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