Projective Geometry and Transformations

7/25/2019 Projective Geometry and Transformations

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2-1Professur Digital- und SchaltungstechnikProf. Dr.-Ing. Gangolf Hirtz

Computer Vis ionVersion 08.05.2015

Chapter 2

Projective Geometry

and TransformationsVersion 08.05.2015


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Content

2.1 Basics of Projective Geometry

2.1.1 Euclidean geometry and beyond

2.1.2 Affine Geometry

2.1.3 Projective Geometry

2.1.4 Projective vs. Affine Interpretations

2.1.5 Inhomogeneous Representations

2.2 Two-dimensional Projective Space

2.2.1 Representation of Points

2.2.2 The World Coordinate System

2.2.3 Homogeneous Coordinates

2.2.4 Euclidean Transformations - Translation , Rotation and Scaling

2.2.5 Projective Transformations - Projectivity

2.2.6 Lines

2.2.7 Points and Lines at Infinity

2.2.8 Duality of Points and Lines

2.2.9 More on Transformations

2.2.10 Specialized Transformations in 2D

2.3 Three-dimensional Projective Space

2.3.1 Points2.3.2 Planes

2.3.3 Transformations in 3D

Table of Contents


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Introduction

Euclid, 300 BC, also known asEuclid of Alexandria, was a Greek

mathematician, often referred to as the "Father ofGeometry

Euclid deduced the principles of what is now called Euclideangeometryfrom a small set of axioms

Euclid also wrote works onperspective,conic sections,spherical

geometryandnumber theory

Euclid stated a very intuitive understanding of forms, eg. Plane

Geometry:

1. One point to another form a straight line

2. Extension of a straight line stays straight

3. A circle is described by its radius and centre point

4. Any right angle is a right angle5. The parallel postulate: Two straight lines are parallel when a third

line intersects one line of them perpendicular and the second will

be intersect perpendicular, too.

Or: If a straight line falls on two straight lines and the interior

angles on the same side are less than two right angles, the two

straight lines will definitely intersect on this side.


http://de.wikipedia.org/wiki/Euklid

http://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Parallel_postulate
http://de.wikipedia.org/wiki/Euklidhttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Parallel_postulatehttp://en.wikipedia.org/wiki/Parallel_postulatehttp://en.wikipedia.org/wiki/Euclidhttp://de.wikipedia.org/wiki/Euklid


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Cartesian Coordinate System

At least known from secondary school, Euclidean space/geometry

uses several forms of coordinates. The most prominent is the

Cartesian coordinate system. For example it states a 2- and 3-dimensional point P as

Transformations

The group of Euclidean transformations consists of translation,

rotation,reflection,scaling

Euclidean, Elliptical and Hyperbolic Geometry As long curvature is excluded theorems of Euclid. geometry hold true

The 5th postulate mentioned before loses its validity as soon as

curvature (i.e. hyperbolic, elliptical) is introduces.

Such spaces are called non-Euclidean.

Hyperbolic geometry after Lobatschewski is widely used in

astrophysics, as they are basis ofEinsteinstheory of relativity.

Both, hyperbolic and elliptical are important in differential

geometry.

The diagrams show how the sum of the inner angles of a triangle isrelated to the curvature of its underlying 3-dimensional space.

Invariant Properties

The transformationsin any geometry (e.g. Euclidean, hyperbolic,

elliptical, affine, projective, ...) form a group which turn out to be

invariant to certain properties:

In Euclidean geometry the group of movements is invariant to

lengthandarea(isogonal).


(2D)

xP ,

y

(3D)

x

P y .

z

Euclidean Geometry specialised form of describing a N-dimensional space without

curvature.


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Introduction

Affine geometryis ageneralisationof the Euclidean geometry.

The parallel postulate holds true

BUT measures of distance and angles are foreign to affine

geometry.

Affine geometry is the study of parallel lines.

