rotation and orientation: affine combination jehee lee seoul national university
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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
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