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 –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 –Simple –Efficient Cons –Linear precision –Singularity: The weighted sum may be zero

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

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

12. De Casteljau Algorithm A procedure for evaluating a point on a Bezier curve t : 1-t P(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 (q i, a i, b i, q i+1 ) of i-th curve segment

17. Multi-Linear Method Slerp is not associative

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

19. Global Linearization

20. Pros –Easy to implement –Versatile Cons –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 –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