# Feature-Preserved Point Cloud Simplification Based on Natural Quadric Shape Models
## 2 The Methodology
### Simplification Entropy of Key Points
#### 2.2.1 Scale-Keeping Simplification Entropy  
Inspired by the SIFT (scale-invariant feature transform) algorithm [40], for scale-keeping, the multi-layer point cloud voxel grids are created by voxelization, like a pyramid. Each layer in the pyramid is composed of the \(point_{SIFT}\) with the same scale value, \(SIFT.Scale=\sigma\). The \(point_{SIFT}\) is the point set that is the local DOG(difference of Gaussians) optimum in one layer with \(SIFT.Scale=\sigma\):  
\[point_{SIFT}(p_{i}^{\sigma})=\left\{p_{i}(x_{i},y_{i},z_{i})\Big{|}p_{i}\in R \wedge D(p_{i},\sigma)>D(p_{j},\sigma)\right\},(i,j=1\dots n\wedge i\
eq j ) \tag{2}\]  
where \(R\) is the data set of point cloud; \(p_{i}\) is the ith element in \(R\); \(\sigma\) is the scale value; and \(D(p_{i},\sigma)\) is the DOG of \(p_{i}\).  
Taking the bunny data set [41] as the example, the \(point_{SIFT}\) in one layer of the pyramid is instantiated in Figure 3. The red points are the \(point_{SIFT}\) and \(\sigma=0.05\). The size of the \(point_{SIFT}\) is 8, and the result can be seen from the red points in Figure 3.  
Then, the \(point_{SIFT}\) of the multi-layers is extracted like a pyramid, as shown in Figure 4a. The extracted \(point_{SIFT}\) elements are shown in Figure 4b with the different colors. The 'Color_bar' in Figure 4b shows the \(point_{SIFT}\) element in different layers of the pyramid.  
Figure 4: The pyramid of scale-keeping simplification entropy: (**a**) the multi-layer pyramid; (**b**) the view of \(point_{SIFT}\) elements in different layers.  
Figure 3: The elements of \(point_{SIFT}(p_{i}^{\sigma})\) in one layer, where \(\sigma=0.05\).  
The \(point_{SIFT}\) elements in one layer with _SIFT.Scale_ = \(\sigma\) represent the local optimum. Therefore, the distribution of \(point_{SIFT}\) elements in the layer is even, as shown in Figure 3. However, as shown in Figure 4b, the \(point_{SIFT}\) elements in different layers show an aggregation phenomenon and the Euclidean distance of points in \(point_{SIFT}\) is short. So, we think that the close \(point_{SIFT}\) elements in different layers have a lower capability to express the scale-keeping feature; therefore, a lower simplification entropy is set in FPPS. For the accuracy simplification entropy, the weight of the simplification entropy is set as shown in Equation (3).  
\[weight=\frac{1}{card(point_{SIFT}(p_{i}^{o}))}\qquad(i=1\dots layers) \tag{3}\]  
where _layers_ represents the layer number of the pyramid, and \(card\) represents the cardinality of \(point_{SIFT}(p_{i}^{o})\).  
The \(\alpha_{\text{sift}}\) is the probability value of \(point_{SIFT}\) in the full set R. Thus, the number of key points for scale keeping is less than \(\alpha_{\text{sift}}\times size(R)\). Assuming that the number of key points for scale keeping is \(\alpha_{\text{sift}}\times size(R)\), the points need to be saved by FPPS. Then, the probability of \(point_{SIFT}\) is limited in the interval [0, \(\alpha_{\text{sift}}\)). We set the weight of scale-keeping simplification entropy as the layer in the pyramid, and then points on the top of the pyramid have the biggest weight values; the points in the same layer are equally capable of describing scale-keeping information. The scale-keeping simplification entropy \(H_{s}\) can be defined as follows:  
\[H_{s}=\log(\alpha_{sift}*weight) \tag{4}\]