Manifold Parameterization

1
Manifold Parameterization

• Lei Zhang, Ligang Liu, Zhongping Ji, Guojin Wang
• Department of Mathematics
• Zhejiang University
• Accepted as regular paper by CGI2006

2
Overview

• Parameterization
• Least-squares Mesh
• Manifold Parameterization
• Similar destination, different way
• Similar way, different destination

3
Reference

• O. Sorkine and D. Cohen-Or. Least-squares meshes.
  In Proceedings of Shape Modeling International,
  2004.
• Lei Zhang, Ligang Liu, Zhongping Ji and Guojin
  Wang. Manifold Parameterization. Accepted as
  regular paper by Computer Graphics International,
  2006.
• V. Kraevoy and A. Sheffer. Cross-Parameterization
  and Compatible Remeshing of 3D Models. SIGGRAPH,
  2004.

4
Reference

• M. Paone and Andrew Yuen. Mesh Fitting to Points.
  Projects Presentations (PPT), Simon Fraser
  University, Canada.
• K. D. Cheng, W. P. Wang, H. Qin, K. K. Wong, H.
  P. Yang and Y. Liu. Fitting Subdivision Surfaces
  to Unorganized Point Data Using SDM. Proceedings
  of the 12th Pacific Conference on Computer
  Graphics and Applications, 2004.

5
Overview

• Parameterization
• Least-squares Mesh
• Manifold Parameterization
• Similar destination, different way
• Similar way, different destination

6
Parameterization

• Concept
• Parameterization is a one-to-one mapping from a
  triangular mesh surface onto a suitable domain.

Domain
Mesh
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Planar Parameterization

• Select a plane as the parameterization domain for
  an open mesh

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Spherical Parameterization

• Select a sphere as the parameterization domain
  for 0-genus mesh

E. Praun and H. Hoppe, SIGGRAPH 04
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Manifold Parameterization

• Select a surface as parameterization domain for
  another surface

Domain
Mesh
10
Manifold Parameterization
11
Overview

• Parameterization
• Least-squares Mesh
• Manifold Parameterization
• Similar destination, different way
• Similar way, different destination

12
Least-squares Meshes

• O. Sorkine, D. Cohen-Or
• Tel Aviv University
• Proceeding of Shape Modeling International 2004

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Introduction

• Mesh

?
?
Connectivity
Mesh surface
Geometry


14
Introduction

• Least-squares mesh
• Using a set of control points, approximate the
  original mesh surface by its connectivity graph.

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Introduction
19851 vertices
200 control points
1000 control points
3000 control points
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Least-squares meshes

• Vertex conditions
• -Smooth condition L(vi)0, vi all vertices
• -Geometry condition vjcj, cj constraint
• L-Laplacian operator

Vi
Vj
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Least-squares meshes

• Laplacian Equation
• Smooth condition
• Geometry condition

vjcj
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Least-squares meshes

• Example

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Least-squares meshes

• Equation Solution
• The system is solved in least-squares sense.

A is sparse, and equation can be solved by TAUCS
library quite fast.
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Weighted Least-squares meshes

• Higher weights for control points

constraints
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Weighted Least-squares meshes
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Overview

• Parameterization
• Least-squares Mesh
• Manifold Parameterization
• Similar destination, different way
• Similar way, different destination

23
Overview

• Parameterization
• Least-squares Mesh
• Manifold Parameterization
• Similar destination, different way
• Similar way, different destination

24
Cross-Parameterization and Compatible Remeshing
of 3D Models

• V. Kraevoy and A. Sheffer
• SIGGRAPH 2004

25
Introduction

• Given two mesh M1 and M2, obtain correspondence
  via base meshes.

f1 F f2-1
M2
M1
f1
f2
F
B1
B2
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Main Steps

• Construct topologically identical path layouts

• No interior intersection
• Cyclical order

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Main Steps

• Get topologically identical base mesh

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Main Steps

• Map patch layout to base mesh

Mean value parameterization
f1
f2
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Main Steps

• Construct mapping between base mesh

F
Barycentric coordinate
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Main Steps

• Result parameterization

f1 F f2-1
M2
M1
f1
f2
F
B1
B2
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Examples
32
Conclusion

• Indirect
• Boring path layout searching
• Time-consuming

33
Overview

• Parameterization
• Least-squares Mesh
• Manifold Parameterization
• Similar destination, different way
• Similar way, different destination

34
Mesh Fitting to Points

• M. Paone and A. Yuen
• Supervisor Richard (Hao) Zhang
• Simon Fraser University, Canada
• Project Report

35
Fitting Subdivision Surfaces to Unorganized Point
Data Using SDM

• K. D. Cheng, W.P. Wang, H. Qin, K. K. Wong, H. P.
  Yang and Y. Liu
• PG 04

36
Introduction

• Reconstruction of smooth surface from point
  clouds
• Tool Loop subdivision surface
• Measure SD (Squared Distance)
• H. Pottmann and M. Hofer. Geometry of the
  Squared Distance Function to Curves and Surfaces.
  Visualization and Mathematics III, Springer,
  2003.

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Loop Subdivision
Edge-Vertex
Vertex-Vertex
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Squared Distance
39
Main Steps
Normalization
Target data points are scaled to .
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Main Steps
Normalization
Pre-computation
Compute distance field and curvatures at all data
points.
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Main Steps
Normalization
Pre-computation
Initial mesh
Use Marching Cubes to obtain an initial control
mesh.
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Main Steps
Normalization
Pre-computation
Initial mesh
Sampling
Get sample points on limit surface.
J. Stam. Evaluation of Loop Subdivision Surfaces.
SIGGRAPH99, course.
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Main Steps
Normalization
Pre-computation
Initial mesh
Sampling
Optimization
SDM error function
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Main Steps
Normalization
Pre-computation
Initial mesh
Optimization
Error evaluation
Sampling
Maximum approximation error
Average error
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Main Steps
Normalization
Pre-computation
Initial mesh
Optimization
Error evaluation
Sampling
Insert new control points to regions of large
errors.
46
Examples
47
Thank You