SMI 2012: Full Paper Spectral computations on nontrivial line bundles $ Alexander Vais n , Benjamin Berger, Franz-Erich Wolter Welfenlab, Division of Computer Graphics, Leibniz University of Hannover, 30167 Hannover, Germany article info Article history: Received 1 December 2011 Received in revised form 15 March 2012 Accepted 17 March 2012 Available online 28 March 2012 Keywords: Spectral geometry processing Vector bundles Computational topology Laplace operator Finite elements abstract 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  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 Z 2 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 b the rank of the first cohomology group, there are 2 b 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. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Curves, surfaces and solids are commonly used in computer graphics, computer vision and computer aided geometric design, where they serve as basic building blocks or data elements. Several continuous and discrete representations exist and many algo- rithms have been proposed to operate on these representations for a variety of tasks or to convert between them. It is often useful to view these objects within the framework of differential geome- try where they become instances of manifolds, with or without boundary. Loosely speaking, a manifold M is a space that locally looks like the Euclidean space. In this setting it becomes possible to transfer the tools and techniques of multivariate calculus onto M and to develop algorithms that benefit from the rich arsenal of techniques available in this mathematical fundament. In this paper we deal with the Laplace–Beltrami operator which is the generalization of the Euclidean Laplacian. As in the Euclidean case it can be defined by Df : ¼div rf , where rf is the gradient of the real-valued function f defined on the manifold M and div is the divergence operator. We follow the above sign convention, making D a positive definite operator. The Laplace operator is abundant throughout mathematics, physics and engineering. It plays an important role in describing physical phenomena such as heat diffusion and wave propagation. Moreover, in the context of shape analysis, the Laplace–Beltrami operator has proven useful for a variety of applications due to the fact that it captures important geometric and topological information about the shape represented by the manifold in an isometrically invariant way. Solving the eigenvalue problem Df ¼ lf results in a sequence of eigenvalues l 1 , l 2 , ..., called spectrum of D and a sequence of eigenfunctions f 1 , f 2 , ... corresponding to the eigenvalues. The eigenfunctions have the useful property that they are orthonor- mal with respect to the L 2 inner product ðf , hÞ¼ Z M f ðxÞhðxÞ dM and form a basis for the corresponding Hilbert space of functions defined on M. For rectangular or spherical manifolds M we obtain the well-known Fourier bases and spherical harmonics, respectively. From this functional-analytic point of view, knowledge of the eigenvalues and eigenfunctions leads to an improved understanding of the Laplacian itself, since it becomes diagonal in the basis of its eigenfunctions. Intuitively this means that the action of D on a function f can be described by the action of an infinite diagonal matrix, the diagonal entries being the eigenvalues, on the infinite vector of coefficients describing f with respect to the eigenfunction basis, see e.g. [1]. The aforementioned decomposition of f provides a Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/cag Computers & Graphics 0097-8493/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cag.2012.03.027 $ If applicable, supplementary material from the author(s) will be available online after the conference. Please see http://dx.doi.org/10.1016/j.cag.2012.03. 027. n Corresponding author. E-mail addresses: vais@welfenlab.de, vais@gdv.uni-hannover.de (A. Vais), bberger@welfenlab.de (B. Berger), few@welfenlab.de (F.-E. Wolter). Computers & Graphics 36 (2012) 398–409