The affine geometry can be described with (freely interpreted)

incidences axioms as follows:

1. Through two points there is only one straight line intersecting

2. On every straight line there lie at minimum two points

3. The parallel relation is reflexive, symmetric and transitive

4. Through every point there is a straight line which is parallel to agiven one

5. A triangle ABC is given. If two pointsA and Bwhich have the

property ofAB||ABexists, then there is one pointCwhich leads

toAC||ACandBC||BC.

Transformations

Transformations from Euclidean Geometry

Shearing


Parallelism of lines,area ratios,line at infinity

2.1.2 Affine geometry

Reflexive

A straight line is g||g

Symmetric

If g is mirrored to a line,then g is parallel to g

Transitive

If g||h and g||f, then f||h

Euclidean Geometry

lines are parallel

circles are circles

Affine Geometry

lines are parallel

angles are distorted

circles are ellipses


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Introduction

Perspective geometry is a generalisation of Affine and Euclidean

geometry E A P.

Geometry in strict sense describes what kind of transformations are

invariant to a geometric primitive or feature.

In Euclidean space a translation of a cube will definitely be a cube

again. Think of throwing a dice, any rotation or translation will not

affect the feature of the dice intuitively.

In projective geometry only lines are preserved, no angles nor

length. The best way to think of projective space is a 3D-sketch

on paper of a dice. So what we describe is a (in Euclidean

geometry) degenerated dice yet not in projective space.

To work with projective geometry, one uses homogeneous

coordinates.

Transformations

Transformations from Affine Geometry

Perspective Transformation


Intersection and tangency

2.1.3 Projective Geometry

lines are parallel

angles are distorted

circles are ellipses

lines are lines

circles are general

(green circle doesnt fit)


7/372-7Professur Digital- und Schaltungstechnik

Prof. Dr.-Ing. Gangolf HirtzComputer Vis ion

Version 08.05.2015


Although it is quite common to watch perspective distortions, one can

also discover affine effects on occation

2.1.4 Projective vs. Affine Interpretations




Version 08.05.2015


Introduction

Cartesian coordinates have difficulties by representing ideal points

such asinfinitywith finite coordinates.

In projective geometry one uses homogeneous coordinates hence, normal representation is also called inhomogeneous

coordinates

2.1.5 Inhomogeneous Representations




Version 08.05.2015

A pointp can be represented in of 2D and 3D coordinates as

follows:

Any point consists ofncomponents, which represent one dimension

each. In this example we call them p1, p2and p3, repectively.

Since this seems to be well known, let us find out more about the

meaning behind this notation! Let us consider an example:

Now try to plot these points in the right pictureAny idea?

Since every point representation is only valid with an coordinate

frame addicted to it, it is necessary to define some coordinate frame

for our points:

Having this information, you are able to plot the points, with respectto their coordinate frames


2.2.1 Representation of Points

122

D

p

p

p

1

23

3

D

p

p pp

1

5

2

p2

8

4

p

1( , )

( , )

5

2i j

i j

p 2( , )( , )

8

4k l

k l

p




Version 08.05.2015

A Coordinate System represents a n-dimensional space in the form

of a bunch of coordinatebasis vectors

frame 1: basis vectoriandj

frame 2: basis vectorkandl

Representing a pointp having components p1and p2let us say with

respect to the frame 1, means to stretch the basis vectors iandj by

the scalar values of p1and p2respectivly:

Now lets define some properties for our coordinate frame Frame 1 defines ourworld coordinate system (WCS), which

means that its origin is thezero-vector

The basis vectorsi and j are orthogonal to each other and both

have unit lengthhence they are orthonormal

Since the two basis vectors are orthogonal to each other we resist

in the euclidean 2-dimensional space

With the knowledge of some basics of Linear algebra one can rewrite

the representation of our pointpas

with theIdentity matrixrepresenting our WCS.


2.2.2 The World Coordinate System

Some remarks

Notice that the Identity itself is provided with respect to our world

coordinate Frame

What about frame 2 or any other frame one likes to define?: Ones a

WCS is defined, any other frame is expressed with respect to it. We

will come back to this later.

Since now every pointp will be defined implicitly with respect to theWCS.

