Image Processing Reference
In-Depth Information
M
E
=[
R
,
t
]
(13.88)
where
R
=
R
R
R
L
T
,
t
=
t
R
−
Rt
L
.
and
(13.89)
The coordinates of vectors marked with
L
and
R
are in the left and right camera
coordinates, respectively, whereas the
L
and
R
marked matrices rotate the world
coordinates to the respective camera coordinates.
Compared with Sect. 13.2, we note that the left camera frame (the reference) here
takes the role of the world frame there, (not to be confused with the world frame
attached to the calibration pattern, at
O
W
in Fig. 13.7). Consistently with this,
t
represents the position of the left camera center (the world center) in the right cam-
era coordinates, whereas the matrix
R
transfers the coordinates of a vector in the
left camera (the world) to coordinates in the Right camera, see Figs. 13.3, 13.7 in
connection with lemmas 13.6 and 13.8.
Lemma 13.9.
Let the intrinsic matrices
M
I
,
M
I
of two cameras and the extrinsic
matrix of the corresponding stereo system
M
E
=[
R
,
t
]
, having the left camera as
reference, be given. Then a TLS estimate of a homogenized world point position,
p
,
in the reference frame is given by the solution of a 4D linear symmetry direction
determination problem,
A
=
T
L
M
I
[
I
,
0
]
T
R
M
I
M
E
,
p
T
A
T
Ap
,
min
p
=1
with
(13.90)
where the image coordinates of a point
P
are encoded as
⎛
⎝
⎞
⎠
,
⎛
⎝
⎞
⎠
1
r
L
10
1
r
R
10
0
−
0
−
T
L
=
T
R
=
−
c
L
−
c
R
(13.91)
−
r
L
c
L
0
−
r
R
c
R
0
for the left and in the right camera views, respectively.
That
M
E
should be independent of our (reasonable but yet) arbitrary placement
of the calibration pattern (the world center at
O
W
) is supported by lemma 13.8. The
extrinsic parameters of a stereo system can be obtained by combining the individual
calibrations of the two cameras using the lemma. Based on individual camera ro-
tation and translation estimations w.r.t. to the current world coordinates, a series of
such measurements can thus be computed from displaced and rotated calibration pat-
terns via Eq. (13.82). The obtained
M
E
, as well as the intrinsic matrices
M
I
,
M
I
,
can subsequently be averaged to reduce the random measurement errors.
Example 13.2. Using known patterns, such as a checkerboard pattern, and corre-
spondence between points, we can estimate the extrinsic and intrinsic matrices of a
stereo camera system which allows us to determine the positions of the cameras in
