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due to the B-spline representation of the contours. The contour indices in the sets
{
corresponding to these epipolar intersections are denoted by
w
(i)
1
and
w
(j
2
, respectively. For each epipolar line
v
, the epipolar intersections are sorted
in ascending order according to their respective
u
(i
1
and
u
(j
2
values. Assuming that
the ordering constraint is valid, the following three cases must be distinguished.
1. The contours in both images have an identical number of epipolar intersections,
i.e.
e
1
(v)
=
e
2
(v)
. Then epipolar intersection
i
on the right image will be as-
signed to epipolar intersection
c
(n)
c
(m)
1
,b
}
and
{
2
,a
}
=
j
, respectively.
2. The contours on both images do not have an identical number of epipolar in-
tersections, i.e.
e
1
(v)
=
e
2
(v)
, and either
e
1
(v)
or
e
2
(v)
is odd. In this case, the
epipolar line is a tangent to the respective B-spline contour and is thus discarded.
3. The contours on both images do not have an identical number of epipolar in-
tersections, i.e.
e
1
(v)
=
e
2
(v)
, and both
e
1
(v)
and
e
2
(v)
are even. Without loss
of generality it is assumed that
e
1
>e
2
. An even intersection index denotes an
inward transition and an odd intersection index an outward transition. Hence, an
intersection with even index
j
on the left image may only be assigned to an in-
tersection with even index
i
on the right image, and analogously for odd indices,
to account for the topology of the segmented blob features. According to the or-
dering constraint, we will always assign pairs of neighbouring intersections in
the right image to pairs of neighbouring intersections in the left image, i.e. if
intersection
j
is assigned to intersection
i
, intersection
j
j
on the left image with
i
+
1 will be assigned to
intersection
i
1
)
assignments are
allowed, for each of which we compute the sum of square differences:
+
1. According to these rules,
((e
1
−
e
2
)/
2
+
e
2
u
(k
+
2
(j
−
1
))
1
−
d
min
2
−
u
(k)
2
S
j
=
for
j
=
1
,...,(e
1
−
e
2
)/
2
+
1
.
k
=
1
(1.106)
Epipolar intersection
i
on the right image will consequently be assigned to in-
tersection
j
on the left image. This heuristic rule helps to avoid
spurious objects situated near to the camera; similar heuristics are used in state-
of-the-art correlation-based blockmatching algorithms (Franke and Joos,
2000
).
The mutual assignment of contour points on epipolar line
v
results in pairs of indices
of the form
(i, j)
. A disparity measure
d
s
that involves a single epipolar line
v
can
be obtained in a straightforward manner by
=
arg min
j
{
S
j
}
u
( j)
2
u
˜
(i)
1
(v).
(1.107)
This disparity measure, however, may significantly change from one epipolar line to
the next, as the contours are often heavily influenced by pixel noise. Furthermore,
d
s
becomes inaccurate for nearly horizontal contour parts. An example of a disparity
image obtained by using (
1.107
) is shown in Fig.
1.20
.
To obtain a less noisy and more accurate disparity value, we define a contour
segment-based disparity measure which relies on an evaluation of
L
S
neighbour-
ing epipolar lines. The two contour segments of length
L
S
are denoted by the sets
{
d
s
=
(v)
−
s
(i)
1
s
(j)
2
}
i
=
1
,...,L
S
and
{
}
j
=
1
,...,L
S
with
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