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.

翻译：网格变形是三维网格重建的核心任务，但定义预测网格与目标网格间的高效差异度量仍是一个开放性问题。当前深度学习中主流方法采用基于集合的方式，通过从两个网格中随机采样点云并利用Chamfer伪距离比较两表面差异。然而，基于集合的方法仍存在局限性，例如缺乏对采样点云数量选择的理论保证，以及Chamfer散度的伪度量特性与二次复杂度。针对这些问题，我们提出一种用于学习网格变形的新度量。该度量将网格表示为概率测度（通过varifold表示可灵活编码连续测度、经验测度及离散测度），并采用切片Wasserstein距离——一种具有线性计算复杂度的高效最优传输距离——对网格进行度量。该度量能提供快速统计率以近似网格表面。在此基础上，我们利用神经常微分方程(ODE)通过建模表面点运动轨迹实现输入曲面到目标形状的变形。在大脑皮层表面重建实验表明，本方法在多数据集及多指标上均优于其他竞争方法。