Game Development Reference
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On the other hand, if all available points are co-planar, that is
z
w
= 0, equation 42
becomes:
[
]
y
ˆ
w
ˆ
y
w
y
ˆ
−
x
ˆ
w
−
x
ˆ
w
⋅
a
'
−
x
ˆ
=
0
,
(43)
T
s
s
s
y
y
y
−
1
1
1
−
1
2
−
1
−
1
1
2
−
1
2
2
a
'
=
t
r
t
r
t
t
t
r
t
r
where
.
y
y
1
y
x
y
y
s
s
s
x
x
x
w
c
a
only
In this case,
R
,
t
x
, and
t
y
can be recovered from the calculated vector
if the aspect ratio
s
y
/
s
x
is known.
In summary, there are two stages in this algorithm:
w
c
1.
Compute the 3-D
pose
R
,
t
x
,
t
y
, and
s
y
/
s
x
(in case enough non-co-planar
points are available); and
2.
Optimize the effective focal length
f
, radial distortion coefficients
k
1
,
k
2
, ...,
and
t
z
, by employing a simple search scheme.
Because the problem has been split into these two stages, the whole computation
becomes much simpler and more efficient.
Further improvements
A requirement of Tsai's algorithm is that the position of the principal point and
the aspect ratio (in case only co-planar points are available) are known
a priori
.
One practical possibility of finding the principal point accurately is to minimize
the left-hand side of equation 42 (in the non-co-planar case) or that of equation
43 (in the co-planar case) (Lenz & Tsai, 1988; Penna, 1991). The horizontal scale
factor (and thus the aspect ratio) can be measured by using the difference
between the scanning frequency of the camera sensor plane and the scanning
frequency of the image capture board frame buffer (Lenz & Tsai, 1988).
However, this scale factor estimation method is not so practical due to the
difficulty of measuring the required frequencies (Penna, 1991). A more direct
way would be to employ the image of a sphere for calculating the aspect ratio
(Penna, 1991). The power spectrum of the images of two sets of parallel lines
can also be utilized for the same purpose (Bani-Hashemi, 1991).
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