Camera Model - Computer Vision - Lecture Slides | CAP 5415, Study notes of Computer Science

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Lecture-

Camera Model

Homework (Due 1/25/01)

• Exercises in section 1.11 in “Fundamentals

of Computer Vision”.

– 1 to 5

– 8 to 9

Thursday’s class Room 103 CSB

Computer Vision Lab

• Digitization

• Image tools in C

– Reading an image

– Writing an image

– Displaying an image

Rotation Around an Arbitrary Axis

(Rodriguez’s Formula)

V =(V .n)n+(V−(V.n)n )

V ⊥′ =cos θ (V−(V.n)n)+sin θ (n×(V−(V.n)n )

V ′^ =V⊥′+V C′

V ′ =V+sin θ n×V+( 1 −cos θ )n×(n×V ) n × (n×V)=(V.n)n−V

VC ′=(V.n) n

V ′=cos θ V+sin θ n×V+( 1 −cos θ )(V.n) n

n ×(n ×V)+V=(V.n) n

Rotation Around an Arbitrary Axis

(Rodriguez’s Formula)

r = ||r||||rr||= θ n



 

 

 

 −

−

−

= + + −

0

0

0 ( )

|| || ( 1 cos ) () || || ( , ) sin () 2

2

y x

z x

z y r r

r r

r r X r

r

X r r R r θ I θ X r θ

Camera Model

Ch =PCR−X φ R −Z θ GW h

4

2

4

1

h

h

h

h

C

y C

C

x C

31 32 33 34

21 22 23 24

11 12 13 14

4

3

2

1

Z

Y

X

a a a a

a a a a

a a a a

a a a a

Ch

Ch

Ch

Ch

C h AWh

4 41 42 43 44

2 21 22 23 24 4

1 11 12 13 14 4

Ch a X a Y a Z a

Ch a X a Y a Z a Chy

Ch a X aY a Z a Chx

Camera Model

21 22 23 24 41 42 43 44

11 12 13 14 41 42 43 44

a X a Y a Z a a Xy a Yy a Zy a y

a X aY a Z a a Xx aYx a Zx a x

4 41 42 43 44

2 21 22 23 24 4

1 11 12 13 14 4

Ch a X a Y a Z a

Ch a X a Y a Z a Chy

Ch a X aY a Z a Chx

Camera Model

21 22 23 24 41 42 43 44

11 12 13 14 41 42 43 44

a X a Y a Z a a Xy a Yy a Zy a y

a X aY a Z a a Xx aYx a Zx a x

21 22 23 24 41 42 43 44

21 2 222 232 24 41 22 422 2 432 2 44 2

21 1 221 23 1 24 41 11 4211 4311 441

11 12 13 14 41 42 43 44

11 2 122 132 14 41 22 4222 43 22 442

11 1 121 131 14 41 11 4211 4311 441

n n n nn nn nn n

n n n nn nn n n n

aX aY a Z a a Xy aYy aZy a y

aX aY a Z a aX y aYy aZy a y

aX aY aZ a aXy aYy a Zy a y

aX aY aZ a a Xx aYx aZx ax

aX aY aZ a a Xx aYx aZx a x

aX aY a Z a a Xx aYx a Zx ax

M

M

One point

n points 2n equations, 12 unknowns

Camera Model Revisited

z

y

x c w

z

y

x c w

Z T

Y T

X T

r r r

r r r

r r r P RP T

Z

Y

X

T

T

T

r r r

r r r

r r r P RTP

31 32 33

21 22 23

11 12 13

31 32 33

21 22 23

11 12 13

[ P T]

R

R

R

P RP T TT w

T

c w  −

3

2

1

3

2

3

1

R P T y fR P T

R P T

x fR P T T w c T w

T w c T w −

im y y

im x x

y y o s

x x o s

3

2

3

1

R P T

y o s fR P T

R P T

x o s fR P T

T w im y y T w

T w im x x T w

image coordinates

camera coordinates

image center (in pixels)

effective size of pixels (in

millimeters) in the horizontal and

vertical directions.

( xim ,yim ) ( x,y ) ( sx ,sy )

( ox ,oy )

Camera Model Revisited

Camera Model Revisited

r r r R T

fr fr fr fRT

fr fr fr fRT

M

r r r RT

r r r RT

r r r RT

M

sf o

sf o

M

T

T

T

T

T

T ext

x y

x x

31 32 33 3

21 22 23 2

11 12 13 1

31 32 33 3

21 22 23 2

11 12 13 1

int

int 3

2

1

Z

Y

X

M M

Ch

Ch

Ch

ext