While dealing with matching shapes to their parts, we often apply a tool known as functional maps. The idea is to translate the shape matching problem into "convenient" spaces by which matching is performed algebraically by solving a least squares problem. Here, we argue that such formulations, though popular in this field, introduce errors in the estimated match when partiality is invoked. Such errors are unavoidable even for advanced feature extraction networks, and they can be shown to escalate with increasing degrees of shape partiality, adversely affecting the learning capability of such systems. To circumvent these limitations, we propose a novel approach for partial shape matching. Our study of functional maps led us to a novel method that establishes direct correspondence between partial and full shapes through feature matching bypassing the need for functional map intermediate spaces. The Gromov Distance between metric spaces leads to the construction of the first part of our loss functions. For regularization we use two options: a term based on the area preserving property of the mapping, and a relaxed version that avoids the need to resort to functional maps. The proposed approach shows superior performance on the SHREC'16 dataset, outperforming existing unsupervised methods for partial shape matching.Notably, it achieves state-of-the-art results on the SHREC'16 HOLES benchmark, superior also compared to supervised methods. We demonstrate the benefits of the proposed unsupervised method when applied to a new dataset PFAUST for part-to-full shape correspondence.

翻译：在处理形状与其部分匹配问题时，我们通常采用称为功能映射的工具。其核心思想是将形状匹配问题转换到"便利"的空间中，通过求解最小二乘问题以代数方式完成匹配。本文指出，尽管此类方法在该领域颇为流行，但在涉及部分性匹配时，会在估计匹配中引入误差。即使采用先进的特征提取网络，此类误差仍不可避免，且可证明其会随着形状部分性程度的增加而升级，对这些系统的学习能力产生不利影响。为规避这些限制，我们提出了一种新颖的部分形状匹配方法。通过对功能映射的研究，我们开发出一种新方法，通过特征匹配直接在部分形状与完整形状之间建立对应关系，从而绕过了功能映射中间空间的需求。度量空间之间的格罗莫夫距离构成了我们损失函数的第一部分。在正则化方面，我们采用两种方案：基于映射保面积特性的正则项，以及避免依赖功能映射的松弛版本。所提方法在SHREC'16数据集上表现出优越性能，超越了现有的无监督部分形状匹配方法。值得注意的是，该方法在SHREC'16 HOLES基准测试中取得了最先进的结果，其性能甚至优于有监督方法。我们通过在新型数据集PFAUST上进行部分到完整形状对应的实验，验证了所提无监督方法的优势。