The Origin of the WCS is usually marked withOW.

The same investigations can be made for the 3- or n-dimensional

spacewithout exception.

If not explicitly mentioned, the representation of any vector relates to

the WCS.

,

1

1 2,

2

pp p

p

i j

i jp i j

1

1 22

1 0with ,

0 1W W WW

pp p

p

p i j i j

1

2

1 0

0 1W

W

p

p

p


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Introduction

An arbitrary point p of the n-dimensional projective space is

described by homogeneous coordinates of vector size (n+1)

x, y, z... Vector components that stretches coordinate frame

vectors

w...scaling factor

To have a projection comparable to the Euclidean spacew=1 such:

Examples:



(2D) (3D),

p p

xx

yy

zw

w

homogeneous Cartesian

(2D)

1

/

/w

xx w x

yy w y

w

p

2 4 82 4 / 2 8 / 4

2.5 5 102.5 5 / 2 10 / 4

1 2 4

h i

p p


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Homogeneous Representation

The origin of a n-dimensional space must not be at w=0:

Points with x(n+1)=0 are ideal points orpoints at infinity and they

cantbe represented in Cartesian coordinates.

A homogeneous coordinate system allows to clearly describe the

position of a finite and infinite point in the projective space.

This coordinates allow to describe a projective transformation via a

transformation matrix as homomorphism (i.e. structure-preserving

map)

So collineation and projection can be represented as linear

transformations.

Formulas with homogenous coordinates are often simpler and more

symmetric than their Cartesian counterparts


Mappings from one to the other coordinate system (withx(n+1)0):

0 0

0 0

O ; O

0 0

0 w

1 2

1 1 1

homogeneousT T T

1 2 1 2 1 1 2

Cartesian/inhomogeneousT T T

1 2 1 1 2

( , , , ) ( , , , , ) ( , , , ,1)

( , , , , ) ( , , , ) ( , , , )nn n n

n n n n

xx x

n n n x x x

x x x x x x x x x x

x x x x x x x


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Translation

Given a pointpit is possible to translate or better move it by another

vector say the translation vectort:

Rotation

A pointp1can be rotated around the origin by an angle in issueing

a so called rotation matrixR:

Remark: Since in the 3D case, there are three axes to rotate around,

the rotation matrix is a little bit more complex and issues three angles

itself:,and.

Scaling

A pointp2can be scaled up/down by an arbitray factor s, to vary the

size of the vector, but do not change its direction:

This can be rewritten to comply to a matrix notation of s:



1 1

1

2 2

p t

p t

p p t

2 1

cos sinwith

sin cos

p R p R

3 2s p p

3 2

0

0

s

s

p p


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Concatination of Transformations

Assume that we wanted to perform all the transformations

(translation, rotation and scaling) in one single transformationH:

The more operations are concatenated, the more complicated the

computation

As Solution one can use Homogeneous Coordinates:



Euclidean Transformation Matrix H

2 1

2 1

21 11 12 11 12 11 1

22 21 22 21 22 12 2

p s s r r p t

p s s r r p t

p H p

p s R p t

2 1 1 ?????? p R R p t t t p

2 1

2 1

21 11

22 12

1

1

1 2

with1

1 1

p p

p p

p H p

p s R p t

s R tH H

0

p H p

11 12 13

3 3 21 22 23

11 12 13 14

21 22 23 24

4 4

31 32 33 34

2D Space:

0 0 1

3D Space:

0 0 0 1

x

x

h h h

h h h

h h h h

h h h h

h h h h

H

H


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2.2.4 Euclidean Transformations - Translation , Rotation and Scaling0001 % Use the following 2-by-11 matrix to draw a simple house

0002

0003 X = [ -6 -6 -7 0 7 6 6 -3 -3 0 0

0004 -7 2 1 8 1 2 -7 -7 -2 -2 -7 ];

0005 no_pts = size(X, 2);

0006 h1 = figure(1), plot(X(1, :), X(2, :), 'ob'), hold on;

0007 title('A Simple House', 'FontWeight', 'bold', 'FontSize', 12);

