Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 53

Fig. 1.16 ( a ) Pair of stereo images, acquired by two cameras with non-parallel optical axes.
( b ) Image pair rectified to standard geometry

W x W c i +

λB − 1
i

S i

= x ˜ and

λ (s) B (s i − 1 S (s)

(1.98)

W x W c i +

= x , ˜

i

which directly implies the rectification equation
S (s)
i

λ

λ (s) B (s)

B − 1
i

S i

˜ = ˜

x x (1.99)

i

(Fusiello et al., 2000 ).
At this point it is necessary to compute a grey value for each pixel of the rec-
tified image, where the pixel coordinates are integer-valued. The pixel coordinates
in the original image obtained according to ( 1.99 ), however, are real-valued. Ac-
cordingly, an interpolation technique such as bilinear interpolation has to be applied
(Fusiello et al., 2000 ). An example of a stereo image pair originally acquired with a
convergent camera setup and rectified to standard geometry is shown in Fig. 1.16 .
The first two coordinates of the projective vector S (s)

˜

x correspond to the rectified

i

pixel values u (s)
i

v (s 1 of image 2
now form a corresponding pair of epipolar lines. As a consequence, the problem of
stereo image analysis becomes a problem of establishing corresponding points along
image rows. We will now regard the computation of three-dimensional scene struc-
ture from the coordinates of image points measured in stereo image pairs rectified
to standard geometry. Without loss of generality, at this point the coordinate system
of camera 1 is used as the world coordinate system, such that the corresponding
rotation matrix R 1 corresponds to the identity matrix and the translation vector t 1 is
zero, and we define t

and v (s)
i

.Therow v (s)
1

of image 1 and the row v (s)
2

=

between the optical centres of the two
cameras corresponds to the baseline distance of the stereo camera system. As the
images are rectified to standard geometry, both optical axes are orthogonal to the
baseline (Horn, 1986 ). Due to the rectification to standard geometry, both cameras
have the same effective principal distance b 0 .
The image coordinates in standard geometry are denoted by u (s)
1

≡ t 2 . The distance t

and v (s)
1

in im-

age 1 and by u (s)

2 and v (s 2 in image 2, respectively. As it is assumed that the world
coordinate system corresponds to the coordinate system of camera 1, a scene point
can be described by (x,y,z) T . We furthermore assume square pixels

with k u = k v , corresponding to a pixel edge length d p =

W x C 1 x

= =

1 /k u , and without skew.
On the basis of Horn ( 1986 ), ( 1.1 ) then implies the equations