## 3 Registration Algorithms Based on Probability Stochastic Distribution Model
### Random Sample Consensus (RANSAC)-Based  
The RANSAC algorithm [72] has been applied universally in various modeling problems, particularly in the field of computer vision. The algorithm is straightforward and well-suited for the problem of robust estimation. Figure 5 illustrates a conceptual diagram of the RANSAC algorithm in a unidimensional modeling problem. The RANSAC algorithm seeks to identify models that are resistant to noise. By extracting the smallest sample, the algorithm calculates the feasible model parameters, then returns the model parameters to all data samples and calculates the associated inlier rate until the number of iterations exceeds the set number. If the current optimum model's interior point rate exceeds a predefined threshold, the current optimal result is taken as the final model and sampling is terminated; otherwise, sampling continues. Since the maximum number of iterations N guarantees that at least one set of sampled data is full of interior points under a certain confidence probability, the calculation formula of \(N\) is derived as follows:  
\[N=\frac{\log(1-p)}{\log(1-1(1-v)^{m})} \tag{2}\]  
where \(v\) is the outlier rate of sample; \(m\) is the minimum amount of data required to calculate the model parameters, \(p\) is the confidence probability set in advance, that is, the probability of \(p\) is guaranteed by \(N\) times of sampling, making the samples drawn are all interior points. However, the performance of the RANSAC algorithm in high-dimensional spaces degrades because the likelihood of sampling a set consisting entirely of interior points decreases exponentially. To overcome this, the NAPSAC algorithm [73] was devised. [PERSON] et al. [74] discovered that selecting more points than the minimum subset necessary for model estimate results in more accurate model parameters (for example, homography parameter estimation requires at least four pairs of matching points). On this premise, the LO-RANSAC algorithm was proposed to raise the registration accuracy by resampling the tentative interior point set. To enhance the effectiveness over time, [PERSON] and [PERSON] [100] designed a pre-check-based RANSAC-Tdd. As a whole, the algorithm can indeed save the time required to execute. If the inlier rate is high, the algorithm does have an ideal effect, but when the inlier rate is very low, the algorithm tends to slip into the infinite sampling and testing process. Unlike RANSAC, which treats all correspondences equally, PROSAC [75] does not uniformly draw random samples from the complete set but rather linearly ranks the corresponding sets by building a similarity function using correspondences. As a result,  
Figure 5: Conceptual of Random Sample Consensus. the PROSAC algorithm provides considerable computational savings under the moderate assumption that similarity measurements predict the validity of a match better than random guesses. Aiming at the problem of low matching accuracy caused by the failure of the PROSAC computational model, [PERSON] et al. [76] improved the PROSAC algorithm and adopted the spectral clustering algorithm to filter the mismatches in the subset, thus accelerating the computational efficiency. Similarly, [PERSON] et al. [77] decreased spatial matching operations within the RANSAC basic framework by generating and matching congruent 4-points, and presented 4-points congruent sets for robust pairwise surface registration, thereby expediting the registration process. Specifically, a coplanar 4-point set is constructed in the input point clouds P and Q of any pose, and the affine invariance constraint is used to match the eligible corresponding point pairs in the coplanar 4-points set. The largest common pointset (LCP) strategy is employed to find the maximum overlap of 4-points after registration to obtain the optimal matching result and accomplish point cloud registration. Subsequently, [PERSON] et al.