Mesh Parameterization: Theory and Practice Non-Planar Domains.
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Transcript of Mesh Parameterization: Theory and Practice Non-Planar Domains.
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