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.

翻译：网格变形在动态模拟、渲染和重建等众多三维视觉任务中扮演着关键角色。然而，定义预测网格与目标网格之间的高效差异度量仍是一个未解决的问题。当前深度学习中一种主流方法是基于集合的方法，通过从两个网格中随机采样点云，并利用Chamfer伪距离比较两个曲面的差异。但基于集合的方法仍存在局限性，例如缺乏选择采样点云中点数的理论保证，以及Chamfer散度的伪度量性和二次复杂度。为解决这些问题，我们提出了一种用于学习网格变形的新型度量。该度量通过对网格表示为概率测度（推广了基于集合方法）上的切片Wasserstein距离进行定义。通过利用概率测度空间，我们能够灵活地使用多种形式的概率测度来编码网格，例如通过varifold表示实现连续测度、经验测度和离散测度。在编码概率测度后，我们可以使用切片Wasserstein距离比较网格，这是一种高效的最优传输距离，具有线性计算复杂度，并能快速实现曲面逼近的统计速率。最后，我们采用神经常微分方程(ODE)通过建模曲面上点的轨迹，将输入曲面变形为目标形状。我们在皮层表面重建上的实验表明，我们的方法在多个数据集和指标上均优于其他竞争方法。