Game Development Reference
In-Depth Information
Suppose there exists a straight 3-D line in the 3-D scene. Correspondingly, a
straight image line should appear in the focal plane of an ideal linear camera. Let
T
im
im
im
x
denote an arbitrary point on that image line. The following
equation should then be satisfied
=
x
y
im
im
,
x
sin
θ
+
y
cos
θ
=
ρ
where
θ
is the angle between the image line and the x-axis of the focal plane,
is the perpendicular distance from the origin to this line.
Suppose that the reconstruction-distortion model is employed, and in the distor-
tion model assume that
f
x
=
f
y
(Brown, 1971; Devernay & Faugeras, 2001).
Substituting equation 12 into equation 14 leads to an expression of the form
and
ρ
(
)
ˆˆ
im
im
Re
Re
Re
Re
Re
Re
Re
fx y xyssk k k P P s s
,
,
,
,
,
,
,
,
,
,
,
,
,
θρ
,
+=
,
0
(15)
00
xy
1
2
3
1
2
1
2
where
x
0
,
y
0
,
s
x
,
s
y
,
k
1
Re
,
k
2
Re
,
k
3
Re
,
P
1
Re
,
P
2
Re
,
s
1
Re
,
s
2
Re
,
θ
and
ρ
are all unknown,
and
is a random error.
If enough colinear points are available,
ε
2
nm
(
)
(
)
=
∑∑
ς
fx y xyssk k k P P s s
ˆˆ
im
,
im
,
,
,
,
,
Re
,
Re
,
Re
,
Re
,
Re
,
Re
,
Re
,
θ
,
ρ
(16)
ij
ij
00
x
y
1
2
3
1
2
1
2
i
i
==
i
11
j
can be minimized to recover
x
0
,
y
0
,
s
x
,
s
y
,
k
1
Re
,
k
2
Re
,
k
3
Re
,
P
1
Re
,
P
2
Re
,
s
1
Re
,
s
2
Re
,
θ
i
[
]
T
and
ρ
i
.
x
ˆ
im
ij
ˆ
y
im
ij
(
i
= 1 ...
n
,
j
= 1 ...
m
) are distorted image points whose
[
]
T
distortion-free correspondences
x
im
ij
y
im
ij
should lie on the same image line
ρ
i
.
However, due to the high inter-correlation between the de-centering distortion
coefficients (
P
1
Re
,
P
2
Re
), and the principal point coordinates
with rotation
θ
i
and polar distance
[
]
T
x
0
,
x
0
a n d
y
0
are always assumed to be known
a priori
. Otherwise, a proper optimization
strategy (e.g., coarse-to-fine [Swaminathan & Nayer, 2000]) has to be adopted
to get a more stable solution.
y
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