## 2 Relevant Methods: State of the Art  
Earlier works on shape representation analysis for object identification can be divided into two broad categories. The first category uses methods such as PCA to align the model in a canonical coordinate system. Then, it defines the representation of the shape by relating it to this orientation. Such methods include, among others, moments for object surfaces [11] and Extended Gaussian Images [12]. [PERSON] [13] (Appendix B) mentions that PCA-based methods are unstable due to the multiplicity of eigenvalues and sensitivity to outliers. The second category of methods defines representation invariants with respect to rotation and includes methods such as Shape Histograms [14] and Shape Distributions [5]. In this work, we adhere to ideas based on decomposing some function given on a sphere into the sum of spherical harmonics. The works of [7; 13; 15; 16] gave the impetus for developing such an approach to the problems of classifying 3D objects and retrieving their information from databases. In [17; 18], tools for searching for 3D objects were presented. The model is represented as a polygonal grid and serves as a query, and similar objects are extracted from a collection of 3D objects. The first stage of the algorithm is normalization (estimation of the pose) when the models are transformed into a canonical coordinate system. Then, feature vectors are extracted and compared with vectors obtained from normalized models in the search space. Using metrics in the feature vector space, the nearest neighbors are calculated and ranked. The objects extracted in this way are displayed for inspection, selection, and processing. Feature vectors are based on rays cast from the object's center of mass. The distance from the center of mass to the surface in the ray's direction sets the function value on the sphere. To evaluate the pose, a modified Karhunen-Loeve transformation is introduced, taking into account not only vertices or polygonal centroids from 3D models but also all points in polygons of objects. In [19], the 3D model's symmetry descriptors are presented as a set of spherical functions that describe the measure of the model's rotational and reflective symmetry with respect to each axis passing through the center of mass. [PERSON] et al. [20] present a methodology for reconstructing a 3D shape based on spherical harmonics, which ensures invariance to scaling and rotation. Rotation normalization is performed using continuous principal component analysis and its application to the model surface normals. The 3D model is represented as a set of spherical functions. This representation comprises the intersections of the surface with rays originating from the origin as well as points in the direction of each ray that are closer to the origin than the farthest intersection point. Ref. [21] presents a 3D shape descriptor that uses a set of panoramic views of a 3D object. The views describe the position and orientation of the object's surface in 3D space. A panoramic view of the 3D object is obtained by projecting it onto the surface of the cylinder parallel to one of its three main coordinate axes and is centered on the object's centroid. The object is projected onto three perpendicular cylinders aligned axes to capture the global shape of the object. The corresponding 2D discrete Fourier transformation and a 2D discrete wavelet transformation are found for each projection. Ref. [22] presents the decomposition of spherical harmonics for spherical functions to represent 3D triangulated star-shaped objects. After splitting the surface into star-shaped sections, this result can be extended to any triangular object. The evaluation of the spherical harmonic coefficients is performed by edge integration using the Monte Carlo method, which distinguishes this work from voxel methods. In [23], after translating and scaling the 3D model, the concentric spheres are used to analyze the object, and a new set of functionals is defined and applied for each sphere.