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, volume rendering, and even improving surface maps of boundary representations--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 to the boundary.
翻译:虽然对等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。 被忽略的体积对应- 与模拟、 医学成像、 体积成像、 体积成像、 甚至改进边界代表图表的形状有关的自然延伸任务, 提出了在二维情况下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色素表示的体积之间绘制地图的方法。 我们的配方最大限度地减少了一种扭曲的能量, 旨在对称地提取地图, 即不依赖于源和目标域的顺序。 我们与我们的方法一起进行理论讨论, 描述这个对称假设的后果, 导致我们选择一个偏向于对称对应的亚行能量。 我们的最终配方在匹配边界时优化了近相色度。 我们用不同的几何数据集展示了我们的方法, 产生与边界相匹配的低扭曲匹配。