Camera Models
A camera is a mapping between the
3D world and a 2D image
The principal camera of interest is
central projection
Central Projection
• Cameras modeling Central projection are
specialization of the general projective camera. It
is examined using the tools of projective
geometry.
• Specialized models fall into two classes:
• (1) camera with a finite centre.
(2) cameras with centre at infinity
Affine camera is an important example
Finite cameras
• The basic pin hole model(most specialized)
Pinhole camera
• A point in space with coordinates
• X =(x,y,z)T is mapped to a point on the
image plane.
• (x,y,z)T ____ ( f x/z , f y/z)T (5.1)
• The centre of projection is called the
camera centre
Principal axis and principal point.
• The line from the camera centre
perpendicular to the image plane is called
the principal axis and point where it
intersects the image plane is called the
principal point.
• The plane parallel to the image plane and
passing through the camera centre is called
the principal plane
Central projection using homogenous
coordinates
• If the world and image points are
represented by homogenous vectors, then
• P = diag( f , f, 1) [I ! 0]
x
 
 fx 
f
 
 y

 z    fy   
z
 




1
 
f
x
0  
 y

0  
z
1 0  
1
Image and camera coordinate systems
Principal point offset
• The expression (5.1) assumes that the origin
of the coordinates in the image plane is the
principal point. In practice, it may not be as
follows.
• (x,y,z)T ____ ( f x/z + px , f y/z + py )T
• Where ( px, py) T are the coordinates of the
principal point.
Principal point offset 2
x
 
 fx 
f
 
 y


f
y

 
z

 z 
 

 
1
 
f
1
x
0  
 y

0  
z
0  
1
• Now writing
f
K



f
px 
py 

1

(5.3)
Principal point offset 3
• Then (5.3) has the concise form
•
x = K [ I ! 0] xcam (5.5)
• (x, y,z, 1) as Xcam as the camera is assumed to be
located at the origin of a Euclidean coordinate
system with the principal axis of the camera
pointing straight down the z-axis. K is called the
camera calibration matrix.
The Euclidean transformation between
the world and camera coordinate frame
Camera rotation and translation

• C represents the coordinates of the camera
centre in the world coordinate frame.
•

X is an inhomogeneous 3-vector in world
coordinate frame
• R is 3x3 rotation axis



Xcam  R( X - C)
Camera Rotation and translation
X

 


 Y  

R
R
C
R
R
C
X cam  
   
X
 0
1  Z   0
1 
1
 
(5.6)
•
• Putting (5.5) and (5.6) together leads to
•

x  KR [ I ! - C] X
(5.7)
Camera Rotation and translation 2
• The parameter contained in K are called internal
camera parameters

• The parameter of R and C
which relate the orientation and position to a
world frame are called the external parameters. A
more convenient form of the camera matrix is
•
•
P = K [ R ! t]

t -RC
(5.8)
CCD Cameras
• The CCD camera may have rectangular pixels,
where unit distances in x and y directions are mx
and my, then
x0 
a x
K  
a y y 0 

 x0 = mx px , and y0 = my py
 ax = f mx , and ay = f my
1 
Finite projective camera
• S is the skew parameter
• A camera

P  KR [ I ! - C]
(5.11)
is called a finite projective camera. It has 11
degree of freedom, same as a 3x4 matrix
defined up to an arbitrary scale
General projective Camera
The projective camera
A general projective camera P maps world
point X to image points x according to
x = PX
Camera centre
Camera centre 2
Column vectors
• The columns of P are pi , i = 1, 2, 3, 4
• Then p1 , p2 , p3 are the vanishing points of the
world coordinate x , y, z axes respectively.
• For example: x axis ahs direction D =(1,0,0,0),
which is imaged at p1 = PD
• The column p4 is the image of the world origin
The three image points defined by the
columns pi, i= 1, 2, 3 of the projection
matrix are the vanishing points of the
directions of the world axes
Row vectors
Principal plane
Axis planes
Summary of the properties of a
projective camera P=[M ! p4]
M 
Summary of the properties of a
projective camera 2
Principle point
Principle axis
Principle axis 2
Principle axis 3
Two of the three planes defined by the
rows of the projection matrix
Action of a projective camera on points
Back projection of points to rays
Back projection of points to rays 2
Back projection of points to rays 3
Depth of points
Depth of points 2
Linear optics
Decomposition of the camera matrix
Finding camera orientation and internal
parameters
Finding camera orientation and internal
parameters 2
Example 5.2 The camera matrix
Euclidean vs Projective space
Euclidean and affine interpretation
Cameras at infinity
Affine cameras
Increasing focal length from left to right
Affine camera 2
Affine camera 3
Affine camera 4
Focal length increases as the object
distance between the camera increases
The image remains the same size, but
perspective effects diminish
Perspective vs weak perspective
projection
Orthographic projection
Orthographic projection 2
Scaled orthographic projection
Weak perspective projection
Affine camera
A general affine camera
A general affine camera 2
More properties of the affine camera
General cameras at infinity
Pushbroom camera
Pushbroom camera 2
Pushbroom camera 3
Pushbroom camera – mapping of line
Line camera
Line camera 2
Acquisition geometry of a pushbroom
camera