2017/07/21 by Yu Wang, Mirela Ben‐Chen, Wang, Yu +6 · 2 citations
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #Advanced Numerical Analysis Techniques #Boundary (topology) #Boundary value problem #Computer Graphics and Visualization Techniques #Dirichlet distribution #Discretization #Eigenvalues and eigenvectors #Geometry #Geometry processing #Laplace operator #Mathematical analysis #Mathematics #Neumann boundary condition #Operator (biology) #Physics #Polygon mesh #Spectral geometry #cs.GR
paper · pdf · doi:10.48550/arxiv.1707.07070
published in arXiv (Cornell University) (Cornell University) · Additional experiments added
openalex publication_date 2017/07/21 · arxiv created 2018/04/24 · arxiv updated 2018/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.