Vision 7CameraModelA S09

8/19/2019 Vision 7CameraModelA S09

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I‐Chen Lin, Assistant Professor

Computer Vision:

7.

Camera

Models

(a)

Dept. of CS, National Chiao Tung University


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• Coordinates x and y

• For

a

more

general

coordinate

representation,

we

usually

use

a vec or orm.

j

⋅=p =(x,y)

T

jop y ⋅=

i

o


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• Line l : ax+by =c (a,b)T ( x,y )=c

j

p=(x,y)Tc

i

(a,b)T


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• Uniform treatment of points and lines

• Line

‐point

incidence:

l T

p=0

j

stay the same when scaled:

p=(x,y,1)T~(kx,kx,k)Tc

i

(a,b)T l=(a,b,‐c)T~(ka,kb,‐kc)T


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.

• Furthermore, …

– We use

homogenous

coordinates

to

combine

rotation

and

translation into same framework: matrix transformation.

–

It allows

easy

transformation

between

“frames”

– common

between computer vision and graphics.


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• To apply the camera model, objects in the scene must be

expressed

in

camera coordinates.

y

Camera

w

c

T x z

World

Coordinates

Coordinates z

Calibration target looks tilted from camera

viewpoint. This can be explained as a

difference in coordinate s stems.

x


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‐ ‐

body.

• .

• R, t : the extrinsic parameters.

y (0, 1, 0)

v (vx, vy, vz)

w (wx, wy, wz)

v’w’

optic axisc

x (1,0,0)

u (ux, uy, uz)

u’


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•

• AP is coordinates of P in frame A

• BP

is

coordinates

of

P

in

frame

B

A

A x

A

⎜ ⎟

A A A= y A

z

⎝

⎜

⎠

⎟= x • A y • A z • A A

A

i A

O A


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B

P=

A

P+

B

O A( )k B

j BO B

A B

A

i A

O A


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• Using homogeneous coordinates, translation can be

expressed

as

a

matrix

multiplication. B A B

AP P O= +

1 0 1 1

B B A

AP I O P=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

• Translation is commutative


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A B x x⎛ ⎞ ⎛ ⎞( ) ( ) A B A A A B B B

A B

OP i j k y i j k y

z z

= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

uuuv

B B A=

means describing

frame

A

in

A R The coordinate system of

frame B


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. . . A B A B A B

B

⎡ ⎤⎢ ⎥

i i j i k i

. . .

. . .

A A B A B A B

A B A B A B⎢ ⎥⎣ ⎦i k j k k k

B B B

A A A⎡ ⎤= ⎣ ⎦i j k

A T

B

A T

⎡ ⎤

⎢ ⎥

i

B

A T

B

=⎢ ⎥k

Orthogonal matrix: R-1 = RT


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What is the rotation matrix?


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’ ’’ , ,

• Heading, pitch roll: world Z, new X, new Y

’ ,

focus on that

⎥⎥

⎢⎢

−

= 0)cos()sin(

s ncos

)( θ θ θ Z R

⎥⎥⎥

⎢⎢⎢ −=

cossin0

)sin()cos(0)( φ φ φ X R

⎥

⎤

⎢

⎡ )sin(0)cos( κ κ

⎥⎦⎢⎣− )cos(0)sin( κ κ Y

Remind: applying coordinate rotation φ is equal to applying rotation – φ to objects.


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• s ng omogeneous coor nates, rotat on can e expresse as a matrix multiplication.

B B A

AP R P=

1 0 1 1 A=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

• Rotation is not commutative


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B B A B A A


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.

.

1 0 B B B AP O R P⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

1 0 1 0 1 1 B B A

=⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 1

A A

T = ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦0

1 1

B

AT =⎢ ⎥ ⎢ ⎥


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• Linear transformation of perspective projection coordinate.

⎤⎡ X

⎥⎥

⎢⎢

⎥⎥

⎢⎢==⎥

⎥⎢⎢= 0010]0[

Z

Y P I v p

⎦⎣ 1

• Recover image (normalized) coordinate by projection.

X uˆ

Y v

Z w

==

==

ˆ Z w


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• eca perspect ve pro ect on

http://commons.wikimedia.org/wiki/

File:Taiwan_HighSpeedRail_Train_Business_Class_Car.JPG


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)!OK(oftenconstantisdepthPretend

=ˆ Z

ur

=ˆ Z v r

⎥

⎤

⎢

⎡

⎤⎡⎤⎡ 0001

X

u

⎥⎥

⎢⎢

⎥⎥⎦⎢

⎢⎣

=

⎥⎥⎦⎢

⎢⎣ 000

0010 Z

Z Z

w

v

r

r


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O


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Pixels are on a grid of a certain dimension

O

[f] = m (in meters)

[k] = pixels/m


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We put the pixel coordinate origin on topleft

O

0


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a r xna ra o

X ⎤⎡

×

0

0

0010]0[ Z

Y vP I K v p

⎥⎥

⎢⎢

⎥⎥

⎢⎢

⎥⎥

⎢⎢==⎥

⎥⎢⎢= β

1

w

⎦⎣

0ˆ

u

sY X u

u +

+

==

α

0ˆ v

Y vv +==

β skew


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cot

a r xna ra o

xu ⎤⎡⎤⎡ −

×

θ α α

0 0010sin

]0[1

z

yvP I K

zv p

⎥⎥

⎢⎢

⎥⎥

⎢⎢

⎥⎢==⎥⎥

⎢⎢=

θ

β

11

⎦⎣⎦⎣

0

cot

u

x

u +

−

=

θ α α

0v y

v += β


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x

.

• Perspective projection mapping (including intrinsic and

extrinsic parameters).

y

x

O R

uuc

w

c

w

0001cot

10 ⎥

⎤⎢⎡

⎤⎡⎥⎤

⎢⎡

⎥⎤

⎢⎡ −

⎥⎤

⎢⎡

β

θ α α

z z1

1001001sin1

0

⎥⎥⎦

⎢⎢⎣

⎦⎣⎥⎦⎢⎣⎥⎦⎢⎣

==

⎥⎦⎢⎣

=θ

[ ] MP z

PO RK

z

c

w

c

w

11==

5+6 DOF = 11 !

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Vision 7CameraModelA S09