Unsupervised Learning of Efficient Geometry-Aware Neural Articulated Representations

We propose an unsupervised method for 3D geometry-aware representation learning of articulated objects. Though photorealistic images of articulated objects can be rendered with explicit pose control through existing 3D neural representations, these methods require ground truth 3D pose and foreground masks for training, which are expensive to obtain. We obviate this need by learning the representations with GAN training. From random poses and latent vectors, the generator is trained to produce realistic images of articulated objects by adversarial training. To avoid a large computational cost for GAN training, we propose an efficient neural representation for articulated objects based on tri-planes and then present a GAN-based framework for its unsupervised training. Experiments demonstrate the efficiency of our method and show that GAN-based training enables learning of controllable 3D representations without supervision.Comment: 19 pages, project page https://nogu-atsu.github.io/ENARF-GAN

Grunbaum colorings of even triangulations on surfaces

A Grunbaum coloring of a triangulation G is a map c : E(G){1,2,3} such that for each face f of G, the three edges of the boundary walk of f are colored by three distinct colors. By Four Color Theorem, it is known that every triangulation on the sphere has a Grunbaum coloring. So, in this article, we investigate the question whether each even (i.e.,Eulerian) triangulation on a surface with representativity at least r has a Grunbaum coloring. We prove that, regardless of the representativity, every even triangulation on a surface F has a Grunbaum coloring as long as F is the projective plane, the torus, or the Klein bottle, and we observe that the same holds for any surface with sufficiently large representativity. On the other hand, we construct even triangulations with no Grunbaum coloring and representativity r=1,2, and 3 for all but finitely many surfaces. In dual terms, our results imply that no snark admits an even map on the projective plane, the torus, or the Klein bottle, and that all but finitely many surfaces admit an even map of a snark with representativity at least3