Projective 2D geometry

Lecture 3

68

A Hierarchy of Transformations

We would like to examine the following hierarchy of

transformations:

Euclidean ⊂ Similarity ⊂ Affine ⊂ Projective

69

Projective Transformations

We have seen that projective transformations under

composition form a group. This group is called the

projective linear group.

In the projective plane P2, projective linear group is

denoted as PL(3). The elements of this group are

non-singular 3× 3 matrices with real entries.

70

An element of PL(3) is of the form

HP =

a11 a12 tx

a21 a22 ty

v1 v2 v

.

In block matrix form HP can be written as

HP =

A t

vT v

where A is a 2×2 non-singular matrix, t a translation

2-vector, and v = (v1, v2)T .

71

• The most fundamental projective invariant is the

cross ratio of four collinear points.

• A projective transformation HP has 8 degrees of

freedom.

• HP can be computed from 4 point correspondence.

• Transforms: rotation, scaling, translation, shear,

prospective projection.

72

Special Cases of a Projective Transformation

73

Affine Transformations

A subgroup of PL(3) consisting of matrices having

last row (0,0,1) is called an affine group. An element

of an affine group is the form

HA =

a11 a12 tx

a21 a22 ty

0 0 1

.

74

In block matrix form HA can be written as

HA =

A t

0T 1

where A is a 2×2 non-singular matrix, t a translation

2-vector, and 0 a null 2-vector. The matrix A can be

decomposed as A = R(θ)R(−φ)D R(φ) where R(θ)

and R(φ) are rotations by θ and φ respectively, and D

is a diagonal matrix

D =

λ1 0

0 λ2

.

75

• Parallel lines, ratio of lengths of parallel line seg-

ments, and ratio of areas are three important invari-

ant under affine transformation.

• An affine transformation HA has 6 degrees of free-

dom.

• HA can be computed from 3 point correspondence.

• Transforms: rotation, scaling, translation, shear.

76

φ

deformationrotation

θ

Distortions from a affine transformation.

Rotation by R(θ), and a deformation by

R(−φ)D R(φ).

77

Similarity Transformations

Similarity transformations are a subset of affine trans-

formations. A similarity transformation is the form

HS =

s cos(θ) −s sin(θ) tx

s sin(θ) s cos(θ) ty

0 0 1

.

78

In block matrix form HS can be written as

HS =

sR t

0T 1

where R is a 2×2 non-singular matrix called a rotation

matrix, s an isotropic scaling, t a translation 2-vector,

and 0 a null 2-vector. The matrix R is given by

R =

cos(θ) − sin(θ)

sin(θ) cos(θ)

.

79

• Angles between lines, ratio of two lengths, ratio of

areas are invariant under similarity transformation.

• A similarity transformation HS has 4 degrees of free-

dom.

• HS can be computed from 2 point correspondence.

• Transforms: rotation, scaling, translation.

80

Euclidean Transformations

Euclidean transformations are a subset of similarity

transformations. A Euclidean transformation is the

form

HE =

cos(θ) − sin(θ) tx

sin(θ) cos(θ) ty

0 0 1

81

Euclidean transformations preserve distance and hence

are also known as isometries.

In block matrix form HE can be written as

HE =

R t

0T 1

where R is a rotation matrix, t a translation 2-vector,

and 0 a null 2-vector.

82

• Angles between lines, length between two points,

and area are invariant under Euclidean transforma-

tions.

• A Euclidean transformation HE has 3 degrees of

freedom.

• HE can be computed from 2 point correspondence.

• Transforms: rotation, translation.

83

Decomposition of H

A projective transformation H can be decomposed

into a chain of transformations, where each matrix

in the chain represents a transformation higher in the

hierarchy than the previous one.

H =

A t

vT v

=

sR t

0T 1

R 0

0T 1

I 0

vT v

= HS HA HP

The decomposition is valid if v 6= 0 and unique if s > 0.84

Example The projective transformation

H =

1.707 0.586 1.0

2.707 8.242 2.0

1.0 2.0 1.0

may be decomposed as

H =

2cos 45o −2 sin 45o 1

2 sin 45o 2cos 45o 2

0 0 1

0.5 1 0

0 2 0

0 0 1

1 0 0

0 1 0

1 2 1

.

85

Number of Invariants

Result 1.7. The number of functionally independent

invariants is equal to or greater than the number of

degrees of freedom of the configuration minus the

number of degrees of freedom of the transformation.

Example 1. The number of invariants for a configu-

ration of 4 points under an affine transformation is at

least 8− 6 = 2.86

Overview of 2D transformations

87

Overview of transformations

1002221

1211

y

x

taa

taa

1002221

1211

y

x

tsrsr

tsrsr

333231

232221

131211

hhh

hhh

hhh

1002221

1211

y

x

trr

trr

Projective

Affine

Similarity

Euclidean .

88

Projective Geometry of 1D

The projective geometry of a line, P, can be developed

similar to P2. A point x on the line is represented by

homogeneous coordinates (x1, x2)T . A point for which

x2 = 0 is called an ideal point or point at infinity. The

point (x1, x2)T will be denoted by the 2-vector x.

