## 2 Registration Algorithms Based on Hierarchical Optimization
### Iterative Closest Point (ICP)-Based  
The point-to-point iterative closest point algorithm is the most classical approach for registering point clouds, which has historically been a significant research topic in the field of computer vision. The P2P-ICP [31], which leverages the transformation connection discovered in two sets of data from distinct coordinate systems to match data, is usually referred to as the ICP algorithm. The basic principle is presented in Figure 2, that is, solving the spatial transformation between the source point cloud \(P\) and the target point cloud \(Q\) in the overlapping region of two pieces of point clouds, to minimize the distance between them. Suppose \(E\) (\(R\), \(t\)) is the error between \(Q\) and \(P\) in the rotation and translation; then the problem of solving the optimal transformation matrix can be transformed into the optimal solution satisfying \(\min\)\(E\) (\(R\), \(t\)). It can be expressed by the following formula:  
\[E(R,t)=\sum_{i=1}^{n}\left\|Q_{i}-(P_{i}\times R+t)\right\|^{2} \tag{1}\]  
where \(R\) is rotation, \(t\) is translation, and the source point cloud \(P_{i}\) corresponds to the target point cloud \(Q_{i}\). Figure 1: Chronological Overview of Registration Algorithms Based on Hierarchical Optimization [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66]. Figure 2: Principle of ICP algorithm. The P2P-ICP algorithm is simple, intuitive, and easy to implement. However, its running speed and the convergence of the global optimum depend largely on the given initial transformation estimation and the establishment of the corresponding relationship throughout the iteration. As a result, establishing the correct corresponding point set during iteration to avoid falling into the local extreme value becomes the key to various ICP improvement algorithms. [PERSON] and [PERSON] [32] determined the corresponding point set by the projection of \(p\) in the source point cloud \(P\) via the viewpoint direction of the target point cloud \(q\) on the target point cloud \(Q.\) This algorithm eliminates the phase of searching for corresponding points, which considerably shortens the calculation time, but it does not achieve high registration accuracy. The iterative closest compatible point (ICCP) [33] is calculated by comparing the curvature of points on the point cloud surface and angle between point normal vectors and other related features. Additionally, it seeks corresponding points between point clouds based on these features, considerably reducing the ICP algorithm's spatial search time. [PERSON] and [PERSON] [34] replace the calculated point-to-point distance with the point-to-surface distance, which can reduce the number of iterations and makes it difficult to fall into the local extremum. To verify the correctness of point-to-point correspondence, [PERSON] et al. [35] introduced the ICP-IF algorithm to linearly weight numerous Euclidean invariant features, such as curvature and spherical hue. This method successfully improved the local optimum problem and decreased the initial value requirements of the method, although the noise was not eradicated. [PERSON] [36] took advantage of the Levenberg-Marquardt algorithm to estimate the transformation by adding a damping factor to the original LK algorithm. On the basis of enhancing the robustness, the speed of this strategy will not slow down, and the effect is particularly noticeable for point clouds with a high overlap rate. [PERSON] and [PERSON] [37] proposed a GP-ICPR method by incorporating RANSAC into the ICP algorithm. It searches for probable matching locations employing changes in the estimated normal vector of the surface generated by the neighborhood, considerably decreasing the computing cost and improving registration speed and accuracy. [PERSON] [38] suggested the PL-ICP by adopting the point-to-line measuring approach, which will enhance the iterative efficiency to a certain extent.