Projective 2D Geometry

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Prof. Kyoung Mu Lee

SoEECS, Seoul National University

Multiple View Geometry (Spring '08)

Projective 2D Geometry 1

Multi View Geometry (Spring '08)

Projective 2D Geometry

K. M. Lee, EECS, SNU Projective 2D Geometry 2

Homogeneous representation of lines and points

• Lineequation:

• Homogeneous representation of lines

• Homogeneous representation of point(x,y) on

• homogeneous point inhomogeneous point 0 )

1 , , ( or , 0 ) 1 , , ( 0 ) 1 , , (

0 0

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⇔

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

= + +

⇔

= + +

kc kb ka y x c

b a k ky kx c

b a y x

kc kby kax c

by ax

0 , ) , ,

( ≠

=k a b c ^T k l

0 , ) 1 , , ( ) , ,

( = ≠

= kx ky k ^T k x y ^T k x

x T

x

x , )

( ₁_, ₂ ₃

=

x x ^T

x x x , ) (

3 2 3 1

equivalence class of vectors =homogeneous vectors c T

b a, , )

=( l The set of all equivalence classes inR³−(0,0,0)^TformsP²

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Multiple View Geometry (Spring '08) K. M. Lee, EECS, SNU Projective 2D Geometry 3

• The point xlies on the line liff x^Tl=0

• The intersection of two lines land l’ is the point (Note ) Ex) intersection of line x=1 (-x+1=0)

and y=1 (-y+1=0) Thus,

Inhomogeneous point (1,1)^T

• The line passing two points xand x’is

(Note )

l l x= × ′

T T, (0, 1,1) )

1 , 0 , 1

(− ′= −

= l

l

x x l= × ′

0 ) ( )

( × ′ = ′• × ′ =

• l l l l l l

0 ) ( )

(x×x′ •x= x×x′ •x′=

=1 x

=1 y

Homogeneous representation of lines and points

⎥⎦

⎢ ⎤

⎣

⎡

= ′

= ^T_T

l L l 0 Lx

L

, of space null the

Ideal points and the line at infinity

• Intersection of parallel lines

• Ideal points(points at infinity):

• Line at infinity: which satisfy

• The parallel lines l and l’intersect l_∞in the ideal point (b,-a,0)^T, where(b,-a)^Tis the line’sdirection

• Thus, the line at infinity is the set of directions of lines in the plane

0 :

) , , (

0 :

) , , (

=

′ + +

′

=

′

= + +

=

c by ax c b a

T T

l l

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

−

′−

=

′

′=

×

=

0 )

( a

b c c c b a

c b a

k j i l l x

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−0 0a b Inhomogeneous representation

x T

x, ,0) ( ₁ ₂ )T

1 , 0 , 0

=(

l∞ (x₁,x₂,0)l_∞ =0

l

Example

=1 x x=2

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A model for the projective plane

exactly one line through two points exactly one point at intersection of two lines

Points and lines of P²can be represented by the intersections of rays and planes through the origin by the plane x₃=1.

Duality

• Duality Principle:

9 To any theorem of 2-D projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem

x l

=0 x l^T

=0 l x^T

' l l

x= × l=x×x'

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Conics

• Conics: Curve described by 2^nd-degree equation in the plane

9 hyperbola, ellipse, and parabola

• Conic equation (inhomogeneous coordinates):

• In homogeneous representation:

• Symmetric, 6 elements but 5 DOF (less one for scale)

3 2 3

1, x y x x

x→ x →

{^a^:^b^:^c^:^d^:^e^:^f}

=0 Cx x^T

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

f e d

e c b

d b a

2 / 2 /

2 / 2

/

2 / 2 /

with C Point conic

Conic coeff. matrix

Determination of a Conic

• How to determine the conic coeff. matricC?

• Using five points on the conic:

By stacking these 5 constraints, we have

Cis the null vector of 5x6 matrix.

f T

e d c b

a, , , , , )

=( c

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Tangent Line to Conics

x l

C

The line ltangent toCat a pointxonCis given byl= Cx.

Dual Conics

• Conics that defines an equation on lines: C^*

• A line ltangent to the conic Csatisfies , where C^* is the adjointmatrix of C.

• In general, for a non-singular symmetric matrix, .

• Dual conics= line conics = conic envelopes

point conic line conic

=0

∗l C l^T

−1

∗=C C

=0

∗l C l^T

=0 Cx x^T

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Degenerate Conics

• A conic is degenerateif matrix Cis not of full rank e.g.two lines (rank 2)

e.g.repeated line (rank 1)

• Degenerate line conics: 2 points (rank 2), double point (rank1)

• Note that for degenerate conics

T

T ml

lm C= +

l m

llT

C=

( )

^C^* ^*^≠^C

Projective Transformations

• Def) A projectivityis an invertible mapping hfrom P²to itself such that three points x₁, x₂, x₃lie on the same line iff h(x₁),h(x₂) and h(x₃) do. (line toline mapping)

• Thm) A mapping h: P²→P²is a projectivity iff there exists a non-singular 3x3 matrix Hsuch that h(x) = Hx.

