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coordinates are also known. By setting up a system of equations that relates these coordinates through
Calibration is performed by imaging a set of points whose global coordinates are known and iden-
tifying the image coordinates of the points and the correspondence between the two sets of points. This
results in a series of five-tuples, (
x
i
,
y
i
,
z
i
,
c
i
,
r
i
) consisting of the three-dimensional global coordinates and
two-dimensional image coordinates for each point. The two-dimensional image coordinates are a map-
ping of the local two-dimensional image plane of the camera located a distance
f
in front of the camera
(
Eq. B.171
)
. The image plane is located relative to the three-dimensional local coordinate system (
u
,
v
,
w
)
of the camera (
Figure B.60
). The imaging of a three-dimensional point is approximated using a pinhole
camera model. The three-dimensional local coordinate system of the camera is related to the three-
dimensional global coordinate system by a rotation and translation (
Eq. B.172
)
; the origin of the camera's
local coordinate system is assumed to be at the focal point of the camera. The three-dimensional coor-
dinates are related to the two-dimensional coordinates by the transformation to be determined.
Equation B.173
expresses the relationship between a pixel's column and row number and the global coor-
dinates of the point. These equations are rearranged and set equal to zero in
Equation B.174
.Theycanbe
put in the form of a system of linear equations (
Eq. B.175
) so that the unknowns are isolated (
Eq. B.176
)
by using substitutions common in camera calibration (
Eqs. B.176
and
B.177
)
. Temporarily dividing
through by
t
3
ensures that
t
3
6¼
0.0 and therefore that the global origin is in front of the camera. This step
results in
EquationB.178
,
where
A
0
is the first 11 columns of
A
;
B
0
is the last column of
A
;and
W
0
is the first
11 rows of
W.
Typically, enough points are captured to ensure an overdetermined system. Then a least-
squares method, such as singular value decomposition, can be used to find the
W
0
that satisfies
by undoing the substitutions made in
Equation B.176
by
Equation B.181
.
Because of numerical impre-
cision, the rotationmatrix recoveredmay not be orthonormal, so it is best to reconstruct the rotationmatrix
first (
Eq. B.182
)
, massage it into orthonormality, and then use the new rotation matrix to generate the rest
of the parameters (
Eq. B.183
).
c
i
c
0
¼ s
u
u
i
r r
0
¼ s
v
v
i
(B.171)
2
4
3
5
¼ R
2
4
3
5
þ T
u
i
v
i
f
x
i
y
i
z
i
2
4
3
5
¼
2
4
3
5
r
0
;
0
r
0
;
1
r
0
;
2
R
0
R
1
R
2
R ¼
r
1
;
0
r
1
;
1
r
1
;
2
(B.172)
r
2
;
0
r
2
;
1
r
2
;
2
2
4
3
5
t
0
t
1
t
2
T ¼
u
i
f
¼
c
i
c
0
s
u
f
c
i
c
0
f
u
R
0
½x
i
y
i
z
i
þt
0
R
2
½x
i
y
i
z
i
þt
2
¼
¼
(B.173)
v
i
f
¼
r
i
r
0
s
v
f
c
i
c
0
fv
R
1
½x
i
y
i
z
i
þt
1
R
2
½x
i
y
i
z
i
þt
2
¼
¼
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