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(note that in (
1.47
) the matrix elements according to (
1.11
) are used). The extrinsic
parameters for each image are then obtained according to
λA
−
1
h
1
r
1
=
λA
−
1
h
2
r
2
=
(1.48)
r
3
=
r
1
×
r
2
λA
−
1
h
3
.
t
=
The matrix
R
computed according to (
1.48
), however, does not necessarily fulfill
the orthonormality constraints imposed on a rotation matrix. For initialisation of
the subsequent nonlinear bundle adjustment procedure, a technique is suggested by
Zhang (
1998
) to determine the orthonormal rotation matrix which is closest to a
given 3
3 matrix in terms of the Frobenius norm.
Similar to the DLT method, the intrinsic and extrinsic camera parameters com-
puted so far have been obtained by minimisation of an algebraic error measure
which is not physically meaningful. Zhang (
1999a
) uses these parameters as ini-
tial values for a bundle adjustment step which is based on the minimisation of the
error term
×
n
m
m
ij
−
A(R
i
M
j
+
t
)
2
.
(1.49)
i
=
1
j
=
1
In the optimisation, a rotation
R
is described by the Rodrigues vector
r
. The di-
rection of this vector indicates the direction of the rotation axis, and its norm
denotes the rotation angle in radians. Zhang (
1999a
) utilises the Levenberg-
Marquardt algorithm (Press et al.,
2007
) to minimise the bundle adjustment error
term (
1.49
).
To take into account radial lens distortion, Zhang (
1999a
) utilises the model
defined by (
1.3
). Tangential lens distortion is neglected. Assuming small radial
distortions, such that only the coefficients
k
1
and
k
3
in (
1.3
) are significantly
different from zero, the following procedure is suggested for estimating
k
1
and
k
3
: An initial solution for the camera parameters is obtained by setting
k
1
=
k
3
=
0, which yields projected control points according to the pinhole model.
The parameters
k
1
and
k
3
are computed in a second step by minimising the av-
erage Euclidean distance in the image plane between the projected and the ob-
served image points, based on an overdetermined system of linear equations.
The final values for
k
1
and
k
3
are obtained by iteratively applying this proce-
dure.
Due to the observed slow convergence of the iterative technique, Zhang (
1999a
)
proposes an alternative approach to determine lens distortion by incorporating the
distortion parameters appropriately into the error term (
1.49
) and estimating them
simultaneously with the other camera parameters.
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