Pf) Letx₁, x₂, x₃lies on a linel, then

Then, for a non-singularH_3x3, all pointsx’_i=Hx_ilie on the same linel’=H^-Tlsuch that

. 3 , 2 , 1 ,

0 =

= i

i Tx l

0 )

( ¹

'

' = ⁻ = ⁻ = i=

T i T i T T i

Tx H l Hx l H Hx l x

l

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Projective Transformations

• Def) A planar projective transformationis a linear transformation on homogeneous 3-vectors represented by a non-singular 3x3 matrix:

• Homogeneous matrixH: 8 DOF (less one for scale)

• Projectivity = collineation = projective transfromation = homography

Hx x ′ =

Mapping between Planes

central projectionmay be expressed by x’=Hx (application of theorem)

Distortions by central projection

similarity affine projective

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Examples of Projective Transformations

Removing the projective distortions

• Select four points in a plane with known coordinates

9Inhomogeneous correspondence (x,y) ↔ (x’,y’)

9 the eight elements are determined by four point correspondences.

(linear in h_ij)

(2 constraints/point, 8DOF ⇒4 points needed)

x x′

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Transform of lines and conics

• Point transform:

• Transformation of lines:

Note:

• Transformation of conics:

Note:

Hx x′=

) (

⁻¹

− ′ =

′=H l l l H

l ^T ^T ^T

0 )

( )

( ¹ ′ = ¹ ′= ′ ′=

=l H⁻x l H⁻ x l x x

l^T ^T ^T ^T

) (

^'

1 T

TCH C HCH

H

C′= ⁻ ⁻ ^∗ = ^∗

Hierarchy of transformations - Isometries

• Class I: Isometries(iso=same, metric=measure)

• Planar Euclideantransform

• rigid body motion

• 3 DOF (1 rotation, 2 translation)

• 2 point correspondences

• Invariants: length, angle and area

ε=1 : orientation-preserving ε=-1: reverse orientation

R : 2x2 rotation matrix (orthogonal) t : translation 2-vector

I R R^T =

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Similarity transformations

• Class II: Similarity transformations(isometry + scale):

• Equi-formtransformation, preserves shape.

• 4 DOF (1 scale, 1 rotation, 2 translation)

• Invariants: angles, ratio of lengths and areas, parallel lines

• Metric structure= structure defined up to a similarity

s : isotropic scaling

I R R^T =

Affine transformations

• Calss III: Affine transformations:

• 6 DOF (2 scale, 2 rotation, 2 translation)

• Three point correspondences

• Invariants: parallel lines, ratio of lengths of parallel line segments, ratio of areas

A : non-singular matrix

non-isotropic scaling

(2DOF: scale ratio and orientation)

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Projective transformations

• Case IV: projective transformations:

• 8 DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity)

• Action non-homogeneous over the plane

• Invariants: cross ratio (ratio of ratios) of four collinear points

( 1, 2)^T

v= v v

Summary

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Action of affinities and projectivities on line at infinity

• Line at infinity (ideal point) stays at infinity for affine transform, but points move along line

9Parallel line are still parallel

• Line at infinity (ideal point) becomes finite for perspective transform, allows to observe vanishing points, horizon

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎥⎦

⎢ ⎤

⎣

⎡

0 0

0 ²

1 2

1

x x x

x v A A

T

t

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛ +

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎥⎦

⎢ ⎤

⎣

⎡

2 2 1 1

2 1 2

1

v 0

x v x v

x x x

x v A A

T

t

Decomposition of a projective transformation

• Hcan be decomposed as

• Ex

A : non-singular matrix, A = sRK + tv^T K : upper-triangular matrix with detK=1 v ≠0, s is positive

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Number of invariants

• The number of functional invariants is equal to, or greater than, the number of degrees of freedom of the configuration less the number of degrees of freedom of the transformation

• e.g. configuration of 4 points in general position has 8 dof (2/pt) and so 4 similarity, 2 affinity and zero projective invariants

Recovery of affine and metric properties from images

• Under a projective transformH, since ideal points are mapped to finite points, the line at infinityl_∞ is mapped to a finite line.

• However, l_∞is a fixed line iffHis an affinity.

