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2 of the rotation matrix, which nevertheless allows us to estimate the full
orthonormal rotation matrix.
The second calibration stage of the method by Tsai (
1987
) is described by Horn
(
2000
) as a minimisation of the reprojection error in the image plane (cf. Sect.
1.3
),
during which the already estimated parameters are refined and the principal point
(u
0
,v
0
)
and the radial and tangential distortion coefficients (cf. Sect.
1.1
) are deter-
mined based on nonlinear optimisation techniques.
1.4.4 The Camera Calibration Method by Zhang (
1999a
)
The camera calibration method by Zhang (
1999a
) is specially designed for utilising
a planar calibration rig which is viewed by the camera at different viewing angles
and distances. This calibration approach is derived in terms of the projective geom-
etry framework.
For a planar calibration rig, the world coordinate system can always be chosen
such that we have
Z
0 for all points on it. The image formation is then described
by Zhang (
1999a
) in homogeneous normalised coordinates by
⎛
=
⎛
⎝
⎞
⎠
=
⎞
⎛
⎞
X
Y
0
1
u
v
1
X
Y
1
⎝
⎠
=
⎝
⎠
,
A
[
R
|
t
]
A
[
r
1
|
r
2
|
t
]
(1.38)
where the vectors
r
i
denote the column vectors of the rotation matrix
R
. A point on
the calibration rig with
Z
(X, Y )
T
. The corresponding vector
in normalised homogeneous coordinates is given by
=
0 is denoted by
M
=
M
(X, Y,
1
)
T
. According
=
˜
to (
1.38
), in the absence of lens distortion the image point
m
can be obtained from its
corresponding scene point
M
by applying a homography
H
. A homography denotes
a linear transform of a vector (of length 3) in the projective plane. It is given by a
3
3 matrix and has eight degrees of freedom, as a projective transform is unique
only up to a scale factor (cf. Sect.
1.1
). This leads to
˜
×
M
.
(1.39)
To compute the homography
H
, Zhang (
1999a
) proposes a nonlinear optimisation
procedure which minimises the Euclidean reprojection error of the scene points pro-
jected into the image plane. The column vectors of
H
are denoted by
h
1
,
h
2
, and
h
3
. We obtain
m
=
H
with
H
=
A
[
r
1
r
2
t
]
[
h
1
h
2
h
3
]=
λA
[
r
r
r
2
t
]
,
(1.40)
(
1
/λ)A
−
1
h
1
and
r
2
=
with
λ
as a scale factor. It follows from (
1.40
) that
r
1
=
(
1
/λ)A
−
1
h
2
with
λ
=
1
/
A
−
1
h
1
=
1
/
A
−
1
h
2
. The orthonormality of
r
1
and
r
2
yields
r
1
0 and
r
1
r
2
·
r
2
=
r
2
, implying
h
1
A
−
T
A
−
1
h
2
=
·
r
1
=
·
0
(1.41)
h
1
A
−
T
A
−
1
h
1
=
h
2
A
−
T
A
−
1
h
2
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