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