Multiple View Geometry Richard Hartley and Andrew Zisserman CVPR June 1999

240 Pages·2001·4.86 MB·English

Multiple View Geometry
Richard Hartley and Andrew Zisserman
CVPR June 1999
Part I: Single and Two View Geometry
The main points covered in this part are:
• ×
A perspective (central) projection camera is represented by a 3 4 matrix.
•
The most general perspective transformation transformation between two planes
(a world plane and the image plane, or two image planes induced by a world
plane) is a plane projective transformation. This can be computed from the cor-
respondence of four (or more) points.
•
The epipolar geometry between two views is represented by the fundamental
matrix. This can be computed from the correspondence of seven (or more)
points.
Imaging Geometry
Perspective projection
Y
   
X
X
x
X
y
   
λ y = x
Y x
O
f
Z
p
Z
where λ = /f .
Z image plane
This can be written as a linear mapping between homogeneous coordinates (the
equation is only up to a scale factor):
 
   
X
x 1 0 0 0
 
     Y 
y = 0 1 0 0
 
Z
f 0 0 1 0
1
×
where a 3 4 projection matrix represents a map from 3D to 2D.
Image Coordinate System
Internal camera parameters
y
cam
−
k x = x x
x 0
cam
−
k y = y y
y p
y 0
cam 0
x
cam
y
where the units of k , k
x y
are [pixels/length].
x x
0
       
x α x x x
x 0
  1    cam   cam 
       
x = y = α y y = K y
y 0
cam cam
f
1 1 f f
where α = f k , α = f k .
x x y y
Camera Calibration Matrix
×
K is a 3 3 upper triangular matrix, called the camera calibration
 
matrix:
α x
x 0
 
 
K = α y
y 0
1
•
There are four parameters:
(i) The scaling in the image x and y directions, α and α .
x y
(ii) The principal point (x , y ), which is the point where the
0 0
optic axis intersects the image plane.
•
The aspect ratio is α /α .
y x
World Coordinate System
External camera parameters
Y
cam
   
Z
X (cid:13) (cid:14) X
 cam   
O Z
  R t   cam cam
Y Y
 cam    R, t
=
  (cid:4)   Xcam
0 1
Z Z
cam O
Y
1 1
X
Euclidean transformation between world and camera coordinates
• ×
R is a 3 3 rotation matrix
• ×
t is a 3 1 translation vector
Concatenating the three matrices,
 
   
(cid:13) (cid:14) X
 
x 1 0 0 0
     
R t
Y
      |
x = y = K 0 1 0 0 = K [R t] X
(cid:4)
 
0 1
Z
1 0 0 1 0
1
×
which defines the 3 4 projection matrix from Euclidean 3-space to
an image as
(cid:4)
| |
x = PX P = K [R t] = KR[I R t]
(cid:4) − (cid:4)
Note, the camera centre is at ( , , ) = R t.
X Y Z
×
In the following it is often only the 3 4 form of P that is important,
rather than its decomposition.
A Projective Camera
The camera model for perspective projection is a linear map between
 
homogeneous point coordinates
   
X
 
x
     
Y
   ×   
y P (3 4)
 
Z
1
1
Image Point Scene Point
x = P X
•
The camera centre is the null-vector of P
| (cid:4)
e.g. if P = [I 0] then the centre is X = (0, 0, 0, 1) .
•
P has 11 degrees of freedom (essential parameters).
•
P has rank 3.
What does calibration give?
•
K provides the transformation between an image point and a ray
in Euclidean 3-space.
•
Once K is known the camera is termed calibrated.
•
A calibrated camera is a direction sensor, able to measure the
direction of rays — like a 2D protractor.
     
x α x
X
x 0
     cam 
     
x = y = α y = Kd
Y
y 0
cam
1 1
Z
cam
Angle between rays
d
1
x
1
C
d .d
1 2 θ
cos θ =
1 1 x
2
(d .d ) (d .d )
2 2
1 1 2 2
d
2
(cid:4) (cid:4) −(cid:4) −
1
d d x (K K )x
1 2 1 2
cos θ = =
(cid:4) (cid:4) (cid:4) −(cid:4) − (cid:4) −(cid:4) −
(d d )1/2(d d )1/2 (x (K K 1)x )1/2(x (K K 1)x )1/2
1 1 2 2 1 1 2 2
(cid:4)
x ωx
1 2
=
(cid:4) (cid:4)
(x ωx )1/2(x ωx )1/2
1 1 2 2
(cid:4) −
1
where ω = (KK ) .