密集网格上可扩展的高效函数映射计算
Scalable and Efficient Functional Map Computations on Dense Meshes
摘要：
谱几何方法给几何处理领域带来了革命性的变化。特别感兴趣的是研究拉普拉斯频谱作为一个紧凑的，等距和置换不变的形状表示。最近的一些工作表明，一个完整的形状的内在几何形状可以从它的频谱中恢复，但也有方法，考虑更具有挑战性的问题，从部分形状的频谱信息恢复的几何形状。在本文中，我们提出了一种可能的方法来填补这一空白。我们引入了一种基于学习的方法来估计部分非刚性3D形状的并集的拉普拉斯谱，而无需实际计算并集的3D几何形状或这些部分形状之间的任何对应关系。我们这样做纯粹是在谱域中操作，并通过定义短序列的特征值之间的联合操作。我们表明，近似的联合谱可以按原样用于重建完整的几何结构[MRC*19]，在模板上执行区域定位[RTO*19]并从数据库中检索形状，推广ShapeDNA [RWP 06]以处理偏性。使用特征值允许我们处理未知的对应关系，不同的采样和不同的离散化（点云和网格），使该操作特别鲁棒和通用。我们的方法是数据驱动的，并且可以推广到表面的等距和非等距变形，只要这些变形保持在相同的语义类内（例如，人的身体或马），以及隐藏在训练时看不到的伪像。
英文摘要：
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.
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