Rigid Geometric Transformations COMPSCI 527 Computer Vision - - PowerPoint PPT Presentation

Rigid Geometric Transformations

COMPSCI 527 — Computer Vision

Outline

1 Motivation 2 Projection 3 Cross Product 4 Triple Product 5 Rotations 6 Rigid Transformations

Motivation

• The relative motion between a camera and an otherwise

static scene is a rigid transformation

• It underlies 3D reconstruction
• A rigid transformation is rotation + translation
• Understanding rotation requires orthogonality and

projection, which we have seen in general

• All vectors in R3
• Reconstruction techniques also require knowing about

cross product and triple product

• Read notes for details

Projection

• Definition of projection of a onto b: the point p on the line

through b that is closest to a

• p is on the line through b:

p = xb for some x

• p is closest to a when (a, p) is orthogonal to b:

bT (a xb) = 0, which yields x = bT a

bT b so that

p = xb = b x = bbT

bT b a

BI

bib

pi

O

c IR

tEf

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MAGNITUDE

• f F

The Projection Matrix

• p = Pba where Pb = bbT

bT b a

• Pb is rank 1, symmetric, and idempotent: P2

b = Pb

• Proof:
• Norm squared of p:

kpk2 =

• When kbk = 1,

Pj

IT

EIS Ps

btp

at

Pb Pba

IP

I

at bgbb a

EIJI

p

sign Eb

Pb

bBT

latte

1lb

The Cross Product

• Geometry: The cross product of two three-dimensional

vectors a and b is a vector c orthogonal to both a and b,

• riented so that the triple a, b, c is right-handed, and with

magnitude kck = ka ⇥ bk = kak kbk sin θ where θ is the smaller angle between a and b

• The magnitude of a ⇥ b is the area of a rectangle with sides

a and b

• Algebra: c = a ⇥ b =
• ax

ay az bx by bz

• = (aybz azby , azbx axbz , axby aybx)T
• Easy to check that a ⇥ b = b ⇥ a

tea

O

ETI UEI LIE

The Cross-Product Matrix

• c = (aybz azby , azbx axbz , axby aybx)T is linear in b
• Therefore, there exists a 3 ⇥ 3 matrix [a]× such that

c = a ⇥ b = [a]×b c = 2 4 cx cy cz 3 5 = 2 4 3 5 2 4 bx by bz 3 5

• The matrix [a]× is skew-symmetric:

[a]T

× = [a]×

a

ay Qx

O

The Triple Product

• Definition: det([a, b, c]) = aT(b ⇥ c)

= ax(bycz bzcy) ay(bxcz bzcx) + az(bxcy bycx)

• Easy to check: aT(b ⇥ c) = bT(c ⇥ a) = cT(a ⇥ b) =

aT(c ⇥ b) = cT(b ⇥ a) = bT(a ⇥ c)

• Signed volume of parallelepiped

E

ta 5

d

g

• i BE

Multiple Reference Frames

• If we associate a reference system to a camera and the

camera moves, or we consider multiple cameras, or we consider one camera and the world, we have multiple reference systems

• Point coordinates are x, y, z
• Left superscript denotes which reference system

coordinates are expressed in: 1y

• Subscripts denote which point or reference system we are

talking about: x2

• 2y3 is the y coordinate of point 3 in reference system 2

Multiple Reference Frames

• A zero left superscript can be omitted: 0z = z
• The origin of a reference system is t (for “translation”)
• We always have iti =

2 4 3 5

• If i, j, k are the unit points of a reference system, we always

have ⇥ iii

iji iki

⇤ = I , the 3 ⇥ 3 identity matrix

if

t

O

se

Rotations

• No translation: 0t1 = t1 =

2 4 3 5

• Both systems right-handed
• i1, j1, k1 are the unit vectors of reference system 1

expressed in reference system 0

• Given p =

0p, what is 1p?

f

O

Rotations

x y z

i x

j y

k1 z

p

p = 1x i1 + 1y j1 + 1z k1

1x = iT 1 p

,

1y = jT 1 p

,

1z = kT 1 p 1p =

2 4

1x 1y 1z

3 5 = 2 4 iT

1 p

jT

1 p

kT

1 p

3 5 = R1 p where R1 = 0R1 = 2 4 iT

1

jT

1

kT

1

3 5 (unit vectors are the rows)

000

O

Rotations in General

• More generally,

bp = aRbap

where

aRb =

2 4

aiT b ajT b akT b

3 5

• Rotations are reversible, so there exists bRa = aR−1

b

• bRa = aRT

b

• Intuition: aRb =

2 4 r11 r12 r13 r21 r22 r23 r31 r32 r33 3 5 = 2 4

aiT b ajT b akT b

3 5

• rmn is the signed magnitude of the projection of the m-th

vector in Sb onto the n-th vector in Sa

• Therefore, vice versa (direction cosines)
• Therefore, what we want in the rows of R−1 is in the

columns of R

O

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Properties of Rotation

• det(R) = iT(j ⇥ k) = iTi = 1

because i ⇥ j = k , j ⇥ k = i , k ⇥ i = j

• Cross-product is covariant with rotations:

(Ra) ⇥ (Rb) = R (a ⇥ b)

e

Coordinate Transformation

• A.k.a. rigid transformation
• First translate, then rotate:

1p = R1(p t1)

• Inverse: p = RT

1 1p + t1

• Generally, if bp = aRb(ap atb) then ap = bRa(bp bta)

where bRa = aRT

b and bta = aRbatb

bp

lap

R

ftp.bR.bpbta

Y p

p t

• bpatap

bp

btabp

l2aTeptbta

bR

fptbReOD0ooDIIi.I

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