## 2 Relevant Methods: State of the Art
This results in a vector of feature descriptors. This vector is invariant to rotation and, hence, suitable for matching 3D models. Weight coefficients are assigned to each descriptor to improve the mapping. In [24], descriptors of 3D shapes are generated using functions on a sphere. A set of vector descriptors invariant to rotation is extracted using spherical harmonics. First, the models' size is normalized, and then a sample of data is taken from the surface of the 3D model and converted into point cloud data. Next, spherical harmonics are applied to this point cloud to obtain invariant rotation descriptor vectors. The method proved resistant to noise on the model's surface and allowed us to compare similarities of 3D models. The essential part of identifying the similarity of two objects is to find a rotation that best combines two sets of vectors (or two clouds of points). Algebraic approaches to solving the problem of optimal rotation were proposed in [25; 26; 27; 28]. These results are exact but require knowing the correspondence among the points in these two sets. In [29], the spherical harmonics of a pair of images are related to each other by a shift theorem, which uses an irreducible representation of the rotation group. Using this theorem, Euler angles are extracted. Regarding the required number of spherical coefficients, it is assumed that they are global image encoders and that rotations can be estimated using a simple reference table of combinations of coefficients and angles. [PERSON] et al. [30] solve the equivariance problem of 3D rotation using convolutional neural networks. The authors model 3D data with spherical functions and propose a spherical convolutional network that implements point convolutions on a sphere. The paper demonstrates that networks with low bandwidth without increasing the dataset size can demonstrate performance comparable to state-of-the-art networks. In [31], the fundamental problem of finding the shape and registering a point cloud for the point clouds with zero point-to-point correspondences is considered. Using the trained optimization process, the authors propose a method to represent 3D point clouds by spherical Gaussians and a rotation-invariant convolution. In [32], spherical harmonic functions are directly used to simulate irregular 3D particles. Discrete surface points of irregularly shaped 3D particles are represented by spherical harmonic functions with a limited number of harmonic coefficients to restore the particle'smorphology. These spherical harmonic functions are used to detect overlap between particles and calculate the interparticle contact forces, moments, and particle motions in various engineering and industrial processes. The analysis of spherical harmonics was initially used to solve problems in geophysics, potential theory, and mathematical physics [33]. Later, it was found to apply to the classification and identification of 3D objects. An increasing number of 3D models of buildings are hosted on web platforms for model sharing. Therefore, based on data reuse, in [34; 35], an approach to coding and searching 3D models of buildings using point clouds obtained by airborne light detection and ranging (LiDAR) systems is proposed. To encode LiDAR point clouds with sparse, noisy, and incomplete sampling, the authors introduce an encoding scheme based on a set of low-frequency spherical harmonic basis functions. Descriptors are extracted from spatial histograms and used to search for 3D models. The state-of-the-art approach to solving the optimization problem of mapping objects to each other is the ICP algorithm. The solution is found by step-by-step optimizing the loss function defined at the sets of points of both objects. Based on gradient search, the ICP iteratively finds the best correspondence among the subset of points of the objects. The latest modification of the ICP is Go-ICP [36], where Local ICP is based on a branch-and-bound (BnB) scheme. In [37; 38], global optimization methods based on implicit enumeration were mentioned.