## 3 Point Cloud Alignment Methods
### 4 PCS Coarse Alignment Based on ISS Feature Points  
When the 4 PCS algorithm is applied to point cloud coarse alignment, due to the need to traverse all the 4-point sets from the target point cloud that have affine invariant consistency features with the source point cloud, which leads to a long time and inefficiency of the coarse alignment. To address the above problems, this paper introduces ISS feature points on the basis of 4 PCS algorithm. Although the time complexity \(O(n^{2}+k)\) of searching for point pairs in the target point cloud remains unchanged, the number of points searched  
Figure 5: 4 PCS algorithm schematic. a, b, c, d are the four points chosen in \(P\) that are coplanar but not collinear; e is the intersection of the line ab with the line cd.  
by the algorithm is reduced due to the ISS feature points extracted from the source point cloud and the target point cloud, which in turn reduces the time of coarse alignment. The coarse alignment process based on the ISS feature points is shown in Figure 6. After inputting the source and target point clouds at the same time, voxel filtering is used to downsample the source and target point clouds. Next, the ISS feature points are extracted to form the source and target point cloud feature points, and these feature points are used for 4 PCS coarse alignment. Finally, the RMSE is computed and the optimal transformation matrix is output.  
Step 1: Due to the large amount of data collected from the source and target point clouds, the two pieces of point clouds are downsampled first, and then the ISS algorithm is utilized to extract the feature points of the downsampled two pieces of point clouds.  
Step 2: The feature points of the target point cloud are traversed by determining the coplanar disjoint 4-point set of the source point cloud feature points. The 4 PCS coarse alignment is used to align the source point cloud feature points with the feature points of the target point cloud. Then, the least squares method is used to calculate the rigid-body transformation matrix of the two feature points.  
Step 3: According to the derived rigid-body transformation matrix, the spatial positional transformation of the source point cloud feature points is performed. After several iterations, the sum of the nearest-point distances between the source and target point cloud feature points after the positional transformation is calculated. The rigid-body transformation matrix of the sum of the nearest-point distances will be used as the final positional  
Figure 6: Flowchart of 4 PCS coarse alignment based on ISS feature points.  
transformation matrix outputted by the 4 PCS algorithm between the source and target point cloud feature points.  
Step 4: To analyze the error of the coarse alignment results, this paper chooses root mean square error (RMSE) as the evaluation index to measure the accuracy of the point cloud alignment. RMSE is used to quantify the size of the error of the alignment by calculating the square root of the ratio of the square sum of the distances of the matched pairs of \(p_{i}\) and \(q_{i}\) to the logarithmic value of \(n\), and the smaller the value, the better the result of the alignment. The expression is as follows:  
\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}{{{(p_{i}-q_{i})}^{2}}}}{n}} \tag{6}\]