# Point Cloud Upsampling Algorithm: A Systematic Review
## 3 Optimization-Based Point Cloud Upsampling  
In 2003, [PERSON] et al. [1] proposed the first algorithm for point cloud upsampling. They upsample a point set by interpolating points as vertices of a Voronoi diagram on the moving least squares (MLS) surface. It takes any three points on the plane and draws a Voronoi diagram. Each Voronoi vertex is the center of a circle that touches three or more of the points without any point inside. After obtaining the Voronoi diagram, it selects the center of the circle with the largest radius and projects it onto the MLS surface. The result is an upsampled point. This process is repeated until the radius of the largest circle is smaller than the specified threshold. The local approximation method is chosen to improve the calculation efficiency. Subsequently, [PERSON] et al. [21] proposed a non-parameterized point resampling and surface reconstruction method and applied it to point cloud upsampling. The locally optimal projection operator (LOP) is introduced to approximate the surface from the point set data, which can be used to project any set of points onto the input point cloud. After performing multiple LOP iterations on the point set, the initial point set can be upsampled. The operator is non-parameterized and does not rely on estimating local normals, fitting local planes, or using any other local parameter representation. This method works well in situations in which the orientation is not clear and the geometry is complex. [PERSON] et al. [22] made modifications and extensions based on LOP. They proposed a weighted locally optimal projection (WLOP), which adds local adaptive density weights to LOP to make the original point cloud distribution more even. The irregular particle distribution produced by the original LOP operator may cause some closed-cell defects when generating the surface, and WLOP can improve this problem. Later, [PERSON] et al. [23] proposed a WLOP operator based on a Gaussian mixture describing the input point density, called Continuous LOP (CLOP). The Gaussian mixture model was used to describe the point cloud density's geometric maintenance method, making it suitable for more compact and continuous point cloud representation. Compared with WLOP, CLOP adopts more particles than input points, generating better point cloud upsampling results. None of the above solutions consider how to deal with sharp features, and some methods require reliable normals as part of the input. Thus, [PERSON] et al. [24] proposed a resampling method, edge-aware resampling (EAR), which relies on the median to deal with noisy and possibly outlier point sets in an edge-aware manner. It resamples from the edge so that a reliable normal can be calculated at the sampling point, based on which the orientation point is inserted and projected onto the potential surface, which is an unknown base surface defined by the input point set. Then, it determines the bottom surface, direction, and distance of the projection. To correctly handle the sharp features, the position information and normal information are added in the above steps to give the projection operator bilateral and edge perception. Repeating the above-mentioned upsampling process and incrementally filling the gaps along the edge, singular points can reconstruct sharp features while increasing or decreasing the point density. [PERSON] et al. [25] defined the concept of deep points and proposed a consolidation method based on deep points. Based on EAR, new samples close to the input data are projected onto the basic surface using bilateral projection. This can effectively restore small geometric details. In addition, bilateral normal smoothing can be performed to adjust the surface points' normals to better retain clear features on the merged surface. Compared with LOP, the above two edge-sensing upsampling methods made specific improvements but still have a certain degree of smooth surface transition. [PERSON] et al. [26] proposed a 3D point cloud super-resolution local algorithm based on the graph total variation (GTV).