Projective 3D Geometry

(1)

Projective 3D Geometry 1

2008-1

Multi View Geometry (Spring '08)

Prof. Kyoung Mu Lee

SoEECS, Seoul National University

Projective 3D Geometry

Points, Planes, Lines and Quadrics in 3D

• 3D points:

• Projective transform in :

9Collinear

9Lines are mapped to lines 0 , ) 1 , , , (

) , , , (

4 4 3 4 2 4 1

3 4 3 2 1

≠

⇔

=

X X X X X X X

in X X X X

T

T P

X

3 4 3 4 2 4

1, , )

( inR

X X X X X

X _T

X H X′= _4×₄

Homogeneous representation Inhomogeneous representation

P3

(2)

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

Planes

• 3D planes:

0 π π π

π₁X+ ₂Y+ ₃Z+ ₄ =

0 π

π π

π₁X₁+ ₂X₂+ ₃X₃+ ₄X₄=

4 3 4 2 4

1, ,

X Z X X Y X X

X= X = =

: homogeneous plane

)T

, , ,

(π₁ π₂ π₃ π₄

= π

X X'=H

π π'=H^-T Transformation

0 X π^T =

X T

X X

X , , , )

( ₁ ₂ ₃ ₄

=

X : homogeneous 3D point

• Euclidean representation:

• Duality:

~ 0 .X+d= n

(

π₁,π₂,π₃

)

^T

= n

(

X,Y,Z

)

^T

~ = X

π4

= d

n / d

points ↔planes lines ↔lines

: plane normal

4=1 X

(3)

Determination of a plane by Points

• Three points on a plane satisfies

• Or, using the coplanarity constraint;

234 1D

X

(

₂₃₄, ₁₃₄, ₁₂₄, ₁₂₃

)

^T

π= D −D D −D

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

0 det

3 4 2 4

1 4 4

3 3 2 3

1 3 3

3 2 2 2 2 1 2

3 1 2 1

1 1 1

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

X X

[ ]

⁰

det XX₁X₂X₃ =

124 3 134 2 234

1D X D X D

X₁D₂₃₄−−X₂D₁₃₄++X₃D₁₂₄−X₄D₁₂₃ =0 X₁D₂₃₄ X₂D₁₃₄

X −

0

3 2 1

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡ π X X X

T T T

πis the null space of

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

T T T

3 2 1

X X X

π Xon

∀

Determination of a plane Points

(4)

Points from Planes

• Three planes meets at a point:

• A plane can be represented by its span:

0 π

π π

3 2 1

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡ X

T T T

X

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

T T T

3 2 1

π π π

Mx X=

[

1 2 3

]

3

4_x = XX X

M 0 M= πT

⎥⎦

⎢ ⎤

⎣

=⎡

3 3

then I x

M p

(

a,b,c,d

)

^T

π

if = ⎟

⎠

⎜ ⎞

⎝

=⎛− − − a d a c a b, , andp

M: 3dim null-space of π

is the null space of

Lines – Null-space and span representation

• The line joining Aand Bis the span of the row space of W.

• The intersecting line of two planes Pand Qis the span of the row space of W^*.

B A μ λ + (4dof)

⎥⎦

⎢ ⎤

⎣

=⎡ _T^T B W A

⎥⎦

⎢ ⎤

⎣

=⎡ ^T_T Q

W^* P ^λ^′^P⁺^μ^′^Q

(5)

Lines – Null-space and span representation

•

2 2

T T

∗ ∗

= = ×

W W WW 0

Points, Lines and Planes

• A plane by a point Xand a line W:

• A point Xby a line Wand a plane :

⎥⎦

⎢ ⎤

⎣

=⎡ _T X

M W Mπ=0

⎥⎦

⎢ ⎤

⎣

=⎡ _T π M W

*

=0 MX

W*

π

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

If the dim N(M) = 2, then Wis on π W

X π

π π

(6)

Lines - Plücker matrix representation

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

j i j i

ij AB BA

l = − L=AB^T −BA^T i) Lis a 4x4 skew-symmetric homogeneous matrix

ii) Lhas rank 2 iii) 4 DOF

iv) generalization of

v) Lis independent of choice Aand B

vi) Transformation

2 4

*

=0×

LW ^T y x l= ×

HLHT

L'=

[ ] [ ]

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

=

0 0 0 1

0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 L T

Ex)x-axis

Dual Plücker Matrix

• Dual Plücker matrix:

• Transformation:

• Correspondence:

• Join and Incidence

1

' − ∗ −

∗ =H LH L ^T

T

T QP

PQ L^*= −

* 12

* 13

* 14

* 23

* 42

* 34 34 42 23 14 13

12:l :l :l :l :l l :l :l :l :l :l

l =

X L π= ^*

Lπ X=

*X=0 L

: plane through point and line : point on line

: intersection point of plane and line : line in plane

=0 Lπ

[

L1^,L2^,K

]

π=⁰ : coplanar lines

(7)

Lines - Plücker line coordinates representation

• Plücker line coordinates:

• From detL=0

[

^l12^,l13^,l14^,l23^,l42^,l34

]

^T

L=

∈ P

⁵

23 0

14 42 13 34

12l +l l +l l = l

Klein quadric constraint

=0

⇔L^TKL

[ ]

