multi view geometry (spring '08)ocw.snu.ac.kr/sites/default/files/note/2765.pdf ·...

1

Projective 3D Geometry 1

2008-1

Multi View Geometry (Spring '08)

Prof. Kyoung Mu LeeSoEECS, Seoul National University

Projective 3D Geometry


Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU

Points, Planes, Lines and Quadrics in 3D

• 3D points:

• Projective transform in :

CollinearLines are mapped to lines15 (4x4-1) DOF

0,)1,,,(

),,,(

44

3

4

2

4

1

34321

≠⇔

=

XXX

XX

XX

inXXXX

T

T PX3

4

3

4

2

4

1 ),,( RinXX

XX

XX T

XHX 44×=′

Homogeneous representation Inhomogeneous representation

3P

2



Planes

• 3D planes:

0ππππ 4321 =+++ ZYX

0ππππ 44332211 =+++ XXXX4

3

4

2

4

1 ,,XXZ

XXY

XXX ===

: homogeneous planeT),,,( 4321 ππππ=π

XX' H=ππ' -TH=

Transformation

0Xπ =T

TXXXX ),,,( 4321=X : homogeneous 3D point



• Euclidean representation:

• Duality:

0~. =+ dXn

( )T321 π,π,π=n

( )TZYX ,,~ =X

4π=d

n/d

points ↔ planeslines ↔ lines

: plane normal

14 =X

3



Determination of a plane by Points

• Three points on a plane satisfies

• Or, using the coplanarity constraint;

2341DX( )T123124134234 ,,,π DDDD −−=

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

0det

4342414

3332313

2322212

1312111

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

XXXXXXXXXXXXXXXX

[ ] 0 det 321 =XXXX

124313422341 DXDXDX +− 01234124313422341 =−+− DXDXDXDX 13422341 DXDX −

0

3

2

1

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

πXXX

T

T

T

π is the null space of ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

T

T

3

2

1

XXX

πX on ∀



Determination of a plane Points

4



Points from Planes

• Three planes meets at a point:

• A plane can be represented by its span:

0πππ

3

2

1

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

XT

T

T

X⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

T

T

3

2

1

πππ

MxX =

[ ]32134 XXX=xM

0M =Tπ

⎥⎦

⎤⎢⎣

⎡=

33

then xI

pM( )Tdcba ,,,π if = ⎟

⎠⎞

⎜⎝⎛= −−−

ad

ac

ab

,, and p

M: 3dim null-space of π

is the null space of



Lines – Null-space and span representation

• The line joining A and B is the span of the row space of W.

• The intersecting line of two planes P and Q is the span of the row space of W*.

BA μλ +(4dof)

⎥⎦

⎤⎢⎣

⎡= T

T

BA

W

⎥⎦

⎤⎢⎣

⎡= T

T

QP

W* QP μλ ′+′

5



Lines – Null-space and span representation

•2 2

T T∗ ∗×= =W W WW 0



Points, Lines and Planes

• A plane by a point X and a line W:

• A point X by a line W and a plane :

⎥⎦

⎤⎢⎣

⎡= TX

WM 0π =M

⎥⎦

⎤⎢⎣

⎡= Tπ

WM

*

0=MX

*W

π

If the dim N(M) = 2, then X is on W

If the dim N(M) = 2, then W is on π

W

X π

π

π

6



Lines - Plücker matrix representation

• The line joining two points can be represented by the Plücker matrix

jijiij ABBAl −= TT BAABL −=

i) L is a 4x4 skew-symmetric homogeneous matrixii) L has rank 2iii) 4 DOFiv) generalization ofv) L is independent of choice A and Bvi) Transformation

24*

×= 0LW T

yxl ×=

THLHL ='

[ ] [ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0001000000001000

1000

0001

0001

1000

TL

Ex) x-axis



Dual Plücker Matrix

• Dual Plücker matrix:

• Transformation:

• Correspondence:

• Join and Incidence

1' −∗−∗ = HLHL T

TT QPPQL −=*

*12

*13

*14

*23

*42

*34344223141312 :::::::::: llllllllllll =

XLπ *=

LπX =

0* =XL

: plane through point and line

: point on line

: intersection point of plane and line

: line in plane0=Lπ

[ ] 0,, 21 =πLL K : coplanar lines

7



Lines - Plücker line coordinates representation

• Plücker line coordinates:• From det L =0

[ ]T,l,l,l,l,ll 344223141312=L 5P∈0231442133412 =++ llllll

Klein quadric constraint

0=⇔ LL KT

[ ] 0

000001000010000100001000010000100000

34

42

23

14

13

12

344223141312 =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

llllll

llllll

( ) ( )B,A,A,B, ˆˆˆ ↔LL

[ ]( )LLLL ˆ|Kˆ

ˆˆˆˆˆˆB,AB,A,det 123413421423231442133412

==

+++++=T

llllllllllll

two lines are intersect iff the 4 points are coplanar



Lines - Plücker line coordinates representation

• Results:

