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1.2 Geometric Aspects of Stereo Image Analysis
The reconstruction of three-dimensional scene structure based on two images ac-
quired from different positions and viewing directions is termed stereo image anal-
ysis. This section describes the 'classical' Euclidean approach to this important field
of image-based three-dimensional scene reconstruction (cf. Sect. 1.2.1 )aswellas
its formulation in terms of projective geometry (cf. Sect. 1.2.2 ).
1.2.1 Euclidean Formulation of Stereo Image Analysis
In this section, we begin with an introduction in terms of Euclidean geometry, fol-
lowing the derivation described by Horn ( 1986 ). It is assumed that the world coordi-
nate system is identical with the coordinate system of camera 1; i.e. the transforma-
tion matrix C W T corresponds to unity while the relative orientation of camera 2 with
respect to camera 1 is given by
C 2
W T and is assumed to be known (in Sect. 1.4 we
will regard the problem of camera calibration, i.e. the determination of the extrinsic
and intrinsic camera parameters). The three-dimensional straight line (ray) passing
through the optical centre of camera 1, which is given by the equation
x 1
y 1
z 1
u 1 s
ˆ
ˆ
C 1 x
=
,
=
v 1 s
bs
(1.6)
v 1 ) T in im-
age 1 for all possible values of s . In the coordinate system of camera 2, according
to ( 1.2 ) the points on the same ray are given by
I 1 x
with s as a positive real number, is projected into the point
=
(
u 1 ,
ˆ
ˆ
x 2
y 2
z 2
(r 11 ˆ
u 1 +
r 12 ˆ
v 1 +
r 13 b)s
+
t 1
C 2 x
=
R C 1 x
=
+
t
=
(r 21 ˆ
u 1 +
r 22 ˆ
v 1 +
r 23 b)s
+
t 2
(1.7)
(r 31 ˆ
u 1 +
r 32 ˆ
v 1 +
r 33 b)s
+
t 3
with r ij as the elements of the orthonormal rotation matrix R and t i as the elements
of the translation vector t (cf. ( 1.2 )). In the image coordinate system of camera 2,
the coordinates of the point I 2 x
v 2 ) T
=
(
u 2 ,
ˆ
ˆ
are given by
ˆ
ˆ
u 2
b =
x 2
z 2
v 2
b =
y 2
z 2 ,
and
(1.8)
assuming an identical principal distance b for both cameras.
For the point
C 1 x is lo-
cated on the ray defined by ( 1.6 ), but its associated value of s is unknown. The
point I 2 x in image 2 which corresponds to the same scene point must be located
on a line which is obtained by projecting the points on the ray into image 2 for
all values of 0
I 1 x in image 1, the corresponding scene point
W x
=
s<
. The point on the ray with s
=
0 corresponds to the op-
C 1 c 1 of camera 1. It projects into the point
I 2 c 1 in image 2 and the
tical centre
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