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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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