  Convolutional neural networks have been extremely successful for 2D images
and are readily extended to handle 3D voxel data. Meshes are a more common 3D
shape representation that quantize the shape surface instead of the ambient
space as with voxels, hence giving access to surface properties such as normals
or appearances. The formulation of deep neural networks on meshes is, however,
more complex since they are irregular data structures where the number of
neighbors varies across vertices. While graph convolutional networks have
previously been proposed over mesh vertex data, in this paper we explore how
these networks can be extended to the dual face-based representation of
triangular meshes, where nodes represent triangular faces in place of vertices.
In comparison to the primal vertex mesh, its face dual offers several
advantages, including, importantly, that the dual mesh is regular in the sense
that each triangular face has exactly three neighbors. Moreover, the dual mesh
suggests the use of a number of input features that are naturally defined over
faces, such as surface normals and face areas. We evaluate the dual approach on
the shape correspondence task on the FAUST human shape dataset and other
versions of it with varying mesh topology. While applying generic graph
convolutions to the dual mesh shows already improvements over primal mesh
inputs, our experiments demonstrate that building additionally convolutional
models that explicitly leverage the neighborhood size regularity of dual meshes
enables learning shape representations that perform on par or better than
previous approaches in terms of correspondence accuracy and mean geodesic
error, while being more robust to topological changes in the meshes between
training and testing shapes.
