When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of whether distances measured between pairs of surface points should influence the matching process. The approach we propose treats surfaces as manifolds equipped with geodesic distances, and addresses the partial shape matching challenge by introducing a novel criterion to meticulously search for consistent distances between pairs of points. The new criterion explores the relation between intrinsic geodesic distances between the points, geodesic distances between the points and surface boundaries, and extrinsic distances between boundary points measured in the embedding space. It is shown to be less restrictive compared to previous measures and achieves state-of-the-art results when used as a loss function in training networks for partial shape matching.

翻译：在将曲面部分区域与其整体进行匹配时，一个根本性问题随之产生：哪些点应被纳入匹配过程？当使用等距变换度量相似性时，该问题尤为突出，因为这需要验证曲面点对之间的测量距离是否应影响匹配过程。我们提出的方法将曲面视为具有测地距离的流形，并通过引入新准则来精细搜索点对之间的一致性距离，从而解决部分形状匹配的挑战。该新准则探究了以下三者的关系：点之间的本征测地距离、点到曲面边界的测地距离，以及嵌入空间中边界点之间的外显距离。相较于先前度量方法，该准则被证明具有更弱的约束性，当作为损失函数训练部分形状匹配网络时，能够取得最先进的结果。