Spectral Mesh Simplification

Abstract

Figure 1: We propose to simplify a mesh using edge collapses while aiming to preserve the input eigenvectors and eigenvalues as much as possible. While different strategies exist to reduce a mesh (here, from 25,727 vertices to 771 vertices, or 3% of its initial size), such as enforcing uniform edge lengths or using the Quadric Error Metric [GH97], they do not focus on keeping the spectral properties of the mesh. Reducing a mesh can be spectrally described using functional maps [OBCS * 12], shown here with the output meshes, and which should ideally be diagonal. We also evaluate functional maps using two norms, the Laplacian commutativity • L and the orthogonality • D .

Chordal decomposition for spectral coarsening

“…We evaluate our solver by comparing against the existing stateof-the-art spectral coarsening [Liu et al 2019] and simplification [Lescoat et al 2020], using functional maps and the quantitative metrics ∥ • ∥ L and ∥ • ∥ D proposed in [Lescoat et al 2020]. We further demonstrate the power of our solver in controlling the sparsity patterns, approximating volumetric behavior using only boundary surface vertices and detaching the differential operator from the mesh.…”

“…Beyond the Laplace operator, Liu et al [2019] propose an algebraic approach to coarsen common geometric operators while preserving spectral properties. Lescoat et al [2020] extend the formulation to achieve spectral-preserving mesh simplification. Our approach is purely algebraic.…”

“…Our convex formulation leads us to have better spectral preservation compared to the similar algebraic approach [Liu et al 2019] in spectral coarsening. Our flexibility in controlling the sparsity allows us to post-process the results of spectral simplification [Lescoat et al 2020] and further improve its quality. In addition, we enable a novel application which independently optimizes the operator for computation purposes and the mesh vertices for preserving the appearance (see Fig.…”

“…Here M is the mass matrix in the coarse domain and R is a restriction operator, encoding the correspondences information from the original mesh to its coarsened counterpart. The restriction operator is computed either during the decimation [Lescoat et al 2020] or simply a subset selection matrix as in [Liu et al 2019]. One can also perceive the matrix C as an inner product matrix between the eigenvectors Φ on the coarsened domain and the restricted eigenvectors RΦ to the coarsened domain.…”

The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems

“…Schemes for the approximate solution of eigenproblems are static condensation [Bathe 2014] in engineering and the Nyström method [Williams and Seeger 2001] and random projections [Halko et al 2011] in machine learning. Approximation schemes for the Laplace-Beltrami eigenproblem on surfaces have been introduced in Chuang et al [2009], Lescoat et al [2020], Liu et al [2019], and Nasikun et al [2018]. In contrast to the eigensolvers we consider in this work, these schemes do not provide any guarantee on the approximation quality of the eigenpairs.…”