Projective geometry- 2D. Homogeneous coordinates x1, x2,

Projective geometry- 2D

Acknowledgements

Marc Pollefeys: for allowing the use of his excellent slides on this topic

http://www.cs.unc.edu/~marc/mvg/

Richard Hartley and Andrew Zisserman, "Multiple View Geometry in Computer Vision"

Homogeneous coordinates

0 = + +by c ax

(

_a,b,c

)

T 0 , 0 ) ( ) (ka x+ kb y+kc= k

(

a,b,c

)

T ~k

(

a,b,c

)

T Homogeneous representation of lines

equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T _{forms P}2

Homogeneous representation of points

0 = + +by c ax

(

_a,b,c

)

T = l

( )

_x,yT x= on if and only if

( )(

x,y,1 a,b,c

) ( )

T = x,y,1 l =0

(

x,y,1

)

T ~k

(

x,y,1

)

T,k0

The point x lies on the line l if and only if xT_l=lT_x

=

₀

Homogeneous coordinates Inhomogeneous coordinates

( )

x,yT

(

)

T 3 2 1,x ,x

Spring 2006 Projective Geometry 2D 3

Points from lines and vice-versa

l' l x= Intersections of lines

The intersection of two lines l and is l' Line joining two points

The line through two points x and is x' l=xx'

Example 1 = x 1 = y

Ideal points and the line at infinity

(

)

T 0 , , l' l = ba

Intersections of parallel lines

(

)

T

(

)

T andl' , , ' , , l= a b c = a bc Example 1 = x x=2 Ideal points

(

x1,x2,0

)

T Line at infinity

( )

T 1 , 0 , 0 l= = 2 l 2 _R

P Note that in P2 there is no distinction between ideal points and others

Spring 2006 Projective Geometry 2D 5

Summary

The set of ideal points lies on the line at infinity,

intersects the line at infinity in the ideal point A line parallel to l also intersects in the same ideal point, irrespective of the value of c’.

In inhomogeneous notation, is a vector tangent to the line. It is orthogonal to (a, b) -- the line normal.

Thus it represents the line direction.

As the line’s direction varies, the ideal point varies over . --> line at infinity can be thought of as the set of directions of lines in the plane.

A model for the projective plane

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

Points represented by rays through origin Lines represented by planes through origin x1x2 plane represents line at infinity

Spring 2006 Projective Geometry 2D 7

Duality

x

_l

0

x

l

T

=

0

l

x

T

=

l'

l

x

=

l

=

x

x'

Duality principle:

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

Conics

Curve described by 2nd_{-degree equation in the plane} 0 2 2_{+bxy+cy +dx+ey+ f =} ax 0 2 3 3 2 3 1 2 2 2 1 2 1 +bxx +cx +dxx +ex x +fx = ax 3 2 3 1 _, x x y x x x or homogenized 0 x xTC = or in matrix form = f e d e c b d b a 2 / 2 / 2 / 2 / 2 / 2 / C with

{

a:b:c:d:e:f

}

5DOF:

Spring 2006 Projective Geometry 2D 9

Conics …

http://ccins.camosun.bc.ca/~jbritton/jbconics.htm

Five points define a conic

For each point the conic passes through 0 2 2_{+bxy +cy +dx +ey + f =} axi i i i i i or

(

_x2,_xy,_y2,_x,_y,_f

)

_c=0 i i i i i i

(

)

T f e d c b a, , , , , = c 0 1 1 1 1 1 5 5 2 5 5 5 2 5 4 4 2 4 4 4 2 4 3 3 2 3 3 3 2 3 2 2 2 2 2 2 2 2 1 1 2 1 1 1 2 1 = c y x y y x x y x y y x x y x y y x x y x y y x x y x y y x x

Spring 2006 Projective Geometry 2D 11

Tangent lines to conics

The line l tangent to C at point x on C is given by l=Cx l x C

Dual conics

0

l

l

T

_C

*

₌

A line tangent to the conic C satisfies

Dual conics = line conics = conic envelopes 1 *

₌

_C

C

In general (C full rank): C* : Adjoint matrix of C.

Spring 2006 Projective Geometry 2D 13

Degenerate conics

A conic is degenerate if matrix C is not of full rank

T

T

_ml

lm

+

=

C

e.g. two lines (rank 2)

e.g. repeated line (rank 1) T

ll

=

C

l

l

m

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

( )

C

* *

C

Note that for degenerate conics

Projective transformations

A projectivity is an invertible mapping h from P2 _{to itself}

such that three points x1,x2,x3lie on the same line if and only if h(x₁),h(x₂),h(x₃) do.

