## 3 Registration Algorithms Based on Probability Stochastic Distribution Model
### Gaussian Mixture Model (GMM)-Based  
Other statistically based methods transform the registration problem of two pieces of point clouds into the estimation problem of probability density function and take one piece of a point cloud as a template to construct a Gaussian mixture model (GMM) through the maximum likelihood estimation method to fit another piece of the point cloud, thus achieving point cloud registration. These approaches have the benefit of not requiring  
Figure 6: Process of basic Normal Distribution Transform [102] (**a**) Construction of the normal distribution with target point clouds. (**b**) Projection of the source point clouds into the normal distribution. (**c**) Score calculation for each source point. point-to-point connection to be established between two pieces of point clouds, unlike ICP. Since the approach for local optimization used in this type of registration still relies on the initial value to determine the parameters for the spatial transformation. They are nevertheless prone to falling into local minima even when there is poor initial alignment or little overlap between two pieces of point clouds. GMM is also a popular method for coping with point cloud registration. The main concept behind the GMM-based approach is to convert the registration problem into a maximum likelihood problem of the input data. Afterwards, calculate the transformation matrix and Gaussian mixture model parameters after optimizing the maximum likelihood problem. Figure 7 illustrates the basic principle of GMM-based algorithm. The benefit of the GMM-based method is that it is insensitive to noise and outliers. Furthermore, the research direction of this method is to develop an optimization strategy to optimize the transition matrix by maximizing the likelihood. To estimate the transformation between point clouds, [PERSON] and [PERSON] [92] proposed the CPD algorithm based on the GMM model [93], which regards the two pieces of point clouds as the centroid and data of the GMM model and aligns the centroids of point clouds by maximizing the posterior probability. The algorithm has strong noise resistance and robustness, but its high computational complexity makes it less efficient for matching data with higher density. To resolve point sets that are disorganized by noise and outliers, [PERSON] et al. [94] presented the joint registration of multiple point clouds (JRMPC) algorithm, which assumes that multiple point sets are produced by a single GMM. Batch and incremental EM optimization algorithms are also employed. A new convex hull indexed Gaussian mixture model (CH-GMM) [95] point cloud registration algorithm is proposed, which combines the convex hull (a set of original point sets) and GMM to lower the computational complexity. The algorithm could address rigid and non-rigid 3D point cloud registration problems. [PERSON] et al. [96] introduced a hierarchical data structure for 3D point clouds that is based on GMM and includes a compact and generative representation. Model creation is sped up by generating constraints on spatial interactions between points and model. Our model produces findings with better fidelity at smaller sizes, with only moderate trade-offs in construction time, even on mobile devices, compared with discrete techniques or hybrid approaches like the NDT. [PERSON] et al. [97] took a probabilistic approach, modeling the problem as a hybrid mixture model with a Von-Mises-Fisher mixture model (FMM) modeling the orientation and a Gaussian mixture model (GMM) modeling the location. [PERSON] et al. [98] presented hierarchical gaussian mixtures for adaptive 3D registration (HGMR), which builds a top-down multi-scale representation of point cloud by recursively performing the likelihood segmentation of several small-scale data in parallel on the GPU. This approach is more effective and flexible than the prior iterative closest point and GMM-based point association algorithms. Deep GMR [99] replaces the E-step (Expectation-step) in the expectation maximization method with a neural network.