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In-Depth Information
•
the point
X
s
lying on the unitary sphere is perspectively projected on the normal-
ized image plane
Z
= 1
−
ξ
into a point of homogeneous coordinates:
x
=
f
(
X
)=
1
X
Z
+
Y
Z
+
(16.2)
ξρ
ξρ
(as it can be seen, the perspective projection model is obtained by setting
ξ
= 0);
and
•
the 2D projective point
x
is mapped into the pixel image point with homogeneous
coordinates
x
i
using the collineation matrix
K
:
x
i
=
Kx
where the matrix
K
contains the conventional camera intrinsic parameters cou-
pled with mirror intrinsic parameters, and can be written as
⎡
⎤
f
u
α
uv
u
0
0
f
v
v
0
001
⎣
⎦
.
K
=
can be obtained after calibration using for
instance the method proposed in [1]. The inverse projection from the image plane
onto the unit sphere can be obtained by inverting the second and last steps. As
a matter of fact, the point
x
in the normalized image plane is obtained using the
inverse mapping
K
−
1
:
The matrix
K
and the parameter
ξ
x
=[
xy
1]
=
K
−
1
x
i
.
(16.3)
The point onto the unit sphere is then obtained by inverting the nonlinear projection
(16.2):
xy
1
−
ξ
η
X
s
=
f
−
1
(
x
)=
η
,
(16.4)
where
+
1 +(1
2
)(
x
2
+
y
2
)
x
2
+
y
2
+ 1
=
ξ
−
ξ
.
η
16.2.2
Scaled Euclidean Reconstruction
Several methods were proposed to obtain the Euclidean reconstruction from two
views [8]. They are generally based on the estimation of the essential or homog-
raphy matrices. The epipolar geometry of cameras obeying the unified model has
been recently investigated [10, 23, 11]. For control purposes, the methods based
on the essential matrix are not well suited since degenerate configurations such as
pure rotation motion can induce unstable behavior of the control scheme. It is thus
preferable to use methods based on the homography matrix.
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