Before revision:  To handle irregular points of a discrete mesh while sharing kernel weights, we make the convolution semi-discrete, i.e. the convolution kernels are polynomial functions, and their convolution with discrete surface points becomes sampling and weighted summation. Pooling and unpooling operations are computed along a mesh hierarchy built through simplification. The presented surface CNNs allow effective deep learning on meshes. We show that for tasks of classification, segmentation and non-rigid registration , surface CNNs using only raw input signals achieve superior performances than previous models using sophisticated input features  .
After revision:  To handle irregular points of a discrete mesh while sharing kernel weights, we make the convolution of PFCNN not only maps local surface patches onto flat tangent planes, but also aligns the tangent planes such that they locally form a flat Euclidean structure, thus enabling recovery of standard convolutions. The alignment is achieved by the tool of locally flat connections borrowed from discrete differential geometry, which can be efficiently encoded and computed by parallel frame fields. In addition, the lack of canonical axis on surface is handled by sampling with the frame directions. Experiments show that for tasks of classification, segmentation and non-rigid registration , surface CNNs using only raw input signals achieve superior performances than previous models using sophisticated input features .