# Triangular Mesh Surface Subdivision Based on Graph Neural Network
## 1 Introduction  
Triangular meshes, a common way of representing 3D objects, can be leveraged by a variety of render pipelines. As the computer graphics and game animation industries continue to grow, the demand for high-quality mesh data is increasing. Three-dimensional mesh generation techniques have a long history, and the main methods used include Delaunay triangulation, mesh subdivision, parametric-based methods, isosurface-based methods, and interactive modeling. The subdivision algorithm used for triangular meshes is designed to smooth the mesh surface and improve the resolution and fineness of the mesh, so as to provide high-quality mesh data for subsequent processing. Mesh subdivision algorithms offer many advantages when generating mesh data. First, the subdivision algorithm can obtain high-quality smooth mesh surfaces by iteratively subdividing polygons and adjusting vertex positions. This makes the subdivision algorithm particularly suitable for the generation and processing of the smooth mesh models required in computer graphics as well as modern games and animations. Second, the subdivision algorithm can retain the geometric and topological characteristics of the original mesh to a certain extent while obtaining a smooth surface. It is also not limited by the initial topology of the mesh. In addition, the subdivision algorithm can adapt to subdivision tasks in environments with various resolutions and meet the computing needs of different resolutions. Thus, it is widely used in the field of real-time rendering and virtual reality. Finally, compared with some complex mesh generation techniques, such as Delaunay triangulation, the mesh subdivision algorithm is easy to implement and has low input data requirements, so it has become a universal mesh generation tool. The traditional method [1; 2; 3] used for mesh subdivision adopts a fixed linear subdivision strategy, which leads to excessive smoothing of the subdivision structure. An analysis of experimental results also shows that, although traditional mesh subdivision can obtain smooth subdivision results, it performs poorly in quantitative analysis experiments. In 2020, [PERSON] et al. [4] proposed a nonlinear subdivision method based on a neural network multi-layer perceptron, imitating the classical LOOP subdivision algorithm. This method achieves a good subdivision effect for closed triangular meshes. However, neural subdivision is a data-driven approach. The features of the input coarse mesh processed by the subdivision network need to be similar to the coarse and fine mesh pairs used as training data. For triangular meshes with too large a face density or small mesh faces, the results of the neural subdivision will contain a large number of artifacts. Also, neural subdivisions cannot be applied to bounded triangular meshes. To solve the above problems, this study proposes a new end-to-end subdivision framework based on the graph neural network method and neural subdivision, namely, graph neural subdivision (GNS). The main contributions of this study are summarized as follows:  
1. The half-flap structure used in neural subdivision is replaced by local triangular mesh patches, and the spatial and topological structures inside the mesh and at the mesh boundary are naturally expressed. This expands the scope of neural-network-based subdivision algorithms. 2. The tensor voting strategy is used to move the local patch from a global space to a local space to ensure the translation and rotation invariance of the algorithm. 3. We introduce a dynamic graph convolutional layer, uncover the relationship between the vertices of the mesh and its adjacent vertices and non-adjacent vertices, and detail the local and global characteristics of the mesh. 4. The average area and the number of the one-ring adjacent faces at the vertices of the mesh are added to the training data. This further improves the robustness of the subdivision network.