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Diffeomorphic Deformation via Sliced Wasserstein Distance Optimization for Cortical Surface Reconstruction
Mesh deformation is a core task for 3D mesh reconstruction, but 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 \textit{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. Furthermore, 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
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