0008

0009 for i=1:no_pts

0010 x1 = X(:, i);

0011 x2 = X(:, mod(i, no_pts) + 1);

0012 plot([x1(1) x2(1)], [x1(2) x2(2)], '-b')0013 end

0014

0015 % Make Points homogeneous

0016 X = [X; ones(1, size(X, 2))];

0017

0018 % Scaling

0019 H1 = [1/2 0 0;

0020 0 1 0;0021 0 0 1];

0022

0023 % Rotation

0024 H2 = [ 0.7071 -0.7071 0;

0025 0.7071 0.7071 0;

0026 0 0 1];

0027

0028 % Translation

0029 H3 = [ 1 0 2;

0030 0 1 -3;

0031 0 0 1];

0032

0033 % Scaling, Mirroing0034 H4 = [1/2 0 0;

0035 0 -1 0;

0036 0 0 1];

0037

0038 disp(H1); disp(H2); disp(H3); disp(H4);

0039

0040 % Apply Transformation

0041 X1 = H1*X;

0042 X2 = H2*X;

0043 X3 = H3*X;

0044 X4 = H4*X;

0045 disp(X1); disp(X2); disp(X3); disp(X4);

0046

0047 % Make Coordinates inhomogeneous

0048 X1 = bsxfun(@rdivide, X1(1:2, :), X1(3, :));




0052

0053 X1 = X1(1:2, :);

0054 X2 = X2(1:2, :);

0055 X3 = X3(1:2, :);

0056 X4 = X4(1:2, :);

00570058 % Plot Transformed Houses

0059 h2 = figure(2)

0060 plot(X1(1, :), X1(2, :), 'or'); hold on;

0061 plot(X2(1, :), X2(2, :), 'og'); hold on;

0062 plot(X3(1, :), X3(2, :), 'om'); hold on;

0063 plot(X4(1, :), X4(2, :), 'oc'); hold on;

0064 title('Some Simple Transformations', 'FontWeight', 'bold', 'FontSize

0065 legend('Scaling', 'Rotation', 'Translation', 'Scaling/Mirroring');0066

0067 for i=1:no_pts

0068

0069 x1 = X1(:, i);

0070 x2 = X1(:, mod(i, no_pts) + 1);

0071 plot([x1(1) x2(1)], [x1(2) x2(2)], '-r');

0072

0073 x1 = X2(:, i);

0074 x2 = X2(:, mod(i, no_pts) + 1);

0075 plot([x1(1) x2(1)], [x1(2) x2(2)], '-g');

0076

0077 x1 = X3(:, i);

0078 x2 = X3(:, mod(i, no_pts) + 1);0079 plot([x1(1) x2(1)], [x1(2) x2(2)], '-m');

0080

0081 x1 = X4(:, i);

0082 x2 = X4(:, mod(i, no_pts) + 1);

0083 plot([x1(1) x2(1)], [x1(2) x2(2)], '-c')

0084 end

0085 axis equal

0086

0087 print(h1, '-r600', '-dtiff', '01.tif')


0089 highlight('m01.m', 'rtf', 'm01.rtf')

0090


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Transformation Matrices

0.5000 0 0

0 1.0000 0

0 0 1.0000

0.7071 -0.7071 0

0.7071 0.7071 0

0 0 1.0000

1 0 2

0 1 -3

0 0 1

0.5000 0 0

0 -1.0000 0

0 0 1.0000



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Introduction

A projectivity is an invertible mapping from points to points that maps

lines to lines

A projectivityHis also calledcollineationorhomography.



2 3 3 1

2 11 12 13 1

2 21 22 23 1

2 31 32 11

x

x h h h x

y h h h y

w h h w

p H p See Matlab Example!


