Graphics Reference
In-Depth Information
r
11
ˆ
r
13
z
1
+
u
1
b
+
r
12
ˆ
v
1
b
+
u
2
b
z
2
ˆ
t
1
=
r
21
ˆ
r
23
z
1
+
u
1
b
+
r
22
ˆ
v
1
b
+
v
2
b
z
2
ˆ
t
2
=
(1.9)
r
31
ˆ
r
33
z
1
+
u
1
b
+
r
32
ˆ
v
1
b
+
t
3
=
z
2
.
Combining two of these three equations yields the three-dimensional scene points
C
1
x
and
C
2
x
according to
⎛
⎞
⎛
⎞
x
1
y
1
z
1
u
1
/b
ˆ
⎝
⎠
=
⎝
⎠
z
1
C
1
x
=
v
1
/b
1
(1.10)
⎛
⎞
⎛
⎞
x
2
y
2
z
2
u
2
/b
ˆ
ˆ
⎝
⎠
=
⎝
⎠
z
2
.
C
2
x
=
v
2
/b
1
Equation (
1.10
) allows one to compute the coordinates
C
i
x
of a scene point in any
of the two camera coordinate systems based on the measured pixel positions of the
corresponding image points, given the relative orientation of the cameras defined
by the rotation matrix
R
and the translation vector
t
. Note that all computations in
this section have been performed based on the metric image coordinates given by
I
i
x
v
i
)
T
, which are related to the pixel coordinates given by
S
i
x
(u
i
,v
i
)
T
=
(
u
i
,
ˆ
ˆ
=
in the sensor coordinate system by (
1.5
).
1.2.2 Stereo Image Analysis in Terms of Projective Geometry
To circumvent the nonlinear formulation of the pinhole model in Euclidean geom-
etry, it is advantageous to express the image formation process in the more general
mathematical framework of projective geometry.
1.2.2.1 Definition of Coordinates and Camera Properties
This section follows the description in the overview by Birchfield (
1998
) [detailed
treatments are given e.g. in the topics by Hartley and Zisserman (
2003
) and Schreer
(
2005
), and other introductions are provided by Davis (
2001
) and Lu et al. (
2004
)].
Accordingly, a point
x
(x, y)
T
=
in two-dimensional Euclidean space corresponds
(X,Y,W)
T
defined by a vector with three coordinates in the two-
dimensional projective space
˜
to a point
x
=
2
. The norm of
x
is irrelevant, such that
(X,Y,W)
T
P
is equivalent to
(βX, βY, βW )
T
for an arbitrary value of
β
=
0. The Euclidean vec-
=
(X/W,Y/W)
T
.
The transformation is analogous for projective vectors in the three-dimensional
space
tor
x
corresponding to the projective vector
x
is then given by
x
˜
3
P
with four coordinates.
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