# Point Cloud Repair Method via Convex Set Theory
## 1 Introduction  
The point cloud is an important means to describe the surface shape of 3D objects. Limited by environmental factors or scanning equipment, a collected 3D point cloud is incomplete, and incomplete point-cloud data reduce the accuracy of downstream work, such as 3D target reconstruction, target recognition, and scene tracking and understanding. Therefore, point-cloud repair is indispensable work in 3D vision. Figure 1 shows a point cloud with holes. The disorder of space points renders point-cloud repair more challenging. Some current methods [1; 2; 3] established a mesh model and repaired the shape of the object surface via the mesh model. The purpose of meshing is to mitigate the influence of the  
Figure 1: Point cloud with holes. disorder of the point-cloud data. [PERSON] et al. [4] utilized the mesh model for repairing. The characteristic curve was restored, and the divided smooth subhole area was repaired. Nonetheless, this method needs to manually select the points needed for fitting the curve, and it cannot adaptively select the required points. The construction of the mesh is a complex process, and the quality of mesh distribution directly reduces the repair effect. There are some methods to repair point clouds through block matching. [PERSON] et al. [5] transformed the repair problem into an optimization problem using nonlocal self-similarity and a local smoothing constraint to achieve better point-cloud repair. [PERSON] et al. [6] proposed a data-driven point-cloud completion method that repairs the point cloud through symmetric relationship. The dataset was used to infer the 3D shape in the absence of a symmetrical relationship. This method could repair some point-cloud models with serious missing areas, but the effect of repairing irregular objects is poor. [PERSON] et al. [7] needed to ensure that there were many symmetrical or repeated structures in the visible region; however, the conditions required by this method were relatively strict, so the scope of application was limited. The effectiveness of block-matching methods usually depends on the large number of models and rich model types in the database. With the successful application of PointNet [8] and PointNet++ [9] on point clouds, the method based on depth learning [10; 11; 12; 13; 14; 15; 16; 17] was applied to point-cloud completion. Point-cloud completion methods implemented with an encoder-decoder framework mainly follow the coarse-to-fine principle to achieve point-cloud completion [18; 19]. However, such methods often add new points in the nonhole area, changing the inherent structure information of the point cloud [20; 21; 22; 23]. At the same time, the effectiveness of these methods is usually limited by the size of the training data [24; 25]. To solve these problems, we propose a novel point-cloud repair method that transforms the point-cloud repair problem into a problem of constructing a convex set. In this paper, we weaken the convex set's general form, so that it could be used in point-cloud repair. In this paper, we weakened the convex set's general form, introduced the MRF to predict the special subunit, and added new 3D points in the specific subunit, so that the subunit would become a convex set. First, we built a 3D unit S with the size of \(L\times W\times H\) in the neighborhood of the hole boundary (S had to contain \(T\) adjacent boundary points, and the value of T was positively related to the density of the boundary neighborhood. In this paper, \(T\geqslant 3\)). Second, we limited the distribution of new filling points according to weakened convex set theory, so that new spatial points were distributed in the hole area. The calculation of the distribution area of new filling points is mainly divided into two stages, namely, coarse screening and fine screening. At the coarse-screening stage, we discretized S into K subunits and set the label of the subunits with 3D points to 1; otherwise, we set it to 0, thus obtaining the initial label field.