Before revision: We extend Convolutional Neural Networks (CNNs) on flat and regular domains (e.g. 2D images) to curved surfaces embedded in 3D Euclidean space that are discretized as irregular meshes and widely used to represent geometric data in Computer Vision and Graphics. We define surface convolution on tangent spaces of a surface domain, where the convolution has two desirable properties: 1) the distortion of surface domain signals is locally minimal when being projected to the tangent space, and 2) the translation equi-variance property holds locally, by aligning tangent spaces with the canonical parallel transport that preserves metric.
After revision: Surface meshes are widely used shape representations and capture finer geometry data than point clouds or volumetric grids, but are challenging to apply CNNs directly due to their non-Euclidean structure. We use parallel frames on surface to define PFCNNs that enable effective feature learning on surface domain signals is locally minimal when being projected to the tangent space, and 2) the translation equi-variance property holds locally, by aligning tangent spaces with the canonical parallel transport that preserves metric.