Surface Networks
This work addresses the need for more powerful 3D shape representations in computer graphics and geometry processing, offering a novel approach that is not invariant to isometric deformations, though it builds incrementally on existing GNN frameworks.
The paper tackled the problem of representing 3D triangle meshes by proposing Surface Networks (SNs), which leverage extrinsic differential geometry via the Dirac operator to overcome limitations of intrinsic methods like Graph Neural Networks, resulting in stable shape representations demonstrated on tasks such as temporal mesh deformation prediction and generative modeling.
We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs.