1

Basic geometric concepts to understand

• Affine, Euclidean geometries (inhomogeneous coordinates)

• projective geometry (homogeneous coordinates)

• plane at infinity: affine geometry

2

prod.dot with nn AE inf.at pts nn RPnA

nR

Naturally everything starts from the known vector space

Intuitive introduction

3

• Vector space to affine: isomorph, one-to-one

• vector to Euclidean as an enrichment: scalar prod.

• affine to projective as an extension: add ideal elements

Pts, lines, parallelism

Angle, distances, circles

Pts at infinity

4

P2 and R2

0 inf.at line and

,0 pts finitefor :

3

3

3

1

3

1

3

2

1

x

x

x

xx

x

x

x

x

22 RP

01 pts finite

1

: 3

xy

x

y

x22 PR

Relation between Pn (homo) and Rn (in-homo):

Rn --> Pn, extension, embedded in

Pn --> Rn, restriction,

5

Examples of projective spaces

• Projective plane P2

• Projective line P1

• Projective space P3

6

Pts are elements of P2

Projective plane

4 pts determine a projective basis 3 ref. Pts + 1 unit pt to fix the scales for ref. pts

Relation with R2, (x,y,0), line at inf., (0,0,0) is not a pt

Pts at infinity: (x,y,0), the line at infinity

Space of homogeneous coordinates (x,y,t)

Pts are elements of P2

7

Line equation:

21TT

21 ,00),,det( xxlxlxxx

Lines:

21 xxx

Linear combination of two algebraically independent pts

Operator + is ‘span’ or ‘join’

8

Point/line duality:

• Point coordinate, column vector

• A line is a set of linearly dependent points

• Two points define a line

• Line coordinate, row vector

• A point is a set of linearly dependent lines

• Two lines define a point

• What is the line equation of two given points?• ‘line’ (a,b,c) has been always ‘homogeneous’ since high school!

0T xl 0T lx

9

Given 2 points x1 and x2 (in homogeneous coordinates), the line connecting x1 and x2 is given by

Given 2 lines l1 and l2, the intersection point x is given by

21T xxl

T2

T1 llx

NB: ‘cross-product’ is purely a notational device here.

10

Conics: a curve described by a second-degree equation0..... T2 Cxxcbxyax

• 3*3 symmetric matrix

• 5 d.o.f

• 5 pts determine a conic

Conics

11

Projective line

Finite pts:

Infinite pts: how many?

A basis by 3 pts

Fundamental inv: cross-ratio

Homogeneous pair (x1,x2)

12

• Pts, elements of P3

• Relation with R3, plane at inf.

• planes: linear comb of 3 pts

• Basis by 4 (ref pts) +1 pts (unit)

321 xxxx

Projective space P3

13

planes

0),,,det( 321 xxxx

044332211 xuxuxuxu

0

3

2

1

u

x

x

x

T

T

T

In practice, take SVD

14

Key points

• Homo. Coordinates are not unique

• 0 represents no projective pt

• finite points embedded in proj. Space (relation between R and P)

• pts at inf. (x,0) missing pts, directions

• hyper-plane (co-dim 1):

• dualily between u and x,

0T xu

0T ux

15

110

1

1222 y

x

y

xtR

2D general Euclidean transformation:

110

1

1222 y

x

y

xtA

2D general affine transformation:

t

y

x

t

y

x

33A

2D general projective transformation:

Introduction to transformation

ColinearityCross-ratio

16

Projective transformation= collineation = homography

Consider all functions nn : f PP

All linear transformations are represented by matrices A

Note: linear but in homogeneous coordinates!

1)(n1)(n A

17

How to compute transformatins and canonical projective coordinates?

18

Geometric modeling of a camera

u

v

X

u

O

X’

u’

P3

P2

How to relate a 3D point X (in oxyz) to a 2D point in pixels (u,v)?

19

Z

Y

f

y

Z

X

f

x ,

X

Y

Z

xy

u

v

X

x

O

f

Camera coordinate frame

20

xo

y

u

v

X

Y

Z

x y

u

v

X

xO

f

Image coordinate frame

21

• Focal length in horizontal/vertical pixels (2) (or focal length in pixels + aspect ratio)• the principal point (2)• the skew (1)

5 intrinsic parameters

one rough example: 135 film

In practice, for most of CCD cameras:

• alpha u = alpha v i.e. aspect ratio=1• alpha = 90 i.e. skew s=0• (u0,v0) the middle of the image• only focal length in pixels?

22

Xw Yw

Zw

Xw

X

Y

Z

xy

u

v

X

x

O

f

World (object) coordinate frame

23

World coordinate frame: extrinsic parameters

1

1

1w

w

w

c

c

c

Z

Y

X

Z

Y

X

0

tR

Relation between the abstract algebraic and geometric models is in the intrinsic/extrinsic parameters!

6 extrinsic parameters

24

11

11

4333 Z

Y

X

Z

Y

X

v

u

pixel

C0

tR0IK

Finally, we have a map from a space pt (X,Y,Z) to a pixel (u,v) by

25

It turns the camera into an angular/direction sensor!

Direction vector: uKd -1

What does the calibration give us?

uKx -1Normalised coordinates:

26

Camera calibration

ii Xu Given

• Estimate C• decompose C into intrinsic/extrinsic

from image processing or by hand

27

Decomposition

• analytical by equating K(R,t)=P

28

Pose estimation = calibration of only extrinsic parameters

33ii , KXu• Given

• Estimate R and t

29

3-point algebraic method

• First convert pixels u into normalized points x by knowing the intrinsic parameters

• Write down the fundamental equation:

• Solve this algebraic system to get the point distances first

• Compute a 3D transformation

222 cos2 ijjiijji dxxxx

3 reference points == 3 beacons

30

given 3 corresponding 3D points:

3D transformation estimation

• Compute the centroids as the origin• Compute the scale • (compute the rotation by quaternion)• Compute the rotation axis• Compute the rotation angle

31

Linear pose estimation from 4 coplanar points

Vector based (or affine geometry) method

O

A

B

CD

x_a

x_d

32

Midterm statistics

Total 71.80392157 16.30953047

Q1: 14.98039216 5.82920302

Q2: 12.03921569 6.141533308

Q3: 14.56862745 4.817696138

Q4: 12.35294118 7.638909685

Q5: 14.90196078 7.105645367

Q6: 7.254901961 4.511510334

0~59 7

60~69 12

70~79 17

80~89 8

90~99 5

100 2