Spectral computations on nontrivial line bundles

Computing the spectral decomposition of the Laplace–Beltrami operator on a manifold M   has proven useful for applications such as shape retrieval and geometry processing. The standard operator acts on scalar functions which can be identified with sections of the trivial line bundle M×RM×R. In this work we propose to extend the discussion to Laplacians on nontrivial real line bundles. These line bundles are in one-to-one correspondence with elements of the first cohomology group of the manifold with Z2Z2 coefficients. While we focus on the case of two-dimensional closed surfaces, we show that our method also applies to surfaces with boundaries. Denoting by ββ the rank of the first cohomology group, there are 2β2β different line bundles to consider and each of these has a naturally associated Laplacian that possesses a spectral decomposition. Using our new method it is possible for the first time to compute the spectra of these Laplacians by a simple modification of the finite element basis functions used in the standard trivial bundle case. Our method is robust and efficient. We illustrate some properties of the modified spectra and eigenfunctions and indicate possible applications for shape processing. As an example, using our method, we are able to create spectral shape descriptors with increased sensitivity in the eigenvalues with respect to geometric deformations and to compute cycles aligned to object symmetries in a chosen homology class.

Graphical AbstractFigure optionsDownload high-quality image (534 K)Download as PowerPoint slideHighlights
► We compute the spectral geometry for nontrivial real line bundles over surfaces.
► Our algorithm uses modified finite element basis functions.
► Each line bundle yields different spectra and eigenfunctions.
► Some spectra are more sensitive with respect to geometric deformations.
► Eigenfunctions yield geometry-aware characteristic cycles within a homology class.