Performing image-like convolution on triangular meshes

Image convolution with a filtering mask is at the base of several image analysis operations. This is motivated by Mathematical foundations and by the straightforward way the discrete convolution can be computed on a grid-like domain. Extending the convolution operation to the mesh manifold support is a challenging task due to the irregular structure of the mesh connections. In this paper, we propose a computational framework that allows convolutional operations on the mesh. This relies on the idea of ordering the facets of the mesh so that a shift-like operation can be derived. Experiments have been performed with several filter masks (Sobel, Gabor, etc.) showing state-of-the-art results in 3D relief patterns retrieval on the SHREC'17 dataset. We also provide evidence that the proposed framework can enable convolution and pooling-like operations as can be needed for extending Convolutional Neural Networks to 3D meshes.