Camras Digitales

Geometric Camera Parameters
• What assumptions have we made so far?
- All equations we have derived for far are written in the camera reference frames.
- These equations are valid only when:
(1) all distances are measured in the camera’s reference frame.
(2) the image coordinates have their origin at the principal point.

- In general, the world and pixel coordinate systems are related bya set of physical
parameters such as:
* the focal length of the lens
* the size of the pixels
* the position of the principal point
* the position and orientation of the camera

-2• Types of parameters (Trucco 2.4)
- Two types of parameters need to be recovered in order for us to reconstruct the 3D
structure of a scene from the pixel coordinates of its image points:
Extrinsic cameraparameters: the parameters that define the location and orientation
of the camera reference frame with respect to a known world reference frame.
Intrinsic camera parameters: the parameters necessary to link the pixel coordinates of
an image point with the corresponding coordinates in the camera reference frame.
Object Coordinates (3D)

World Coordinates (3D)
extrinsic camera
parameters
CameraCoordinates (3D)

Image Plane Coordinates (2D)

Pixel Coordinates (2D, int)

intrinsic camera
parameters

-3-

• Extrinsic camera parameters
- These are the parameters that identify uniquely the transformation between the
unknown camera reference frame and the known world reference frame.
- Typically, determining these parameters means:
(1) finding the translation vector betweenthe relative positions of the origins of
the two reference frames.
(2) finding the rotation matrix that brings the corresponding axes of the two
frames into alignment (i.e., onto each other)

-4- Using the extrinsic camera parameters, we can find the relation between the coordinates of a point P in world ( P w ) and camera ( P c ) coordinates:
 r 11
P c = R( P w − T ) where R =  r 21

 r31

r 12
r 22
r 32

r 13 
r 23 

r 33 

 Xw 
 Xc 
- If P c =  Y c  and P w =  Y w , then



 Zw 
 Zc 
 X c   r 11
Y  = r
 c   21
 Z c   r 31

r 12
r 22
r 32

r 13   X w − T x 
r 23   Y w − T y 


r 33   Z w − T z 

or
X c = RT ( P w − T )
1
Y c = RT ( P w − T )
2
Z c = RT ( P w − T )
3
where RT corresponds to thei -th row of the rotation matrix
i

-5-

• Intrinsic camera parameters
- These are the parameters that characterize the optical, geometric, and digital characteristics of the camera:
(1) the perspective projection (focal length f ).
(2) the transformation between image plane coordinates and pixel coordinates.
(3) the geometric distortion introduced by the optics.
From CameraCoordinates to Image Plane Coordinates
- Apply perspective projection:
Xc
RT ( P w − T )
1
x= f
=f T
,
Zc
R3 ( P w − T )

Yc
RT ( P w − T )
2
=f T
y= f
Zc
R3 ( P w − T )

From Image Plane Coordinates to Pixel coordinates

x im

pixel frame

P

yim

Yc
Camera
Frame

y

Xc

Yw
optical axis

Zc
center of
perspective projection

o ,oy
x
image
plane
frame

xWorld
Frame
principal point

Xw
Zw

x = − ( x im − o x ) s x or x im = − x / s x + o x
y = − ( y im − o y ) s y or y im = − y / s y + o y
where (o x , o y ) are the coordinates of the principal point (in pixels, e.g., o x = N /2,
o y = M /2 if the principal point is the center of the image) and s x , s y correspond
to the effective size of the pixels in the horizontal and verticaldirections (in millimeters).

-6- Using matrix notation:
x
 −1/ s x
 im  
 y im  =  0
1  0

0
−1/ s y
0

ox  x

oy   y 

1  1

Relating pixel coordinates to world coordinates
RT ( P w − T )
1
,
−( x im − o x ) s x = f T
R3 ( P w − T )

RT ( P w − T )
2
−( y im − o y ) s y = f T
R3 ( P w − T )

or
x im

RT ( P w − T )
1
= − fs x T
+ ox,
R3 ( P...