Siggraph Course Mesh Parameterization: Theory and Practice Barycentric Mappings.

Siggraph Course

Mesh Parameterization: Theory and Practice

Siggraph Course

Mesh Parameterization: Theory and Practice

Barycentric MappingsBarycentric Mappings

Triangle Mesh ParameterizationTriangle Mesh Parameterization

• triangle mesh

– vertices

– triangles

• parameter mesh

– parameter points

– parameter triangles

• parameterization

– piecewise linear map

The Spring ModelThe Spring Model

• replace edges by springs

• fix boundary vertices

• relaxation process

• energy of spring between and :

– spring constant

– spring length

• total energy

Energy MinimizationEnergy Minimization

• interior vertices

• ’s neighbours

• overall spring energy

• partial derivative

Energy MinimizationEnergy Minimization

• minimum of spring energy

for all interior points

• is a convex combination of its neighbors

with weights

The Linear SystemThe Linear System

• separation of variables

unknown parameter points fixed

• linear system

The Linear SystemThe Linear System

• solve system twice

for and coordinates of interior parameter points

• matrix is

– sparse

– diagonally dominant

– nonsingular

as long as all

Choice of WeightsChoice of Weights

• uniform spring constants

– ,

• chordal spring constants

– ,

• no fold-overs for convex boundary

• no linear reproduction

– planar meshes are distorted

Choice of WeightsChoice of Weights

• suppose is a planar mesh

• specify weights such that

• barycentric coordinates of

• then solving

reproduces

Barycentric CoordinatesBarycentric Coordinates

• Wachspress coordinates

• discrete harmonic coordinates

• mean value coordinates normalization

• fold-overs for negative coordinates

– affine combinations ,

• numerically unstable if

• mean value coordinates guaranteed to be positive

Example – PyramidExample – Pyramid

Wachspressdiscrete harmonicmean value

The Boundary MappingThe Boundary Mapping

• chordal parameterization around convex shape

– circle

– rectangle

• projection into least squares plane

– may lead to fold-overs