  We propose DualConvMesh-Nets (DCM-Net) a family of deep hierarchical
convolutional networks over 3D geometric data that combines two types of
convolutions. The first type, geodesic convolutions, defines the kernel weights
over mesh surfaces or graphs. That is, the convolutional kernel weights are
mapped to the local surface of a given mesh. The second type, Euclidean
convolutions, is independent of any underlying mesh structure. The
convolutional kernel is applied on a neighborhood obtained from a local
affinity representation based on the Euclidean distance between 3D points.
Intuitively, geodesic convolutions can easily separate objects that are
spatially close but have disconnected surfaces, while Euclidean convolutions
can represent interactions between nearby objects better, as they are oblivious
to object surfaces. To realize a multi-resolution architecture, we borrow
well-established mesh simplification methods from the geometry processing
domain and adapt them to define mesh-preserving pooling and unpooling
operations. We experimentally show that combining both types of convolutions in
our architecture leads to significant performance gains for 3D semantic
segmentation, and we report competitive results on three scene segmentation
benchmarks. Our models and code are publicly available.
