Functional Maps Representation On Product Manifolds

By Emanuele Rodolà, Zorah Lähner, A. Bronstein, Michael M. Bronstein, Justin M. Solomon. 2018

19 citations recorded by the source corpus.

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

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Citations in the corpus

DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking (2022, background): “Furthermore, since P is a product manifold, the forward diffusion proceeds independently in each manifold [9], and the tangent space is a direct sum: where g = ( r , R, θ ) .”