Mesh Parametrization and Its Applications

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Mesh Parametrization and Its Applications

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Computer Graphics

• Definition
• all technologies related to producing pictures or
  images using a computer
• Computer animation, VR(virtual reality),
• Goal Reality and Real time
• Reality
• Mapping(texture) / Rendering(light)

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Computer Graphics
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Computer Graphics
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Polygonal Objects(Mesh)
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Parametrization

• Embedding 3D mesh to 2D parameter space
• Requirements
• distortion minimization
• one-to-one mapping

(s,t)
v
parametrization in 2D
triangular mesh in 3D
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ParameterizationLevy
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Previous Work

• Energy functional minimization
• Green-Lagrange tensor Maillot93
• orthogonality and homogeneous spacing Lévy98
• Dirichlet energy Hormann99
• Convex combination approach
• shape-preserving parametrization Floater97
• harmonic embedding Eck95

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Convex Combination Approach (1)

• Convex combination and boundary condition
• determine shape of parameter space
• map boundary vertices onto a convex polygon
• determine coefficients for the inner vertices
• solve a linear system Ax b

1-ring neighborhood in parametric space
parameter space
3D mesh
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Convex Combination Approach (2)

• Benefit
• simple and fast, one-to-one embedding
• Drawback
• high distortions near the boundary

parameterization with fixed boundary
3D mesh
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Reducing Distortion near Boundary

• Floating boundary for the parameter space
• non-linear system Maillot93 Lévy98
  Hormann99
• linear system Lévy01
• heavy computation and/or non-one-to-one mapping

parameterization with floating boundary
3D mesh
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Motivation

• Extension of convex combination approach
• distortion minimization near the boundary
• simple and fast
• one-to-one mapping

3D mesh
floating boundary
fixed boundary
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Our Approach (1)

• Virtual boundary
• virtual vertices attached to the real boundary
• virtual boundary is fixed but real boundary can
  move to reduce the distortion in parameterization

virtual boundary
parametrization with virtual boundary
3D mesh
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Our Approach (2)

• Parametrization process

Compute coefficients ?i,j (inner vertices
boundary vertices)
Determine shape of parameter space (convex
polygon)
making virtual boundary
Map virtual vertices to the polygon
Solve linear system
parametrization
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Virtual Boundary

• Virtual vertices
• of virtual vertices 2 ? of real boundary
  vertices
• boundary vertex is adjacent to three virtual
  vertices
• no 3D positions are required for virtual vertices

virtual boundary
real boundary
connectivity of virtual vertices
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Coefficient Computation (1)

• Shape-preserving parametrization Floater97
• conformal mapping of 1-ring neighborhood
• average of barycentric coordinates

conformal mapping onto 2D
1-ring neighborhood in 3D
averaging barycentric coord.
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Coefficient Computation (2)

• Coefficients of real boundary vertices

map to 2D while preserving angles and lengths
place virtual vertices in 2D
1-ring neighborhood in 3D
1-ring neighborhood in 2D
1-ring neighborhood virtual vertices in 2D
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2D Positions of Virtual Vertices

• Mapping virtual vertices onto convex polygon
• using edge lengths between real boundary vertices

virtual boundary
real boundary
relation of real and virtual boundary
mapping virtual boundary
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Shape of Parameter Space

• Strong influence on the parameterization
• simple choices such as circle and rectangle?
• Convex hull of the projection of real boundary

circle
rectangle
convex polygon from projected boundary
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Extended Virtual Boundary (1)

• More virtual vertices in multi-layered structure
• to reduce distortions near the real boundary

3D mesh
parametrization
region far from the boundary
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Extended Virtual Boundary (2)

• Structure
• each layer has the same of virtual vertices
• Coefficients for virtual vertices

2nd virtual layer
1st virtual layer
real boundary
connectivity an coefficients of virtual vertices
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Extended Virtual Boundary (3)

• Effect for concave real boundary

3D mesh
one layer
no virtual vertices
two layers
three layers
four layers
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Results (1)
rectangle
circle
projected polygon

map the boundary
3D mesh
map virtual boundary
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Results (2)

• Texture mapping

3D mesh
circle, virtual boundary
projected polygon, virtual boundary
rectangle
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Applications(Texture)Levy
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Applications(Texture) )Levy
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Applications(Texture) )Levy
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Conclusion and Future Work

• Extension of convex combination approach
• distortion minimization near the boundary
• Virtual boundary
• fixed instead of the real boundary
• multi-layered structure
• Future work
• connectivity and coefficients of virtual vertices
• speed up with multilevel approach