0

0 0 0 0 0 1

0 0 0 0 1 0

0 0 0 1 0 0

0 0 1 0 0 0

0 1 0 0 0 0

1 0 0 0 0 0

34 42 23 14 13 12

34 42 23 14 13

12 =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

l l l l l l

( )

^A,B ^,

( )

^A^,^B

,Lˆ↔ ˆ ˆ

L

[ ]

( )

^L ^L

L

Lˆ K | ^ˆ

ˆ ˆ ˆ ˆ ˆ Bˆ ˆ

, Aˆ B, A,

det ₁₂₃₄ ₁₃₄₂ ₁₄₂₃ ₂₃₁₄ ₄₂₁₃ ₃₄₁₂

=

+ + + + +

=

T

l l l l l l l l l l l l

two lines are intersect iffthe 4 points are coplanar

Lines - Plücker line coordinates representation

• Results:

( )

^L^|^L^ˆ ⁼^det

[

^A,B,^A^ˆ^,^B^ˆ

]

⁼⁰

( )

^L^|^L^ˆ ⁼^det

[

^P,Q,^P^ˆ^,^Q^ˆ

]

⁼⁰

( )

L^|L^ˆ ⁼

( )( ) ( )( )

^P^T^A ^Q^T^B ⁻ ^Q^T^A ^P^T^B ⁼⁰

( )

L|L =0

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) Qis a 4x4 symmetric matrix ii) 9 DOF

iii) in general 9 points define quadric iv) detQ=0 ↔degenerate quadric v) pole – polar

vi) (plane ∩quadric)=conic vii) transformation

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

•

= o o o

o o

Q o

X π=Q

QM M

C= ^T π:X=Mx

-1 -TQH H Q'=

0 π π^TQ^* =

-1

* Q

Q =

i) relation to quadric (non-degenerate) ii) transformation Q'^*=HQ^*H^T

π 0

X X^TQ =

X

Classification of quadrics

•

• Drepresents Qup to projective equivalence.

• Signature:

U D U

Q= T~ Scale normalization of DQ=H^TDH

) ( ) (Q σ D

σ = Independent of H

Single plane X²=0

(1,0,0,0) 1

1

Two planes X²= Y²

(1,-1,0,0) 0

Single line X²+ Y²= 0

(1,1,0,0) 2

2

Cone X²+ Y²= Z²

(1,1,-1,0) 1

Single point X²+ Y²+ Z²=0

(1,1,1,0) 3

3

Hyperboloid (1S) X²+ Y²= Z²+1

(1,1,-1,-1) 0

Sphere X²+ Y²+ Z²=1

(1,1,1,-1) 2

No real points X²+ Y²+ Z²+1=0

(1,1,1,1) 4

4

Realization Equation

Diagonal Sign.

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 paraboloid

sphere ellipsoid

Degenerate ruled quadrics:

cone two planes

Twisted cubics

• A conic in Π^2:

• A twisted cubic inΠ^3:

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

+ +

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

2 33 32 31

2 23 22 21

2 13 12 11

2 3

2

1 1

A

θ θ

θ θ θ

θ

a a a

a a a x

x x

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

+ + +

=

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

=

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

3 44 2 43 42 41

3 34 2 33 32 31

3 24 2 23 22 21

3 14 2 13 12 11

3 2

4 3 2

1 1

A

θ θ θ

a a a a

x x x x

(10)

The hierarchy of transformations

⎥⎦

⎢ ⎤

⎣

⎡

T v v

t A Projective

15dof

Affine 12dof

Similarity 7dof

Euclidean 6dof

Intersection and tangency

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

The absolute conic Ω∞

Volume

⎥⎦

⎢ ⎤

⎣

⎡ 1 0

t A

T

⎥⎦

⎢ ⎤

⎣

⎡ 1 0

t R

T

s

⎥⎦

⎢ ⎤

⎣

⎡ 1 0

t R

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 Hiff His an affinity

X T

X X , , ,0)

( ₁ ₂ ₃

= D

)T

1 , 0 , 0 , 0

=( π∞

∞

−

− ∞

∞ =

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎥⎦

⎢ ⎤

⎣

⎡

= −

′ = π

1 0 0 0 1 t π 0

π A

H A

T T

A

π∞

(11)

The absolute conic

• the absolute conic Ω_∞is defined on π_∞s.t.

• Thus, a conic with C= I_3x3

• The absolute conic Ω_∞is a fixed conic under the projective transformation Hiff His a similarity

0

4 2 3 2 2 2

1 =

⎭⎬ + ⎫ +

X X X

X

(

X₁,X₂,X₃

) (

I X₁,X₂,X₃

)

^T

i) Ω_∞is only fixed as a set

ii) Circle intersect Ω_∞in two points iii) 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_∞⁾

2 0

1 Ω_∞d = d^T

( ) ( )(

1 1 2 2

)

2

cos 1

d d d d

d d

T T

= T

θ

( )

(

1 1

)(

2 2

)

2

cos 1

d d d d

d d

∞

Ω Ω

= Ω

T T

θ T

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 Hiff His a similarity

⎥⎦

⎢ ⎤

⎣

=⎡

∞ 0 0

0

* I Q T

i) 8 DOF

ii) plane at infinity is the null-vector of Q_∞ iii) Angles between and :

( )(

2

)

* 2 1

* 1

2

*

cos 1

π Q π π Q π

π Q π

∞

= ∞

T T

θ T

π∞

π1 π₂