## 2 Materials and Methods
### Proposed Multi-Sensor-Based Detection Model
#### 2.2.2 Preprocessing with the Gaussian-Sobel Filter  
Object detection with RGB cameras suffers from significant performance degradation in night-time environments compared with daytime environments. To address this limitation, we propose a method that leverages depth maps obtained from LiDAR to enhance pedestrian detection in low-light conditions. Given that the depth map is derived from low-resolution PCD, object contours exhibit reduced sharpness. Consequently, an enhancement process is necessary, and a sharpening filter is additionally applied to address this issue. First, a Gaussian filter is applied to suppress the variance of depth information caused by the different sampling intervals of PCD depending on the LiDAR channel. Subsequently, a Sobel filter is utilized to accentuate the line edges of objects whose boundaries with the background become blurred owing to Gaussian blurring, thereby facilitating feature extraction of objects. By doing so, the model most effectively preserves pixel values of the depth information while further accentuating object contours. A Gaussian filter generates a natural blurring effect on images by using a filter mask created by approximating a Gaussian distribution [23; 24]. A 2D Gaussian distribution function with a mean of \((0,\ 0)\) and standard deviations of \(\sigma_{\alpha}\) and \(\sigma_{\beta}\) along the \(\ \alpha\) and \(\beta\) axes, respectively, is defined as follows:  
\[\Lambda_{\sigma_{\alpha}\sigma_{\beta}}(\alpha,\beta)=\frac{1}{2\pi\sigma_{ \alpha}\sigma_{\beta}}e^{-(\frac{\alpha^{2}}{2\sigma_{\alpha}^{2}}+\frac{ \beta^{2}}{2\sigma_{\beta}^{2}})} \tag{5}\]  
The depth map, \(L_{d}\), can be expressed as follows after undergoing preprocessing by a Gaussian filter:  
\[L_{d,\beta}=\Lambda_{\sigma_{\alpha}\sigma_{\beta}}(\alpha,\beta)*L_{d} \tag{6}\]  
The Gaussian function peaks at \(\Lambda_{\sigma_{\alpha}\sigma_{\beta}}(0,0)\) and decreases as the distance from the center increases. Furthermore, in the filter mask, pixels near the filtering target receive higher weights, whereas those further away receive lower weights, which helps mitigate the effects of noise pixels in the depth map. Figure 2: Process for generating a depth map for image registration: (**a**) RGB image; (**b**) PCD projected on RGB image; (**c**) depth map. Thus, the Gaussian filter suppresses the effects of noise in the depth map and blurs the overall image, resulting in a decrease in contrast. To compensate for this drawback, a Sobel filter is directly applied to preserve as much depth information of the object as possible while still accentuating object contours. The Sobel filter uses two 3 \(\times\) 3 kernels to detect edges: one is for finding changes along the horizontal direction, while the other is for finding changes in the vertical direction. Edges are points where the instantaneous rate of change of a function is large. Therefore, if a value exceeding a certain threshold is derived by calculating the differential value at each pixel of the image, the corresponding point is determined as an edge [25; 26]. The two kernels are convolved with the original image to approximate the rate of change; if we define \(  
As mentioned earlier, Figure 3a illustrates the variance of depth information in PCD, which varies with the data generation interval. To suppress this, we applied a Gaussian filter (Figure 3b), which suppresses the effects of noise and smooths out the contour information of the object. Moreover, we applied an additional Sobel filter, resulting in a clearer delineation of object contours, as shown in Figure 3c. Notably, the use of Gaussian blurring and Sobel operators is also a process included in the Canny edge filter. Furthermore, the reasons for the mandatory use of the Gaussian-Sobel filter are as follows: after Gaussian blurring and Sobel masking, the Canny edge filter [27] further applies NMS and hysteresis edge tracking.