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7.2.2
Camera Model and Two Views Geometry
X
The standard perspective camera model maps all scene points
with homogeneous
coordinates in the camera frame
X
=[
XYZ
1]
T
from a line passing through the
optical center of the camera to one image point with homogeneous coordinates
m
=
[
m
x
m
y
1]
T
in the normalized image plane:
λ
m
=
PX
.
(7.5)
3
×
4
P
0
3
×
1
]. The 2D projective point
m
is then mapped into the pixel image point with homogeneous coordinates
p
=
[
xy
1]
using the collineation matrix
K
:
∈
R
is the projection matrix, that is
P
=[
I
3
×
3
|
p
=
Km
(7.6)
where
K
contains the intrinsic parameters of the camera:
⎡
⎤
fp
u
−
fp
u
cot(
α
)
u
0
⎣
⎦
.
0
fp
v
/
sin(
α
)
v
0
K
=
0
0
1
u
0
and
v
0
are the pixels coordinates of principal point,
f
is the focal length,
p
u
and
p
v
are the magnifications respectively in the
u
and
v
directions, and
α
is the angle
between these axes.
Consider now two views of a scene observed by a camera (see Figure 7.1). A
3D point
with homogeneous coordinates
X
=[
XYZ
1]
T
is projected under per-
spective projection to a point
p
in the first image (with homogeneous coordinates
measured in pixel
p
=[
xy
1]
T
) and to a point
p
f
in the second image (with homo-
geneous coordinates measured in pixel
p
f
=[
x
f
y
f
1]
T
). It is well-known that there
exists a projective homography matrix
G
related to a virtual plane
X
Π
, such that for
1
,
all points
X
belonging to
Π
Gp
f
.
When
p
and
p
f
are expressed in pixels, matrix
G
is called the collineation matrix.
From the knowledge of several matched points, lines or contours [10, 21], it is possi-
ble to estimate the collineation matrix. For example, if at least four points belonging
to
p
∝
are matched,
G
can be estimated by solving a linear system. Else, at least
eight points (3 points to define
Π
) are necessary to estimate the
collineation matrix by using for example the linearized algorithm proposed in [20].
Assuming that the camera calibration is known, the Euclidean homography can be
computed up to a scalar factor
2
:
Π
and 5 outside of
Π
K
+
GK
H
∝
.
(7.7)
1
p
∝
Gp
f
⇐⇒
α
p
=
Gp
f
where
α
is a scaling factor.
2
K
+
denotes the inverse of
K
.
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