1. 7)
lidate
kind
q. 9)
n of
rect’
ated
didate
The probability for this third kind of ambiguity is
C(n-7Dbfa 4. (n=) re? a
Pacs = F T SF sno —— (En. 19)
With (Eq. 8), (Eq. 9) and (Eq. 10) the probability of an
unsolvable ambiguity in the method of intersection of
epipolar lines becomes
P, = Pan) tPa + Pac)
4 - (n—1) -e b b
= (n : ) £^ N1+ 212 + 212 : (Eq. 11)
and the expectable numiber of remaining ambiguities
becomes
4. (i?- n): e bi by
a tz (Eq. 12)
An optimum (i.e. a minimum number of remaining ambi-
guities) is achieved with by; = by3 = bys, which means a
configuration of the three projective centers in a equilateral
triangle. Other than in a two camera model the number of
ambiguities does not depend on the length of the epipolar
lines (i.e. on the depth range in object space resp. the base-
length) any longer. In total the number of ambiguities is
being reduced by at least a factor of 10 (see table 1).
2.2 Collinear arrangement of three
cameras
The method of intersection of epipolar lines may be the
most evident, but it is not the only way of exploiting a third
camera. Using a different algorithm one can also work with
three cameras which are arranged in a way that their projec-
tive centers are lying on a straight line as shown in Figure
8. In this case possible correspondences between the first
and the second image have to be verified by a propagation
into the third image.
Zmax
liz
bi; bis
Figure 8: Proceeding with three collinearly arranged cameras
For all possible matches (1-2) a point in object space is
being calculated
^
C* bi
Px
X=Z: y= 0 (Eq. 13)
SYR
Depending on an assumed maximum error £ of the parallax
py the thus established point(s) will have an error mainly in
depth; this leads to a reduced search space Za, Z4 in object
space:
C: by, c: b: Z
3 p,-€£ C-b,-(Z-e)
b
X231 15
Z4 et eee ide
Z Z max + Zmin
C: bj, Cc: bi: Z
7. = +
4 np, +6 2-DatZ'E
X bis
X=, aZ nu
2 s Z 4 Zinar™ Zin (Eq. 14)
which is being imaged into image 3, where the length /;,3
of the search window becomes
= sr re
1,23 =x 4 x 3
A. b - X, 5 b13 — X3
Za Z4
eren. n
n Zu Z4
ut c: by Z Cb, 7
Ps
= 2 € :
bi,
(Eq. 15)
This way one receives a short epipolar search area in image
3 for all the candidates in image 2. If exactly one valid
candidate is found in these search spaces the necessary and
sufficient condition for a correct correspondence is
fulfilled.
A similar proceeding is used by Lotz/Fróschle (1990); they
suggest a strongly asymmetric arrangement of cameras as
shown in Figure 9 to reduce the probability of occurence of
ambiguities.
Figure 9: asymmetric camera arrangement (Lotz/Fróschle, 1990)
