Image Processing Reference
In-Depth Information
•
The corresponding points can be searched along lines that can be effectuated as
a 1D search instead of a search in two dimensions. For example, if the point
(
−−−−−→
O
L
P
L
)
LD
is known, then by substituting it in Eq. (13.116) one obtains the
search line on which the corresponding unknown point (
−−−−−→
O
R
P
R
) must lie.
•
The establishment of corresponding points in images is a crucial problem for 3D
geometry reconstruction from stereo.
•
The epipolar lines can be utilized to implement triangulation where the rays
−−−−→
O
L
P
L
and
−−−−→
O
R
P
R
are guaranteed to intersect.
13.6 The Fundamental Matrix by Correspondence
Assume that the correspondences of (at least) eight points in the digital image co-
ordinates are known. Then Eq. (13.116) can be written for a known pair of points,
expressed in homogeneous coordinates, as
0=(
p
R
)
T
Fp
L
=
c
R
c
L
F
11
+
c
R
r
L
F
12
+
c
R
F
13
+
+
r
R
c
L
F
21
+
r
R
r
L
F
22
+
r
R
F
23
+
+
c
L
F
31
+
r
L
F
32
+
F
33
=0
(13.121)
where the unknown matrix elements,
⎛
⎞
F
11
F
12
F
13
F
21
F
22
F
23
F
31
F
32
F
33
⎝
⎠
F
=
(13.122)
and the known pair of points,
p
L
=(
c
L
,r
L
,
1)
T
p
R
=(
c
R
,r
R
,
1)
T
,
and
(13.123)
have been utilized. Equation (13.121) is a scalar product between two 9D vectors
that reduces to the homogeneous equation
q
T
f
=0
(13.124)
with
q
=(
c
R
c
L
,c
R
r
L
,c
R
,r
R
c
L
,r
R
r
L
,r
R
,
L
,r
L
,
1
T
f
=(
F
11
,F
12
,F
13
,F
21
,F
22
,F
23
,F
31
,F
32
,F
33
)
T
(13.125)
where the vectors
q
and
f
encode the known data and the unknown parameters,
respectively. Geometrically, Eq. (13.124) represents the equation of a hyperplane in
E
9
, and we are interested once again in finding the direction of this hyperplane,
f
,
when we know a set of correspondences,
Q
=
{
q
}
, of image points in a pair of stereo