Definition:

A mapping h:P2_P2_{is a projectivity if and only if there} exist a non-singular 3x3 matrix H such that for any point in P2 _{represented by a vector x it is true that h(x)=Hx}

Theorem:

Definition: Projective transformation

= 3 2 1 33 32 31 23 22 21 13 12 11 3 2 1 ' ' ' x x x h h h h h h h h h x x x x x'=H or 8DOF projectivity=collineation=projective transformation=homography

Spring 2006 Projective Geometry 2D 15

Mapping between planes

central projection may be expressed by x’=Hx

(application of theorem)

Removing projective distortion

33 32 31 13 12 11 3 1 ' ' ' h y h x h h y h x h x x x + + + + = = 33 32 31 23 22 21 3 2 ' ' ' h y h x h h y h x h x x y + + + + = =

(

31 32 33

)

11 12 13 ' h x h y h h x h y h x + + = + +

(

31 32 33

)

21 22 23 ' h x h y h h x h y h y + + = + +

select four points in a plane with know coordinates

(linear in h_ij) (2 constraints/point, 8DOF 4 points needed)

Spring 2006 Projective Geometry 2D 17

Transformation of lines and conics

Transformation for lines

l

l'

₌

_H

-T

Transformation for conics

-1 -T

_CH

H

C

'

=

Transformation for dual conics T

H

HC

C

_'

*

₌

*

x

x'

=

H

For a point transformation

Distortions under center projection

Similarity: squares imaged as squares.

Affine: parallel lines remain parallel; circles become ellipses. Projective: Parallel lines converge.

Spring 2006 Projective Geometry 2D 19

Class I: Isometries

(iso=same, metric=measure) = 1 1 0 0 cos sin sin cos 1 ' ' y x t t y x y x 1 ± = 1 = 1 = orientation preserving: orientation reversing: x 0 x x' = = 1 t T R H_E RTR₌I

special cases: pure rotation, pure translation 3DOF (1 rotation, 2 translation)

Invariants: length, angle, area

Class II: Similarities

(isometry + scale) = 1 1 0 0 cos sin sin cos 1 ' ' y x t s s t s s y x y x x 0 x x' = = 1 t T R HS s RTR=I

also know as equi-form (shape preserving)

metric structure = structure up to similarity (in literature) 4DOF (1 scale, 1 rotation, 2 translation)

Invariants: ratios of length, angle, ratios of areas, parallel lines

Spring 2006 Projective Geometry 2D 21

Class III: Affine transformations

= 1 1 0 0 1 ' ' 22 21 12 11 y x t a a t a a y x y x x 0 x x' = = 1 t T A HA

non-isotropic scaling! (2DOF: scale ratio and orientation) 6DOF (2 scale, 2 rotation, 2 translation)

Invariants: parallel lines, ratios of parallel lengths, ratios of areas

( ) ( ) ( )

R DR R A= = 2 1 0 0 D

Class VI: Projective transformations

x v x x' = = v P T t A H

Action non-homogeneous over the plane

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

Invariants: cross-ratio of four points on a line (ratio of ratio)

(

)

T 2 1,

Spring 2006 Projective Geometry 2D 23

Action of affinities and projectivities on line at infinity

+ = 2 2 1 1 2 1 2 1 0 v x v x v x x x x v A A T t = 0 0 0 2 1 2 1 x x x x v A A T t

Line at infinity becomes finite,

allows to observe vanishing points, horizon. Line at infinity stays at infinity,

but points move along line

Decomposition of projective transformations

= = = v v s P A S T T T _vT t v 0 1 0 0 1 0 t K I A R H H H H T tv + = RK A s Kupper-triangular,detK=1

decomposition unique (if chosen s>0)

= 0 . 1 0 . 2 0 . 1 0 . 2 242 . 8 707 . 2 0 . 1 586 . 0 707 . 1 H = 1 2 1 0 1 0 0 0 1 1 0 0 0 2 0 0 1 5 . 0 1 0 0 0 . 2 45 cos 2 45 sin 2 0 . 1 45 sin 2 45 cos 2 H Example:

Spring 2006 Projective Geometry 2D 25

Overview transformations

1 0 0 22 21 12 11 y x t a a t a a 1 0 0 22 21 12 11 y x t sr sr t sr sr 33 32 31 23 22 21 13 12 11 h h h h h h h h h 1 0 0 22 21 12 11 y x t r r t r r Projective 8dof Affine 6dof Similarity 4dof Euclidean 3dof Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio

Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids).

The line at infinity l

Ratios of lengths, angles.

The circular points I,J