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。

关于部分形状对应与功能映射的研究

On Partial Shape Correspondence and Functional Maps

Finding correspondences between 3D shapes is a crucial problem in computer
vision and graphics, which is for example relevant for tasks like shape
interpolation, pose transfer, or texture transfer. An often neglected but
essential property of matchings is geometric consistency, which means that
neighboring triangles in one shape are consistently matched to neighboring
triangles in the other shape. Moreover, while in practice one often has only
access to partial observations of a 3D shape (e.g. due to occlusion, or
scanning artifacts), there do not exist any methods that directly address
geometrically consistent partial shape matching. In this work we fill this gap
by proposing to integrate state-of-the-art deep shape features into a novel
integer linear programming partial shape matching formulation. Our optimization
yields a globally optimal solution on low resolution shapes, which we then
refine using a coarse-to-fine scheme. We show that our method can find more
reliable results on partial shapes in comparison to existing geometrically
consistent algorithms (for which one first has to fill missing parts with a
dummy geometry). Moreover, our matchings are substantially smoother than
learning-based state-of-the-art shape matching methods.

该论文提出了一种整合先进的深度形状特征到新型整数线性规划部分形状匹配公式的优化方法，能够在低分辨率的形状上获得全局最优解，并通过粗到精的过程进行改进，相比于现有的几何一致算法，我们的方法在处理部分形状时能够找到更可靠的结果且匹配更为平滑，同时能够避免使用补齐几何形状的方法。