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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 operators for polygonal meshes

Discrete Laplace Operatorsfor Polygonal MeshesΔ

Marc Alexa Max Wardetzky

TU Berlin U Göttingen

laplace operators
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
Geometry Processing
  • Smoothing / fairing

[Desbrun et al. ’99]

geometry processing1
Geometry Processing
  • Smoothing / fairing
  • Parameterization

[Gu/Yau ’03]

geometry processing2
Geometry Processing
  • Smoothing / fairing
  • Parameterization
  • Mesh editing

[Sorkine et al. ’04]

geometry processing3
Geometry Processing
  • Smoothing / fairing
  • Parameterization
  • Mesh editing
  • Simulation

[Bergou et al. ’06]

polygon
Polygon
  • Polygons are not planar
    • Not clear what surface the boundary spans
    • Integration of basis function unclear / slow
laplace on polygon meshes1
Laplace on Polygon Meshes
  • Triangulating the polygons?
laplace on polygon meshes2
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 as Area Gradient
  • Laplace flow = area gradient [Desbrun et al.]
  • Triangle
    • cotan
laplace as area gradient1
Laplace as Area Gradient
  • Laplace flow = area gradient [Desbrun et al.]
  • Triangle
    • cotan
laplace as area gradient2
Laplace as Area Gradient
  • Laplace flow = area gradient [Desbrun et al.]
  • Triangles
    • Same plane
laplace as area gradient3
Laplace as Area Gradient
  • Laplace flow = area gradient [Desbrun et al.]
  • Flat polygon
non planar polygons1
Non-planar polygons
  • Vector area

x2

x1

x0

0

non planar polygons2
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 polygons3
Non-planar polygons
  • Vector area gradient
    • Is in the plane of maximalprojection
    • As before, orthogonal to
    • Simply use cross product with a
non planar polygons6
Non-planar polygons
  • Differences along oriented edges
    • “Co-boundary” operator
properties of
Properties of
  • is symmetric by construction as
  • Consequently, L is symmetric
properties of1
Properties of
  • L is linearly precise
properties of2
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
Main result
  • Laplace operator for any mesh
    • Symmetric, Linearly precise, PSD
    • Reduces to standard ‘cotan’ for triangles
implementation
Implementation
  • Very simple!
  • For each face, compute
    • and (differences, sums of coordinates)
    • , , (matrix products)
    • from (SVD)
implementation1
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
  • Planarization
conclusions future work
Conclusions / Future work
  • Laplace operator all meshes
    • Symmetric, PSD,linear precision
    • Reduces to cotan
  • Make non-planar part geometric