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Spectral geometric methods have brought revolutionary changes to the field ofgeometry processing. Of particular interest is the study of the Laplacianspectrum as a compact, isometry and permutation-invariant representation of ashape. Some recent works show how the intrinsic geometry of a full shape can berecovered from its spectrum, but there are approaches that consider the morechallenging problem of recovering the geometry from the spectral information ofpartial shapes. In this paper, we propose a possible way to fill this gap. Weintroduce a learning-based method to estimate the Laplacian spectrum of theunion of partial non-rigid 3D shapes, without actually computing the 3Dgeometry of the union or any correspondence between those partial shapes. We doso by operating purely in the spectral domain and by defining the unionoperation between short sequences of eigenvalues. We show that the approximatedunion 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 adatabase, generalizing ShapeDNA [RWP06] to work with partialities. Working witheigenvalues allows us to deal with unknown correspondence, different sampling,and different discretizations (point clouds and meshes alike), making thisoperation especially robust and general. Our approach is data-driven and cangeneralize to isometric and non-isometric deformations of the surface, as longas these stay within the same semantic class (e.g., human bodies or horses), aswell as to partiality artifacts not seen at training time.

Learning Spectral Unions of Partial Deformable 3D Shapes