Computer Vision Geometric Camera Models and Camera Calibration

Computer Vision

Geometric Camera Models and

Camera Calibration

Bahadir K. Gunturk 2

Coordinate Systems

Let O be the origin of a 3D coordinate system spanned by the unit vectors i, j, and k orthogonal to each other.

i

j

kO

Px OP i

��

y OP j��

z OP k��

OP x y z i j k��

x

y

z

PCoordinate vector


Homogeneous Coordinatesn

a

b

c

HH

O

P

x

y

z

P

0HP OH ��

2 2 2( ) 0ax by cz a b c

0ax by cz d

0

1

x

ya b c d

z

Homogeneous coordinates 0 H P

P

TH


Coordinate System Changes

Translation

BPAP



Rotation

where

Exercise: Write the rotation matrix for a 2D coordinate system.

ˆ

ˆ

ˆ

AX B

AY B

AZ B

B i P

B j P

B k P



Rotation + Translation

In homogeneous coordinates

Rigid transformation matrix


Perspective Projection

Perspective projection equations

' ' 'x y z

x y z


Intrinsic Camera Parameters

Perspective projection

( , )u v x

P y

z



We need take into account the dimensions of the pixels.

CCD sensor array



The center of the sensor chip may not coincide with the pinhole center.



The camera coordinate system may be skewed due to some manufacturing error.




These five parameters are known as intrinsic parameters



In a simpler notation: 1K

zp 0 P

With respect to the camera coordinate system


Extrinsic Camera Parameters

Translation and rotation of the camera frame with respect to the world frame


1K

zp 0 PUsing , we get 1

11

1

X

C CYW W

TZ

Wu

WR Ov K

Wz

00


Combine Intrinsic & Extrinsic Parameters

We can further simplify to

1

11

1

X

C CYW W

TZ

Wu

WR Ov K

Wz

00

1

11

X

YC CW W

Z

Wu

Wv K R O

Wz

3x4 matrix with 11 degrees of freedom: 5 intrinsic, 3 rotation, and 3 translation parameters.


Camera Calibration

Camera’s intrinsic and extrinsic parameters are found using a setup with known positions in some fixed world coordinate system.


Camera Calibration

courtesy of B. Wilburn

XZ

Y i

i

u

v

wiwiwi

x

y

z


Camera Calibration

Mathematically, we are given n points

We want to find M

i

i

u

v

wiwiwi

x

y

z

1,...,i nand where


Camera Calibration

We can write


Camera Calibration

Scale and subtract last row from first and second rows

to get


Camera Calibration

Write in matrix form for n points

to get

Let m34=1; that is, scale the projection matrix by m34.


Camera Calibration

The least square solution of is

From the matrix M, we can find the intrinsic and extrinsic parameters.


Camera Calibration

Consider the case where skew angle is 90. Since we set m34=1, we need to take that into account at the end.

Notice that

Since R is a rotation matrix,

Therefore,


Camera Calibration

We get

See Forsyth & Ponce for details and skew-angle case.


Applications

courtesy of Sportvision

First-down line


ApplicationsVirtual advertising

courtesy of Princeton Video Image


Parameters of a Stereo System Intrinsic Parameters

Characterize the transformation from camera to pixel coordinate systems of each camera

Focal length, image center, aspect ratio

Extrinsic parameters Describe the relative

position and orientation of the two cameras

Rotation matrix R and translation vector T

pl

pr

P

Ol Or

Xl

Xr

Pl Pr

fl fr

Zl

Yl

Zr

Yr

R, T


Calibrated Camera

( , ,1)[ ( ')] 0 with

' ( ', ',1)

T

T

p u vp t Rp

p u v

' 0Tp Ep

' 0 with Tp Ep E t R Essential matrix


Uncalibrated Camera

' 0 pE p

scoordinate camerain ' and toingcorrespond scoordinate pixelin points ' and pppp

p'MppMp '' and 1int

1int

1int int

' 0

with '

T

T

p F p

F M E M

Fundamental matrix

Documents

Computer Vision Geometric Camera Models and Camera Calibration