Discrete Laplace–Beltrami operators for shape analysis and segmentation

Abstract

Shape analysis plays a pivotal role in a large number of applications, ranging from traditional geometry processing to more recent 3D content management. In this scenario, spectral methods are extremely promising as they provide a natural library of tools for shape analysis, intrinsically defined by the shape itself. In particular, the eigenfunctions of the Laplace-Beltrami operator yield a set of real valued functions that provide interesting insights in the structure and morphology of the shape. In this paper, we first analyze different discretizations of the Laplace-Beltrami operator (geometric Laplacians, linear and cubic FEM operators) in terms of the correctness of their eigenfunctions with respect to the continuous case. We then present the family of segmentations induced by the nodal sets of the eigenfunctions, discussing its meaningfulness for shape understanding.

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

“…In [15] a very similar Laplace operator is used, however, constructed by only linear finite elements with a lumped (diagonal) mass matrix (see also [16] for a comparison of different Laplace Beltrami discretizations). Also a simple segmentation example is presented in [15] again based on k-means clustering of the spectral embedding of near isometric shapes.…”

“…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. Zero level sets, however, do not generally align to shape features, therefore we will not use them here.…”

“…-the smoothness of bounding curves of the segments, -the fact that no sub-segmentations are created if no geometric features are available there, -the high accuracy of the eigenfunctions via cubic FEM as opposed to many mesh Laplacians (see [16,32]) 1 , -the capability to do hierarchical pose invariant segmentations and part registrations at the same time, -the robustness to different poses, -the robustness with respect to mesh quality, density and noise -the detailed treatment of the difficulties (sign flips, switching) -and the detailed overview on the background material (FEM, Morse-Smale complex, size functions, separated persistence diagrams) and description of the implementation.…”

Anisotropic Laplace-Beltrami Operators for Shape Analysis

“…Computing the LB operator directly on meshes with approaches such as [18,15,20], numerous works exploit the information contained in its eigen-decomposition to build a new representation of a shape, either by explicit formulas [23,24,1] or by learning [13,22]. Based on these descriptors, several methods have been developed for segmentation purposes, for instance [14,21].…”

“…Combining (7), (14) and (16), the result (10) comes easily. Regarding the expression for the mass matrix A (9), we refer to [15,20]. In practice, we used the method described in [8] to compute the Voronoi areas.…”