1. You can check broken videos in this slide here : https://vimeo

2. Nearly-Rigid Deformation by Linear Optimization
SoHyeon Jeong
Korea University

3. Contents Introduction Deformation Algorithm Line Segment Handle
Topological Weight
Results
Conclusion

4. 1. Introduction

5. Goal Fast Nearly-rigid Deformation for 3D Shapes Goal Application
To deform 10,000s points in real-time
To preserve rigidity of a shape as far as possible
Goal Application
An interactive manipulation system

6. Related Work As-rigid-as-possible Deformation
Preserving global rigidity as far as possible by local rigid transformations
Triangle : [Igarashi 2005]
Rigid transformations
(Rotation & Translation)
Point: [Shaefer et al. 2006]

7. Motivation (1) 3D Nearly-rigid Deformation Key Issue
Based on Moving Least Squares[Schaefer et al. 2006]
Key Issue
Approximating a 3D rotation in a rigid transformation
Solved by
 More efficient approximation for plausible interactivity
Closed-form Solution
Eigenvector Problem 2D 3D
Fast
Computational costs

8. Motivation (2) + α ?? Regarding Topology Additional Constraints
Point handles only constrain the position
Regarding Topology
To deform topologically complex mesh models
(x, y z)
+ α ??

9. Contributions Improving Deformation Quality Improving
Nearly-rigid deformation
Improving
Deformation Quality
Nearly-rigid deformation
Reducing Computational Costs
Linear optimization
Reducing Computational Costs
Linear optimization
More Delicate Manipulation
Point&Line segment handles
More Delicate Manipulation
Point&Line segment handles
Considering Topological Features
Topological weight
Considering Topological Features
Topological weight

10. 2. Deformation Algorithm

11. Overview Nearly-rigid deformation Main Issue
Performing a rigid transformation for each point locally
Main Issue
Approximating each best rigid transformation of a point
Maximizing efficiency

12. Approximation (1) Moving Least Squares Approximation
The best transformation 2 1 2 1
minimizes
(1) 1
(2)
where
(3)

13. Approximation (2) Constraint : and are rigid transformations
Point handles
vector handles 1
Translations are eliminated
is only a 3D rotation 1
The 3x3 rotation matrix
minimizes
(4)

14. Linear Optimization (1)
Rotation Approximation
Eigenvector Problem
(4)
Linear Optimization
Replace R by log(R)
(5)
A rotation vector

15. Linear Optimization (2)
By the optimal rotation vector
(5) 2 2 1
can be obtained by 1
(6)
 Fast linear closed-form solution

16. 3. Line Segment Handle

17. Overview Line Segment Handle
Constrain position & rotation of point set explicitly
Twisting & bending
Movement of an articulated model

18. Line Segment Handle Idea
To find the most influence point on the line segment
To regard a line segment handle as a point v 1 2 3
Iso-surfaces of a line segment

19. Ellipsoidal Mapping a c V Vproj V pi V pi [ Closest Point ]

20. Line Segment Handles Equation
The rotation vectors : The movements of handles
Translation
Translation + Rotation

21. Bending & Twisting
bending
Twisting
Bending
Twisting

22. 4. Topological Weight

23. Overview Topological Weight Euclidean Weight
To integrate topology & connectivity of a mesh
Euclidean Weight
Consider only coordinates of points
Disregard topology and connectivity of a mesh
Not suitable for topologically complex models

24. Topological Weights Adjusting the Weight Function
Spatial distance  Topological distance of a shape
: topological shortest distance
on the mesh

25. Euclidean VS Topological
Euclidean Weight
Topological Weight

26. Euclidean VS Topological
Euclidean Weight
Topological Weight

27. 5. Results

28. Experimental Environment
Hardware
Intel Core2Quad 2.40 GHz
2G main memory
Implementation
C++
MFC/STL/OpenGL Libraries

29. Computation Times - Time Complexity : O(mn) # points #handles
#time (ms)
10,000 5
11.9
20,000
24.6
40,000
50.6
80,000
99.8 10
190.7 15
281.4
3,000,000
3273.0 x2 x2 x2 x2
- Time Complexity : O(mn)

30. Original
Affine
Rigid
Similarity

31. Euclidean Point Handles
Woman
points
- 11 point handles
- Euclidean Weight

32. Topological Point Handles
Dragon
25418 points
Topological Weight

33. Articulated Models
Horse
48485 points
19 Line Handles

34. Articulated Models Armadillo 172,974 points 19-line handles
2-point handles

35. 6. Conclusions

36. Conclusions 2D MLS Deformation Nearly-rigid deformation 3D Extension
+ Linear Optimization
Fast & interactive deformation
Additional Manipulation
+ Topological Weight
+ Line Segment Handle
Regarding topology
More delicate rotation control

37. Thank You!!