• Note however that since

a point on l_∞is not mapped to the same point on l_∞ unless

− ∞

−

− ∞

∞ =

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟=

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎥⎦

⎢ ⎤

⎣

⎡

= −

′ = l

A t

0 l A

H l

1 0 0

T 1

T T T

A

T

T k x x

x

x, ) ( , )

( ₁ ₂ = ₁ ₂

A

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Recovery of affine properties from images

• If is the imaged line at infinity with l₃≠0, following H’^l⁼⁽^l¹^,^l²^,^l³_p⁾maps lback to l_∞= (0,0,1)^T

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

3 2 1

' 0 1 0

0 0 1

l l l

A

p H

H ∞

− = =l

H^'_p ^T(l₁,l₂,l₃)^T (0,0,1)^T projection rectification

Determining imaged line at infinity – vanishing line

v₁ v₂

l₁

l₂ l₄

l₃

l_∞

2

1 v

v l_∞= ×

2 1

1 l l

v = ×

4 3

2 l l

v = ×

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Distance ratios

( ) (d ) a b d a′,b′ : b′,c′ = ′: ′

( )1,0^T v'=H

( ) ( ) (0,1^T, a,1^T, a+b,1)^T

c , b , a′ ′ ′

H

The circular points

• Circular points:

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The circular points

• Any circle intersects l_∞in the circular points

“circular points”

2 0

3 3 2 3 1 2 2 2

1 +x +dxx +ex x + fx =

x

2 0

2 2

1 +x =

x

l_∞

( )

^T

T

0 , , 1 J

0 , , 1 I

i i

−

=

(

¹^,⁰^,⁰

)

^T

(

⁰^,¹^,⁰

)

^T

I= +i

Algebraically, encodes orthogonal directions

3 =0 x

Conic dual to the circular points

• The conic dual to the circular points:

•

(17)

Angles

• For lines the angle between them is

• This can be rewritten by

• This is invariant to projective transformsince for and

•

T

T m m m

l l

l, , ) and ( , , )

(₁ ₂ ₃ = ₁ ₂ ₃

= m

l

Hx x′= )

(

⁻¹

− ′ =

′=H l l l H

l ^T ^T ^T

orthogonal

*_∞m =0 ⇒ C

l^T

l=(a,b,c) (a,b) (b,-a) Euclidean:

Projective:

Recovery of metric properties from images

• We can find a projective transform that maps the imaged circular points to their canonical positions (1,±i,0)^T, then rectify the image using it.

• OR, metric rectification using :

9For point transform

•

∗

C∞

Hx x′=

( ) ( )

⎥⎦

⎢ ⎤

⎣

=⎡

=

∞

v v

v

v '

*

KKT

KK

KK KK

H H C H H

H H H C H H H

H H H C H H H C

T T T

T T

T T T

T

HA HP HA

HP

HA HP HS HS HA HP

HS HA HP HS

HA HP

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Recovery of metric properties from images

• Rectifying homography using SVD:

9Taking SVD of

9Then the rectifying projectivity is up to a similarity, since

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

=

⇒

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=H H^T H U

0 0 0

0 1 0

0 0 1

∗'

C∞

∗∞

∗∞ =

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=U U U U C

H C H

0 0 0

0 1 0

0 0 1 0

0 0

0 1 0

0 0 1

' T T T

r r

)

( ¹ ¹ ^T

r H U U

H = ⁻ = ⁻ =

Metric from affine

(s11,s12,s22)^T

= s

( )

0

0 0

0

3 2 1 3

2

1 =

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

′

⎥⎦

⎢ ⎤

⎣

′ ⎡

′

m m m l

l l

KKT

(

l₁′m₁′,l₁′m₂′ +l₂′m₁′,l₂′m₂′

)(

s₁₁,s₁₂,s₂₂

)

^T =0

•Affine to metric rectification using 2 orthogonality constraints

• is the null-vector of 2x3 matrix

•s

⎥⎦

⎢ ⎤

⎣

=⎡

22 12

12 11

s s

s

T s KK

∗'

C∞

SVDUT H=U

UC^*_∞

KKT _K C^∗∞^'

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Metric from projective

( ) ( ) ( )

(l₁′m₁′,0.5l₁′m₂′+l₂′m₁′,l₂′m₂′,0.5l₁′m₃′+l₃′m₁′,0.5l₂′m₃′+l₃′m₂′ ,l₃′m₃′)c=0

( )

0

v v v

v

3 2 1 3

2

1 =

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

′

⎥⎦

⎢ ⎤

⎣

′ ⎡

′

m m m l

l

l _T _T

T T

K K KK

•General metric rectification using 5 orthogonality constraints

•c is the null-vector of 5x6 matrix

•c

∗'

C∞

' SVD

∗∞

C UC^*_∞UT H=U

Pole-polar relationship

The polar line l=Cxof the point xwith respect to the conic Cintersects the conic in two points. The two lines tangent to Cat these points intersect at x

Polar of x

Pole of l Conjugate

=0 Cx y^T