Laplacian2Mesh: Laplacian-Based Mesh Understanding

Abstract

Geometric deep learning has sparked a rising interest in computer graphics to perform shape understanding tasks, such as shape classification and semantic segmentation on three-dimensional (3D) geometric surfaces. Previous works explored the significant direction by defining the operations of convolution and pooling on triangle meshes, but most methods explicitly utilized the graph connection structure of the mesh. Motivated by the geometric spectral surface reconstruction theory, we introduce a novel and flexible convolutional neural network (CNN) model, called Laplacian2Mesh, for 3D triangle mesh, which maps the features of mesh in the Euclidean space to the multi-dimensional Laplacian-Beltrami space, which is similar to the multi-resolution input in 2D CNN. Mesh pooling is applied to expand the receptive field of the network by the multi-space transformation of Laplacian which retains the surface topology, and channel self-attention convolutions are applied in the new space. Since implicitly using the intrinsic geodesic connections of the mesh through the adjacency matrix, we do not consider the number of the neighbors of the vertices, thereby mesh data with different numbers of vertices can be input. Experiments on various learning tasks applied to 3D meshes demonstrate the effectiveness and efficiency of Laplacian2Mesh.

A Survey on Graph Neural Networks and Graph Transformers in Computer Vision: A Task-Oriented Perspective

“…The polygonal mesh discretely represents the surface of a 3D shape with faces and vertices. Since it can be viewed as an undirected graph, mainstream GNNs or graph Transformers can be used for 3D mesh analysis [247], [248], [249], [250], [251], [252], [253], [254], [255], [256], [257], [258], [259].…”

“…More recently, SubdivNet [257] performs representation learning on meshes with loop subdivision sequence connectivity by constructing hierarchical mesh pyramids. Laplacian2Mesh [258] considers the graph convolution and pooling on triangular meshes in the spectral domain, which maps the mesh features in the Euclidean space to the multidimensional Laplacian-Beltrami space. Different from the custom of constructing graphs with regular neighborhood structures, MeshWalker [259] applies a similar way as in [224] to mesh analysis, where random walks along edges are conducted for 3D shape information extraction.…”