Camera model
Relation between pixels and rays in space
?
Pinhole camera
Pinhole camera model
Camera is a mapping between 3D world (object space) and 2D image;
Camera model: matrix representing camera mapping; interested in central projection
Plane through camera
Center C, parallel to image
Plane is called principle
Plane of the camera
P = Intersection of principle axis
With image plane
Focal plane
Z=f
Line from camera
Center perp. To image
plane
T
( X , Y , Z )  ( fX / Z , fY / Z )
X
   fX   f
Y    
 Z    fY   
   Z  
1
 
f
T
X
0  
Y 

0  
Z
1 0 
1
linear projection in homogeneous coordinates!
Pinhole camera model
X
X
 fX   f
 10  0 
  
Y 
Y 




0
1
0
 fY x  PX f


  Z
 Z 
 Z  
1  0 1 0 
  
1
1
P  diag ( f , f ,1)I | 0
Principal point offset
So far, assumed the origin of the coordinate in the image
plane is the Same as the principal point
T
T
( X , Y , Z )  ( fX / Z + px , fY / Z + p y )
( px , p y )
T
principal point
X
   fX  Zp x   f
 
Y  
 Z    fY  Zp x   
 
  
Z
 
1
 
f
px
py
1
X
0  
Y 

0  
Z
0 
1
Principal point offset
px
 fX  Zp x   f

 
py
 fY  Zp x x  KI | 0fXcam

 
Z
1

 
f
K  

f
X
0  
Y 

0  
Z
0 
1
px 
p y  calibration matrix
1 
Camera rotation and translation
Points in space are usually expressed in terms of world coordinate frame, i.e. Euclidean
~
X Inhomogeneous vector in world coordinate
~
C Coordinates of camera center in world coord
X camInhomogeneous 3 vector in camera coord.
Xcam

~ ~
 R X-C

X
~
~  
R - RC Y  R - RC 
X cam  
X
   

1 
1  Z   0
0
General mapping for pinhole camera
1
 
~
x  KI | 0Xcam
x  KR I | -C X
~
x = PX
P  KR | t 
t  -RC
 
9 deg of freedom: f, px,py,3 rot, 3 trans
CCD camera
Pinhole camera model assumes image coordinates are Euclidean coordinates with equal scales in both axial
directions; CCD cameras does not have square pixels
p x f
mx x





KK
my y p y

11
f
px 

py 
1 
mx is the number of pixels per unit distance in image coordinates in the x direction
my is the number of pixels per unit distance in image coordinates in the y direction
αx = f mx and αy = f my : focal length of the camera in terms of the pixel dimensions in x and y directions
10 degrees of freedom for CCD camera
When is skew non-zero?
 x s
K  
x

px 
p y 
1 
arctan(1/s)
1
g
for CCD/CMOS, always s=0
Example of skew: negative enlarging
General projective camera
 x s
K  
x

~
P  KR I | C


px 
p y 
1 
11 dof (5+3+3)
non-singular
P  KR | t 
intrinsic camera parameters
extrinsic camera parameters
Camera parameters
A camera is described by several parameters
•
•
•
•
Translation T of the optical center from the origin of world coords
Rotation R of the image plane
focal length f, principle point (x’c, y’c), pixel size (sx, sy)
blue parameters are called “extrinsics,” red are “intrinsics”
X 
Y 
   ΠX
Z 
 
1
The projection matrix models the cumulative effect of all parameters
Useful to decompose into a series of operations
Projection equation
•
•
 sx  * * * *
x  sy   * * * *
 s  * * * *
identity matrix
 fsx

   0

 0
0
 fsy
0
intrinsics
•

x'c 1 0 0 0

 R
y'c 0 1 0 0 3x 3

 0

0 0 1 0
1 
 1x 3
projection
rotation
0 3x1I 3x 3

1 

01x 3


1 

T
3x1
translation
The definitions of these parameters are not completely standardized
–
especially intrinsics—varies from one book to another
Projection matrices for Orhographic and scaled
orthographic projections
Orthographic projection
r1T t1 
P  r1T t 2 
(5dof)


0
1


Scaled orthographic projection
r1T t1 
P  r1T t 2 


 0 1 / k 
(6dof)
Radial distortion
• Due to spherical lenses (cheap)
• Model:
R
R
 x
( x, y )  (1  K1 ( x 2  y 2 )  K 2 ( x 4  y 4 )  ...)  
 y
straight lines are not straight anymore
http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpg
Camera model
Relation between pixels and rays in space
?
Projector model
Relation between pixels and rays in space
(dual of camera)
?
Affine cameras
Track back while zooming in
Keep object of interest same size