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scheme is not equivalent to bundle adjustment. While bundle adjustment minimises
the reprojection error in the image plane, (
1.32
) illustrates that the DLT method min-
imises the error of the backprojected scaled pixel coordinates
(u
i
/Q
i
,v
i
/Q
i
)
.Itis
not guaranteed that this somewhat arbitrary error measure is always a reasonable
choice.
1.4.3 The Camera Calibration Method by Tsai (
1987
)
Another important camera calibration method is introduced by Tsai (
1987
), which
estimates the camera parameters based on a set of control points in the scene (here
denoted by
W
x
(x,y,z)
T
) and their corresponding image points (here denoted by
=
I
x
v)
). According to the illustrative presentation by Horn (
2000
) of that ap-
proach, in the first stage of the algorithm by Tsai (
1987
) estimates of several extrin-
sic camera parameters (the elements of the rotation matrix
R
and two components
of the translation vector
t
) are obtained based on the equations
ˆ
=
(
u,
ˆ
ˆ
u
b
=
s
r
11
x
+
r
12
y
+
r
13
z
+
t
x
(1.35)
r
31
x
+
r
32
y
+
r
33
z
+
t
z
+
+
+
v
b
=
ˆ
r
21
x
r
22
y
r
23
z
t
y
(1.36)
r
31
x
+
r
32
y
+
r
33
z
+
t
z
following from the pinhole model (cf. Sect.
1.1
), where
s
is the aspect ratio for
rectangular pixels, the coefficients
r
ij
are the elements of the rotation matrix
R
, and
t
=
(t
x
,t
y
,t
z
)
T
. Following the derivation by Horn (
2000
), dividing (
1.35
)by(
1.36
)
leads to the expression
ˆ
+
+
+
u
ˆ
r
11
x
r
12
y
r
13
z
t
x
v
=
s
(1.37)
r
21
x
+
r
22
y
+
r
23
z
+
t
y
which is independent of the principal distance
b
and the radial lens distortion, since
it only depends on the direction from the principal point to the image point. Equa-
tion (
1.37
) is then transformed into a linear equation in the camera parameters. This
equation is solved with respect to the elements of
R
and the translation components
t
x
and
t
y
in the least-squares sense based on the known coordinates of the control
points and their observed corresponding image points, where one of the translation
components has to be normalised to 1 due to the homogeneity of the resulting equa-
tion.
Horn (
2000
) points out that the camera parameters have been estimated indepen-
dently, i.e. the estimated rotation matrix is generally not orthonormal, and describes
a method which yields the most similar orthonormal rotation matrix. The orthonor-
mality conditions allow the determination of
s
and the overall scale factor of the
solution. The principal distance
b
and the translation component
t
z
are then ob-
tained based on (
1.35
) and (
1.36
). For the special case of a planar calibration rig, the
world coordinate system can always be chosen such that
z
0 for all control points,
and (
1.35
)-(
1.37
) are applied accordingly. This special case only yields a submatrix
=
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