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Discrete Laplace Operators for Polygonal Meshes Δ. Marc Alexa Max Wardetzky TU Berlin U Göttingen. Laplace Operators. Continuous Symmetric, PSD, linearly precise, maximum principle Discrete (weak form) Cotan discretization [ Pinkall/Polthier,Desbrun et al.]
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Discrete Laplace Operatorsfor Polygonal MeshesΔ Marc Alexa Max Wardetzky TU Berlin U Göttingen
Laplace Operators • Continuous • Symmetric, PSD, linearly precise, maximum principle • Discrete (weak form) • Cotandiscretization [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 • Polygons are not planar • Not clear what surface the boundary spans • Integration of basis function unclear / slow
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 • Vector area x2 x1 x0 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 maximalprojection • As before, orthogonal to • Simply use cross product with a
Non-planar polygons e1 e0 b0 0
Non-planar polygons • Differences along oriented edges • “Co-boundary” operator
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 ofmay 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 (differences, sums of coordinates) • , , (matrix products) • from (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 M1d(weak form) • Strong form requires normalization by M0
Planarization • Planarization
Conclusions / Future work • Laplace operator all meshes • Symmetric, PSD,linear precision • Reduces to cotan • Make non-planar part geometric