For two points there is one line which is incident with them

Two lines intersect in exactly one incident point. Thus there are noparallel lines. (see later)

There are four points such that no line is incident with more than two

of them.

non-zero elements


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2.2.5 Projective Transformations - Projectivity0001 % Use the following 2-by-11 matrix to draw a simple house

0002

0003 X = [ -6 -6 -7 0 7 6 6 -3 -3 0 0

0004 -7 2 1 8 1 2 -7 -7 -2 -2 -7 ];

0005 no_pts = size(X, 2);

0006 h1 = figure(1); plot(X(1, :), X(2, :), 'ob'), hold on;

0007 title('A Simple House', 'FontWeight', 'bold', 'FontSize', 12);

0008

0009 for i=1:no_pts

0010 x1 = X(:, i);

0011 x2 = X(:, mod(i, no_pts) + 1);

0012 plot([x1(1) x2(1)], [x1(2) x2(2)], '-b')0013 end

0014

0015 % Make Points homogeneous

0016 X = [X; ones(1, size(X, 2))];

0017

0018 % Projectivity 1

0019 H1 = [ 1 0 0;

0020 0 1 0;0021 0.04 0.00 1];

0022

0023 % Projectivity 2

0024 H2 = [ 1 0 0;

0025 0 1 0;

0026 0.04 0.04 1];

0027

0028 disp(H1); disp(H2);

0029

0030 % Apply Transformation

0031 X1 = H1*X;

0032 X2 = H2*X;

0033 disp(X1); disp(X2);0034

0035 % Make Coordinates inhomogeneous


0037 X2 = bsxfun(@rdivide, X2(1:2, :), X2(3, :));0038

0039 X1 = X1(1:2, :);

0040 X2 = X2(1:2, :);

0041

0042 % Plot Transformed Houses

0043 h2 = figure(2);

0044 plot(X1(1, :), X1(2, :), 'or'); hold on;

0045 plot(X2(1, :), X2(2, :), 'og'); hold on;

0046 title('Some Simple Transformations', 'FontWeight', 'bold', 'FontSize

0047 legend('Projectivity 1', 'Projectivity 2');

0048

0049 for i=1:no_pts

0050

0051 x1 = X1(:, i);

0052 x2 = X1(:, mod(i, no_pts) + 1);0053 plot([x1(1) x2(1)], [x1(2) x2(2)], '-r');

0054

0055 x1 = X2(:, i);

0056 x2 = X2(:, mod(i, no_pts) + 1);

0057 plot([x1(1) x2(1)], [x1(2) x2(2)], '-g');

0058 end

0059 axis equal

0060



0063 highlight('m02.m', 'rtf', 'm02.rtf')

0064


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Transformation Matrices

1.0000 0 0

0 1.0000 0

0.0400 0 1.0000

1.0000 0 0

0 1.0000 0

0.0400 0.0400 1.0000



20/37




As for image processing homogeneous coordinates are so important

(due to projections of the 3D world to a 2D image), we continue to

talk about representations with these coordinates.

A line in a plane in its general form (normal representation):

wherea,bandcgive rise to different lines.

So a line can be naturally represented by the vector [a b c]T and also

its multiplesk[a b c]T for any non-zerok.

From another point of view: The pointx=[x y]T

in the 2D Euclideanspace belongs to a linel, represented by three valuesa,bandc,so

l= [a b c]T if the condition holds true:

Due to homogeneous coordinates a symmetry is achieved (duality

principle)

xis said to be theright null-spaceoflT.Why?

2.2.6 Lines

Common point of two lines can be found by:

Note that also parallel lines have a common pointapoint at infinity.

It is also possible to find the point by using a skew-symmetric matrix:

0ax by c

T 01

x

a b c y ax by c

l x

T T 0 x l l x

1 1 2

1 2 1 1 2

1 1 2

0

0

0

c b a

c a b

b a c

x l l

( )a cb b

y x m x n

1 2 l x x

2.2


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Example1

Determine the intersection of the linesl1andl2. Where the first line

corresponds tox=1 and the second toy=1:

Example2

Determine if the pointx1orx2lie on the linel1:

See Matlab Example!


