  Geometric deep learning has sparked a rising interest in computer graphics to
perform shape understanding tasks, such as shape classification and semantic
segmentation. When the input is a polygonal surface, one has to suffer from the
irregular mesh structure. Motivated by the geometric spectral theory, we
introduce Laplacian2Mesh, a novel and flexible convolutional neural network
(CNN) framework for coping with irregular triangle meshes (vertices may have
any valence). By mapping the input mesh surface to the multi-dimensional
Laplacian-Beltrami space, Laplacian2Mesh enables one to perform shape analysis
tasks directly using the mature CNNs, without the need to deal with the
irregular connectivity of the mesh structure. We further define a mesh pooling
operation such that the receptive field of the network can be expanded while
retaining the original vertex set as well as the connections between them.
Besides, we introduce a channel-wise self-attention block to learn the
individual importance of feature ingredients. Laplacian2Mesh not only decouples
the geometry from the irregular connectivity of the mesh structure but also
better captures the global features that are central to shape classification
and segmentation. Extensive tests on various datasets demonstrate the
effectiveness and efficiency of Laplacian2Mesh, particularly in terms of the
capability of being vulnerable to noise to fulfill various learning tasks.
