# Point Cloud Denoising Algorithm via Geometric Metrics on the Statistical Manifold
## 5 Conclusions  
To conclude, a novel point cloud denoising algorithm is proposed in combination with the K-means algorithm. By calculating the expectation and covariance of the data points, this algorithm utilizes geometric metrics on the statistical manifold to map the original data point cloud to the Gaussian distribution family manifold, forming a parameter point cloud. Additionally, different measure structures are constructed on the Gaussian distribution family manifold, and the K-means method is used for clustering the parameter point cloud, aiming to cluster the corresponding original data. The robustness of different means with various metrics is analyzed by calculating their approximate influence functions. Simulations show that Algorithm 1 with the geometry metrics can obtain a better denoising \begin{table}
\begin{tabular}{c c} \hline \hline
**Metric** & **Complexity** \\ \hline FM (Proposition 1) & \(O((m+l-2)n^{2})\) \\ AIRM (Proposition 2) & \(O((2m+1)ln^{4})\) \\ LEM (Proposition 3) & \(O(2(m+1)ln^{4})\) \\ KLD (Proposition 4) & \(O(((2m+3)l+m+1)n^{3})\) \\ SKLD (Proposition 5) & \(O((2m+3)ln^{3})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational complexity of influence functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Metric** & **Complexity** \\ \hline FM & \(O((m-1)n^{2})\) \\ AIRM & \(O((m-1)n^{4})\) \\ LEM & \(O(mn^{4})\) \\ KLD & \(O((m+1)n^{3})\) \\ SKLD & \(O((m+2)n^{3})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computational complexity of means. effect than that with the Euclidean metric, and the geometric means are more robust than the arithmetic mean. We use three criteria (TPR, FPR, and SNRG) to evaluate the denoising effect of Algorithm 1 corresponding to different metrics. Simulations indicate that the algorithm corresponding to the Euclidean metric has a lower TPR and a higher FPR in most cases. SNRG is the most important criteria, which represents the promotion of signal-to-noise ratio through Algorithm 1. Table 2 shows that the SNRG of Algorithm 1 with geometric metrics is generally higher than that with the Euclidean metric. However, it should be noted that although denoising algorithms based on geometric metrics have the advantages of high TPR, low FPR, and high SNRG, it faces higher computational complexity than those utilizing the Euclidean metric, reaching \(O(n^{4})\). Additionally, computational complexity according to KLD and SKLD also reaches \(O(n^{3})\). This represents a drawback of denoising algorithms employing geometric metrics. In this manuscript, Algorithm 1 is only used for numerical simulation of point cloud denoising, and can also be applied to specific tests in the future. Ultrasonic testing with a manipulator is an important non-destructive testing method used to detect the internal defects of materials with complex structures. Before the manipulator detects the work-piece, the laser instrument is used to collect the point cloud data of the workpiece for 3D reconstruction. However, the point cloud data collected by laser instruments often contains a significant amount of redundant or invalid points. Algorithm 1 can be used to these redundancies and improve the quality of point cloud data. This will result in reduced noise and more accurate models, ultimately enhancing subsequent processing effects. Investigation, X.D. ; Methodology, X.D. and L.F.; Software, X.Z. ; Writing--original draft, X.D. ; Writing--review and editing, X.D., L.F. and X.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China grant number 61401058. The experimental data employ 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. The authors would like to thank the anonymous reviewers for their detailed and careful comments, which helped improve the quality of presentation. The authors declare no conflict of interest.