Spectral geometric methods have brought revolutionary changes to the field of geometry processing. Of particular interest is the study of the Laplacian spectrum as a compact, isometry and permutation-invariant representation of a shape. Some recent works show how the intrinsic geometry of a full shape can be recovered from its spectrum, but there are approaches that consider the more challenging problem of recovering the geometry from the spectral information of partial shapes. In this paper, we propose a possible way to fill this gap. We introduce a learning-based method to estimate the Laplacian spectrum of the union of partial non-rigid 3D shapes, without actually computing the 3D geometry of the union or any correspondence between those partial shapes. We do so by operating purely in the spectral domain and by defining the union operation between short sequences of eigenvalues. We show that the approximated union spectrum can be used as-is to reconstruct the complete geometry [MRC*19], perform region localization on a template [RTO*19] and retrieve shapes from a database, generalizing ShapeDNA [RWP06] to work with partialities. Working with eigenvalues allows us to deal with unknown correspondence, different sampling, and different discretizations (point clouds and meshes alike), making this operation especially robust and general. Our approach is data-driven and can generalize to isometric and non-isometric deformations of the surface, as long as these stay within the same semantic class (e.g., human bodies or horses), as well as to partiality artifacts not seen at training time.

翻译：谱几何方法为几何处理领域带来了革命性变化。其中，将拉普拉斯谱作为一种紧致且具有等距和置换不变性的形状表示进行研究尤为引人关注。近期一些工作展示了如何从全形状的谱中恢复其内在几何结构，但针对从部分形状的谱信息恢复几何这一更具挑战性的问题，现有方法仍存在局限。本文提出了一种填补此空白的可行方案：我们引入基于学习的方法来估计多个非刚性部分3D形状的并集拉普拉斯谱，而无需实际计算并集的3D几何或这些部分形状之间的对应关系。该方法完全在谱域中操作，通过定义特征值短序列之间的并集运算来实现。我们证明，该近似并集谱可直接用于重构完整几何结构 [MRC*19]、在模板上进行区域定位 [RTO*19] 以及从数据库中检索形状，并将ShapeDNA [RWP06] 扩展至处理部分形状。基于特征值的操作使我们能够应对未知对应关系、不同采样和不同离散化（点云与网格皆可），使该运算具有极强的鲁棒性和通用性。该方法为数据驱动型，可泛化至同一语义类别（如人体或马匹）内曲面的等距与非等距变形，以及训练中未见的局部缺失情况。