( ) [ ] 0ˆˆdetˆ == B,AA,B,|LL

( ) [ ] 0ˆˆdetˆ == Q,PP,Q,|LL

( ) ( )( ) ( )( ) 0ˆ =−= BPAQBQAP| TTTTLL

( ) 0=LL|

i) Plücker internal constraint:

ii) two lines intersect or coplanar:

4 points

4 planes

2 planes and 2 points

8



Quadrics and dual quadrics

• Quadric:

• Dual quadric:

i) Q is a 4x4 symmetric matrixii) 9 DOFiii) in general 9 points define quadriciv) det Q=0 ↔ degenerate quadricv) pole – polar vi) (plane ∩ quadric)=conicvii) transformation

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

••••••••••

=

ooo

oo

oQ

Xπ Q=QMMC T= xX:π M=

-1-TQHHQ ='

0ππ * =QT

-1* QQ =i) relation to quadric (non-degenerate)ii) transformation THHQQ **' =

π

0XX =QT

X



Classification of quadrics

•• D represents Q up to projective equivalence.• Signature:

UDUQ ~T= DHHQ T=Scale normalization of D

)()( DQ σσ = Independent of H

Single planeX2=0(1,0,0,0)11

Two planesX2= Y2(1,-1,0,0)0

Single lineX2+ Y2= 0(1,1,0,0)22

Cone X2+ Y2= Z2(1,1,-1,0)1

Single pointX2+ Y2+ Z2=0(1,1,1,0)33

Hyperboloid (1S)X2+ Y2= Z2+1(1,1,-1,-1)0

SphereX2+ Y2+ Z2=1(1,1,1,-1)2

No real pointsX2+ Y2+ Z2+1=0(1,1,1,1)44

RealizationEquationDiagonalSign.Rank

9



Classification of quadrics•

•

•

Nun-ruled quadrics: projectively equivalent to sphere:

Ruled quadrics: contains straight lines

hyperboloids of one sheet

hyperboloid of two sheets

paraboloidsphere ellipsoid

Degenerate ruled quadrics:

cone two planes



Twisted cubics

• A conic in Π2:

• A twisted cubic inΠ3:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++++++

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

2333231

2232221

2131211

23

2

1 1A

θθθθθθ

θθ

aaaaaaaaa

xxx

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

++++++++++++

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

344

2434241

334

2333231

324

2232221

314

2131211

3

2

4

3

2

1 1

A

θθθθθθθθθθθθ

θθθ

aaaaaaaaaaaaaaaa

xxxx

10



The hierarchy of transformations

⎥⎦

⎤⎢⎣

⎡vTvtAProjective

15dof

Affine12dof

Similarity7dof

Euclidean6dof

Intersection and tangency

Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞

The absolute conic Ω∞

Volume

⎥⎦

⎤⎢⎣

⎡10tA

T

⎥⎦

⎤⎢⎣

⎡10tR

T

s

⎥⎦

⎤⎢⎣

⎡10tR

T

Group Matrix Distortion Invariant properties



The plane at infinity

• The plane at infinity:• Contains directions (points)• two planes are parallel ⇔ line of intersection in • line // line (or plane) ⇔ point of intersection in

• The plane at infinity is a fixed plane under a projective transformation H iff H is an affinity

TXXX )0,,,( 321=D

T)1,0,0,0(=∞π

∞

−

∞−

∞ =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡

−==′ π

1000

1t0

ππA

AH

TT

A

∞π

∞π

∞π

11



The absolute conic

• the absolute conic Ω∞ is defined on π∞ s.t.

• Thus, a conic with C = I3x3

• The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity

04

23

22

21 =

⎭⎬⎫++

XXXX ( ) ( )T321321 ,,I,, XXXXXX

i) Ω∞ is only fixed as a setii) Circle intersect Ω∞ in two pointsiii) Spheres intersect π∞ in Ω∞



Metric properties

• Once is identified in projective 3-space, angles and relative lengths can be measured.

• orthogonal (conjugacy)

∞Ω

Invariant to projective transform

d1, d2 : directions of two lines

(Intersection points of lines on π∞)

021 =Ω∞dd T

( )( )( )2211

21cosdddd

ddTT

T

=θ

( )( )( )2211

21cosdddd

dd

∞∞

∞

ΩΩ

Ω=

TT

T

θ

Euclidean:

Projective:

plane

normal

12



The absolute dual quadric

• The Absolute dual quadric:

• The absolute dual quadric Q*∞ is a fixed quadric under the

projective transformation H iff H is a similarity

⎥⎦

⎤⎢⎣

⎡=∞ 00

0I*TQ

i) 8 DOFii) plane at infinity is the null-vector of Q∞

iii) Angles between and :

( )( )2*

21*

1

2*

1cosπQππQπ

πQπ

∞∞

∞=TT

T

θ

∞π1π 2π

multi view geometry (spring '08)ocw.snu.ac.kr/sites/default/files/note/2765.pdf ·...

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