# Triangular Mesh Surface Subdivision Based on Graph Neural Network
## 2 Related Work
### Neural-Network-Based Mesh Generation  
In recent years, due to the rapid development of neural network technology [5; 6; 7], mesh generation and neural network technology have become more closely combined. Mesh subdivision algorithms belong to the field of mesh generation, and other mesh generation techniques can inspire mesh subdivision tasks. Neural-network-based mesh generation (IMG) can be broadly divided into six categories: deformation-based, classification-based, isosurface-based, parameterization-based, Delaunay triangle-based, and Advancing-Front-based methods. The deformation-based IMG algorithm mainly applies neural networks to predict the displacement vector of the vertex position or directly predict the vertex position. As data of the non-Euclidean kind, mesh data do not have the regular organization form of a one-dimensional signal or two-dimensional image, and it is not easy to directly apply the existing network model to the mesh [8; 9; 10; 11; 12; 13]. Therefore, the IMG algorithm based on deformation has attracted more and more attention. The common feature of this technique is the existence of the initial mesh. The algorithm only needs to calculate the position of the vertices without changing the connection relationship between the vertices, leading to a simple algorithm implementation. However, this is also a disadvantage of this technique, which can only generate meshes with the same topology as the initial mesh. [PERSON] et al. [14] proposed a straightforward solution that provides multiple basic template meshes for the algorithm to select automatically. [PERSON] et al. [15] used face-pruning operations to iteratively adapt to the mesh's topology while maintaining the template's primary properties. The subdivision algorithm based on a neural network changes the topology of the initial mesh to a certain extent by iteratively inserting vertices. Classification-based IMG algorithms [16] can be divided into occupancy prediction and mesh basic-element prediction. Occupancy prediction refers to voxelizing three-dimensional space and determining whether a voxel belongs to an object. Mesh basic-element prediction treats mesh generation as a combination of basic elements, so the generative model only needs to determine whether the essential elements to be detected should be present. Both of them can be seen as binary classification problems. In addition to binary classification problems, multiple classification can also be used to achieve mesh generation [17]. The isosurface-based IMG algorithm [18; 19; 20] uses neural networks for implicit function fitting and zero isosurface extraction. It has the advantage of representing complex geometries and topologies and is not limited to predefined resolutions. However, this method is sometimes limited by its susceptibility to noise, outliers, and non-uniform sampling. In addition, implicitly solving large-scale equations can be time consuming. In parameterization-based IMG algorithms [21], neural networks learn parametric mappings or process parametric data. Parameterization refers to the correspondence be tween a mesh in 3D space and a 2D parameter space. For example, a surface can be mapped to a two-dimensional parameter space by parameterization to obtain a geometric image, which can be further converted into a mesh using a neural network [22]. Alternatively, a mapping of the parameter space to the 3D space is established, and then a mesh is generated in the parameter space by a neural network [23; 24]. Delaunay triangulation can connect vertices in 3D point clouds to generate 3D meshes, which maximizes the minimum angle. The IMG algorithm based on the Delaunay triangle predicts vertex positions and generates a 3D mesh. In practice, it is more about 3D representations, so [PERSON] et al. [25] used the 3D Delaunay method to generate tetrahedral meshes from 3D point clouds. Finally, the Advancing-Front-based method [26] is a greedy algorithm that gradually generates a mesh from the boundary inward in a recursive manner.