Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence--a natural extension relevant to shapes extracted from simulation, medical imaging, and volume rendering--presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align closely to the boundary.

翻译：虽然对称是几何处理中的一个中心问题,但这项任务的大多数方法只适用于二维表面。 被忽略的体积对称任务 — — 与模拟、医学成像和体积成像的形状相关的自然扩展任务 — — 与模拟、医学成像和体积成像的形状相关的自然延伸任务 — — 呈现出二维案例所没有出现的独特挑战。 在这项工作中,我们提出一个在以四面形色素表示的体积之间绘制图案的方法。 我们的配方最大限度地减少了一种扭曲能量,旨在对称地提取地图,即不依赖源和目标域的顺序。 我们用这个方法来进行理论讨论,描述这一对称假设的后果, 导致我们选择了偏重于对称对应对应的亚光谱能量。 我们的最终配方在匹配边界的同时优化了接近的近光度。 我们在不同的几何数据集上展示了我们的方法, 产生与边界相近的低扭曲匹配。