Discrete Laplace Operators for Polygonal Meshes Δ
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Transcript of Discrete Laplace Operators for Polygonal Meshes Δ
Discrete Laplace Operatorsfor Polygonal Meshes
ΔMarc 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.]
– 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
x0
x1
0
x2
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
e0
e1
0
b0
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 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 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