  This paper introduces a generative model for 3D surfaces based on a
representation of shapes with mean curvature and metric, which are invariant
under rigid transformation. Hence, compared with existing 3D machine learning
frameworks, our model substantially reduces the influence of translation and
rotation. In addition, the local structure of shapes will be more precisely
captured, since the curvature is explicitly encoded in our model. Specifically,
every surface is first conformally mapped to a canonical domain, such as a unit
disk or a unit sphere. Then, it is represented by two functions: the mean
curvature half-density and the vertex density, over this canonical domain.
Assuming that input shapes follow a certain distribution in a latent space, we
use the variational autoencoder to learn the latent space representation. After
the learning, we can generate variations of shapes by randomly sampling the
distribution in the latent space. Surfaces with triangular meshes can be
reconstructed from the generated data by applying isotropic remeshing and spin
transformation, which is given by Dirac equation. We demonstrate the
effectiveness of our model on datasets of man-made and biological shapes and
compare the results with other methods.
