# A Unified Denoising Framework for Restoring the LiDAR Point Cloud Geometry of Reflective Targets
## 3 Experiments and Results
### Evaluation Metrics  
To evaluate the denoising performance of the proposed algorithm, we employed four metrics derived from the confusion matrix of the denoising results (see Table 1), which have been commonly applied in previous studies [44]. These metrics include the type I error (T.I), type II error (T.II), total error (T.E. ), and kappa coefficient (\(\kappa\)). The type I error (Equation (8)) represents the percentage of target points incorrectly classified as noise points, while the type II error (Equation (9)) reflects the proportion of noise points mistakenly classified as target points. The total error (Equation (10)) quantifies the overall proportion of misclassified points. Meanwhile, the kappa coefficient (Equation (11)) provides an alternative metric for accessing overall classification accuracy by accounting for chance agreement and quantifying the improvement in classification accuracy over a random classification. \[Type\ I\ error=\frac{FN}{TP+FN} \tag{8}\]  
\[Type\ II\ error=\frac{FP}{FP+TN} \tag{9}\]  
\[Total\ error=\frac{FN+FP}{S} \tag{10}\]  
\[Kappa\ coefficient=\frac{p_{0}-p_{e}}{1-p_{e}} \tag{11}\]  
where \(S=TP+FN+FP+TN\), \(p_{0}=\frac{TP+TN}{5}\), \(p_{e}=\frac{(TP+FN)(TP+FP)+(FP+TN)(FN+TN)}{5^{2}}\). Additionally, to evaluate the performance of the proposed algorithm in restoring the geometric features of reflective targets, four metrics were introduced: mean square error (MSE), mean city-block distance (MCD), relative height error (R.H), and relative width error (R.W). These metrics are designed to evaluate the geometric fidelity of the point cloud, with a focus on spatial accuracy and dimensional consistency [45]. Suppose the ground-truth point cloud and the restored point cloud are \(P=\{\mathbf{p}_{i}\}_{i=1}^{N_{1}}\), \(Q=\{\mathbf{q}_{i}\}_{i=1}^{N_{2}}\), where \(\mathbf{p}_{i},\mathbf{q}_{i}\in\mathbb{R}^{3}\). It should be noted that \(N_{1}\) and \(N_{2}\) are not necessarily equal. First, we calculate the average squared Euclidean distance from each ground-truth point to its closest restored point and vice versa: from each restored point to its closest ground-truth point. The final MSE value (Equation (12)) is derived by averaging these two measures, \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Denoised Data**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Target Points**} & \multicolumn{1}{c}{**Noise Points**} \\ \hline Ground-Truth & Target points & TP & FN \\  & Noise points & FP & TN \\ \hline \hline \end{tabular} Note: TP, TN, FP, and FN refer to True Positive, True Negative, False Positive, and False Negative, respectively, representing the correctly and incorrectly classified targets and noise points.  
\end{table} 
Table 1: Confusion matrix for the LiDAR point cloud denoising results. providing a quantitative measure of the discrepancy between the ground-truth points and the restored points. The MCD (Equation (13)) is calculated in a similar manner to the MSE, but it employs the \(l_{1}\) norm in place of the \(l_{2}\) norm. \[MSE=\frac{1}{2}\left(\frac{\sum\limits_{\mathbf{p}_{i}\in P}\underset{\mathbf{q}_{j}\in Q }{\text{min}}\Big{\|}\mathbf{p}_{i}-\mathbf{q}_{j}\Big{\|}_{2}^{2}}{N_{1}}+\frac{\sum \limits_{\mathbf{q}_{j}\in Q}\underset{\mathbf{p}_{j}\in P}{\text{min}}\Big{\|}\mathbf{q}_ {i}-\mathbf{p}_{j}\Big{\|}_{2}^{2}}{N_{2}}\right) \tag{12}\]  
\[MCD=\frac{1}{2}\left(\frac{\sum\limits_{\mathbf{p}_{i}\in P}\underset{\mathbf{q}_{j} \in Q}{\text{min}}\Big{|}\mathbf{p}_{i}-\mathbf{q}_{j}\Big{|}}{N_{1}}+\frac{\sum \limits_{\mathbf{q}_{j}\in Q}\underset{\mathbf{p}_{j}\in P}{\text{min}}\Big{|}\mathbf{q}_ {i}-\mathbf{p}_{j}\Big{|}}{N_{2}}\right) \tag{13}\]  
The actual height (H) and width (W) of the target are first measured using a ruler. Subsequently, the height (h) and width (w) of the restored point cloud are calculated. The relative error (Equations (14) and (15)) is defined as the ratio of the absolute difference between the calculated and actual values to the actual values.