# Feature-Preserved Point Cloud Simplification Based on Natural Quadric Shape Models
## 2 The Methodology
### Simplification Rules  
The simplification entropy is used for estimating the geometric features of point clouds. When combined with the shape features, we define the simplifying rules, which are shown in Table 1. In this paper, according to the different simplification actions, we divide the simplification rules into four parts, i.e., the rule for planes, the rule for spheres, the rule for ellipsoids and the rule for cylinders/cones. According to the rules, adaptive adjustment of the simplification ratio can be accomplished, and eventually the point clouds data is divided into two categories: the reserved set \(S\) and the reduced set \(C\). (1) **Rule for planes:** Because planes are flat and the location information is important for the local plane, we set the rule and actions for planes by creating voxels. The simplification algorithm is implemented by using voxelization at resolution \(\alpha\), which is represented as the symbol \(voxel(\alpha)\), to downsample the point cloud. The center and boundary points are useful for determining the location information of a plane. The \(center(voxel)\) is the central voxel in the plane, and the \(alpha\_shape\) is the function to extract the boundary. Based on actions of (b) and (c) in the first row of Table 1, the center and boundary of the plane are selected and saved in \(S\). The point cloud \(R\) of the standard plane are shown in the third column of Table 1, and the simplified point cloud \(S\) are shown in the last column of Table 1.  
(2) **Rule for spheres:** If the area is matched with a sphere, the point cloud curvature is similar in this area. The radius and the center of the sphere are important location information for the sphere. The center is saved in \(S\) according to the action _center_(_sphere_) in the second row of Table 1. As for the sphere's shape feature, we consider that the points which are not satisfied by the sphere function are also important. The points are classified into three types, i.e., the points which are sharper than the sphere (\(\Delta_{curvature}<\varepsilon_{\text{cur}}\)), the points which are next to the sphere(\(\|p_{i}-\emph{center}(\emph{sphere})\|=\frac{1}{\varepsilon_{cur}}\pm\varepsilon\)) and the points which are more smooth than the sphere (\(\Delta_{curvature}\geq\varepsilon_{cur}\)). By the way, the \(c_{cur}\) is the curvature of the sphere; the \(\Delta_{curvature}\) is the variance of point clouds' curvature; the \(\varepsilon\) and \(\varepsilon_{cur}\) are the threshold for the classification. The example of standard sphere simplification is shown in the second row of Table 2.  
(3) **Rule for ellipsoids:** If the area is matched with an ellipsoid, the shape features can be described as the center and the three axes of the ellipsoid. The degree of importance of the points for shape-keeping changes with the simplification entropy. Thus, the voxels are created with the variable \(\overline{h}\). \(\overline{h}\) is the average of \(h_{s}\) (i.e., the scale-keeping simplification entropy), \(h_{p}\) (i.e., the profile-keeping simplification entropy), and \(h_{c}\) (i.e., the curve-keeping simplification entropy), which are computed by the Equations (4), (5) and (9), respectively. The simplification algorithm is implemented by using voxelization with variable resolution \(\alpha\times\overline{h}\),(i.e.,\(\emph{xoxel}(\alpha\times\overline{h})\)) to downsample the point cloud. The boundary and the center of the ellipsoid are also important to the location of the shape features. Thus, the boundary is extracted by the function _alpha-shape_; and the center of the ellipsoid can be implemented by the action _center_(_ellipsoid_). The example of standard ellipsoid simplification is shown in the fourth row of Table 1.