1. Rotation and Orientation: Affine Combination
Jehee Lee
Seoul National University

2. Applications What do we do with quaternions ? Curve construction
Keyframe animation

3. Applications
What do we do with quaternions ?
Filtering
Convolution

4. Applications What do we do with quaternions ? Statistical analysis
Mean

5. Applications What do we do with quaternions ?
Curve construction
Keyframe animation
Filtering
Convolution
Statistical analysis
Mean
It’s all about weighted sum !

6. Weighted Sum How to generalize slerp for n-points Methods
Affine combination of n-points
Methods
Re-normalization
Multi-linear
Global linearization
Functional Optimization

7. Inherent problem Weighted sum may have multiple solutions
Spherical structure
Antipodal equivalence

8. Re-normalization Expect result to be on the sphere Weighed sum in R
Project onto the sphere 4

9. Re-normalization Pros Cons Simple Efficient Linear precision
Singularity: The weighted sum may be zero

10. Multi-Linear Method
Evaluate n-point weighted sum as a series of slerps
Slerp
Slerp

11. Multi-Linear Method
Evaluate n-point weighted sum as a series of slerps
Slerp
Slerp

12. De Casteljau Algorithm
A procedure for evaluating a point on a Bezier curve
t : 1-t
P(t)
t : 1-t
t : 1-t

13. Quaternion Bezier Curve
Multi-linear construction
Replace linear interpolation by slerp
Shoemake (1985)

14. Quaternion Bezier Spline
Find a smooth quaternion Bezier spline that interpolates given unit quaternions
Catmull-Rom’s derivative estimation

15. Quaternion Bezier Spline
Find a smooth quaternion Bezier spline that interpolates given unit quaternions
Catmull-Rom’s derivative estimation

16. Quaternion Bezier Spline
Find a smooth quaternion Bezier spline that interpolates given unit quaternions
Catmull-Rom’s derivative estimation
Bezier control points (qi, ai, bi, qi+1) of i-th curve segment

17. Slerp is not associative
Multi-Linear Method
Slerp is not associative

18. Multi-Linear Method Pros Cons Simple, intuitive
Inherit good properties of slerp
Cons
Need ordering
Eg) De Casteljau algorithm
Algebraically complicated

19. Global Linearization

20. Global Linearization Pros Cons Easy to implement Versatile
Depends on the choice of the reference frame
Singularity near the antipole

21. Functional Optimization
In vector spaces
We assume that this weighted sum was derived from a certain energy function

22. Functional Optimization
In vector spaces
Functional
Minimize
Weighted sum

23. Functional Optimization
In orientation space
Buss and Fillmore (2001)
Spherical distance
Affine combination satisfies

24. Functional Optimization
Pros
Theoretically rigorous
Correct (?)
Cons
Need numerical iterations (Newton-Rapson)
Slow

25. Summary Re-normalization Multi-linear method Global linearization
Practically useful for some applications
Multi-linear method
Slerp ordering
Global linearization
Well defined reference frame
Functional optimization
Rigorous, correct

26. Summary We don’t have an ultimate solution
An appropriate solution may be determined by application
More specific problems may have better solutions
For convolution filters, points have an ordering