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.

翻译：几何深度学习在计算机图形学中引发了形状理解任务（如形状分类和语义分割）的日益关注。当输入为多边形表面时，必须应对不规则网格结构的挑战。受几何谱理论启发，我们提出了Laplacian2Mesh——一种新颖且灵活的卷积神经网络（CNN）框架，用于处理不规则三角形网格（顶点可具有任意度数）。通过将输入网格表面映射到多维拉普拉斯-贝尔特拉米空间，Laplacian2Mesh无需处理网格结构的不规则连接性，即可直接利用成熟的CNN进行形状分析任务。我们进一步定义了网格池化操作，使得网络感受野得以扩展，同时保留原始顶点集及其连接关系。此外，我们引入通道级自注意力模块来学习特征成分的个体重要性。Laplacian2Mesh不仅将几何与网格结构的不规则连接性解耦，还能更好地捕捉对形状分类和分割至关重要的全局特征。在多种数据集上的广泛测试证明了Laplacian2Mesh的有效性和高效性，尤其是在对噪声敏感以完成各类学习任务的能力方面表现突出。