# Point Cloud Denoising Algorithm via Geometric Metrics on the Statistical Manifold
## 4 Simulations and Results  
The following section presents numerical experiments to demonstrate the denoising effect of Algorithm 1. It also compares the norms of the mean influence functions with FM, AIRM, LEM, KLD, and SKLD. In these simulations, all samples on \(\mathrm{P}(n)\) can be generated using the equation:  
\[\exp(\frac{V+V^{T}}{2}), \tag{33}\]where \(V\in\mathbb{R}^{n\times n}\) refers to a random matrix generated by MATLAB. A. Numerical Simulations  
In this example, the distance function induced by FM, AIRM, LEM, KLD, or SKLD is used in Algorithm 1, and the denoising effect caused by these metrics is compared, where the experimental data employ the 3D point cloud of Teapot.ply. Teapot.ply is one of the built-in data in MATLAB and can also be obtained through some open-source projects, such as the Stanford 3D Scanning Repository. In MATLAB, Teapot.ply is an example model that is often used to demonstrate and test 3D graphics processing and visualization functions. It is stored in PLY file format and contains three-dimensional geometric information representing a teapot. As shown in Figure 2a, background noise can be distributed uniformly with a signal-to-noise ratio (SNR) of 4137:1000. Taking \(K=2,q=5\) and \(\epsilon_{0}=0.01\), Figure 2b-f present the denoised images of the original point cloud using Algorithm 1, according to FM, AIRM, LEM, KLD, and SKLD, respectively. From Figure 2b, it can be observed that Algorithm 1, which is based on the Euclidean metric, still has a significant amount of noise points remaining in the image. Figure 2c,d show that Algorithm 1 performs well in removing noise points when using AIRM and LEM. Figure 2e,f illustrate that Algorithm 1, utilizing KLD or SKLD, effectively denoises the image but also removes some real data points from the teapot. It is evident that Algorithm 1 on the basis of the Euclidean metric is not as effective as other geometric metrics. Figure 2: Comparison of point cloud before and after denoising. B. Results and Discussions  
To evaluate the denoising effects of these metrics, true positive (TP), false positive (FP), false negative (FN), and true negative (TN) are adopted. Then, the true positive rate (TPR), the false positive rate (FPR), and the signal-noise rate growing (SNRG) are defined by  
\[\text{TPR}=\frac{\text{TP}}{\text{N}_{\text{data}}},\text{FPR}=\frac{\text{FP }}{\text{N}_{\text{noise}}},\text{SNRG}=\frac{\text{TP}}{\text{FP}}\frac{ \text{N}_{\text{noise}}}{\text{N}_{\text{data}}}-1, \tag{34}\]  
where \(\text{N}_{\text{data}}\) represents the number of real data, and \(\text{N}_{\text{noise}}\) stands for the number of noises. In Table 2, we have an SNR of 4137:1000 and 4137:2000. As higher TPR, higher SNRG, and lower FPR indicate that the algorithm can better distinguish between real data and noise, the highest TPR, the lowest FPR, and more than 99% SNRG are displayed in bold. From Table 2, it is evident that Algorithm 1 based on the Euclidean metric generally has a higher FPR, lower TPR, and lower SNRG. Conversely, geometric metrics tend to lower FPR, higher TPR, and higher SNRG. Among them, Algorithm 1 with KLD and SKLD exhibits a lower FPR due to the removal of some valid data points in Figure 2e,f. Therefore, Table 2 demonstrates the advantages of using geometric metrics (AIRM, LEM, KLD, and SKLD) over the Euclidean metric. To compare the robustness of influence functions with different metrics, we consider simulations involving 100 randomly generated symmetry positive-definite matrices along with \(l\)-injected outliers. We examine four scenarios: \(l=10,40,70\), and 100 to analyze the effect of outlier numbers on influence function robustness.