## 2 Methods
### Unordered Point Cloud Normal Vector Estimation
Then the plane \(H\) can be obtained by locally minimizing Equation (2),  
\[\sum_{i=1}^{N}{(\langle n,p_{i}\rangle-D)^{2}\theta(\parallel p_{i}-q\parallel)} \tag{2}\]  
where \(\theta\) is a smooth monotonically decreasing, constant positive function, and \(q\) is the projection of \(r\) on the \(H\), \(r\) moves to the plane \(H\) along the direction of the plane normal vector \(n\), which denotes that \(q\) is the projection of \(r\) on the \(H\). Here, we could define \(q=r+tn\) where \(t\in\mathbb{R}\), Equation (3) is redefined as follows:  
\[\sum_{i=1}^{N}{\langle n,p_{i}-r-tn\rangle^{2}\theta(\parallel p_{i}-r-tn \parallel)} \tag{3}\]  
where the smaller \(r\) is, the closer the local tangent plane \(H\) is to \(r\), and \(q\) is the origin of the standard orthogonal coordinate system on the local tangent plane \(H\). Then, the surface in \(r\) domain is fitted by binary polynomial (as illustrated in Figure 2) to make \(q_{i}\) be the projection of \(p_{i}\) on \(H\), and \(f_{i}\) is the distance between \(p_{i}\) and \(H\). We can obtain the approximate coefficients of the polynomial \(g\) by minimizing the Equation (4). \[\sum_{i=1}^{N}{(g(x_{i},y_{i})-f_{i})^{2}\theta(\parallel p_{i}-q\parallel)} \tag{4}\]  
where \((x_{i},y_{i})\) is the local coordinate representation of \(q_{i}\) in \(H\). The Gaussian function proposed in [25]:  
\[\theta(d)=e^{-\frac{\theta^{2}}{\theta^{2}}} \tag{5}\]  
is used as the weight function, and the constant parameter \(h\) determines the selection range of the neighborhood point set. The features with dimensions smaller than \(h\) are smoothed out. The larger the \(h\) is, the smoother the projected point cloud will be. Figure 2: Schematic of discrete group point projection in the node neighborhood. In this study, we utilized satellite model point cloud data collected by a ToF camera and applied the mpp-MLS algorithm for processing the input point cloud. As shown in Figure 3, the initial point cloud is depicted in blue, while the point cloud processed by the algorithm is shown in red. It is evident that the red point cloud preserves the geometric features of the satellite surface while eliminating redundant and noisy points. By observing the XZ planar view, we further verified that the mpp-MLS algorithm accurately represents the planar features of the satellite point cloud. These findings indicate that mpp-MLS not only eliminates redundant noise points in the point set, but also more precisely captures the curvature and geometric features of the point set's surface, thereby enhancing alignment accuracy. The point cloud data acquired from 3D sensors typically lack normal vector information. In such cases, a parameter is utilized to estimate the approximate normal vector of the input point cloud. The mpp-MLS processing technique is applied to the input point cloud, which maintains its structural integrity and effectively smooths the surface while preserving key features. Figure 4 illustrates this process, where (a) shows the normal vector estimation directly on the original point cloud with noticeable dispersion in normal vector directions. In contrast, (b) depicts the normal vector estimation results after applying the mpp-MLS method, showcasing highly consistent normal vector directions in the point cloud's planar neighborhood. The normal vector consistency of the neighboring points in the two point clouds is calculated separately. The results are illustrated in Figure 5. Approximately 80% of the normal vector angle difference values estimated from the original point cloud fall within the range from 20\({}^{\circ}\) to 160\({}^{\circ}\), whereas around 80% of the normal vector angle difference values estimated from the point cloud after mpp-MLS processing are distributed between 0\({}^{\circ}\) and 20\({}^{\circ}\). This indicates that the algorithm proposed in this study effectively enhances the uniformity of the normal vector direction in the disordered point cloud, thereby improving registration accuracy. Figure 4: Unordered point cloud normal vector estimation results.