Spectral Geometry Processing with Manifold Harmonics

Abstract

We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices.In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.

Hierarchical Shape Segmentation and Registration via Topological Features of Laplace-Beltrami Eigenfunctions

“…Note that [22] employs level sets of eigenfunctions to implictly construct correspondence of brain structures for statistical shape analysis. Other work employing spectral entities of the Laplace Beltrami operator includes retrieval [23,24], medical shape analysis [25][26][27]22], filtering/smoothing [28][29][30] of which specifically [28] mentions the use of zero level sets of specific eigenfunctions for mesh segmentation. This idea is later analyzed in more detail in [16], where also a comparison of different common discrete Laplace Beltrami operators is given.…”

Posture-invariant statistical shape analysis using Laplace operator

“…The Laplace operator has previously been used for mesh smoothing and regularization [12], for surface editing [13], and for 3D mesh fitting [14]. Furthermore, the spectrum of the Laplace operator has been used for mesh processing in a similar way to the Fourier transforms of images [15], and for shape matching and dissimilarity computation of volumetric data [16] and of triangle meshes [17]. Baran et al [18] used differential coordinates over local connected patches of a mesh to define a shape representation that is used to transfer mesh deformations from one character to another one while preserving the semantic meaning.…”

Height and Tilt Geometric Texture

“…For example, a few steps of mean curvature flow [16] provides a good vertex-to-vertex correspondence between the original textured surface and a smoother version, used as base shape. For more intricate geometries, a multiresolution smoothing strategy such as [17] or a spectral approach such as [18] are preferable (see Fig. 4, middle).…”