2.2.6 Lines

1 2

0 1 0 0 0 0 1 1 0 1 11 0 1 1 0 1 0 1 1 1 1

0 1 0 1 0 0 1 1 0 1 1

x l l

1

1

1 1 1 0 0

1

x x

l

2

0

1 1 1 0 1

1

y y

l

1 2 1

1 0 1

1 , 1 , 0

1 1 1

x x l

!

T

1 1

1

0 1 1 1 0 1 1 1 0 1 1 0

1

x l

!

T

2 1

1

0 0 1 1 0 0 1 1 0 1 1 0

1

x l


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2.2.6 Lines

0001 clear all, close all

0002

0003 [x, y] = meshgrid(-10:1:+10, -10:1:+10);

0004

0005 h1 = figure(1); hold on0006 plot(x, y, 'r.');

0007 X = [

0008 reshape(x, 1, []);

0009 reshape(y, 1, []);0010 ones(1, size(reshape(x, 1, []), 2));

0011 ];

0012

0013 l1 = [-1; 0; 1];

0014 l2 = [ 0; -1; 1];

0015

0016 result1 = l1'*X;

0017 result2 = l2'*X;

00180019 h2 = figure(2), hold on

0020 title('2D Line Evaluation (Line 1)', 'FontWeight', 'bold');

0021 surf(x, y, reshape(result1, size(x)), 'EdgeColor', 'none');0022 xlabel('x'), ylabel('y'), zlabel('result of dot product')

0023 view(45, 45), colorbar

0024 h3 = figure(3), hold on

0025 title('2D Line Evaluation (Line 2)', 'FontWeight', 'bold');

0026 surf(x, y, reshape(result2, size(x)), 'EdgeColor', 'none');

0027 xlabel('x'), ylabel('y'), zlabel('result of dot product')

0028 view(45, 45), colorbar

0029

0030

0031 figure(1);0032 plot(x(reshape(result1, size(x)) == 0), ...

0033 y(reshape(result1, size(y)) == 0), 'g.');

0034 plot(x(reshape(result2, size(x)) == 0), ...

0035 y(reshape(result2, size(y)) == 0), 'g.');

0036 xlabel('x'), ylabel('y')

0037 title('2D Lines', 'FontWeight', 'bold');

0038





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Consider two pairs of parallel lines:

l1:x1=-1

l2:x2=+1

l3:y3=-1

l4:y4=+1

Letscalculate their intersection:

This is directly comparable to the general understanding of two

parallel lines only meeting at infinity. So the resulting point is anideal

pointorpoint at infinity.

The set of all ideal points may be written as:

Since two points form a line, we can compute theline at infinity l



inhomogeneous

1 1 2 1

inhomogeneous

2 1 2 2

1 1 00 / 0

0 0 22 / 0

1 1 0

0 0 22 / 0

1 1 00 / 0

1 1 0

x l l x

x l l x Trying to solve for any ideal point [x1,x2,0]

T reveals a line

with the property:

This line in the is calledline at infinity. In other words this isstating that all ideal points lie on this line.

From this we can deduce following two statements:

Parallel lines meet at an ideal point or the point at infinity.

Different sets of parallel lines intersect in different points at infinity.

To find the line at infinity one needs simply different sets of parallel

lines.

1x

2x

l

inhomogeneous

1 2 1

0 2 00 / 4

2 0 00 / 4

0 0 4

l x x x

T 0x l

1 23 3

0 0

0 0 0 0x x

l l

Tx l l

2

1

2

0

x

x

x

1l 2l3l

4l

2 2 S


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Under a projective transformation ideal pointsxmay be mapped to

finite pointsxand consequentlylis mapped to a finite linel.



x l

3

1 1

2 2

3 00

x

x x

x x

x

x H x

H

1 1 2

1

2 3

3 0, 0

0 0

T

l l

l

l l

l

l l H

H

2 2 T di i l P j ti S


25/37





Metric Properties can be recovered from projectively distorted planes

be mappingfiniteideal points back to infinity

2 2 T di i l P j ti S


26/37



In a linelis defined uniquely by the join of two pointsxi

In a pointxis defined uniquely by the join of two linesli

2.2.8. Duality of Points and Lines


T

1

1 2 T

2

xl x x l 0x

2

T

1

1 2 T

2

lx l l x 0

l

2

zero vector

2 2 T di i l j ti


27/37



2.2 Two-dimensional projective space

Any homogeneous point in is represented as a 3-vectorxin

Hx is a invertible, linear mappingof homogeneous coordinates,

calledprojectivity.

