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

Rotation and Orientation:Affine Combination

Jehee Lee

Seoul National University

Applications

• What do we do with quaternions ?– Curve construction

• Keyframe animation

33221100 )()()()()( qqqqq tBtBtBtBt

Applications

• What do we do with quaternions ?– Filtering

• Convolution

kikikiki aaaF qqqq 0)(

iq

)( iF q

Applications

• What do we do with quaternions ?– Statistical analysis

• Mean

Applications

• What do we do with quaternions ?– Curve construction

• Keyframe animation– Filtering

• Convolution– Statistical analysis

• Mean

• It’s all about weighted sum !

Weighted Sum

• How to generalize slerp for n-points– Affine combination of n-points

• Methods– Re-normalization– Multi-linear– Global linearization – Functional Optimization

Inherent problem

• Weighted sum may have multiple solutions– Spherical structure– Antipodal equivalence

Re-normalization

• Expect result to be on the sphere– Weighed sum in R– Project onto the sphere

4

nn

nn

www

www

qqq

qqqq

1100

1100

Re-normalization

• Pros– Simple– Efficient

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

Multi-Linear Method

• Evaluate n-point weighted sum as a series of slerps

321 4

1

4

1

2

1pppc 321 4

1

4

1

2

1pppc

Slerp

Slerp

Multi-Linear Method

• Evaluate n-point weighted sum as a series of slerps

321 4

1

4

1

2

1pppc 321 4

1

4

1

2

1pppc

Slerp

Slerp

)),,,31(,4

1( 321 qqqq slerpslerpc

De Casteljau Algorithm

• A procedure for evaluating a point on a Bezier curve

t : 1-t

t : 1-t

t : 1-t

P(t)

Quaternion Bezier Curve

• Multi-linear construction– Replace linear interpolation by slerp– Shoemake (1985)

)log()1()1()1(

)log()0()0()0(

)1(

)0(

312

1

110

1

3

0

bbqq

bbqq

bq

bq

k

k

3b

1b 2b

0b

Quaternion Bezier Spline

• Find a smooth quaternion Bezier spline that interpolates given unit quaternions– Catmull-Rom’s derivative estimation

1iq

)log(2

1)( 1

11

iiit qq

1iq

iq

2iq

Quaternion Bezier Spline

• Find a smooth quaternion Bezier spline that interpolates given unit quaternions– Catmull-Rom’s derivative estimation

1iq

1iq

iq

2iq

)log(3

)log(2

1)(

1

111

ii

iiit

aq

qq

ia

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 segm

ent

1iq

1iq

iq

2iq

ia

ib

Multi-Linear Method

321 4

1

4

1

2

1pppc

321 4

1

4

1

2

1pppc

321 4

1

4

1

2

1pppc

Slerp is not associative

)),,,31(,4

1( 321 qqqq slerpslerpc )),,21(,,2

1( 321 qqqq slerpslerpc

Multi-Linear Method

• Pros– Simple, intuitive– Inherit good properties of slerp

• Cons– Need ordering

• Eg) De Casteljau algorithm– Algebraically complicated

Global Linearization

)logloglogexp( 1100 nnwww qqqq

1q

1log q

3q

2q

2log q 3log q

Global Linearization

• Pros– Easy to implement– Versatile

• Cons– Depends on the choice of the re

ference frame– Singularity near the antipole

Functional Optimization

• In vector spaces– We assume that this weighted sum was derived from

a certain energy function

nnwww pppc 2211

Functional Optimization

• In vector spaces

i

iiwf2

2

1)( ppp

0)( i

iiwfd

dppp

p

nnwww pppp 2211

Functional

Minimize

Weighted sum

Functional Optimization

• In orientation space– Buss and Fillmore (2001)

• Spherical distance

• Affine combination satisfies

i

iiwf 2,dist2

1)( qqq

)log(),dist( 21121 qqqq

)0,0,0()log()( 1 i

iiwf qqq

q

q

Functional Optimization

• Pros– Theoretically rigorous– Correct (?)

• Cons– Need numerical iterations (Newton-Rapson)– Slow

Summary

• Re-normalization– Practically useful for some applications

• Multi-linear method– Slerp ordering

• Global linearization– Well defined reference frame

• Functional optimization– Rigorous, correct

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