We present a systematic treatment to 3D shape analysis based on the well-established de Rham-Hodge theory in differential geometry and topology. The computational tools we developed are widely applicable to research areas such as computer graphics, computer vision, and computational biology. We extensively tested it in the context of 3D structure analysis of biological macromolecules to demonstrate the efficacy and efficiency of our method in potential applications. Our contributions are summarized in the following aspects. First, we present a compendium of discrete Hodge decompositions of vector fields, which provides the primary building block of the de Rham-Hodge theory for computations performed on the commonly used tetrahedral meshes embedded in the 3D Euclidean space. Second, we present a real-world application of the above computational tool to 3D shape analysis on biological macromolecules. Finally, we extend the above method to an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. Our work on the decomposition of vector fields, spectral shape analysis on static shapes, and evolving shapes has already shown its effectiveness in biomolecular applications and will lead to a rich set of features for machine learning-based shape analysis currently under development.
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