Game Development Reference
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arise: First, for a high degree of distortion, the scale of the model increases
dramatically; Second, it is very difficult to take into account projective con-
straints, as addressed in a previous section for DLT, in the calibration; Third, only
when distortion is not considered, can the original camera parameters be
computed linearly with this model (Wei & Ma, 1994). Therefore, a more efficient
way of estimating the distortion coefficients is needed. For this purpose, Tsai's
algorithm is a good and representative example.
Tsai's Algorithm
Assuming that only the radial distortion occurs in the camera and the principal
point
T
xy
is known (or can be approximated) in advance, Tsai (1987)
proposed a two-stage algorithm for explicitly calibrating the camera. In this case,
the imaging equations used are exactly the same as those in equation 12, except
that the possible distortion is limited as follows:
[
]
0
0
k
k
1
∆
xr
ˆ
2
ˆ
xr
4
xr
ˆ
6
x
2
=
∆
.
yr
ˆ
2
yr
ˆ
4
yr
ˆ
6
k
y
3
With this radial distortion, a pixel in the image is only distorted along the radial
direction, thus a
radial alignment constraint
(
RAC
) can be formed (Tsai,
1987) (ref. Equation 12):
s
y
c
ˆ
ˆ
⋅
x
x
s z
ˆ
im
x
xx
−
×=− ⋅
yc
0
×=
0
s
.
(40)
x
y
ˆ
f
ˆ
im
y
ˆ
yy
−
c
0
y
Expanding equation 2 gives:
1
2
3
w
rr r x
t
1
1
1
x
()(
)()
T
T
xR xt
c
=
w
w
−
w
==
Rxt
w
w
+
=
rr r y
1
2
3
w
+
t
,
(41)
c
c
c
2
2
2
y
rr r z
1
2
3
w
t
3
3
3
z
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