89

• A projective transformation on a line is represented

by a 2 × 2 homogeneous matrix H2×2 and has 3 de-

grees of freedom.

• A projective transformation on a line can be deter-

mined from a three points correspondence.

90

Cross Ratio

Given four points x1, x2, x3 and x4, the cross ratio is

defined as

Cross(x1,x2,x3,x4) =|x1 x2| |x3 x4||x1 x3| |x2 x4|

where

|xi xj| =∣∣∣∣∣∣xi1 xj1

xi2 xj2

∣∣∣∣∣∣ .91

Cross ratio simplifies to

Cross(x1,x2,x3,x4)

=(x11 x22 − x21 x12)(x31 x42 − x41 x32)

(x11 x32 − x31 x12)(x21 x42 − x41 x22)

=(x12 x22)(x32 x42)

(x12 x32)(x22 x42)

(x11x12− x21

x22

) (x31x32− x41

x42

)(

x11x12− x31

x32

) (x21x22− x41

x42

)

=

(x11x12− x21

x22

) (x31x32− x41

x42

)(

x11x12− x31

x32

) (x21x22− x41

x42

)

=(x1 − x2)(x3 − x4)

(x1 − x3)(x2 − x4).

92

Some Facts about Cross Ratio

• The value of cross ratio is independent of which

homogeneous coordinates is used.

• If the second coordinate is 1 for each of the point

xi and xj, then |xi xj| represents the signed distance

from xi and xj.

• The value of the cross ratio is invariant under any

projective transformation of the line.93

Projective transformations between lines

Each set of four collinear points is related to the oth-

ers by a line-to-line projectivity, and the value of the

cross ratio of each set is same.

94

Concurrent lines

In projective projective plane P2 we have seen that

lines are dual to points.

In projective line P, lines are also dual to points.

95

ll

l

l

l

xx

x

x 1

1

2

2

3

4

4

3

A configuration of concurrent lines is a dual to collinear

points on a line. Hence the value of the cross ratio of

these lines is given by the cross ratio of the points.

96

1

2

3

l

x

xx

xc

x 1

x2

x3

x 4

4

If c represents a camera center, and the line ` repre-

sents an image line, then the points xi are the projec-

tions of points xi into the image line.

97

• The cross ratio of the points xi characterizes the

the projective configuration of these image points xi.

• The actual position of the image line is not relevant

as far as the projective configuration of the four im-

age points xi is concerned.

• The projective geometry of concurrent lines is im-

portant to the understanding of the projective geom-

etry of epipolar lines (in Chapter 8).

98

Recovering affine and metric

properties from images

• The main affine properties are parallelism of lines,

and ratio of areas.

• The main metric properties (that is up to similarity

properties) are angles between lines, and the ratio

of two lengths.99

A projective transformation has 8 degrees of freedom.

A similarity transformation has 4 degrees of freedom.

So we need only four (that is, 8 − 4 = 4) degrees of

freedom to determine metric properties. In projective

geometry these 4 degrees of freedom are given by the

line at infinity `∞, and the two circular points.

100

The projective distortion may be removed once the

image of `∞ is specified. Similarly, the affine distortion

may be removed once the image of the circular points

is specified.

Thus we examine the image of the line at infinity, and

then the image of two circular points.

101

The line at infinity

Under a projective transformation the line at infinity

`∞ is mapped to a finite line. However, if the trans-

formation is an affine, then `∞ is mapped to `∞. This

follows from the following:

`′∞ = H−TA `∞ =

A−T 0

−t−TA−T 1

001

=

001

= `∞.

102

Result 1.8. The line at infinity, `∞, is a fixed line

under the projective transformation H if and only if H

is an affine transformation.

The identification of `∞ allows one to recover the

affine properties (parallellism, ratio of areas).

103

l

/

π ππ1 2 3

H H

H

P P

A

l = H ( )P

A projective transformation HP maps `∞ on π1 to a

finite line ` on π2. A projective transformation H′P is

constructed such that ` is mapped back to `∞.104

105

Circular points and their dual

Under any similarity transformation there are two points

on `∞ which are fixed. These are the circular points

I,J with canonical coordinates

I = (1, i,0)T and J = (1,−i,0)T .

106

These circular points are fixed under an orientation-

preserving similarity transformation:

I′ = Hs I =

s cos(θ) −s sin(θ) tx

s sin(θ) s cos(θ) ty

0 0 1

1i0

= s e−iθ

1i0

∼ I.

Result 1.9. The circular points I,J are fixed points

under the projective transformation H if and only if H

is a similarity.

107

I and J are called circular points because every circle

intersect `∞ at the circular points.

The equation of a circle is

x21 + x2

2 + dx1x3 + ex2x3 + fx23 = 0.

The `∞ intersects the circle if x3 = 0. Hence we get

x21+x2

2 = 0 with solution I = (1, i,0)T , J = (1,−i,0)T .

Hence every circle intersects `∞ at circular points.

108