Lecture05 Camera Param

8/8/2019 Lecture05 Camera Param

1/21

CAP5415 Computer Vision

Spring 2003

Khurram Hassan-Shafique


2/21

Camera Parameters

X

Y

Z C

Image plane

Focal plane

M

m

CCC ZYX ,,

fvu CC ,,

From Perspective Projection

S

U

Z

Yfv

S

U

Z

Xfu

C

CC

C

CC

!!

!!

-

-

!

-

1

0100

000

000

C

C

C

Z

Y

X

f

f

S

V

U


3/21

Camera Parameters

X

Y

Z C

Image plane

Focal plane

M

m

CCC ZYX ,,

u

v

i

j

I

JJkjIkiv

u

!

!

-

-

!

-

-

-

!

-

1100

0

0

01100

00

00

1

0

0

0

0

c

c

v

u

c

c

v

u

I

I

v

u

vk

uk

v

u

v

u

k

k

v

u

II vu ,

fvu CC ,,


4/21

Camera Parameters

-

-

!

-

1100

0

0

1

0

0

c

c

v

u

I

I

v

u

vk

uk

v

u

-

-

!

-

S

V

U

vk

uk

S

V

U

v

u

new

new

100

0

0

0

0

Equation 3

S

Vv

S

Uu

newI

newI

!

!

S

Vv

S

Uu

!

!


5/21

Camera Parameters

-

-

-

!

-

-

!

-

10100

000

000

100

0

0

100

0

0

0

0

0

0

C

C

C

v

u

v

u

new

new

ZY

X

f

f

vk

uk

S

V

U

vk

uk

S

V

U

Equation 1:

-

-

!

-

10100

000

000

C

C

C

Z

Y

X

f

f

S

V

U


6/21

Camera Parameters

-

-

!

-

10100

00

00

0

0

C

C

C

v

u

ne

ne

Yvfk

ufk

S

-

-

!

-

1010000

00

0

0

C

C

C

v

u

new

new

Z

Y

X

vf

uf

SV

U

vv

uu

fkf

fkf

!

!

Equation 4


7/21

Camera Parameters

Intrinsic Parameters (Do not depend on camera position):

0

0

.4

.3

.2

.1

v

u

fkf

fkf

vv

uu

!!


8/21

Intrinsic Parameters

-

-

!

-

10100

00

00

0

0

C

C

C

v

u

new

new

Z

Y

X

vf

uf

S

V

U

-

!

-

!!

CT

CT

CT

C

T

T

T

CI

MQMQ

MQ

MQQ

Q

PMm3

2

1

3

2

1


9/21

Calculating Camera Center (from

Intrinsic Parameters)

-

!

-

!

CT

CT

CT

C

T

T

T

I

MQ

MQ

MQ

M

Q

Q

Q

m

3

2

1

3

2

1

? A 011

!

-

d!

-

CpP

CP

0!pCP

pPC 1d!


10/21

Extrinsic Parameters

X

Y

Z C

Image plane

Focal plane

M

m

CCC

ZYX ,,

u

v

i

j

I

JJkj

Iki

v

u

!

!

II vu ,

fvu CC ,,Z

Y

X

WWW ZYX ,,

By Rigid Body Transformation:

WC

W

W

W

C

C

C

DMMZ

YR

Z

Y!

-

-

!

-

v

vv

1

10

1

31

1333


11/21

Camera Model WIWCCI

PDMmDMMPMm !!! ,

-

-

-

!

-

v

vv

1

100100

00

00

31

1333

0

0

W

W

W

v

une

ne

Z

YRvf

uf

S

-

!

-

!

Z

Y

rrr

rrrrrr

R andLet

333231

232221

131211


12/21

Camera Model

Z

WWW

Y

WWW

v

I

Z

WWW

X

WWW

u

I

TZrYrXr

TZrYrXrfvv

TZrYrXr

TZrYrXrfuu

!

!

333231

2322210

333231

1312110


13/21

Suggested Reading Chapter 3, Olivier Faugeras, "Three Dimensional

Computer Vision", MIT Press, 1993

Chapter 2, David A. Forsyth and Jean Ponce, "ComputerVision: A Modern Approach"

Chapter 1, Mubarak Shah, "Fundamentals of Computer

Vision"


14/21

Estimating Camera Parameters

11

,

vu

22 ,vu

33,vu

NN vu ,

111

,,Z

YX

222 ,, ZYX

333 ,, ZYX

NNN ZYX ,,


15/21


Z

W

i

W

i

W

i

Y

W

i

W

i

W

iv

I

i

Z

W

i

W

i

W

i

X

W

i

W

i

W

iu

I

i

TZrYrXr

TZrYrXrfvv

TZrYrXr

TZrYrXr

fuu

!

!

333231

2322210

333231

1312110

iiiii ZYXvu ,,and,pairingcorrespondeachor

secondbyequationirstthedividingand0andet 00 !!! vuff

v

uE

X

W

i

W

i

W

i

I

iY

W

i

W

i

W

i

I

iTZrYrXrvTZrYrXru ! 131211232221 E

0131211232221 ! X

I

i

W

i

I

i

W

i

I

i

W

i

I

iY

I

i

W

i

I

i

W

i

I

i

W

i

I

i TvrZvrYvrXvTurZurYurXu EEEE


16/21

Estimating Camera Parameters iiiii ZYXvu ,,and,pairingcorrespondeachor

087654321 ! vvvZvvYvvXvvuvZuvYuvXu

I

i

W

i

I

i

W

i

I

i

W

i

I

i

I

i

W

i

I

i

W

i

I

i

W

i

I

i

XY TvTv

rvrv

rvrv

rvrv

E

E

E

E

!!

!!

!!

!!

84

137233

126222

115211

0!Av? A

Tvvvvvvvvv 87654321!

Compute vby SVD decomposition ofA=UDVT (The solution

vectoris the column ofVcorresponding to null(or smallest)

singular value) in D.


17/21


Computing Scale Factor

TXYTrrrTrrrv

vEEEE 131211232221

thenector,

solution vobtainedthebeLet!

KK !!

!

223

222

221

223

22

21

2

23

2

22

2

21 havewe,1Since

rrrvvv

rrr


18/21


Computing E

KEEK !!!

2

23

2

22

2

21

222

7

2

6

2

5

223

222

221 havee,1ince

rrrvvv

rrr

Computing Sign of K

signstheeverse

0I 131211 " xWWW TZrYrXrx


19/21


Computing third row of Rotation Matrix

213 rrr v!

ComputingZvu Tff and

XWiWiWiuZWiWiWiIi TZrYrXrfTZrYrXru ! 131211333231 iiiii

ZY

Xvu

,,

and,

pairingcorrespondeachFor

bf

TA

u

Z!

-

olve


20/21


Computing Image Center ,vu

Let Tbe a triangle on the image plane defined by the threeVanishing points of three mutually orthogonal sets of

Parallel lines in space. The image center is the orthocenter of

T


21/21

Suggested Readings: Chapter 6, Emanuele Trucco, Alessandro Verri,

"Introductory Techniques for 3-D Computer Vision",

Prentice Hall,

1998 Chapter 1, Mubarak Shah, "Fundamentals of Computer

Vision"

Documents

Lecture05 Camera Param