# 3-D Point Cloud Registration Using Convolutional Neural Networks
## 4 The Proposed Descriptors for 3-D Point Clouds
### Simplified Spatial Point Distribution Descriptor  
A 3-D data point cloud \(\mathbf{Q}\) employed as an input of the descriptors can be given as follows  
\[\mathbf{Q}=\left[ \begin{array}{c}\mathbf{q}_{1}\\ \mathbf{q}_{2}\\ \vdots\\ \mathbf{q}_{m}\end{array} \right]=\left[ \begin{array}{ccc}x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\\ &\vdots\\ x_{m}&y_{m}&z_{m}\end{array} \right] \tag{3}\]  
where \(\mathbf{q}_{j}\) denotes the \(j^{th}\) data point. In the first stage, the rough registration stage, features extracted from the point set \(\mathbf{Q}\) should generally describe the orientation of \(\mathbf{Q}\) such that a successful and fast convergence in the learning process can be guaranteed. Because the rough registration covers a full range of uncertainty from \(-180^{\circ}\) to \(180^{\circ}\) that leads to significant disparities in characteristics of these features, it is hard to obtain optimal weights with a small enough loss for a CNN model after training. Therefore, a Simplified Spatial Point Distribution (SSPD) descriptor is proposed in this paper to describe coarse characteristics of the orientation of a 3-D data point cloud in 3-D space. The key point is to divide a spatial region involving the point set into several sub-regions and count the number of points in each sub-region, followed by a normalization with the total quantity of points of the point cloud. To begin with, a cubic bounding box with the length of an edge \(l_{cube}\) is created that consists of the point set. \[l_{cube}=\max\left\{\text{norm}_{2}\left(\mathbf{q}_{i},\mathbf{q}_{j}\right)|i\
eq j\right\} \tag{4}\]  
where \(norm_{2}\) denotes the Euclidean distance between two points \(\mathbf{q}_{i}\) and \(\mathbf{q}_{j}\). Secondly, this cube is divided into several sub-cubes as shown in Figure 4 with the size of \(s\) x \(s\) x \(s\) where the length of an edge of a sub-cube can be computed as follows. \[l_{sub-cube}=\frac{l_{cube}}{s} \tag{5}\]  
Finally, the quantity of points in each sub-cube is counted and then divided by the total number of points of the point cloud. Because each sub-cube has one value, SSPD is a 3-D matrix with the size of \(s\times s\times s\) where the parameter \(s=15\) is selected based on experiments. Algorithm 2 illustrates steps in details to compute SSPD based on a given 3-D point cloud. Figure 4: A demonstration about the establishment of cube and sub-cubes in Simplified Spatial Point Distribution (SSPD) descriptor. ```
Input: Data point cloud \(\mathbf{Q}\); length of an edge of a sub-cube \(l_{sub-cube}\), the number of slices \(s\);
1:for\(i=1\) to \(m\)do - Shift all points to the first quadrant; \[x_{\mathbf{q}_{i}}=x_{\mathbf{q}_{i}}-\min\big{\{}x_{\mathbf{q}_{i}}|i=1,2,...,m\big{\}};y_{ \mathbf{q}_{i}}=y_{\mathbf{q}_{i}}-\min\big{\{}y_{\mathbf{q}_{i}}|i=1,2,...,m\big{\}};\] \[z_{\mathbf{q}_{i}}=z_{\mathbf{q}_{i}}-\min\big{\{}z_{\mathbf{q}_{i}}|i=1,2,...,m \big{\}};\] - Bin points into sub-cubes by converting coordinate values to according positions; \[w=\frac{x_{\mathbf{q}_{i}}}{l_{sub-cube}};d=\frac{y_{\mathbf{q}_{i}}}{l_{sub-cube}}; h=\frac{z_{\mathbf{q}_{i}}}{l_{sub-cube}};\] \[b(w,d,h)=b(w,d,h)+1;\] where \(b(w,d,h)\) means the number of points in the sub-cube at position \((w,d,h)\) in 3-D space;  end for
2: Normalize \(b(w,d,h)\) with the total number of points \(m\) of the data point cloud; 3: Compute SSPD, a 3-D matrix with the size of \(s\) x \(s\) x \(s\); \[\small\small\mathbf{SSPD}=\frac{1}{m}\left[\begin{array}{ccccc}\left[ \begin{array}[] {ccccc}b(w_{1},d_{1},h_{1})&b(w_{1},d_{2},h_{1})&\cdots&b(w_{1},d_{s},h_{1})\\ b(w_{2},d_{1},h_{1})&b(w_{2},d_{2},h_{1})&\cdots&b(w_{2},d_{s},h_{1})\\ &\vdots&&&\\ \underbrace{b(w_{s},d_{1},h_{1})}&b(w_{s},d_{2},h_{1})&\cdots&b(w_{s},d_{s},h_{ 1})\\ &&&\vdots\\ &&&\\ \left[\begin{array}{ccccc}b(w_{1},d_{1},h_{s})&b(w_{1},d_{2},h_{s})&\cdots&b(w _{1},d_{s},h_{s})\\ b(w_{2},d_{1},h_{s})&b(w_{2},d_{2},h_{s})&\cdots&b(w_{2},d_{s},h_{s})\\ &\vdots&&&\\ b(w_{s},d_{1},h_{s})&b(w_{s},d_{2},h_{s})&\cdots&b(w_{s},d_{s},h_{s})\\ \end{array}\right]\end{array}