Deep functional maps have recently emerged as a successful paradigm for non-rigid 3D shape correspondence tasks. An essential step in this pipeline consists in learning feature functions that are used as constraints to solve for a functional map inside the network. However, the precise nature of the information learned and stored in these functions is not yet well understood. Specifically, a major question is whether these features can be used for any other objective, apart from their purely algebraic role in solving for functional map matrices. In this paper, we show that under some mild conditions, the features learned within deep functional map approaches can be used as point-wise descriptors and thus are directly comparable across different shapes, even without the necessity of solving for a functional map at test time. Furthermore, informed by our analysis, we propose effective modifications to the standard deep functional map pipeline, which promote structural properties of learned features, significantly improving the matching results. Finally, we demonstrate that previously unsuccessful attempts at using extrinsic architectures for deep functional map feature extraction can be remedied via simple architectural changes, which encourage the theoretical properties suggested by our analysis. We thus bridge the gap between intrinsic and extrinsic surface-based learning, suggesting the necessary and sufficient conditions for successful shape matching. Our code is available at https://github.com/pvnieo/clover.

翻译：深度函数映射近年来已成为非刚性三维形状对应任务的成功范式。该流程的关键步骤在于学习特征函数，这些函数作为约束条件用于在网络内部求解函数映射。然而，目前对这些函数中学习并存储的信息的精确性质尚未充分理解。具体而言，一个核心问题是：这些特征除了在求解函数映射矩阵中扮演纯粹代数角色外，是否还能用于其他目标？本文证明，在温和条件下，深度函数映射方法中学习到的特征可作为逐点描述符使用，因此即使测试时无需求解函数映射，也能在不同形状间直接比较。此外，基于我们的分析，我们提出对标准深度函数映射流程的有效修改，这些修改增强了学习特征的结构属性，显著改善了匹配结果。最后，我们证明先前尝试使用外在架构进行深度函数映射特征提取的失败案例，可通过简单的架构变更得到修正，这种变更促进了我们分析所建议的理论特性。我们由此弥合了基于内蕴与外蕴的曲面学习之间的鸿沟，提出了成功形状匹配的必要与充分条件。我们的代码已开源至 https://github.com/pvnieo/clover。