Partial Shape Similarity by Multi-metric Hamiltonian Spectra Matching

Estimating the similarity of non-rigid shapes and parts thereof plays an important role in numerous geometry analysis applications. We propose a method for evaluating the similarity and matching of shapes describing articulated objects that gracefully handles partiality. The correspondence between a part and a whole is formulated as the alignment of spectra of operators closely related to the Laplace-Beltrami operator (LBO). The proposed approach considers multiple metrics defined on the same surface, which provide a compact description of the underlying geometric structure from different perspectives. Specifically, we study the scale-invariant metric and the corresponding scale-invariant Laplace-Beltrami operator (SI-LBO) together with the regular metric and the regular LBO. We demonstrate that, unlike the regular LBO, the low pass part of the SI-LBO eigen-structure is sensitive to regions with high Gaussian curvature which are of semantic importance in articulated objects. Thus, the low part of the SI-LBO’s spectrum better captures curved regions and complements the information encapsulated in the lower part of the regular LBO’s spectrum. A two spectra matching loss lends itself to a method that outperforms state of the art axiomatic and learning based techniques when evaluated on the task of partial matching on well established benchmarks (Code and results are available at: https://github.com/davidgip74/DualSpectraAlignmnent ).