Transcript Mesh Parameterization: Theory and Practice Non-Planar Domains Limitations of planar domains • so far … parameter domain = topological disk – one connected component – one.

Mesh Parameterization:
Theory and Practice
Non-Planar Domains
Limitations 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 Practice
Non-Planar Domains
seams
Texture atlases: distortion or seams?
distortion
Mesh Parameterization: Theory and Practice
Non-Planar Domains
Beyond 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 Practice
Non-Planar Domains
Generating base complexes
• surface triangulation of seed points
[Eck et al. 1995]
• successive simplification
[Lee et al. 1998]
Mesh Parameterization: Theory and Practice
Non-Planar Domains
Computing 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 Practice
Non-Planar Domains
Applications and limitations
• applications
–
–
–
–
remeshing
compression
surface fitting
morphing
• limitations
– not good for texture mapping
– where to store the color data?
Mesh Parameterization: Theory and Practice
Non-Planar Domains
Spherical 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
Mesh Parameterization: Theory and Practice
Non-Planar Domains
[Gotsman et al. 2003]
Alternatives
• successive simplification
Mesh Parameterization: Theory and Practice
Non-Planar Domains
[Shapiro & Tal 1998]
[Praun & Hoppe 2003]
Applications and limitations
• applications
– remeshing
[Praun & Hoppe 2003]
– compression, morphing, …
• cube maps
– texture mapping
• limitations
– only spherical meshes
Mesh Parameterization: Theory and Practice
Non-Planar Domains
Polycubes
• 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] Po·ly·cube: n. (Geom.) A solid
composed of multiple unit
– automatic [Lin et al. 2008]
cubes attached face to face
Mesh Parameterization: Theory and Practice
Non-Planar Domains
Polycube-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
Mesh Parameterization: Theory and Practice
Non-Planar Domains
MIPS
Area-MIPS
Texture mapping with Polycube-maps
a fragment
objectwith
space
interpolated
texture coord
not necessarily
on the
polycube
texture
space
(3D!)
surface: project
stored in texture RAM
w
final texel value
for the fragment
v
u
map to 2D
a packed texture image
mesh
polycube
Mesh Parameterization: Theory and Practice
Non-Planar Domains
plus a tiny structure
to store polycube layout
Summary
• non-planar domains
– base complexes
– spherical domains
– polycubes
• applications
– remeshing
– texture mapping
– morphing
Mesh Parameterization: Theory and Practice
Non-Planar Domains