# Point Cloud Denoising Algorithm via Geometric Metrics on the Statistical Manifold
###### Abstract  
A denoising algorithm was proposed for point cloud with high-density noise. The algorithm utilized geometric metrics on the statistical manifold and applied the idea of clustering K-means based on local statistical characteristics between noise and valid data. First, by calculating the expectation and covariance matrix of the data points, the point cloud with high-density noise was projected onto the Gaussian distribution family manifold, aiming to form the parameter point cloud. Afterwards, the geometry metrics were assigned to the manifold, and the K-means algorithm was applied to cluster the parameter point cloud, so as to classify the valid data and noise. Furthermore, in order  
algorithm to distinguish between random noise and valid data [3; 4]. To address this issue, a clustering algorithm is proposed based on the neighborhood density of each data in [5; 6]. On the manifold, the local geometric structure of noise and valid data are inconsistent, and thus the data with similar local geometric characteristics need to be considered as the same cluster. In [7; 8], a K-means clustering algorithm is used to denoise point cloud, but only a few metrics are adopted to calculate the distance on the manifold. Meanwhile, the influence function, which aims to evaluate the robustness of the mean induced by each metric to outliers, is not analyzed in these studies. This manuscript extends our previous ideas in [8] by proposing a point cloud denoising method based on the geometry of the Gaussian distribution family manifold. The manifold is endowed with five metrics, including the Euclidean metric, the affine-invariant Riemannian metric, the log-Euclidean metric, the Kullback-Leibler divergence as well as the symmetrized Kullback-Leibler divergence [9]. To evaluate the robustness of the mean endowed by each metric to outliers, the influence functions of geometric metrics are calculated. As stated in [10], it can be very difficult to solve the matrix equations for the influence functions directly, so we calculate the approximate values of these influence functions. To evaluate the denoising effects of these metrics, the true positive rate (TPR), the false positive rate (FPR), and the signal-noise rate growing (SNRG) are adopted. TPR refers to the proportion of correctly predicted positive cases among all actual positive cases. A higher TPR means less real data loss. In fact, higher TPR, higher SNRG, and lower FPR indicate that the algorithm can better distinguish between real data and noise. Our proposed algorithm will be evaluated by the above three standards. The contributions in this paper are summarized as follows. (1) A K-means clustering algorithm is proposed to denoise point cloud with high-density noise by leveraging the difference in local statistical characteristics between noise and valid data. The algorithm operates on a Gaussian distribution family manifold. (2) By calculating the expectation and the covariance matrix of the data points, we map the original point cloud onto the Gauss distribution family manifold to form the parameter point cloud. Next, the metrics are assigned to the manifold, and the K-means method is applied to cluster the parameter point cloud, aiming to classify the original point cloud. (3) With the purpose of analyzing the robustness of the means with different metrics, the approximate values of their influence functions are calculated, respectively. The simulation results demonstrate that using geometry metrics yields better denoising effects compared to using the Euclidean metric. Additionally, mean influence functions with geometry metrics exhibit greater robustness than that with the Euclidean metric. The rest of the current work is structured as follows. Briefly, Section 2 introduces the Riemannian framework of the Gaussian distribution family manifold. Section 3 proposes a K-means clustering algorithm to denoise point cloud with high-density noise and calculates the approximate values of their influence functions with different metrics.