There are eight independent ratios amongst the nine elements of H,

and it follows that a projective transformation has eight degrees of

freedom(dof).

2.2.9. More on Transformations

2

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

'

' H '

'

x h h h x

x h h h x

x h h h x

x x

3

image from [2]

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

'' H '

'

T

T

l h h h l l h h h l

l h h h l

l l

3D Euclidean space

linear transformation in 2D projective space

2 2 T di i l j ti


28/37




2.2.10. Specialized Transformations in 2D

Rotation

As again indicated by the index E ofH this transformations also

conforms to Euclidean metrics.

It hasonedegree of freedom and can hence be computed from one

point correspondence.

Here againlength,angleandorientationare preserved (invariant).

Translation

A translation is one of the most basic transformation, so they are

conform with the Euclidean interpretation of geometry

Regarding the matrix HE (the index E indicates it is conform toEuclids theorems) a translation hastwo degrees of freedom. Thus

two parameters must be specified in order to define the

transformation.

The transformation can be computed fromone point correspondence

Invariant to this transformation arelength,angleandorientation

1

E 2

1 0

' 0 1 , with

0 0 1

t

t

x H x xE T 1

I tH

0

Iis the 2x2 identity matrix

tis the translation vector

0T

is the null vector

E

cos sin 0

' sin cos 0 , with0 0 1

x H x x

E T

cos sin, and

1 sin cos

R 0H R

0

2 2 T o dimensional projecti e space


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Similarity - Translation, Rotation and Scaling

Similarity is an isometry composed with an isotropic scaling

It hasfourdegrees of freedom, accounting one more than Euclidean

for the scaling.

Preserved areangles,parallel linesbut not length whereas the ratio

of lengths is.

Isometries - Translation and Rotation

Also known as Euclidean transformation, rigid body motion or

displacement

The form reminds to the formula of a l ine in Euclideanspace

Preserved arelengthandanglefor sure.

It hasthreedegrees of freedom.

' x Rx t

1

E 2

cos sin

' sin cos , without

0 0 1

t

t

x H x x E T 1

R tH

0

1

S 2

cos sin

' sin cos , with

0 0 1

s s t

s s t

x H x xS T 1

s

R tH

0

srepresent the isotropic scaling

=1 (If it is negative the isometry is not orientation-preserving

and no longer Euclidean Reflection).

2 2 Two dimensional projective space


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Projective Transformation

It is a general non-singular linear transformation of homogeneous

coordinates, a so called collineation.

A projective transformation is also known ashomography.

From the nine elements of the matrix only their ratios are significant,

thus the transformation is specified by eight parameters. Concluding

it haseightdegrees of freedom. A projective transformation between two planes can be computed

from four point correspondences, with no three collinear on either

plane.

Invariant is thecross ratio(ratio of ratios)of four collinear points

Affine Transformation

Affine transformation (aka affinity) is a non-singular linear

transformation followed by a translation.

It has six degrees of freedom, corresponding to the six matrixelements (2 translation, 2 rotation, 2 scale (scale ratio and

orientation)).

This transformation can be computed from three point

correspondences.

The new feature added to similarity is the non-isotropic scaling

which is shear-mapping.

Invariant to affinities areparallel lines,ratios of lengthsof parallel line

segments andratio of areas.

Ais a 2x2 non-singular matrix which combines a rotation and a

non-isotropic scaling

11 12 1

A 21 22 2' , with

0 0 1

a a t

a a t

x H x xA T

1

A tH

0

11 12 1

P 21 22 2

1 2

' , with

1

x H x x

a a t

a a t

v v

P T

1

A tH

v

2 2 Two dimensional projective space


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Conclusion


2 3 Three dimensional projective space


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2.3 Three-dimensional projective space

Introduction

Points in the three-dimensional projective space are described by a

4-vector of homogeneous coordinates.

