Rotational Symmetry Field Design on Surfaces

1
Rotational Symmetry Field Design on Surfaces

• Jonathan Palacios and Eugene Zhang
• Oregon State University

2
Introduction

• N-way Rotational Symmetry
• N-RoSy

2-way
3-way
4-way
6-way
2-RoSy
3-RoSy
4-RoSy
6-RoSy
Image courtesy of the CMU Near-Regular Texture
Database
3
Introduction

• Member vectors

Snoldelevs three interlaced horns
4
Introduction

• N-RoSy has N member vectors.
• When N1, it is a vector.
• When N2, it is a line segment.
• When N4, it is a cross.
• N-RoSy field is a continuous N-RoSy-valued
  function over the domain.

5
N-RoSy Fields and 3D Shapes

• Pen-and-ink sketching

line field (2-RoSy)
cross field (4-RoSy)
Image courtesy of Aaron Hertzmann and Dennis Zorin
6
N-RoSy Fields and 3D Shapes

• Quad remeshing

2-RoSy
4-RoSy
1-RoSy
Image courtesy of Nicolas Ray, Wan Chiu Li, Bruno
Lévy, Alla Sheffer, and Pierre Alliez
7
N-RoSy Fields and 3D Shapes

• Quad remeshing

Image courtesy of Nicolas Ray, Wan Chiu Li, Bruno
Lévy, Alla Sheffer, and Pierre Alliez
8
Our Goals

• N-RoSy field design
• Singularity and separatrix extraction.
• Control over the number and location of
  singularities.
• Control over field smoothness.
• Need to define arithmetic operations on N-RoSys.
• How do we add two N-RoSys?

9
Representation with Member Vectors

10
Representation with Member Vectors

11
Representation with Member Vectors

12
Representation with Member Vectors

• This is problematic for
• Interpolation.
• Field smoothing.
• Field blending.
• Singularity extraction.
• Separatrix computation.

13
Rotational Symmetry

• Video

14
Outline

• Related work
• Representation
• Analysis
• Design
• Applications
• Conclusion and future work

15
Outline

• Related work
• Representation
• Analysis
• Design
• Applications
• Conclusion and future work

16
Related Work

• Vector field analysis
• See Post et al. 2003.
• Vector field design
• See Zhang et al. 2006.
• Latest development
• Chen et al. 2007 (periodic orbit design).
• Fisher et al. 2007 (discrete exterior calculus)

17
Related Work

• Tensor field analysis
• Delmarcelle and Hesselink 1994.
• Tricoche et al. 2001.
• Alliez et al. 2003.
• Tensor field design
• Zhang et al. 2007.

18
Related Work

• 4-RoSy field analysis and design
• Hertzmann and Zorin 2000.
• Ray et al. 2006.
• Tong et al. 2006.

19
Related Work

• N-RoSy field (Ngt2) analysis and design
• Ray et al. 2006b.

20
Outline

• Related work
• Representation
• Analysis
• Design
• Applications
• Conclusion and future work

21
Representation

• Performing arithmetic operations on member
  vectors can cause inconsistencies.
• Vector fields cannot represent all the features
  that are in the N-RoSy fields (Ngt1).

22
Representation
?1
23
Representation
?1
?2
24
Representation
?1
?2
25
Representation
?1
?2
26
Representation

• N-RoSys are mapped onto a subset of symmetric
  N-th order tensors.
• Too many numbers!
• Fortunately, only two independent coefficients.

27
Representation
?
representation vector
28
Summation using Representation Vectors

29
Representation

• We perform design and most analysis on the
  representation vector field.

4-RoSy field
Representation vector field
30
Outline

• Related work
• Representation
• Analysis
• Design
• Applications
• Conclusion and future work

31
Analysis

• Singularities
• Vector field
• Jacobian
• Linearization

32
Analysis

• Singularities
• Poincarè index
• A topological measure of singularities.
• 1 for sources, sinks, centers.
• -1 for saddles.
• 0 for regular points.

33
Analysis
4-RoSy field
Representation vector fields
34
Analysis
1/4
2/4
3/4
1/3
2/3
35
Analysis Separatrices

• N is even

f
?
36
Analysis Separatrices

• 3-RoSy 1

37
Analysis Separatrices

• N is odd
• Outgoing
• Incoming

f
outgoing
?
incoming
38
Analysis

• Computational setup
• Vertex-based.
• Interpolation scheme
• Plane piecewise linear scheme (Tricoche et al.
  2001).
• Surface Zhang et al. 2006.

39
Analysis
40
Outline

• Related work
• Representation
• Analysis
• Design
• Applications
• Conclusion and future work

41
Design

• Two-stage pipeline
• Initialization
• Editing
• Relative straightforward adaptation from the
  vector field design system of Zhang et al.
  2006.
• Operations are performed using representation
  vector fields.

42
Design

• Singularity pair cancellation and movement.

original
after pair cancellation
after movement
43
Outline

• Related work
• Representation
• Analysis
• Design
• Applications
• Conclusion and future work

44
Applications

• Pen-and-ink sketching
• Most relevant previous work
• Hertzmann and Zorin 2000.
• Praun et al. 2001.
• Zhang et al. 2007.

45
Applications Pen-and-ink Sketching

• Topological editing original

46
Applications Pen-and-ink Sketching

• Topological editing after cancellation

47
Applications Pen-and-ink Sketching

• Topological editing after movement

48
Applications Pen-and-ink Sketching

• Topological editing vs. smoothing

Original
Edited
Smoothed
49
Applications

• Quad-remeshing
• Related work
• Alliez et al. 2003 .
• Marinov and Kobbelt 2004.
• Dong et al. 2005.
• Ray et al. 2006.
• Tong et al. 2006.
• Dong et al. 2006.

50
Applications Quad Remeshing
Original
Edited
Smoothed
51
Conclusion

• Contributions
• A coherent and compact representation.
• Analysis on N-RoSy fields including singularity
  and separatrix extraction.
• Interactive design system with control over
  number and location of singularity as well as the
  smoothness in the field.

52
Future Work

• Triangulation using 6-symmetry.
• Analysis of mixed symmetries and symmetry
  hierarchies.

53
Acknowledgement

• Models
• Marc Levoy and Stanford Graphics Group
• Cyberware
• AIM_at_SHAPE Shape Repository

• Discussions
• Greg Turk
• Video production
• Guoning Chen
• Patrick Neill
• Our reviewers
• Funding Agency
• NSF

54
Thank you!