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camera model is defined by its center of projection
C
, which corresponds to the po-
sition of the camera, and the
retinal plane
where points in three-dimensional space
are projected onto. The distance from the center of projection to the image plane is
called
focal length
. A point
M
with coordinates (
X
,
Y
,
Z
) in three-dimensional space
is projected onto a point
m
with coordinates (
x
,
y
) on the retinal plane by the inter-
section of this latter with the ray (
CM
). This projection is defined by the following
linear equation
(
xy
1)
T
=
P
(
XY Z
1)
T
,
λ
λ
∈
R
(1)
where
P
is the perspective transformation matrix which depends upon the camera
parameters.
2.2
The Stereo Camera Setup
We consider now the case of binocular stereovision, where two images, commonly
referred as left and right images, are acquired by two cameras observing the same
scene from two different positions. Each camera is characterized by its optical center
and a perspective transformation matrix. The stereo camera system is assumed to be
fully calibrated, i.e. the camera parameters as well as the positions and orientations
of the cameras are known. Both cameras capture the scene point
M
whose projec-
tions,
m
and
m
, onto the left and right images are given by the intersection of the
lines (
CM
) and (
C
M
) with the corresponding image planes. Using the perspective
transformation (1), we can derive
m
=
PM
,
(2)
m
=
P
M
,
(3)
where
P
and
P
are the left and right camera projection matrices, respectively. Sup-
pose that the origin is located at the left camera, projection matrices are given by:
P
=
A
(
R
)
,
P
=
A
(
I
3
0)
and
(4)
where
A
and
A
are the internal camera parameters and
R
and
t
are the rotation and
translation operating between both cameras.
2.2.1
Epipolar Geometry
Epipolar geometry describes the geometrical relation between two images of the
same scene, taken from two different viewpoints. It is independent of scene struc-
ture, and only depends on the internal cameras parameters [8]. Epipolar geometry
establishes a geometric constraint between a point in one image and its correspond-
ing in the other image, resulting in a very powerful restriction in correspondence
estimation. Let two images be taken by two left and right cameras with optical cen-
ters
C
and
C
, respectively. The point
C
projects to the point
e
in the right image,
