Report

Discrete Laplace Operators for Polygonal Meshes Δ Marc Alexa TU Berlin Max Wardetzky U Göttingen Laplace Operators • Continuous – Symmetric, PSD, linearly precise, maximum principle • Discrete (weak form) – Cotan discretization [Pinkall/Polthier,Desbrun et al.] – Linearly precise, PSD, symmetric, NO maximum principle – No discrete Laplace = smooth Laplace [Wardetzky et al.] Geometry Processing • Smoothing / fairing [Desbrun et al. ’99] Geometry Processing • Smoothing / fairing • Parameterization [Gu/Yau ’03] Geometry Processing • Smoothing / fairing • Parameterization • Mesh editing [Sorkine et al. ’04] Geometry Processing • Smoothing / fairing • Parameterization • Mesh editing • Simulation [Bergou et al. ’06] Polygon meshes Polygon meshes Polygon • Polygons are not planar – Not clear what surface the boundary spans – Integration of basis function unclear / slow Laplace on Polygon Meshes Laplace on Polygon Meshes • Triangulating the polygons? Laplace on Polygon Meshes • Goal: ‘cotan-like’ operator for polygons – Symmetric (weak form) – Linearly precise – Positive semidefinite (positive energies) – Reduces to cotan on all-triangle mesh Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Triangle – cotan Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Triangle – cotan Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Triangles – Same plane Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Flat polygon Non-planar polygons Non-planar polygons • Vector area x2 x0 x1 0 Non-planar polygons • Properties of vector area – Projecting in direction yields largest planar polygon – Area is independent of choice of origin or orientation Non-planar polygons • Vector area gradient – Is in the plane of maximal projection – As before, orthogonal to – Simply use cross product with a Non-planar polygons e0 e1 b0 0 Non-planar polygons Non-planar polygons • Differences along oriented edges – “Co-boundary” operator Non-planar polygons Non-planar polygons Properties of • is symmetric by construction as • Consequently, L is symmetric Properties of • L is linearly precise Properties of • Is L PSD with only constants in kernel? – Co-boundary d behaves right – Kernel of may be too large – spans kernel of Main result • Laplace operator for any mesh – Symmetric, Linearly precise, PSD – Reduces to standard ‘cotan’ for triangles Implementation • Very simple! • For each face, compute – and – , – – from (differences, sums of coordinates) , (matrix products) (SVD) Implementation • Write M into large sparse matrix M1 – M1 has dimension halfedges × halfedges • Build the d-matrices – Have dimension halfedges × vertices • Then L = dT M1 d (weak form) – Strong form requires normalization by M0 Smoothing Parameterization Parameterization Parameterization Planarization • Planarization Planarization Conclusions / Future work • Laplace operator all meshes – Symmetric, PSD, linear precision – Reduces to cotan • Make non-planar part geometric