We describe here the camera model which consists of central projection specialised to planes.
Figure:
(a) Plane Camera Model: a point
on the world plane is imaged as
. Euclidean coordinates
X
-
Y
and
x
-
y
are used for the world and image planes, respectively.
is the camera centre. (b) One-dimensional Camera Model: The camera
centre is a distance
f
(the focal length) from the image line. The ray at the principal point
p
is perpendicular to the image line, and intersects the world line at
P
, with world ordinate
t
.
w
is the angle between the world and image lines.
Figure
1
a shows the imaging process. The notation used is that points on the
world plane are represented by upper case vectors,
, and their corresponding images are represented by lower case vectors
. Under perspective projection corresponding points are related by [
9
,
13
]:
where
H
is a
homogeneous matrix, and ``='' is equality up to scale. The world and
image points are represented by homogeneous 3-vectors as
and
. The scale of the matrix does not affect the equation, so only the
eight degrees of freedom corresponding to the ratio of the matrix
elements are significant.
The camera model is completely specified once the matrix is determined. The matrix can be computed from the relative positioning of the two planes and camera centre. However, it can also be computed directly from image to world point correspondences. This computation is described in section 3 .
A one-dimensional version of the plane to plane homography is described
here. This model is used for the second order analysis in section
4
. Equation (
1
) reduces to
where
is a
homography matrix. For the geometry shown in figure
1
b the matrix is given by
with parameters
and
.
Antonio Criminisi