# Feature-Preserved Point Cloud Simplification Based on Natural Quadric Shape Models
## 2 The Methodology
### The Simplification Model Based on the Shape Feature  
In the process of reverse reconstruction of point clouds, due to the dispersivity of point clouds, it is difficult to find the relationship between the data sets and to estimate the geometric features of the point clouds. In different application fields, there are different focuses on geometrical features. Therefore, there are many descriptions of the geometrical features of point clouds, for example, normal, curvature, flat, peak, cylinder, minimal surface, and ridge, etc. The local geometric features are important for simplification, especially because having an accurate description of them directly determines the effect of the reverse reconstruction or measured precision. To accurately describe the geometric features, the shape feature is brought into FPPS and estimated and measured by the point clouds matching model (PCMM).  
In order to find the point clouds shape features during the process of data simplification, this paper provides PCMM based on the conicoid. By matching the shapes of features between the raw point clouds and PCMM, the data can be simplified by a certain shape feature, while the effective data feature can be reserved.  
Because the conicoid has been clearly described, and the shape feature of the conicoid can be completely controlled, we created PCMM based on the conicoid, as shown in Equation (10).  
It is assumed that there is a point data set \(p(x,y,z)\), and that \(p(x,y,z)\) belongs to the raw point cloud \(R\), that is  
\[f(x,y,z)=C_{0}+C_{1}x+C_{2}y+C_{3}z+C_{4}x^{2}+C_{5}xy+C_{6}xz+C_{7}y^{2}+C_{8} yz+C_{9}z^{2}=0. \tag{10}\]  
If point \(p\) is considered to be the one element in \(PCMM\), the point \(p(x,y,z)\) meets function \(f(x,y,z)\) in Equation (10).  
Because the whole conicoid model database is too large, the matching operation between the model and data sets consumes more time. Therefore, we revised and simplified the \(PCMM\). For the convenience of algorithm implementation, in this paper, the data model is simplified based on the Nature Quadrics Model (NQM). The NQM model includes ball, ellipsoid, cylinder, and cone shapes. Additionally, the plane is an important and frequently used model in reconstruction, and the plane satisfies Equation (11):  
\[f(x,y,z)=ax+by+cz+d=0 \tag{11}\] \[\Rightarrow f(x,y,z)=C_{0}+C_{1}x+C_{2}y+C_{3}z=0\ \.\]  
So, in this paper, the plane and the NQM are considered the basic models in the data matching model database \(PCMM\). Using the NQM and the plane, the \(PCMM\) can be simplified considerably. Based on the \(PCMM\), similar and redundant data can be retrieved, and then the simplification algorithm can be implemented by reducing the data with similar shape features. Then, to exploit the points/area with similar shape features, a shape operator, \(shape(PCMM,Neighbors(p_{i}))\), is used. The neighbors of point cloud \(p_{i}\) are denoted as \(Neighbors(p_{i})\), and \(p_{i}\) is one key point. The data matching model database is \(PCMM\), and the shape measurement between \(Neighbors(p_{i})\) and \(PCMM\) can be computed as shown in Equation (12). The function \(\phi\) is the estimation value of the shape feature, which is computed by the distance of the point q and the projection of the point q. Thus, the function \(\phi\) is important to the shape information description, and the value of \(\phi\) is proportional to the distinctiveness of the shape feature:  
\[\left\{\begin{array}{l}shape(PCMM,Neighbors(p_{i}))=\frac{1}{\sqrt{2\pi \sigma}}\exp(-\frac{\|\phi(Neighbors(p_{i}))-\phi(PCMM)\|^{2}}{2\sigma^{2}}) \ \.\\ \phi=\sum\|q_{i}-q_{i}\prime\|\end{array}\right. \tag{12}\]  
where \(\sigma\) is the scale of neighbor points; \(\phi\) is the function for the estimation of the shape feature;\(q\) is one element in the \(Neighbors(p_{i})\); and \(q_{i}\prime\) is the projection of \(q\) in \(PCMM\).