Mesh Parameterization:Theory and Practice

Mesh Parameterization:Theory and Practice

Non-Planar DomainsNon-Planar Domains

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Limitations of planar domainsLimitations of planar domains

• so far … parameter domain = topological disk– one connected component– one boundary

• parameterization bijective⇒ surface = topological disk

• what about other surfaces?

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Texture atlases: distortion or seams?Texture atlases: distortion or seams?seams

disto

rtion

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Beyond planar domainsBeyond planar domains

• alternative: adapt the parameter domain– same topology as the mesh

• base complexes– simplified triangle mesh

• spherical domains– limited to genus-zero meshes

• polycubes– quadrilateral domain elements

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Generating base complexesGenerating base complexes

• surface triangulation of seed points

[Eck et al. 1995]

• successive simplification

[Lee et al. 1998]

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Computing the parameterizationComputing the parameterization

• initial parameterization– parameter points for mesh vertices– inherit correspondences during simplification– piecewise linear map per mesh triangle

• optimization– Loop smoothing– global minimization of distortion

with transition functions[Khodakovsky et al. 2003]

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Applications and limitationsApplications and limitations

• applications– remeshing– compression– surface fitting– morphing

• limitations– not good for texture mapping– where to store the color data?

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Spherical parameterizationsSpherical parameterizations

• projected Gauss-Seidel iterations [Kobbelt et al. 1999]

– project all points onto sphere– compute barycentric average– reproject onto sphere

• problems– does not guarantee bijectivity– diverges close to solution [Saba et al. 2005]

• solution– spherical barycentric coordinates [Gotsman et al. 2003]

Mesh Parameterization: Theory and PracticeNon-Planar Domains

AlternativesAlternatives

• successive simplification

[Shapiro & Tal 1998]

[Praun & Hoppe 2003]

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Applications and limitationsApplications and limitations

• applications– remeshing

[Praun & Hoppe 2003]

– compression, morphing, …

• cube maps– texture mapping

• limitations– only spherical meshes

Mesh Parameterization: Theory and PracticeNon-Planar Domains

PolycubesPolycubes

Po·ly·cube: n. (Geom.) A solid composed of multiple unit cubes attached face to face

• polycubes as parameter domains [Tarini et al. 2004]

– square domain elements– matching topology– similar coarse shape– not too many elements

• construction– interactively [Tarini et al. 2004]

– automatic [Lin et al. 2008]

Mesh Parameterization: Theory and PracticeNon-Planar Domains

Area-MIPS

Polycube-mapsPolycube-maps

• computing the parameterization– initial projection onto the polycube– global optimization (Gauss-Seidel iterations)

• applications– quadrilateral remeshing– texture mapping– shading textures– level-of-detail rendering

projection MIPS

Mesh Parameterization: Theory and PracticeNon-Planar Domains

stored in texture RAMtexture space (3D!)object space

Texture mapping with Polycube-mapsTexture mapping with Polycube-maps

a packed texture image

polycubeplus a tiny structure

to store polycube layoutmesh

uv

w

map to 2D

a fragmentwith interpolated

texture coord

final texel valuefor the fragment

not necessarilyon the polycubesurface: project

Mesh Parameterization: Theory and PracticeNon-Planar Domains

SummarySummary

• non-planar domains– base complexes

– spherical domains

– polycubes

• applications– remeshing– texture mapping– morphing