## 4 VRFormer
### Data Fusion Techniques: Integrating VR and V2X Inputs
#### 4.3.2 Probabilistic Data Fusion Using Bayesian Inference  
The VRFormer model leverages Bayesian inference within the VRDeepSafety platform to enhance the robustness of its data fusion process. Unlike traditional deterministic methods, the Bayesian approach treats sensor and V2X data as probabilistic, assigning reliability scores and dynamically adjusting their weights based on real-time environmental conditions. This adaptability is critical for AD, as sensor accuracy varies significantly with factors such as weather, lighting, and occlusion. For instance, in low-visibility conditions like fog or heavy rain, the model reduces the weight of camera data and increases the weight of radar data, which are more reliable under these circumstances. Similarly, when V2X data are delayed or incomplete due to network issues, the model relies more on onboard sensor data. This dynamic adjustment allows the model to consistently maintain high accuracy, even in challenging conditions. This is a significant improvement over traditional deterministic methods that do not account for sensor uncertainty.  
To synchronize VR and V2X inputs spatially, VRFormer uses dynamic feature alignment by employing landmark-based adjustments. By using fixed environmental markers like signposts and intersections, VRFormer accurately aligns data and ensures precise positioning and coherence. An environmental adaptation layer further refines sensor weightings in response to real-time changes, such as weather and lighting. This adaptive process enables VRFormer to consistently keep high performance and reliability in diverse and shifting environments within VRDeepSafety.  
Detailed Mathematical Formalism for CNNs and LSTMs  
Beyond sensor fusion, VRDeepSafety uses CNNs for motion prediction. CNNs are effective at extracting spatial features from camera and LiDAR sensor data, which is important for predicting the movement of surrounding objects, especially in bad weather. The CNN model applies filters to the input data, the convolution operation is defined as follows:  
\[\mathbf{f}_{k}(\mathbf{X})=\mathbf{X}\times\mathbf{W}_{k}+\mathbf{b}_{k} \tag{24}\]  
where \(\mathbf{f}_{k}(\mathbf{X})\) is the represents of the convolution operation, and \(\mathbf{W}_{k}\) denotes the filters used. The ReLU activation function is applied to introduce non-linearity:  
\[g(\mathbf{f}_{k}(\mathbf{X}))=\max(0,\mathbf{f}_{k}(\mathbf{X})) \tag{25}\]  
These CNN-based predictions allow the system to predict the future motion of nearby vehicles, thereby enhancing the AV's informed navigation decisions, particularly in dynamic and unpredictable environments.  
Besides motion prediction, VRDeepSafety uses LSTM networks, a type of recurrent neural network (RNN), to predict accidents by analyzing temporal patterns in sensor data. LSTM networks are adept at modeling sequential data and capturing the dependencies between past and current readings. The internal state of the LSTM is updated using a series of gates, such as the forget gate:  
\[f_{t}=\sigma\left(\mathbf{W}_{f}\cdot\left[\mathbf{h}_{t-1},\mathbf{X}_{t} \right]+\mathbf{b}_{f}\right) \tag{26}\]  
And the input gate:  
\[i_{t}=\sigma(\mathbf{W}_{i}\cdot\left[\mathbf{h}_{t-1},\mathbf{X}_{t}\right] +b_{i})\text{, }\widetilde{c}_{t}=\tanh(\mathbf{W}_{c}\cdot\left[\mathbf{h}_{t-1}, \mathbf{X}_{t}\right]+\mathbf{b}_{c}) \tag{27}\]  
The cell state \(\mathbf{c}_{t}\) is updated to incorporate new and retained information:  
\[\mathbf{c}_{t}=f_{t}\cdot c_{t-1}+i_{t}\cdot\widetilde{c}_{t} \tag{28}\]  
This architecture enables the system to predict potential accidents by recognizing patterns in the data that signal impending collisions, further improving the AV's decision-making process.