Partial Shape Similarity via Alignment of Multi-Metric Hamiltonian Spectra

Abstract: Evaluating the similarity of non-rigid shapes with significant partiality is a fundamental task in numerous computer vision applications. Here, we propose a novel axiomatic method to match similar regions across shapes. Matching similar regions is formulated as the alignment of the spectra of operators closely related to the Laplace-Beltrami operator (LBO). The main novelty of the proposed approach is the consideration of differential operators defined on a manifold with multiple metrics. The choice of a metric relates to fundamental shape properties while considering the same manifold under different metrics can thus be viewed as analyzing the underlying manifold from different perspectives. Specifically, we examine the scale-invariant metric and the corresponding scale-invariant Laplace-Beltrami operator (SI-LBO) along with the regular metric and the regular LBO. We demonstrate that the scale-invariant metric emphasizes the locations of important semantic features in articulated shapes. A truncated spectrum of the SI-LBO consequently better captures locally curved regions and complements the global information encapsulated in the truncated spectrum of the regular LBO. We show that matching these dual spectra outperforms competing axiomatic frameworks when tested on standard benchmarks. We introduced a new dataset and compare the proposed method with the state-of-the-art learning based approach in a cross-database configuration. Specifically, we show that, when trained on one data set and tested on another, the proposed axiomatic approach which does not involve training, outperforms the deep learning alternative.