1

2

Motivation (Horn87)

• Registration of 3D Shapes

Assume correspondence

has been determined…

n

iii

bRqbRp

1

2

,min

n

iii

bRqbRp

1

2

,min

Any other better solutions for this least-square problem?

3

Exercise

• Configuration A– p1 = (3,0,0)

– p2 = (0,3,0)

– p3 = (0,0,3)

• Configuration B– q1 = (3,3,-3)

– q2 = (3,0,0)

– q3 = (6,0,-3)

Find the rotation and translation that takes configuration A to configuration B

Solution

4

010

001

100

030

003

300

333

000

333

303

003

633

300

030

003

3

0

3

1

1

1

2

1

4

303

003

633

300

030

003

R

R

d

dddR

This is a badly chosen example, where the three points are linearly independent.

If this is not so (such as three coplanar points), this simple approach cannot be used.

5

Recap in CG Course (link)

• New algebra defined (addition, multiplication)• Length; unit quaternion as rotation• Perform rotations• Converting to/from rotation matrices• slerp

33221100cos

sin

sin

sin

)1(sin)(

rqrqrqrq

rt

qt

ts

q

r

)(22)(

...

020

*

xqqxqqxqqq

qxqx

****

*

)()( qpxqpqpxpqqxqx

pxpx

nqn ˆsincos),ˆ(Rot 22

6

History• Quaternions were introduced by Irish mathematician Sir

William Rowan Hamilton in 1843. Hamilton was looking for ways of extending complex numbers to higher spatial dimensions. He could not do so for 3 dimensions, but 4 dimensions produce quaternions.

• According to the story Hamilton told, on October 16, he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge

7

Definitions1

Addition

Multiplication

8

Definitions2

qppq

Complex conjugate

Norm (of quaternion)

9

Alternative Formulation

A quaternion can be conveniently thought of as either:

• A vector with four components;

• A scalar plus a vector with three components; or

• A complex number with three different “imaginary” parts

10

Product in Matrix Form (Horn)

Expand first quaternion Expand second quaternionNote the matrices are different!

“R is orthogonal”!

RRrIrRR

rrrr

rrrr

rrrr

rrrr

R

rrrr

rrrr

rrrr

rrrr

R

TT

xyz

xzy

yzx

zyx

T

xyz

xzy

yzx

zyx

0

0

0

0

0

0

0

0

,

TRR *

Treat quaternion as a vector in R4Notation: we use to denote quaternion

This is the “matrix way” to compute

quaternion product

:R R:

11

Details

0

0

0

0

0

0

0

0

*

0*

0

0

0

0

0

0

0

0

0

)()()(

)()()(

)()()(

)()()(

,,,

,,,

rrrr

rrrr

rrrr

rrrr

rrrr

rrrr

rrrr

rrrr

R

rrrrr

rrrr

rrrr

rrrr

rrrr

R

rrrr

rrrr

rrrr

rrrr

R

rrrrr

xyz

xzy

yzx

zyx

xyz

xzy

yzx

zyx

zyx

xyz

xzy

yzx

zyx

T

xyz

xzy

yzx

zyx

zyx

TRR *

12

Dot Product of Quaternions

qppq

This is new!

Def:

4D vector viewpoint

*** rqprqprQp

rQprpQrpQrpqTT

TTT

p.9

13

1. Length preserved after operation

Unit Quaternion as Rotation1

Observations: (why Lq(v) is a rotation…)

2. If v is along q, it is left unchanged.

[Horn] dot product and triple product preserved …

14

Unit Quaternion as Rotation2

Lq is linear over R3

Lq(v) = Lq(n+a) = Lq(n) + Lq(a) = Lq(n) + a

v=Lp(u) , w=Lq(v)w= Lq(v)=qvq*=q(pup*)q*=(qp)u(qp)*=Lqp(u)

v

a

nqu

15

Application: Data Registration

Iterative (least square) solutionor Closed form solution

Find the most appropriate rotation R and translation b

16

0

Similarly,

01

;

1 ,

1 centroid

1

11111

11

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

iiii

n

ii

n

ii

q

pn

nppnpppp

qqqppp

qn

qpn

p

Convert global coordinates to local coordinate (from centroid)

17

Objective function (minimization)

First, determine the rotation R, then the optimal translation by:

Optimal rotation R: 2

22

2

2

2

wwvv

wwwvvv

wvwvwv

18

Convert to quaternion:

is symmetric, with real eigenvalues 1, 2, 3, 4 and corresponding orthogonal unit eigenvectors v1, v2, v3, v4 in R4

This correspond to R and R

in p.8

19

11rotation is 2

423

22

21

24

23

22

214433221144332211

qq

vvvvvvvvqq

1: largest eigenvalue

Simplified to: MqqT

qmax

1

4321

1241

231

221

211

244

233

222

211

0,1 :at occurs maximum

vq

MqqT

Therefore,

Represent q in eigen basis

The optimum rotation is the eigenvector with largest eigenvalue. (It is already normalize).

20

Details

332211322313312112

322333221112213113

133112213322113223

211231132332332211

123

132

231

321

123

132

231

321

0

0

0

0

0

0

0

0

qpqpqpqpqpqpqpqpqp

qpqpqpqpqpqpqpqpqp

qpqpqpqpqpqpqpqpqp

qpqpqpqpqpqpqpqpqp

QP

qqq

qqq

qqq

qqq

Q

ppp

ppp

ppp

ppp

P

iT

i

iT

i

symmetricQP iT

i is

21

The Real Exercise

• Configuration A– p1 = (3,0,0)

– p2 = (0,3,0)

– p3 = (3,3,0)

• Configuration B– q1 = (1,4,1)

– q2 = (1,1,4)

– q3 = (1,4,4)

Find the rotation and translation that takes configuration A to configuration B

Quaternion in SVL

22