  Mesh deformation plays a pivotal role in many 3D vision tasks including
dynamic simulations, rendering, and reconstruction. However, defining an
efficient discrepancy between predicted and target meshes remains an open
problem. A prevalent approach in current deep learning is the set-based
approach which measures the discrepancy between two surfaces by comparing two
randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance.
Nevertheless, the set-based approach still has limitations such as lacking a
theoretical guarantee for choosing the number of points in sampled
point-clouds, and the pseudo-metricity and the quadratic complexity of the
Chamfer divergence. To address these issues, we propose a novel metric for
learning mesh deformation. The metric is defined by sliced Wasserstein distance
on meshes represented as probability measures that generalize the set-based
approach. By leveraging probability measure space, we gain flexibility in
encoding meshes using diverse forms of probability measures, such as
continuous, empirical, and discrete measures via varifold representation. After
having encoded probability measures, we can compare meshes by using the sliced
Wasserstein distance which is an effective optimal transport distance with
linear computational complexity and can provide a fast statistical rate for
approximating the surface of meshes. To the end, we employ a neural ordinary
differential equation (ODE) to deform the input surface into the target shape
by modeling the trajectories of the points on the surface. Our experiments on
cortical surface reconstruction demonstrate that our approach surpasses other
competing methods in multiple datasets and metrics.
