Graphics Reference
In-Depth Information
L
1
x
+
L
2
y
+
L
3
z
+
L
4
L
9
x
+
L
10
y
+
L
11
z
+
u
=
1
(1.29)
L
5
x
+
L
6
y
+
L
7
z
+
L
8
v
=
1
.
L
9
x
+
L
10
y
+
L
11
z
+
If we use the abbreviations
b
u
=
b/k
u
,
b
v
=
b/k
v
, and
D
=−
(x
0
r
31
+
y
0
r
32
+
z
0
r
33
)
, the parameters
L
1
...L
11
can be expressed as
u
0
r
31
−
b
u
r
11
D
L
1
=
u
0
r
32
−
b
u
r
12
L
2
=
D
u
0
r
33
−
b
u
r
13
L
3
=
D
(b
u
r
11
−
u
0
r
31
)x
0
+
(b
u
r
12
−
u
0
r
32
)y
0
+
(b
u
r
13
−
u
0
r
33
)z
0
L
4
=
D
v
0
r
31
−
b
v
r
21
L
5
=
D
v
0
r
32
−
b
v
r
22
D
L
6
=
(1.30)
v
0
r
33
−
b
v
r
23
L
7
=
D
(b
v
r
21
−
v
0
r
31
)x
0
+
(b
v
r
22
−
v
0
r
32
)y
0
+
(b
v
r
23
−
v
0
r
33
)z
0
L
8
=
D
r
31
D
L
9
=
r
32
D
L
10
=
r
33
D
.
It is straightforward but somewhat tedious to compute the intrinsic and extrinsic
camera parameters from these expressions for
L
1
...L
11
.
Radial and tangential distortions introduce offsets
u
and
v
with respect to
the position of the image point expected according to the pinhole model. Using the
polynomial laws defined in (
1.3
) and (
1.4
) and setting
ξ
L
11
=
=
−
=
−
u
u
0
and
η
v
v
0
,
these offsets can be formulated as
u
ξ
L
12
r
2
L
14
r
6
+
L
15
r
2
2
ξ
2
+
L
13
r
4
=
+
+
+
L
16
ηξ
(1.31)
η
L
12
r
2
L
14
r
6
+
L
16
r
2
2
η
2
.
L
13
r
4
v
=
+
+
L
15
ηξ
+
+
The additional parameters
L
12
...L
14
describe the radial and
L
15
and
L
16
the tan-
gential lens distortion, respectively.
Kwon (
1998
) points out that by replacing in (
1.29
) the values of
u
by
u
+
u
and
v
by
v
+
v
and defining the abbreviation
Q
i
=
L
9
x
i
+
L
10
y
i
+
L
11
z
i
+
1,
where
x
i
,
y
i
and
z
i
denote the world coordinates of scene point
i
(
i
1
,...,N
), an
equation for determining the parameters
L
1
...L
16
is obtained according to
=
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