The rule of duality in is slightly different to the one which exist in

the : In the 3Dpoints are dual to planes(instead of lines)

The three-dimensional projective space resides in the four-

dimensional Euclidean space

2.3.1. Points

1

1 4

2

(3 ) (3 ) 2 4

3

3 4

4

/

/

/

D D

XX X

XX X

XX X

X

X X

23

34



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Introduction

A plane can be described in homogeneous coordinates with 3

degrees of freedom.

The first 3 components of correspond to the plane normal ofEuclidean geometry

See Matlab Example!

2.3.2. Planes

1 1 2 2 3 3 4 4 0X X X X

1

2

3

4

T 0 X

0048 % Draw a Plane

0049 figure_2 = figure(2); clf; grid on; hold on;

0050 [X, Y, Z] = meshgrid(-5:0.1:5, -5:0.1:5, -5:0.1:5);

0051

0052 Plane_0 = 0*X + 0*Y + 1*Z - 1;

0053 Plane_1 = 1*X + 1*Y + 1*Z + 0;

00540055 p1 = patch(isosurface(X, Y, Z, Plane_0, 0));

0056 p2 = patch(isosurface(X, Y, Z, Plane_1, 0));

0057

0058 set(p1, 'FaceColor', 'red', 'EdgeColor', 'none')

0059 set(p2, 'FaceColor', 'green', 'EdgeColor', 'none')0060 alpha(0.5); view(45, 45);

0061 xlabel('x'), ylabel('y'), zlabel('z');

0062 title ('Planes in 3D', 'FontWeight', 'bold');



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Duality of Points and Planes

In a plane is defined uniquely by the join of three pointsXi

In a pointXis defined uniquely by the join of three planes i

2.3.2. Planes

Points and Planes at Infinity

In 3 dimensions (analogous to 2 dimensions) there exist the point at

infinityXand the plane at infinity .

Two planes are parallel if their intersection line lies on .

See Matlab Example!

T

1

T

2

T

3

0

X

X

X

3

T

1

T

2

T

3

0

X

3

1

2

3

0

0, with the direction given as

0

1 0

X

X

X

D

2 3 Three-dimensional projective space


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Introduction

Projective transformations of 3-space are analogous to those of the

planar transformations.

His the arbitrary 4x4 homography matrix.

We have 16 elements which account for15 degrees of freedom

(DOF)and one for the matrix scaling


' x Hx

11 12 13 141 1

21 22 23 242 2

31 32 33 343 3

41 42 43 444 4

'

'

'

'

h h h hx x

h h h hx x

h h h hx x

h h h hx x

2 3 Three-dimensional projective space


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Conclusion


Ais an invertible 3x3 matrix

Ris a 3D rotation matrix, sis a scalar

tis a 3D translation vector t=(t1,t2,t3)T

va general 3-vector, va scalar

0T is the null 3-vector

vTv

tAProjective(15dof)

Affine

(12dof)

Similarity

(7dof)

Euclidean

(6dof)

Intersection and tangency

Parallelism of planes,

Volume ratios,

The plane at infinity

The absolute conic

Volume

10

tAT

10

tR

T

s

10

tRT

Transformation

(degree of freedom)

Shape Invariants propertiesH

Literature


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P f Di it l d S h lt t h ik C t Vi i

Literature

[1] http://www.cs.unc.edu/~marc/tutorial/node3.html

[2] Richard Hartley, Andrew Zisserman. Multiple View Geometry in computer vision.

Cambridge university press, 2003

[3] Boguslaw Cyganek, J. Paul Siebert. An Introduction to 3D Computer Vision

Techniques and Algorithms. John Wiley & Sons, Ltd, 2009

[4] Richard Szeliski. Computer VisionAlgorithms and Applications. Springer-Verlag

London Limited, 2011

[5] http://www.cs.clemson.edu/~dhouse/courses/405/

Documents

Projective Geometry and Transformations