While dealing with matching shapes to their parts, we often utilize an
instrument 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 when
considering 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 of it without the
need to compute a functional map.
The proposed approach shows superior performance on the SHREC'16 dataset,
outperforming existing unsupervised methods for partial shape matching. In
particular, it achieves state-of-the-art result on the SHREC'16 HOLES
benchmark, superior also compared to supervised methods.

通过提出一种新的部分形状匹配方法，该方法可以通过特征匹配建立偏差和完整形状的直接对应关系，并使用功能地图中间空间。Gromov 距离是构建我们损失函数的第一部分的依据，另外我们还使用了基于映射保持面积属性的项以及无需计算功能地图的放松版本进行正规化，该方法在 SHREC'16 数据集上表现出卓越的性能，并且在 SHREC'16 HOLES 基准测试中超过了现有的无监督方法，并与有监督方法相比也具有卓越的结果。