Graphics Reference
In-Depth Information
(undistorted) ones
(
x
,
y
)
by
x
dist
y
dist
x
0
y
0
˜
x
dist
/
d
x
=
+
˜
y
dist
/
d
y
(6.6)
x
0
y
0
x
−
x
0
x
2
y
2
x
2
y
2
2
=
+
(
1
+
κ
(
˜
+ ˜
)
+
κ
(
˜
+ ˜
)
)
1
2
y
−
y
0
This suggests a simple method for estimating the lens distortion coefficients if
we know the other internal parameters. We obtain a set of ideal points
(
x
,
y
)
and
corresponding observed points
(
x
dist
,
y
dist
)
; each one generates two equations in the
two unknown
κ
's:
x
dist
−
x
2
y
2
x
2
y
2
2
(
x
−
x
0
)(
˜
+ ˜
)(
x
−
x
0
)(
˜
+ ˜
)
κ
x
1
=
(6.7)
x
2
y
2
x
2
y
2
2
(
y
−
y
0
)(
˜
+ ˜
)(
y
−
y
0
)(
˜
+ ˜
)
κ
y
dist
−
y
2
When we have a large number of ideal points and corresponding distorted obser-
vations, the simultaneous equations given by Equation (
6.7
) can be solved as a linear
least-squares problem. The lens distortion parameters for the image in Figure
6.5
were estimated to be
0.18.
In Section
6.3.2
we'll discuss methods for estimating the other internal parameters
for a camera. In the remainder of the chapter, we assume that the images have already
been compensated for any lens distortion, so that the image formation process is well
modeled by Equation (
6.4
).
κ
=−
0.39,
κ
=
1
2
6.2.3
External Parameters
Expressing points in the camera coordinate system makes it easy to determine their
projections. However, for matchmoving we want to determine the locations of such
points—as well as the locations and orientations of the cameras—in a
world coordi-
nate system
independent of the cameras. For example, the world coordinate system
could be defined with respect to the walls of a movie set or surveyed 3D points in
the scene. Therefore, we need to transform points measured in the world coordinate
system
prior to computing the
projection. This transformation is assumed to be a
rigid motion
; that is, a 3D rota-
tion and translation defined by a 3
(
X
,
Y
,
Z
)
to the camera coordinate system
(
X
c
,
Y
c
,
Z
c
)
×
3 rotation matrix
R
and 3
×
1 translation vector
t
such that
X
c
Y
c
Z
c
X
Y
Z
=
+
R
t
(6.8)
The rotation matrix
R
(a function of 3 parameters, as described in Section
6.5.3.1
)
and the translation vector
t
are called the
external
or
extrinsic parameters
of the
camera, since they define its position in an environment regardless of its internal
functioning.
We now can combine the internal Equation (
6.4
) with the external Equation (
6.8
)
to obtain a 4
×
3
camera matrix
P
that contains all the camera parameters:
P
=
K
[
R
|
t
]
(6.9)
