# Feature-Preserved Point Cloud Simplification Based on Natural Quadric Shape Models
## 2 The Methodology  
We provide the formulized description of FPPS. The data set that is organized as the point cloud is described as \(R\), \(R=\{p_{i}(x,y,z)\}\), \((1\leq i\leq n)\). In data set \(R\), \(p_{i}\) is the \(i\)th point, and \(n\) is the size of \(R\). We hypothesize that the geometric features of data set \(R\) can be quantified. Based on the quantified features, the formulized description of FPPS is defined as a multi-tuple _simple_ which is represented as follows:  
\[\textit{simple}=\{R,H,PCMM,C,S,\textit{rules}\}\]  
where \(R\) is the data set of the point cloud, which is the full set of data, \(R=C\cup S\); \(H\) is the set of point cloud simplification entropy; \(PCMM\) is the set of point cloud matching models, which is composed of natural quadric shape models (e.g., ellipsoid, cylinder, cone); \(C\) is the data set that is reduced by FPPS; \(S\) is the data set that is saved by FPPS, which is \(C\)'s complementary set; and _rules_ represent the mapping rules, which are set in terms of the \(PCMM\).  
The purpose of FPPS is to divide the full set, \(R\), into two subsets, \(S\) and \(C\). Ideally, set \(S\) includes all of the geometric features, and set \(C\) is the redundant data. So, \(H\) describes and measures the geometric features, and then the elements in set \(R\) can be mapped one-by-one to the tuple \(H\). \(PCMM\) is independent of the point cloud set \(R\) and is the regular point cloud shape's model set. The similarity between _PCMM_ and subset \(R\) is useful for recognizing the local shape features in the FPPS. So, the mapping of _PCMM_ and subset \(R\) is one-to-many. By comparing the similarity, the _rules_ can be inferred, and _rules_ are important for simplification in FPPS, for example, the adjustment of the self-adaptive reduction ratio.  
The proposed point cloud simplification framework is composed of four stages, as shown in Figure 1. The point clouds are collected by the laser scanner, which only provides the position of the points. Thus, the data attributes are scattered and simple. No topological and relative positional relations can be found in the data. Most of the point clouds just have the points' coordinates. Obviously, the simple data attributes are deficient, which makes data simplification with geometric features difficult. FPPS tries to retrieve more geometric features. In FPPS, the first stage of point cloud simplification is data pre-processing, which is able to estimate the normal vector of point clouds (see Section 2.1). Then, we divide the geometric features into three types and estimate each of them using the entropy which quantifies the geometric features. Like the criterion, the entropy is used to extract the key points. Thus, the key points will include more information about geometric features in the second stage. Additionally, the simplification entropy is defined (see Section 2.2). In the third stage, the neighbors of key points are created, and then the point clouds are divided into different segments, following which we evaluate the shape features of each segment. In the fourth stage of point cloud simplification, we match the shape features of each segment to the natural quadric surface in _PCMM_. Based on the shape features, we set the _rules_ and simplify the point clouds (see Sections 2.3 and 2.4).