1 siggraph 2006, 7/31/2006 cmg/ triangular manifold splines xianfeng david gu, ying he, hong qin smi...
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1 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Triangular Manifold Splines
Xianfeng David Gu, Ying He, Hong Qin
SMI 2005, “Manifold Splines”GMP 2006, “Manifold T-Splines” (Kexiang Wang, Hongyu Wang)
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2 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Specification
•Input: reasonably dense, manifold triangular mesh•Construct a global affine parameterization
• Affine transformations between groups of triangles
• (Except for a few points)•Construct charts on groups of triangles
• Transition functions trivially affine•Embedding functions are triangular splines
• Splines with triangular mesh as “knot vector”
• “Fix” holes by patching
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3 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Global parameterization
•Conformal parameterization• Checkerboard, except for octagon
•Cut by removing center point•Can now “unfold” locally into plane
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4 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Embedding functions
•Triangular splines• Built on 2D triangular mesh• Affine invariant
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5 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Avoiding blend functions
•Affine invariant embedding functions + affine transition functions means chart functions agree where they overlap
• Use same control points
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6 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Using T-splines, Powell-Sabin splines
•Can also use T-splines for embedding function• Globally parameterize to a square
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7 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Advantages
•Simple, affine transformations for transition functions•Ck
•Triangular splines can handle sharp features
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8 Siggraph 2006, 7/31/2006www.cs.wustl.edu/~cmg/
Disadvantages
•Need to fix holes in the parameterization•Triangular splines require optimization
• Also expensive to compute•Limited control over parameterization