# A Unified Denoising Framework for Restoring the LiDAR Point Cloud Geometry of Reflective Targets
## 2 Materials and Methods
### Methods
#### 2.2.2. Range Anomalies Correction  
Random sample consensus (RANSAC) [41] identifies inlier data and fits a model by searching for the minimum cost under the given model and distance threshold. A point is classified as an inlier with a weight of zero if its distance to the model is less than the  
Figure 3: Illustration of veiling effect denoising: (**a**) center of peak probability density of the original LiDAR point cloud intensity; (**b**) coarse denoising to remove low-intensity noise points and extract potential target points. Note: The area of the point cloud after coarse denoising from the X-axis perspective is still larger than the actual target region, indicating that the blooming effect has not been eliminated. Figure 2: Workflow of UDF for denoising LiDAR point cloud of reflective target. threshold; otherwise, it is classified as an outlier. Since all inliers have a weight of zero, the algorithm selects the optimal model with the maximum number of inliers and the smallest total weight. However, when the threshold is set too large, especially for datasets containing distinct outliers, it may result in a significant increase in fitting errors. MSAC [42] is an improved variant of the RANSAC algorithm designed to reduce the algorithm's sensitivity to the selection of the distance threshold T. The modified cost function is shown in Equation (1). When the distance e from a point to the model is less than the threshold T, the point is classified as an inlier and assigned a weight of e. Otherwise, the point is classified as an outlier and assigned a weight of T.  
\[\rho_{2}(e^{2})= \begin{cases}e^{2}&e^{2}<T^{2}\\ T^{2}&e^{2}\geq T^{2}\end{cases}  \tag{1}\]  
Due to the signal saturation and distortion of the pulsed laser echo when measuring the surface of reflective targets, measurement errors cause a shift in the distance along the direction of laser emission. This results in the loss of certain surface features of the target. Therefore, although the spatial distribution characteristics of outliers are similar to those of noise points caused by the veiling effect, their intensity values are nearly identical to those of the point cloud on the surface of the reflective target. Based on the principles of range anomalies described above, the outliers are processed as follows: first, the laser emission vector is obtained by connecting the origin of the LiDAR coordinate system to the range-anomalous point; then, the intersection of this vector with the plane fitted by MSAC is calculated. The intersection point (Equation (2)) represents the corrected position of the outlier. The illustration of the range-anomalous points' correction is shown in Figure 4; the procedure of the algorithm is presented in Algorithm 1. \[ \begin{cases}x_{0}t=-\frac{d_{0}}{a_{0}x_{0}+b_{0}y_{0}+c_{0}z_{0}}\cdot x_{0} \\ y_{0}t=-\frac{d_{0}}{a_{0}x_{0}+b_{0}y_{0}+c_{0}z_{0}}\cdot y_{0}\\ z_{0}t=-\frac{d_{0}}{a_{0}x_{0}+b_{0}y_{0}+c_{0}z_{0}}\cdot z_{0}\end{cases}  \tag{2}\]  
where \((x_{0},y_{0},z_{0})\) and \((x_{0}t,y_{0}t,z_{0}t)\) represent the coordinates of the range-anomalous point and the range-corrected point, respectively. The parameters \(a_{0}\), \(b_{0}\), \(c_{0}\), and \(\mathrm{d}_{0}\) are the coefficients of the plane equation \(a_{0}x+b_{0}y+c_{0}z+d_{0}=0\). Figure 4: Illustration of range-anomalous points’ correction: **(a)** original points with range anomalies; **(b)** points with corrected ranges and coordinates. Note: The fitted plane in the figure is presented from a perspective parallel to the plane. ```
Input: Potential target points \(P\) and the maximum distance from an inlier point to the plane \(d_{\max}\) Output: Projected 2D points \(Q\) (1) Fit a plane to \(P\) using the MSAC algorithm to obtain the plane equation.