towards the sensor vehicle matches with the anchor at 0 rad. A vehicle driving in front of the vehicle would match with the same anchor. The angle classification head intuitively distinguishes between these cases. Therefore, there is no need to compute additional anchors at \(\pi\) and \(-\pi/2\).  
Due to the subdivision of the angular space by the classification head, the yaw regression needs to regress smaller angle differences, which leads to a fast learning progress. A simple yaw regression would instead need to learn a rotation of 180 degrees to match the ground truth bounding box. It has been shown that high regression values and discontinuities negatively impact the network performance [30]. The regression and classification losses used to estimate the yaw angle are visualized in Figure 3.  
The SECOND architecture [31] introduces a sine loss as well. Their subdivision of the positive and negative half-space, however, comes with the drawback that both bounding box configurations shown in Figure 3 would result in the same regression and classification loss values. Our loss is able to distinguish these bounding box configurations.  
As the training does not learn the angle parameter directly, the regression difference is added to the anchor angle under consideration of the classification interval output to get the final value of the yaw angle during inference.